Rising bubbles and falling drops
Manoj Kumar Tripathi
A Thesis Submitted to
Indian Institute of Technology Hyderabad
in Partial Fulfillment of the Requirements for
the Degree of
Doctor of Philosophy
Department of Chemical Engineering
Indian Institute of Technology Hyderabad
February 2015
Declaration
I declare that this written submission represents my ideas in my own words, and where ideas or
words of others have been included, I have adequately cited and referenced the original sources. I
also declare that I have adhered to all principles of academic honesty and integrity and have not
misrepresented or fabricated or falsified any idea/data/fact/source in my submission. I understand
that any violation of the above will be a cause for disciplinary action by the Institute and can also
evoke penal action from the sources that have thus not been properly cited, or from whom proper
permission has not been taken when needed.
————————–
(Signature)
—————————
( Manoj Kumar Tripathi)
—————————–
(Roll No.)
Approval Sheet
This Thesis entitled Rising bubbles and falling drops by Manoj Kumar Tripathi is approved for
the degree of Doctor of Philosophy from IIT Hyderabad
————————–
(———-) Examiner
Dept. of Chem Eng
IITM
————————–
(———-) Examiner
Dept. Math
IITH
————————–
(Dr. Kirti Chandra Sahu) Adviser
Dept. of Chem Eng
IITH
————————–
(Dr. Rama Govindarajan) Co-Adviser
Tata Institute of Fundamental Research
Center for Interdisciplinary Sciences
————————–
(———) Chairman
Dept. of Mech Eng
IITH
Acknowledgements
Thanks to the inspirations which affected my choices and others’ actions to bring me where I am.
Thanks to all the wonderful people I have come in contact with, starting from my parents, my
brother and my sister. I will always be grateful to my mother and father for the sacrifices they have
made for me. I could never imagine writing a PhD thesis without their hard work.
I have been very lucky to get good teachers who taught me many things including the things that
were outside school curricula. Also, I have been lucky to come to the Indian Institute of Technology
Hyderabad and stay here for a PhD, as these have probably been the most defining years for me.
I have been blessed with really good association for which I am very grateful. Ashwani, Chhavikant,
Priyank and Varun, who were practically my roommates, entertained and pulled legs of each other,
sang weird stuff on the tune of famous songs, made diaries for counting cuss words uttered by us,
and had philosophical discussions among many other things. My colleagues, Prasanna didi and
Ashima, who had their tables next to mine were the people I talked to about many things, took
help in plotting, helped in scripting among other useless (or was it?) chit-chat. All of this made my
PhD seem so smooth and memorable.
I am very grateful to have associated with Prof. Rama Govindarajan and Prof. Kirti Sahu.
Thanks, Rama Madam, for allowing me to be your student and to teach me many important things
just by being yourself. Thank you, Sahu sir, for pushing me when I got lazy.
Thanks to Professor Mahesh Panchagnula for inviting me to his lab to conduct experiments on
bubbles and drops. This experience was like a crash-course in experimental methods for me, and
the discussions with Prof. Panchagnula, his students and other lab staff have been very beneficial.
Special thanks to Stephane Popinet and others for developing such a wonderful fluid flow solver
- gerris, and other open-source community members who submitted important patches to the code
and gave their inputs for the development of this code.
iv
Dedication
To my parents who made me able to write this.
v
Abstract
The fascinating behaviour of bubbles and drops rising or falling under gravity, even without the
presence of any impurities or other forces (such as electric, magnetic and marangoni forces), is still
a subject of active research. Let alone a unified description of the dynamics of bubbles and drops, a
full description of a single bubble/drop is out of our reach, as of now. The thin skirted bubbles, for
instance, may rise axisymmetrically or may have travelling waves in azimuthal or vertical direction;
may or may not remain axisymmetric; may eject satellite bubbles, or they may form wrinkles in
their skirt. The length scales may vary across 3 or more orders of magnitude. A rising bubble may
change its topology to become a toroidal bubble and become unstable to break into smaller bubbles,
which may further break into even smaller bubbles. Bubbles which attain a terminal shape and
velocity may change their final behaviour depending on the initial conditions of release. Ellipsoidal
bubbles, released axisymmetrically, may often take a zigzag or a spiral path as they rise. On the
other hand, drops have a completely different dynamics. Drops have been studied due to their
importance in atomization, rain drop size distribution, emulsification and many other problems of
industrial importance. Apart from the low Reynolds number regime and density ratios close to 1, any
literature seldom compares bubbles and drops because of the inherent difference in their dynamics.
The reason for this difference has been investigated in the first part of this thesis.
We show that a bubble can be designed to behave like a drop in the Stokes flow limit when the
density of the drop is less than 1.2 times that of the outer fluid. It has been shown that Hadamard’s
exact solution for zero Reynolds number yields a better condition for equivalence between a bubble
and a drop than the Boussinesq condition. Scaling relationships have been derived for density ratios
close to unity for equivalence at large inertia. Numerical simulations predict a similar equivalence
for large inertia as well. For density ratios far from unity, bubbles and drops are very different.
Axisymmetric numerical simulations show that the vorticity tends to concentrate in lighter fluid,
which manifests into a totally different dynamics for bubbles and drops. This is the reason for thin
trailing end of the drops and thick base of bubbles, which result in a peripheral breakup of drops,
but a central breakup of bubbles at large inertia and low surface tension.
The three dimensional nature of the bubbles and drops has been studied next. We present
the results of one of the largest numerical study of three-dimensional rising bubbles and falling
drops. We show that as the size of the bubble is increased, the dynamics goes through three
abrupt transitions from one known class of shapes to another. A small bubble will attain an axially
symmetric equilibrium shape dictated by gravity and surface tension, and travel vertically upwards
at a terminal velocity thereafter. A bubble larger than a first critical size loses axial symmetry. We
show that this can happen in two ways. Beyond the next critical size, it breaks up into a spherical cap
and many satellite bubbles, and remarkably, the cap regains axial symmetry. Finally, a large bubble
will prefer not to break up initially, but will change topologically to become an annular doughnut-like
structure, which is perfectly axisymmetric. A central result of this work is to characterise the bubble
motion according to their mode of asymmetry, and mode of breakup. Some preliminary results of
three-dimensional drop simulations show that the effect of density ratio is to increase the inertia of
the drop which changes the way a drop breaks up. The effect of viscosity ratio was found to delay
the breakup of a drop. Also, this study confirms that a drop breaks up from the sides while a bubble
breaks up from the center for high inertia and low surface tension.
Next, we examined the buoyancy-driven rise of a bubble inside an infinite viscoplastic medium,
vi
assuming axial symmetry. Our results indicate that in the presence of inertia and in the case of weak
surface tension the bubble does not reach a steady state and the dynamics may become complex for
sufficiently high yield stress of the material. Past researchers had assumed the motion to be steady
or in the creeping flow regime, whereas we show that for low surface tension and large yield stresses,
the bubble exhibits a periodic motion along with oscillations in bubble shape. These oscillations are
explained by the periodic formation and destruction of an unyielded ring around the bubble.
Another physics often encountered in bubble/drop motion is that of heat transfer. A curious
case is that of self-rewetting fluids which have been reported to increase the heat transfer rate
significantly in heat-pipes. Rising bubble in a self-rewetting fluid with a temperature gradient
imposed on the container walls has been studied. To account for the non-monotonicity of surface
tension we consider a quadratic dependence on temperature. We examine the Stokes flow limit first
and derive conditions under which the motion of a spherical bubble can be arrested in self-rewetting
fluids even for positive temperature gradients. Our results indicate that for self-rewetting fluids,
the bubble motion departs considerably from the behaviour of ordinary fluids and the dynamics
may become complex as the bubble crosses the position of minimum surface tension. Under certain
circumstances, motion reversal and a terminal location is observed. The terminal location has been
found to agree well with the analytical result obtained from the Stokes solution. Also, a taylor
bubble is formed when the confinement is increased, thus implying a higher heat transfer rate to the
gas slug inside the tube.
Finally, the effect of evaporation in ambient conditions was examined. To this end, a phasechange model has been incorporated to gerris (open source fluid flow solver) in order to handle the
complex phenomena occurring at the interface. We found that the vapour is generated more on
the regions of the interface with relatively high curvature, and the vapour generation increases with
breakup of the drop. Furthermore, a competition between volatility and the dynamics governs the
vapour generation in the wake region of the drop. This is an ongoing work, and only a few results
have been presented.
vii
List of Publications
Journal Papers (Published/Accepted)
1. “Dynamics of an initially spherical gas bubble rising in a quiescent liquid (2015)”, M. K.
Tripathi, K. C. Sahu and R. Govindarajan, Nature Communications, 6, 6268..
2. “Non-isothermal bubble rise: non-monotonic dependence of surface tension on temperature
(2015)”, M. Tripathi, K. C. Sahu, G. Karapetsas, K. Sefiane and O. K. Matar, Journal of
Fluid Mechanics, 763, 82-108.
3. “Why a falling drop does not in general behave like a rising bubble (2014)”, M. Tripathi, K.
C. Sahu and R. Govindarajan, Scientific Reports (Nature Publishing Group), 4, 4771.
4. “Bubble rise dynamics in a viscoplastic material”, M. K. Tripathi, K. C. Sahu, G. Karapetsas
and O. K. Matar, Journal of Non-Newtonian Fluid Mechanics, accepted - 2015.
Conference Proceeding
5. “Evaporating falling drops”, M. K. Tripathi and K. C. Sahu, IUTAM Symposium on multiphase
flows with phase change: Challenges and opportunities, 8-11 December 2014, in Hyderabad,
India.
Journal Papers (submitted/under preparation)
6. “Bubble rise dynamics in viscosity stratified medium”, Premlata A. R., M. K. Tripathi and K.
C. Sahu, submitted to Physics of Fluids.
7. “Solutal marangoni effects on an octanoic acid drop rising in water”, K. Swaminathan, M. K.
Tripathi, K. C. Sahu, M. V. Panchagnula and R. Govindarajan, under preparation.
8. “Effect of evaporation on falling drop dynamics”, M. K. Tripathi, K. C. Sahu and R. Govindarajan, under preparation.
9. “Stability of double-diffusive displacement flow in three-dimensions”, K. Bhagat, M. K. Tripathi and K. C. Sahu, under preparation.
10. “Bubble dynamics in a pressure driven wavy-walled channel”, H. Konda, M. K. Tripathi and
K. C. Sahu, under preparation.
viii
Contents
Declaration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ii
Approval Sheet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
iii
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
iv
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
vi
Nomenclature
x
1 Introduction and previous work
1
1.1
Background and motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.2
Literature review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
1.3
Outline of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
1.4
Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12
2 Formulation and numerical methods
13
2.1
Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13
2.2
Numerical methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16
2.2.1
Diffuse-interface method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17
2.2.2
Volume of fluid method: Gerris . . . . . . . . . . . . . . . . . . . . . . . . . .
18
Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
2.3.1
Grid convergence test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
2.3.2
Effect of domain size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19
2.3.3
Comparison with numerical simulations . . . . . . . . . . . . . . . . . . . . .
21
2.3.4
Comparison with the experimental result of Bhaga & Weber [1] . . . . . . . .
21
2.3.5
Comparison with analytical results . . . . . . . . . . . . . . . . . . . . . . . .
23
Effect of regularization parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25
2.3
2.4
3 Bubbles and drops: Similarities and differences
28
3.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
28
3.2
In Hadamard flow regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
28
3.3
Bigger bubbles and drops . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
3.4
Differences in bubble and drop dynamics . . . . . . . . . . . . . . . . . . . . . . . . .
37
3.5
Before breakup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
44
3.5.1
Effects of viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
46
3.5.2
Drop breakup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
46
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
47
3.6
ix
4 Three dimensional bubble and drop motion
49
4.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
49
4.2
Bubbles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
50
4.2.1
Regimes of different behaviours . . . . . . . . . . . . . . . . . . . . . . . . . .
50
4.2.2
4.2.3
Path instability and shape asymmetry . . . . . . . . . . . . . . . . . . . . . .
Breakup regimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
53
57
4.2.4
4.3
Upward motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
59
Determination of the behaviour type . . . . . . . . . . . . . . . . . . . . . . . . . . .
60
4.3.1
Shape analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
60
4.3.2
Energy analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
61
4.4
Drops . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
61
4.5
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
63
5 Bubble rise in a Bingham plastic
66
5.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
66
5.2
Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
67
5.3
Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
68
5.3.1
Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
70
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
74
5.4
6 Non-isothermal bubble rise
78
6.1
Effect of temperature gradients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
78
6.2
Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
78
6.3
Analytical results: Stokes flow limit
. . . . . . . . . . . . . . . . . . . . . . . . . . .
81
6.4
Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
86
6.5
Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
97
7 Evaporating falling drop
98
7.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
98
7.2
Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
98
7.2.1
Evaporation model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
99
7.2.2
Model implementation in gerris . . . . . . . . . . . . . . . . . . . . . . . . . . 100
7.3
7.4
Results: Evaporating falling drops . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
8 Conclusions
104
References
106
x
List of Figures
1.1
Figure showing a variety of bubbles and drops observed in experiments. (a) A train
of air bubbles rising in water for a constant flow rate of air in the nozzle; (b) single octanoic-acid bubble in distilled water exhibiting a spiralling motion (images at
different times merged into a single image). These experiments were performed in
collaboration with Prof. Mahesh Panchagnula in his lab at IIT Madras. (c) A falling
water drop, from Edgerton’s book [2]; (d) a falling water drop breaking in a bagbreakup mode, courtesy E. Villermaux [3]. . . . . . . . . . . . . . . . . . . . . . . . .
1.2
2
Visible internal circulation in a glycerine drop falling in castor oil (from the experimental study by Spells [4]). The parameter values corresponding to this experiment
are: Ga = 0.792, Bo = 0.1, ρr = 1.3 and µr = 1.24. This was the first published
evidence of the internal circulation in falling drops. . . . . . . . . . . . . . . . . . . .
1.3
3
Density and viscosity ratios of about 1650 pairs of fluids. Blue (open) and red (filled)
symbols represent liquid-liquid and liquid-gas systems, respectively. It shows that the
density and viscosity ratios range across 8 and 10 orders of magnitude, respectively.
2.1
4
Schematic diagram of the simulation domains considered to solve (a) axisymmetric
(dot-dashed line represents the axis of symmetry), and (b) three-dimensional rising
bubble problem. Bubble size is not to scale. Similar domains are considered for
falling drop problem with inverted gravity. The domain is considered to have a square
base of size, L in three-dimensions and a circular base of diameter, L in cylindrical
coordinates. The outer and inner fluids are designated by ‘o’ and ‘i’, respectively.
The height of the domain, H is chosen according to the expected dynamics of the
bubble/drop. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2
14
Effect of grid refinement on the shape of the bubble at (a) t = 4, and (b) t = 7 for
Ga = 3.09442, Bo = 29, ρr = 7.4734 × 10−4 and µr = 8.1536 × 10−6 . The solid
and dot-dashed lines correspond to the results obtained using Δx = Δz = 0.015 and
0.029, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3
19
Grid convergence test. The shapes of the bubble for two different grid sizes at t = 3
is shown. The parameter values are Ga = 70.7, Bo = 200, ρr = 10−3 and µr = 10−2 .
The smallest grid sizes in panels (a) and (b) are about 0.029 and 0.015, respectively.
The grid refinement criteria used here are based on the vorticity magnitude and the
gradient of volume fraction (ca ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xi
19
2.4
Effect of domain size on upward velocity, w of a bubble exhibiting a spiralling motion. The dashed and solid lines represent domains of base width, L = 30 and 60,
respectively. The dimensionless parameters used for the simulations are: Ga = 100,
Bo = 0.5, ρr = 10−3 and µr = 10−2 . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5
20
Effect of domain size on the bubble shape at t = 4, and t = 7 (left to right) for
Ga = 3.09442, Bo = 29, ρr = 7.4734 × 10−4 and µr = 8.1536 × 10−6 . The solid and
dot-dashed lines correspond to computational domains 8×24 and 16×48, respectively.
The results are generated using square grid of Δx = Δz = 0.015. . . . . . . . . . . .
2.6
20
Comparison of the shape of the bubble obtained from our simulation (shown by solid
red line) with those from the level-set simulations of Sussman & Smereka [5] (dashed
line) at various times: (a) t = 0, (b) t = 0.8, (c) t = 1.6 and (d) t = 2.4. The
parameter values are Ga = 100, Bo = 200, ρr = 0.001 and µr = 0.01. The transition
to toroidal bubble (topological change) is observed at t = 1.6, which matches exactly
with the result of Sussman & Smereka [5]. . . . . . . . . . . . . . . . . . . . . . . . .
2.7
21
Variation of upward velocity of center of gravity of the drop with time for Ga = 219.09,
Bo = 240, ρr = 1.15 and µr = 1.1506. The dashed line is the result due to Han &
Tryggvason [6] and the solid line is the result of the present simulation. The figure is
plotted till breakup. It could be seen that the results match to a very good accuracy,
however small deviations can be seen which may be attributed to the differences in
the interface tracking/capturing methods in the two simulations. . . . . . . . . . . .
2.8
22
Comparison of the shape of the bubble obtained from the present diffuse interface
simulation (shown by red line) with that of Bhaga and Weber [1]. The parameter
values are Ga = 3.09442, Bo = 29, ρr = 7.4734 × 10−4 and µr = 8.1536 × 10−6 .
The dimple is not clearly visible in the experimental result because it is hidden by
the periphery of the bubble. Also, the apparent dimple seen from the side view is
different from the actual dimple because of the refraction due to the curved bubble
surface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.9
22
Comparison of terminal shape of the bubble with Bhaga & Weber [1] (left to right):
Ga = 2.316, Bo = 29, Ga = 3.094, Bo = 29, Ga = 4.935, Bo = 29, and Ga =
10.901, Bo = 84.75. The results in the bottom row are obtained from the present
three-dimensional simulations. The results from the corresponding axisymmetric simulations are shown by red lines in the top row. . . . . . . . . . . . . . . . . . . . . .
23
2.10 Comparison of streamline pattern in the wake of the bubble with Bhaga & Weber [1]
for the following dimensionless parameters: (a) Ga = 7.9, Bo = 17, (b) Ga = 9,
Bo = 21, (c) Ga = 12.6, Bo = 17, (d) Ga = 17.8, Bo = 27, (e) Ga = 21.9, Bo = 17,
and (f) Ga = 33.2, Bo = 11. The rest of the parameter values are ρr = 7.747 × 10−3
and µr = 10−2 . The results on the left hand side and right hand side of each panel
are from the present simulations and Bhaga & Weber’s [1] experiments, respectively.
24
2.11 Streamlines obtained from the analytical result for Hadamard flow (Re → 0) in a
spherical bubble (left hand side), and volume of fluid simulation with gerris (right
hand side) for a domain of half-width 16R. The dimensionless parameters are: Ga =
0.1, Bo = 0.1, ρr = 10−3 and µr = 10−2 . . . . . . . . . . . . . . . . . . . . . . . . . .
xii
25
2.12 Comparison of present numerical result with Hadamard-Rybczynski [7] theory. The
terminal velocity agrees well for a domain size of 30 × 30 × 120 and for the parameter
values: Ga = 0.1, Bo = 0.1, ρr = 10−3 and µr = 10−2 . The center of the vortex is
well predicted by our numerical simulation. . . . . . . . . . . . . . . . . . . . . . . .
25
2.13 The unyielded region in the non-Newtonian fluid (shown in black) at time, t = 10
for different values of the regularized parameter, �: (a) � = 0.01, (b) � = 0.001, (c)
� = 0.0001. The rest of the parameter values are Ga = 70.71, Bn = 14.213, µr = 0.01,
ρr = 0.001, m = 1 and Bo = 30. The unyielded regions for � = 0.001 and 0.0001 are
visually indistinguishable. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
26
2.14 (a) Temporal variation of the center of gravity (zCG ), (b) the aspect ratio (h/w) of
the bubble for different values of the regularization parameter, �. The rest of the
parameter values are the same as those used to generate Fig. 2.13. . . . . . . . . . .
26
2.15 The bubble shape (shown by red line) and unyielded region in the non-Newtonian
fluid (shown in black) at time, t = 2 for (a) regularised model, (b) Papanastasiou’s
model. The rest of the parameter values are � = 0.001, Ga = 70.71, Bn = 14.213,
µr = 0.01, ρr = 0.001, m = 1 and Bo = 30. The aspect ratios of the bubble in (a)
and (b) are the same (h/w = 1.018). . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1
27
Figure taken from the poster presented on Prof. Roddam Narasimha’s 80th birthday.
Rising bubble and falling drop for parameter values: (a) Ga = 50, Bo = 29, ρr =
3.2
3.3
7.47 × 10−4 and µr = 8.15 × 10−6 , and (b) Ga = 30, Bo = 29, ρr = 10 and µr = 10.
29
axis of symmetry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
Theoretical streamlines in a spherical bubble for the Hadamard flow(Re << 1). The
√
stagnation ring (center of the spherical vortex) lies at a distance of 1/ 2 from the
A sketch showing a spherical body falling under gravity and the forces acting on it,
where z represents the vertical coordinate, and Fb , Fg and FD denote the gravitational,
buoyancy and drag forces, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4
30
Vertical location of the center of gravity as a function of time for a drop (ρr = 1.214
and µr = 76), an equivalent bubble based on Boussinesq approximation (ρr = 0.786
and µr = 76) and an equivalent bubble based on conditions (3.21) and (3.22) (ρr =
0.85 and µr = 0.1). The rest of the parameters are Ga = 6 and Bo = 5 × 10−4 . The
bubble designed using the Hadamard’s solution is shown to be better than the one
derived using the often employed Boussinesq condition.
3.5
. . . . . . . . . . . . . . . .
33
(a) Evolution of bubble shape with time for ρr = 0.9, µr = 0.5, Ga = 50, Bo = 50; (b)
evolution of drop shape with time for ρr = 1.125, µr = 0.625, Ga = 50, Bo = 50. The
direction of gravity has been inverted for drop to compare the respective shapes with
those of the bubble. Even for high Ga and Bo, the dynamics can be made similar if
density ratios are close to unity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.6
37
Dynamics in the absence of gravity: (a)evolution of bubble shape with time for ρr =
0.9, µr = 0.5, Re = 50, Bo = 50, (b) evolution of drop shape with time for ρr = 1.125,
µr = 0.625, Re = 50, Bo = 50. The initial shape of both drop and bubble was kept
spherical and the initial velocity given to the fluid blobs is U0 = 1 for both. The
shapes of bubble and drop tend to be similar for density ratios close to unity. . . . .
xiii
38
3.7
Dynamics in the absence of gravity: (a) evolution of bubble shape with time for
ρr = 0.52, µr = 0.05, Re = 50, Bo = 50, (b) evolution of drop shape with time for
ρr = 13, µr = 1.25, Re = 50, Bo = 50. The initial shape of both drop and bubble was
kept spherical and the same initial velocity U0 given to both fluid blobs. The bubble
regains a spherical shape, whereas the drop breaks up in the bag-breakup mode. . .
3.8
39
Evolution of (a) bubble shape with time for ρr = 0.9, µr /ρr = 0.56. (b) drop shape
with time for ρr = 1.125, µr /ρr = 0.56. (c) drop shape with time for ρr = 1.125, with
viscosity obtained from Eq. (3.21). The direction of gravity has been inverted for the
drop in order to compare the respective shapes with those of the bubble. In all three
simulations, Ga = 50, Bo = 50, and the initial shape was spherical. . . . . . . . . . .
3.9
40
Evolution of (a) bubble and (b) drop shapes with time, when densities of outer and
inner fluid are significantly different. As before, for the drop (b), the direction of
gravity has been inverted. In both simulations Ga = 50 and Bo = 10. The other
parameters for the bubble system are ρr = 0.5263 and µr = 0.01, while for the drop
system ρr = 10 and µr = 0.19. Note the shear breakup of the drop at a later time.
Shown in color is the residual vorticity [8]. . . . . . . . . . . . . . . . . . . . . . . . .
41
3.10 Streamlines in the vicinity of a bubble for t = 1, 2, 3 and 4 for parameter values
Ga = 50, Bo = 10, ρr = 0.5263 and µr = 0.01. The bubble is shown in grey and a
red outline. The circulation can be seen lying inside the bubble, which does not allow
the bubble to thin out at its base. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
42
3.11 Streamlines in the vicinity of a drop for t = 1, 2, 3 and 4 for parameter values Ga = 50,
Bo = 10, ρr = 10 and µr = 0.19. The bubble is shown in grey and a red outline. The
direction of gravity has been inverted to compare the shapes with those in Fig. 3.10.
The circulation is seen to move out of the drop, making the drop to thin out at its
trailing end. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
42
3.12 Evolution of (a) bubble and (b) drop (gravity reversed) shapes with time. Parameters
for both bubble and drop systems are: Ga = 100, Bo = 50 and µr = 10. The density
ratio for the bubble and drop are ρr = 0.52 and ρr = 13 respectively, based on Eq.
(3.22). The figure shows that the density, rather than viscosity, decides the location
of vortical structures, which results in altogether different deformation in bubbles and
drops. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
43
3.13 Variation of dimple distance versus time for different Bond numbers for Ga = 50,
ρr = 7.4734 × 10−4 , µr = 8.5136 × 10−6 . The tendency of a bubble to break from the
center is evident. However, a bubble may form a skirt for intermediate Bond numbers
(Bo = 15), which may lead to breakup or shape oscillations in certain cases.
. . . .
45
3.14 Streamlines in and around the bubble at time, t = 1, 1.5, 2.0 and 2.5 respectively,
for Ga = 50, Bo = 29, ρr = 7.4734 × 10−4 and µr = 8.5136 × 10−6 . The shape
of the bubble is plotted in red. The toroidal vortex inside the bubble maintains the
thickness of its base as the liquid jet penetrates the remaining air film at the top. . .
45
3.15 Streamlines in and around the bubble at time, t = 2.5, 5, 7, 9 and 11 respectively,
for Ga = 50, Bo = 15, ρr = 7.4734 × 10−4 and µr = 8.5136 × 10−6 . Three toroidal
vortices form inside and outside the bubble which compete with the surface tension
force to make the bubble shape oscillate. . . . . . . . . . . . . . . . . . . . . . . . . .
xiv
45
3.16 Variation of dimple distance versus time for different Gallilei numbers for Bo = 8, ρr =
7.4734 × 10−4 , µr = 8.5136 × 10−6 . The bubble shapes are shown at corresponding
times for Ga = 5 (top) and 125 (bottom). The shape oscillations ensue after a
threshold in outer fluid’s viscosity i.e. Ga. . . . . . . . . . . . . . . . . . . . . . . . .
−4
3.17 Variation of dimple distance Dd versus time for Bo = 29, ρr = 7.4734 × 10
8.5136 × 10
−6
46
, µr =
. Bubble shapes are shown for non-oscillating (top, black), oscillating
(blue) and breaking (bottom, black) bubbles. . . . . . . . . . . . . . . . . . . . . . .
47
3.18 Streamlines in and around the drop at time, t = 4.5, 6 and 7.5, respectively (from
left to right), for Ga = 50, Bo = 5, ρr = 10 and µr = 10. The circulation zones form
outside the drop, as observed in Fig. 3.9. . . . . . . . . . . . . . . . . . . . . . . . .
47
3.19 Variation of break-up time with Bond number for Ga = 50, ρr = 10 and µr = 10.
A typical bag breakup mode is shown in this figure. Shapes of the drop just before
breakup are shown for various Bond numbers. . . . . . . . . . . . . . . . . . . . . . .
4.1
48
Different regimes of bubble shape and behaviour. The different regions are: axisymmetric (circle), asymmetric (solid triangle) and breakup (square). The axisymmetric
regime is called region I. The two colors within the asymmetric regime represent
non-oscillatory region II (shown in green), and oscillatory region III (blue) dynamics.
The two colors within the breakup regime represent the peripheral breakup region IV
(light yellow), and the central breakup region V (darker yellow). The red dash-dotted
line is the M o = 10−3 line, above which oscillatory motion is not observed in experiments [1, 9]. Typical bubble shapes in each region are shown. In this and similar
figures below, the bubble shapes have been made translucent to enable the reader to
get a view of the internal shape. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2
51
Dynamics expected for bubbles in different liquids. Constant Morton number lines,
each corresponding to a different liquid, are overlayed on the phase-plot to demonstrate that our transitions can be easily encountered and tested in commonly found
liquids. The initial radius of the air bubble increases from left to right on a given line.
Circles, triangles and squares represent air bubbles of 1 mm, 5 mm and 20 mm radii,
respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3
52
Agreement and contrast between present and previous results for different flow regimes.
Comparison between the onset of asymmetric bubble motion obtained in the numerical stability analysis of Cano-Lozano et al. [10] (solid black line), and the present
boundary between regions I and II. Also given in this figure are five different conditions (diamond symbols) studied by Baltussen et al. [11]. The dynamics they obtain
are as follows: A - Spherical, B - Ellipsoidal, C - Boundary between skirted and
ellipsoidal, D - Wobbling and E - Peripheral breakup. The correspondence between
present results and [11] is excellent. Grace et al. [12] obtained spherical bubbles below
the solid blue line shown. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4
53
Dynamics and shapes of region III bubbles: trajectory of the bubble centroid for (a)
Ga = 70.7, Bo = 10, and (b) Ga = 100, Bo = 4, and (c) shape evolution of bubble
corresponding to the latter case. In panel (c), the radial distance of the center of
gravity (rs ) of the bubble measured in the horizontal plane from the original location
is shown below the shapes at each time. . . . . . . . . . . . . . . . . . . . . . . . . .
xv
54
4.5
Differences between two dimensional and three dimensional bubble shapes: (a) A
region III bubble at t = 20 for Ga = 100 and Bo = 0.5, (b) at t = 30 for Ga = 100
and Bo = 4, again in reign III, and (c) a region IV bubble at t = 5 for Ga = 70.71
and Bo = 20. The second row shows the side view of the three-dimensional shapes of
bubbles rotated by 90 degrees about the x = 0 axis with respect to the top row.
4.6
. .
55
Characteristics of a region III bubble of Ga = 100 and Bo = 0.5. (a) Oscillating
upward velocity, with different behaviour at early and late times, (b) trajectory of
the bubble centroid. The two regions corresponding to two different behaviours in
the rise velocity correspond to the inline oscillations and zig-zagging motion. . . . .
4.7
55
Region III bubble corresponding to that shown in Fig. 4.6 (Ga = 100 and Bo = 0.5).
(a) Iso-surfaces of the vorticity component in the z direction at time t = 15 (ωz =
±0.0007) and 26 (ωz = ±0.006), (b) The evolution of the shape of the bubble. The
radial distance of the center of gravity (rs ) of the bubble measured in the horizontal
plane from the original location is shown below the shapes at each time. . . . . . . .
4.8
56
Time evolution of bubbles exhibiting a peripheral and a central breakup. Threedimensional and cross-sectional views of the bubble at various times (from bottom
to top the dimensionless time is 1, 2, 4 and 5). (a) region IV, a bubble breaking
into a spherical cap and several small satellite bubbles, Ga = 70.7 and Bo = 20,
and (b) region V, a bubble changing in topology from dimpled ellipsoidal to toroidal,
Ga = 70.7 and Bo = 200. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.9
57
A new breakup mode in region IV for Ga = 500 and Bo = 1. Bubble shapes are
shown at dimensionless times (from left to right) t = 2, 4, 6, 7, 8, 9 and 9.1). . . . .
57
4.10 Comparison of our 3D results with those of Bonometti & Magnaudet [13] for bubble
breakup. The light yellow and dark yellow colours represent the regions for peripheral
and central breakup. The corresponding data points from the present numerical
simulation are shown as blue and black squares, respectively. . . . . . . . . . . . . .
58
4.11 Rise velocity for bubbles having markedly different dynamics. (a) region I: axisymmetric (Ga = 10, Bo = 1) (b) region II: skirted (Ga = 10, Bo = 200), (c) region
III: zigzagging (Ga = 70.7, Bo = 1), (d) region IV: offset breaking up (Ga = 70.7,
Bo = 20) and (e) region V: centrally breaking up bubble (Ga = 70.7, Bo = 200). In
addition to the upward velocity, the in-plane components are unsteady too in regions
III to V. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
60
4.12 Variation of dimensionless terminal velocity with Bo for different Ga. The terminal
velocity tends to decrease with decreasing surface tension because of the increased
drag on the bubble. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
61
4.13 Variation of the sum of kinetic and surface energies (T E) for (a) Bo = 20, and (b)
Ga = 100. The peak in energy corresponds to the boundaries of the regions shown in
Fig. 4.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
62
4.14 Time evolution of drops for different values of density ratios (ρr ) for parameter values:
Ga = 40, Bo = 5 and µr = 10.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
63
4.15 A large liquid drop violently breaking up while falling in the air at times t = 4 and 5
(from left to right) for parameter values: Ga = 40, Bo = 5, ρr = 1000 and m = 10 . .
xvi
64
4.16 Time evolution of drops for different values of viscosity ratios (µr ) for parameter
values: Ga = 40, Bo = 5 and ρr = 10. . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1
65
Schematic diagram of a bubble of fluid ‘B’ rising inside a Bingham fluid ‘A’ under
the action of buoyancy. The bubble is placed at z = zi ; the value of H, L and zi
are taken to be 20R, 48R, and 10.5R, respectively. Initially the aspect ratio of the
bubble, h/w is 1, wherein h and w are the maximum height and width of the bubble.
5.2
68
The shape of the bubble along with the mesh at t = 1.5 are shown for (a) finer and (b)
coarser grids. Adaptive grid refinement has been used in the interfacial and yielded
regions. The smallest mesh size in the finer and coarser grids are 0.015 and 0.0625,
respectively. Note that the finer grid has been used to generate the results presented
in the subsequent figures. The parameter values are Ga = 70.71, Bn = 14.213,
µr = 0.01, ρr = 0.001, m = 1 and Bo = 30. The aspect ratios of the bubble obtained
using the finer and courser grids are 1.002 and 1.003, respectively. . . . . . . . . . .
5.3
69
(a) Temporal variation of the center of gravity, (b) the aspect ratio of the bubble for
different values of Bn. The parameter values are Ga = 7.071, µr = 0.01, ρr = 0.001,
m = 1 and Bo = 10. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.4
71
(a) Temporal variation of the center of gravity, (b) the aspect ratio of the bubble for
different values of µr . The parameter values are Ga = 7.071, Bn = 0.99, ρr = 0.001,
m = 1 and Bo = 10. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.5
71
The evolution of the shape of the bubble (shown by red lines) and the unyielded region
in the non-Newtonian fluid (shown in black) for different values of Bingham number.
The results of the Newtonian case are shown for the comparison purpose. The rest of
the parameter values are the same as those used to generate Fig. 5.3.
5.6
. . . . . . . .
73
Contour plots for the radial (right) and axial (left) velocity components for (a) Bn = 0
at t = 6 (Newtonian case), (b) Bn = 0.354 at t = 6, (c) Bn = 0.99 at t = 20 and (d)
Bn = 1.34 at t = 20. In each panel the shape of the bubble is shown by red line. The
rest of the parameter values are the same as those used to generate Fig. 5.3.
5.7
. . . .
74
(a) Temporal variation of the center of gravity, (b) the aspect ratio of the bubble for
different values of Bo. The rest of the parameter values are Re = 70.71, Bn = 14.213,
µr = 0.01, ρr = 0.001, and m = 1.
5.8
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
75
The evolution of the shape of the bubble (shown by red lines) and the unyielded
regions in the Bingham fluid (shown in black) for different values of Bo. The rest of
the parameter values are the same as those used to generate Fig. 5.7.
5.9
. . . . . . . .
76
Contour plots for the radial (right) and axial (left) velocity components for (a) Bo = 1
at t = 6, (b) Bo = 1 at t = 8.5, (c) Bo = 30 at t = 6 and (d) Bo = 30 at t = 8.5. In
each panel the shape of the bubble is shown by red line. The rest of the parameter
values are the same as those used to generate Fig. 5.7. . . . . . . . . . . . . . . . . .
6.1
77
Schematic diagram of a bubble moving inside a Newtonian fluid under the action of
buoyancy. The initial location of the bubble is at z = zi ; unless specified, the value
of H, L and zi are 6R, 48R, and 10.5R, respectively. The acceleration due to gravity,
g, acts in the negative z direction. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xvii
79
6.2
Variation of the liquid-gas surface tension along the wall of the tube for Γ = 0.1 and
various values of M1 and M2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.3
80
Temporal variation of the center of gravity of the bubble for the parameter values
Ga = 10, Bo = 10−2 , ρr = 10−3 , µr = 10−2 , Γ = 0.1 and αr = 0.04. The plots for the
isothermal (M1 = 0 and M2 = 0), linear (M1 = 0.4 and M2 = 0) and self-rewetting
(M1 = 0.4 and M2 = 0.2) cases are shown in the figure. The horizontal dotted line
indicates the prediction of Eq. (6.51) for the self-rewetting case. . . . . . . . . . . . .
6.4
86
(a) The terminal velocity of the center of gravity of the bubble along with the aspect
ratio for different values of M1 for M2 = 0; (b) temporal variation of the center of
gravity of the bubble for M2 = M1 /2; (c) variation of the time at which zCG reaches
its maximum for different values of M1 . The rest of the parameter values are Ga = 10,
Bo = 10−2 , ρr = 10−3 , µr = 10−2 , Γ = 0.1 and αr = 0.04. The numerical predictions
of Eq. (6.51) are shown by the filled square symbols on the right vertical axis. . . . .
6.5
Effect of Ga on the temporal evolution of the bubble centre of gravity for Bo = 10
−2
88
,
ρr = 10−3 , µr = 10−2 , M1 = 0.2, M2 = 0.1, Γ = 0.1 and αr = 0.04. The prediction
of Eq. (6.51) is shown by the dotted line. . . . . . . . . . . . . . . . . . . . . . . . .
6.6
89
Effect of Bo on bubble motion for (a) Ga = 10 and (b) Ga = 5; effect of Bo on the
(c,d) length of the bubble, lB , (e,f) aspect ratio of the bubble, Ar for Ga = 5. The
rest of the parameters values ρr = 10−3 , µr = 10−2 , M1 = 0.4, M2 = 0.2, Γ = 0.1
and αr = 0.04. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.7
90
Evolution of bubble shape (blue line), streamlines (lines with arrows), and temperature contours (shown in color) with time for (a) Bo = 10 and (b) Bo = 10−2 . The
initial location of the bubble, zi = 10. The inset at the bottom represents the colormap for the temperature contours. The rest of the parameter values are Ga = 10,
ρr = 10−3 , µr = 10−2 , M1 = 0.4, M2 = 0.2, Γ = 0.1 and αr = 0.04. . . . . . . . . . .
6.8
92
The effect of initial location of the bubble on the temporal evolution of the center of
gravity, zCG . The rest of the parameter values are Ga = 10, Bo = 10−2 , ρr = 10−3 ,
µr = 10−2 , M1 = 0.2, M2 = 0.1, Γ = 0.1 and αr = 0.04. The prediction of Eq. (6.51)
is shown by the dotted line. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.9
93
(a) Evolution of the length of the bubble, lB for two values of Bo when he initial
location of the bubble zi = 8. (b) The effects of initial location of the bubble on
elongation of the bubble for Bo = 100. The radius of the tube, H = 2.5. The rest of
the parameters are Ga = 10, ρr = 10−3 , µr = 10−2 , M1 = 0.4, M2 = 0.2, Γ = 0.1 and
αr = 0.04. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
93
6.10 Evolution of bubble shape (blue line), streamlines (lines with arrows), and temperature contours (shown in color) with time for (a) Bo = 10 and (b) Bo = 100, and
H = 2.5. The initial location of the bubble zi = 8. The inset at the bottom represents
the colormap for the temperature contours. The rest of the parameters are Ga = 10,
ρr = 10−3 , µr = 10−2 , M1 = 0.4, M2 = 0.2, Γ = 0.1 and αr = 0.04. . . . . . . . . . .
94
6.11 Evolution of (a) the length of the bubble, lB , (b) the location of center of gravity, in
a tube having H = 2.1. The initial location of the bubble zi = 8. The rest of the
parameters are Ga = 5, ρr = 10−3 , µr = 10−2 . The non-isothermal curve is plotted
for Γ = 0.1 and αr = 0.04. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xviii
95
6.12 Evolution of bubble shape with time for (a) isothermal case, and (b) M1 = 1.8, M2 =
0.9 (temperature contours shown in color). The inset at the bottom represents the
colormap for the temperature contours. The rest of the parameters are the same as
those used to generate Fig. 6.11. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.1
96
Vapour mass source calculated only in the interfacial cells. Normal to the interface
(yellow, dashed line) and its components (yellow, solid lines) are shown. . . . . . . . 100
7.2
Drop shape and vapour volume fraction contours with minimum and maximum levels
as 0 and 10−3 , for a water drop falling in air at time, t = 1, 3, 4 and 5 (from left
to right). The other parameters are: Ga = 500, Bo = 0.025, ρrb = 1000, ρrv = 0.9,
µrb = 55, µrv = 0.7, P e = 200, λrb = 26, λrv = 1.0, cp,rb = 4, cp,rv = 2, MT = 0.2,
Tc = 293K, and Th = 343K. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
7.3
Drop shape and vapour volume fraction contours with minimum and maximum levels
as 0 and 3 × 10−3 , for a chloroform drop falling in air at time, t = 1, 3, 5 and 7 (from
left to right). The other parameters are: Ga = 100, Bo = 0.1, ρrb = 1480, ρrv = 0.9,
µrb = 281.2, µrv = 0.7, P e = 230, λrb = 6, λrv = 1.0, cp,rb = 1.05, cp,rv = 2,
7.4
MT = 0.2, Tc = 293K, and Th = 343K. . . . . . . . . . . . . . . . . . . . . . . . . . 102
Drop shape and vapour volume fraction contours with minimum and maximum levels
as 0 and 3 × 10−3 , for a chloroform drop falling in air at time, t = 1, 3, 5 and 7 (from
left to right). The other parameters are: Ga = 100, Bo = 0.1, ρrb = 1480, ρrv = 0.9,
µrb = 281.2, µrv = 0.7, P e = 230, λrb = 6, λrv = 1.0, cp,rb = 1.05, cp,rv = 2,
MT = 0.2, Tc = 293K, and Th = 343K. . . . . . . . . . . . . . . . . . . . . . . . . . 102
xix
List of Tables
2.1
Frequently used dimensionless groups relevant to the present work. . . . . . . . . . .
2.2
Comparison of the terminal velocities by Joseph [14] and the present work for the
17
parameter values: (a) Ga = 50, and (b) Ga = 100. The rest of the parameter values
are Bo = 10, ρr = 0.001 and µr = 0.01. . . . . . . . . . . . . . . . . . . . . . . . . . .
xx
24
Chapter 1
Introduction and previous work
1.1
Background and motivation
Blobs of a fluid in another fluid are commonly known as bubbles and drops. Since, there is no
strict definition of a bubble or a drop, it would be helpful for this work to begin by defining a
bubble as a blob of fluid having lower density (ρr < 1) than its surrounding medium and a drop
as a fluid blob surrounded by a lower density fluid (ρr > 1), where ρr is the ratio of inner fluid
density to the surrounding fluid density. Most of the flows in nature involve multiple phases, which
may get disconnected from their respective streams to form blobs of a phase dispersed in another
phase. Bubbles and drops may form as a result of encapsulation of a fluid in another fluid, for
example exhaled air by sea creatures, vapour bubbles in boiling water, molten glass globules in
air at a glass marble manufacturing industry, air in molten glass in glass-blowing factories, fuel
droplets from a fuel injector, clouds floating in air, and bubbles formed by active galactic nuclei
which rise due to buoyancy [15]. The length scale for bubble motion may vary from micrometers to
kiloparsecs (1 kiloparsec ≈ 3.0857 × 1019 m), and the time scales may range from nanoseconds [16] to
a million years [17]. The earliest documented mention of a study of bubble motion has been found
in a manuscript, Codex Leicester, by Leonardo Da Vinci, discovered by Prosperetti [18]. Da Vinci
reported the paradoxical spiral motion of bubbles when released axisymmetrically from bottom of a
container filled with water. This is now known as path instability. A few pictures of rising bubbles
and falling drops are shown in Fig. 1.1, and a few animations and movies of rising bubbles are also
available in the supplementary material of [19]. Bubble dynamics is of huge importance in heat and
mass transfer processes, in natural phenomena like aerosol transfer from sea, oxygen dissolution in
lakes due to rain and electrification of atmosphere by sea bubbles [20], in bubble column reactors, in
petroleum industries, for the flow of foams and suspensions and in carbon sequestration [21], to name
just a few. An important property of bubbles and drops is the internal circulation, which enhances
mixing which results in greater heat and mass transfer. Fig. 1.2 shows the internal circulation
within a glycerine drop released in castor oil [4]. The internal circulation inside fluid bubbles/drops
is responsible for a reduction in drag which causes them to move faster than solid ones. Furthermore,
this circulation affects the flow field in the wake which differentiates a fluid bubble/drop from a solid
one. A large part of this thesis is contained in our published papers [19, 22–24] .
A bubble or drop is commonly influenced by gravitational, surface tension, and viscous forces,
1
(a)
(b)
(c)
(d)
Figure 1.1: Figure showing a variety of bubbles and drops observed in experiments. (a) A train of air
bubbles rising in water for a constant flow rate of air in the nozzle; (b) single octanoic-acid bubble in
distilled water exhibiting a spiralling motion (images at different times merged into a single image).
These experiments were performed in collaboration with Prof. Mahesh Panchagnula in his lab at
IIT Madras. (c) A falling water drop, from Edgerton’s book [2]; (d) a falling water drop breaking
in a bag-breakup mode, courtesy E. Villermaux [3].
although there may be electric, magnetic and other forces depending on the types of fluids and
their environment. The interplay of these forces results in different bubble and drop behaviours,
depending on bubble/drop size, density and viscosity of the fluids involved. Even without the
consideration of surfactant, thermal, magnetic, miscibility effects and so on, the parameter space
consists of (R, ρi , ρo , µi , µo , σ, g), wherein the parameters are radius of a volume equivalent sphere,
density of the inner fluid, density of the outer fluid, viscosity of the inner fluid, viscosity of the outer
fluid, interfacial tension at bubble/drop interface, and the acceleration due to gravity, respectively.
This makes it difficult to perform a parametric study of the problem. As a result, in a number of
studies, a few parameters are considered to be negligible and the dynamics is studied with respect
to one or two of these parameters. However, by applying Buckingham-π theorem, we find that the
number of parameters required to describe the system can be brought down to only four dimensionless
numbers instead of seven dimensional ones. These four dimensionless numbers can be chosen as the
2
Figure 1.2: Visible internal circulation in a glycerine drop falling in castor oil (from the experimental
study by Spells [4]). The parameter values corresponding to this experiment are: Ga = 0.792,
Bo = 0.1, ρr = 1.3 and µr = 1.24. This was the first published evidence of the internal circulation
in falling drops.
√
Gallilei number Ga(≡ ρo R gR/µo ), the Bond number Bo(≡ ρgR2 /σ), the density ratio ρr (≡ ρi /ρo ),
and the viscosity ratio µr (≡ µi /µo ). Eötvös number, which has the same definition as the Bond
number, is also commonly used in bubble literature, however in this thesis we have used Bond
number to represent this dimensionless quantity. It is to be noted that a number of experimental
and numerical works [1, 25] also employ other dimensionless numbers in their studies, such as the
Reynolds number Re(≡ ρo V R/µo ), Weber number W e(≡ |ρo −ρi |V 2 /R) and Morton number M o(≡
gµ4o |ρo −ρi |/ρ2o σ 3 ). Although, these numbers are better indicators of the ratios of various forces in the
system, the rise velocity V is not known a-priori. Therefore these dimensionless numbers (dependent
on the rise velocity V ) do not provide one with a set of conditions which could be controlled before
performing the experiment/simulation. Furthermore, a Morton number defined as Bo3 /Ga4 yields
straight lines of constant Morton numbers on a log-log plot of Ga versus Bo, which makes it easy
for researchers to study the dynamics of bubbles/drops with respect to the fluid properties.
In the past several decades, thousands of published works have attempted to fit various regimes
of bubble motion into simple models. The number of parameters, the nonlinearity and the fully
three-dimensional nature of the problem makes it vast and daunting. The viscosity and density
ratios of nearly 1650 pairs of fluids used in industries and households are presented in Fig. 1.3. The
red circles on the left and right hand sides of the ρr = 1 line represent high density contrast bubbles
(air in liquid) and drops (liquid in air), respectively. These bubbles and drops with a high contrast in
their densities are very different from each other in their behaviour, however the fluid pairs marked
with blue circles (liquid in liquid systems) may behave in a similar fashion even for high inertia.
A popular shape regime chart for low density and viscosity ratio fluid blobs has been presented
by Clift et al. [25]. It should be noted that inclusion of temperature or concentration (of some
species like sugar) often changes the viscosity drastically. Such effects along with the consideration
of non-Newtonian behaviour of other fluids push the boundaries of, or give new dimension to Fig.
1.3. The behaviour regimes of these different bubbles and drops spread across decades of density
and viscosity ratios is one of the objectives of the present work.
As stated above, bubbles and drops have been a subject of active research for more than a century,
and in all probability a lot longer (for example see the representative review articles [26–28]); there
are yet many unsolved problems, which are the subject of recent research (see e.g. [10, 29–34]).
Appealing introductions to the complexity associated with bubble and drop phenomena can also be
found in Refs. [35, 36]. A vast majority of the earlier experimental and theoretical studies have had
3
Figure 1.3: Density and viscosity ratios of about 1650 pairs of fluids. Blue (open) and red (filled)
symbols represent liquid-liquid and liquid-gas systems, respectively. It shows that the density and
viscosity ratios range across 8 and 10 orders of magnitude, respectively.
one of the following goals (i) to obtain the rise velocity (ii) to evaluate the path instability (iii) to
understand bubbly flows, (iv) to make quantitative estimates for particular industrial applications,
and (v) to derive models for estimating different bubble parameters. Most of these restrict themselves
to only a few Ga and Bo. Our study, in contrast, is focussed on the dynamics of a single bubble.
Starting from the initial condition of a spherical stationary bubble, we are interested in delineating
the physics that can happen. We cover a range of several decades in the relevant parameters.
The review of the research work conducted in the related areas is presented below. For readability
this has been classified into few sub-areas: (i) axisymmetric bubble and drop motion, (ii) bubble rise
in non-Newtonian media, (iii) bubble rise in non-isothermal media, (iv) three-dimensional bubble
and drop dynamics and (v) phase change of liquid drops. A brief literature is presented next.
1.2
Literature review
Bubbles and drops: Similarities and differences
Bubbles and drops have often been studied separately, for instance see [1, 33, 37–42] for bubbles
and [4, 43–47] for drops. However, there is also a considerable amount of literature which discusses
both together, e.g. [7, 25, 48–51]. The parameter space for this problem is very large as mentioned
above. Therefore bubble/drop dynamics has been investigated in limiting conditions, for instance
Taylor [52] derived the Oseen’s approximation for small inertia and deformation and showed that
the bubbles and drops deform differently. An introduction to the complexity associated with bubble
and drop phenomena can be found in [28].
When the bubble or drop is tiny, it merely assumes a spherical shape, attains a terminal velocity,
and moves up or down, respectively, under the action of gravity. An empirical formula for the
terminal velocity of small air bubbles was found by Allen [53] in 1900. Later, two independent
4
studies by Hadamard [7] and Rybczynski [54] led to the first solution for the terminal velocity and
pressure inside and outside of a slowly moving fluid sphere in another fluid of different density and
viscosity. The spherical vortex solution due to Hill [55] has been a keystone for most of the analytical
studies on the subject. Later studies [52] showed that at low Reynolds (a measure of the ratio of
inertial to the viscous forces) and Weber numbers (a measure of the ratio of inertial to the surface
tension forces), drops and bubbles of same size behave practically the same way as each other, both
displaying an oblate ellipsoidal shape.
Bigger drops and bubbles are different. A comparison of bubble and drop literature will reveal
that in the typical scenario, bubbles dimple in the centre [1, 5], while drops more often attain a
cup-like shape [6, 34]. This difference means that drops and bubbles which break up would do
so differently. The dimples of breaking bubbles run deep and pinch off at the centre to create a
doughnut shaped bubble, which will then further break-up, while drops will more often pinch at
their extremities. A general tendency of a drop is to flatten into a thin film which is unlike a bubble.
There could be several other modes of breakup (see [6]) like shear and bag breakup for drops falling
under gravity and catastrophic breakup at high speeds.
Flow past fluid blobs has not been studied for the complete ρr − µr phase plane. Researchers
interested in cloud physics [56–58] have studied the hydrodynamic behaviour of water drops in
air in great detail. It was already established by Spells [4] and others [59, 60] that the internal
circulation shown by Hadamard-Rybczynski formula really does exit. Pruppacher & Beard [61],
and Le Clair et al. [62] found the surface velocity and thus the strength of internal circulation to
quantify this phenomenon. They found the maximum surface velocity to be about wT /25, wherein
wT is the terminal velocity of the drop. Thus it was concluded that the Hadamard-Rybczynski
formula under-predicts the strength of internal circulation. This motivated Le Clair et al. [62] to
consider the boundary layer effects in the vicinity of the drop. However the modification did not
work well above Re ≈ 0.5 due to a wrong assumption in their theory. They assumed the boundary
layer thickness variation to be same as that for a rigid sphere, i.e. δ ∝ Re−0.5 , which does not agree
with the experimental findings. An interesting review on this subject has been presented in a book
by Pruppacher et al. [63].
Three dimensional dynamics
While most earlier computational studies have been axisymmetric or two-dimensional, several threedimensional simulations have been done as well, see e.g. [11, 64–71]. A remarkable set of papers
[65, 66, 72–74] study bubbly flows in which the interaction between the flow and a large number of
bubbles is studied. In particular, turbulent flows can be significantly affected by bubbliness. These
studies typically used one or two sets of Bond number and Galilei number. There have also been
several studies in which the computational techniques needed to resolve this complicated problem
have been perfected [11, 67–69, 71, 75]. Furthermore, [76] reported a numerical technique which
combines volume of fluid and level-set methods and limits the interface to three computational cells.
It is remarkable in its relative simplicity in the extension from two to three-dimensions.
This problem has attracted a large number of experimental studies as well, see e.g. [41, 77].
A library of bubble shapes is available, including skirted, spherical cap, and oscillatory and nonoscillatory oblate ellipsoidal. Approximate boundaries between the regimes where each shape is
displayed are available in [1,25] for unbroken bubbles. In experiments on larger bubbles, the shapes at
5
release are designed to be far from spherical. Secondly, experiments which give a detailed description
of the flow field are few, and accurate shape measurements are seldom available. An important point
is that bubble shapes and dynamics are significantly dependent on initial conditions at release, which
are difficult to control in experiments. One of the objectives of our work is to standardise the initial
conditions, a luxury not easily available to experimenters!
A curious phenomenon, the path instability, has been the subject of a host of experimental
[9,42,78,79], numerical [10,80] and analytical [81,82] studies. This is the name given to the tendency
of the bubble, under certain conditions, to adopt a spiral or zigzagging path rather than a straight
one. After Prosperetti (2004) discovered it in the books of Leonardo Da Vinci, he termed the path
instability as Leonardo’s paradox, since it was not known then why an initially axisymmetric bubble
would take up a spiral or zigzag path. We will demonstrate that path and shape-symmetry are
intimately connected, but only the former has been measured experimentally. It is not easy to
measure the evolving bubble shape [79] and flow field accurately in this highly three-dimensional
regime. Most of the workers embarking on this study find it satisfactory to investigate the effect
of initial bubble diameter on the rising dynamics, and no experimental investigations are available
to our knowledge which study the effects of just the surface tension or viscosity of water on the
bubble rise. We mention one study [77] here where the effect of surfactant concentration on the
oscillatory motion of bubbles is evaluated. The path instability of bubbles was obtained by numerical
stability analysis of a fixed axisymmetric bubble shape by Cano-Lozano et al., and Magnaudet &
Mougin [10,80]. An interesting numerical study due to Gaudlitz & Adams [83] shows hairpin vortices
in the wake of an initially zigzagging bubble.
Bubble rise in non-Newtonian media
The motion of droplets in fluids that exhibit yield stress is important in many engineering applications, including food processing, oil extraction, waste processing and biochemical reactors. Yield
stress fluids or viscoplastic materials flow like liquids when subjected to stress beyond some critical
value, the so-called yield stress, but behave as a solid below this critical level of stress; detailed
review on yield stress fluids can be found in the publications by Bird et al., and Barnes [84, 85].
As a result the gravity-driven bubble rise in a viscoplastic material is not always possible as in the
case of Newtonian fluids but occurs only if buoyancy is sufficient to overcome the material’s yield
stress [86, 87]; the situation is also similar for the case of a settling drop or solid particle [88].
The first constitutive law proposed to describe this material behavior is the Bingham model [89]
which was later extended by Herschel & Bulkley [90] to take into account the effects of shear-thinning
(or thickening). According to this model the material can be in two possible states; it can be either
yielded or unyielded, depending on the level of stress it experiences. As the common boundary of
the two distinct regions the so-called yield surface is approached, the exact Bingham model becomes
singular. In simple flows this singularity does not generate a problem, but, in more complex flows
the discontinuous behaviour of the Bingham model may pose significant difficulties due to the fact
that in most cases the yield surface is not known a priori but must be determined as part of the
solution. Nevertheless, there are examples of successful analysis of two-dimensional flows using this
model at the expense of relatively complicated numerical algorithms [88, 91–93]. A simpler way to
overcome these difficulties is to modify the Bingham constitutive equation in order to produce a
non-singular constitutive law, by introducing a ‘regularization’ parameter [94]. This method has
6
been used with success by several researchers in the past [87,95–98] and when used with caution can
give significant insight in the behaviour of viscoplastic materials.
The motion of air bubbles in viscoplastic materials has attracted the interest of many research
groups in the past. The first reported experimental study on rising bubbles in a viscoplastic material
was done by Astarita & Apuzzo [99] who reported bubble shapes and velocities in Carbopol solutions.
They observed that curves of bubble velocity vs bubble volume for viscoplastic liquids had an abrupt
change in slope at a critical value of bubble volume that depended on the concentration of Carbopol
in the solution, i.e. the yield stress of the material. Many years later, Terasaka & Tsuge [100] used
xanthan gum and Carbopol solutions to examine the formation of bubbles at a nozzle and derived
an approximate model for bubble growth. Dubash & Frigaard [101] also performed experiments
with Carbopol solutions and were able to comfirm the observations of Astarita & Apuzzo [99] on
the existence of a critical bubble radius required to set it in motion and noted that the entrapment
conditions are affected significantly by surface tension. It is also noteworthy that the observed
bubble shapes inside a vertical pipe were different from [99] exhibiting a cusped tail, resembling
much the inverted teardrop shapes often found inside a viscoelastic medium [99, 102, 103]. Similar
bubble shapes have been found in the experimental studies by Sikorski et al. [104] and Mougin et
al. [105], using Carbopol solutions of different concentrations. The latter authors also studied the
significant role of internal trapped stresses within a Carbopol gel on the trajectory and shape of the
bubbles; their findings were in agreement with an earlier study presented by Piau [106].
From a theoretical point of view, Bhavaraju et al. [107] performed a perturbation analysis in the
limit of small yield stress for a spherical air bubble. Stein & Buggish [108] were interested on the
mobilization of bubbles by setting an oscillating external pressure and provided analytical solutions
along with some experimental data; the latter suggested that larger bubbles tend to rise faster than
smaller bubbles at similar amplitudes. Dubash & Frigaard [86] employed a variational method to
estimate the conditions under which bubbles should remain static. These estimations, however,
were characterized as conservative, in the sense that they provide a sufficient but not necessary
condition. A detailed numerical study of the steady bubble rise, using the regularized Papanastasiou
model [109], has been performed by Tsamopoulos et al. [87]. These authors presented mappings of
bubble and yield surface shapes for a wide range of dimensionless parameters, taking into account
the effects of inertia, surface tension and gravity. Moreover, they were able to evaluate the conditions
for bubble entrapment. Their work was followed by the study of Dimakopoulos et al. [93] who used
the augmented Lagrangian method to obtain a more accurate estimation of the stopping conditions.
It was shown that the critical Bingham number, Bn, does not depend on the Archimedes number
in accordance with Tsamopoulos et al. [87], but depends non-monotonically on surface tension. We
should note that in both studies the shape of the bubble near critical conditions could not reproduce
the inverted teardrop shapes seen in experiments [101, 104, 105] and raised questions whether this
is due to elasticity, thixotropy or wall effects. Besides the steady solutions it is also interesting to
investigate the bubble dynamics through time-dependent simulations. This was done by Potapov et
al. [110] and Singh & Denn [111] using the VOF method and the level-set method, respectively. Singh
& Denn [111] considered creeping flow conditions and performed simulations for single and multiple
bubbles. It was shown that multiple bubbles and droplets can move inside the viscoplastic material
under conditions that a single bubble or droplet with similar properties would have been trapped
unable to overcome the yield stress. Potapov et al. [110] also studied the case of a single or two
7
interacting bubbles but also took into account the effect of inertia, albeit for a low Reynolds number.
For the parameter range that they have used the single bubble always reached a quasi-steady state.
We should note at this point that for some cases (e.g. for high values of the Archimedes number)
Tsamopoulos et al. [87] were not able to calculate steady shapes which is probably an indication
that the flow may become time-dependent.
Bubble rise in non-isothermal systems
The variation in temperature of a liquid-gas interface results in the formation of surface tension
gradients which induce tangential stresses, known as Marangoni stresses, driving flow in the vicinity
of the interface. This mechanism is always present in non-isothermal interfacial flows and can be
important in a great variety of technological applications. A characteristic problem where thermal
Marangoni stresses play a significant role is the thermocapillary migration of drops and bubbles.
Much of the work in this field has been reviewed by Subramanian and co-workers [112, 113].
The first reported study on the thermal migration of bubbles can be found in the pioneering
work of Young et al. [114]. These authors conducted experiments on air bubbles in a viscous fluid
heated from below and showed that under the effect of the induced Marangoni stresses small bubbles
move downwards, whereas larger bubbles move in the opposite direction as buoyancy overcomes the
effect of thermocapillarity. Young et al. [114] also provided a theoretical description of the bubble
motion assuming a spherical shape and creeping flow conditions and were able to derive an analytical
expression for the terminal velocity. Following this work, a series of theoretical analyses took into
account the effect of convective heat transfer in the limit of both small and large Reynolds numbers
[115–119]. Balasubramaniam & Chai [120] showed that the solution of Young et al. [114] is an exact
solution of the momentum equation for arbitrary Reynolds number, provided that convective heat
transfer is negligible. These authors also calculated the small deformations of a drop from a spherical
shape.
The main motivation for the aforementioned studies came from microgravity applications and
buoyancy was considered to be negligible. The effect of combined action of buoyancy and thermocapillarity was studied by Merritt et al. [121] employing numerical simulations. Balasubramaniam [122]
presented an asymptotic analysis in the limiting case of large Reynolds and Marangoni numbers,
including the buoyant contribution as well as a temperature varying viscosity. It was shown that
the steady migration velocity, at leading order, is a linear combination of the velocity for purely
thermocapillary motion and the buoyancy-driven rising velocity. Later, Zhang et al. [123] performed
a theoretical analysis for small Marangoni numbers under the effect of gravity and showed that
inclusion of inertia is crucial in the development of an asymptotic solution for the temperature field.
The asymptotic analysis presented by these authors is based on the assumption of a finite velocity
and cannot be used for the case of a stationary bubble. The latter problem has to be analyzed
separately as was done by Balasubramaniam & Subramaninan [124]. The solution of this problem
is complicated by the presence of a singularity failing to satisfy the far-field condition. Yariv &
Shusser [125] introduced an exponentially small artificial bubble velocity as a regularization parameter to account for the inability of the asymptotic expansion to satisfy the condition of exact
bubble equilibrium. They were able to evaluate the correction for the hydrodynamic force exerted
on the bubble including convective heat transfer; this correction was shown to be independent of
the regularization parameter.
8
A great variety of numerical methods have been proposed in order to take into account the
effect of surface deformation. These range from boundary-fitted grids [126, 127], to the level-set
method [128, 129], the VOF method [130], diffuse-interface methods [131] and hybrid schemes of the
Lattice-Boltzmann and the finite difference method [132]. It was shown by Chen & Lee [126] that
surface deformation of gas bubbles reduces considerably their terminal velocity. The same effect was
found also in the case of viscous drops by Haj-Hariri et al. [128]. Later, Welch [127] demonstrated
that as the capillary number increases and the bubble deformation becomes important, the bubbles
do not reach a steady state terminal velocity. As was shown by Hermann et al. [133], the assumption
of quasi-steady-state is not valid also for large Marangoni numbers. The latter finding was very
recently confirmed by Wu & Hu [134, 135].
Most of the studies mentioned above concern the motion of a single bubble or drop in an unconfined medium. Acrivos et al. [136] studied systems of multiple drops in the creeping flow limit and
showed that the drops do not interact. However, when inertial effects are included it was shown by
Nas & Tryggvason [137] and Nas et al. [138] that there are strong interactions between the droplets.
The thermocapillary interaction between spherical drops in the creeping flow limit was discussed by
several authors [139, 140]. In the vicinity of a solid wall, the drop migration velocity is affected by
the hydrodynamic resistance due to the presence of the wall as well as by the thermal interaction
between the wall and the drop. Meyyappan & Subramanian [141] examined the motion of a gas
bubble close to a rigid surface with an imposed far-field temperature gradient and found that the
surface exerts weaker influence in the case of parallel motion than in the case of motion normal to
it. Keh et al. [142] investigated the motion of a spherical drop between two parallel plane walls
and found that the wall effect could speed up or slow down the droplet depending on the thermal
conductivity of the droplet and the imposed boundary conditions at the wall. Chen et al. [143] considered the case of a spherical drop and studied the thermocapillary migration inside an insulated
tube with an imposed axial temperature gradient. They found that the migration velocity in the
tube never exceeds the value in an infinite medium due to the hydrodynamic retarding forces that
are being developed.
Very recently, Mahesri et al. [144] extended the work of Chen et al. [143] to take into account
the effect of interfacial deformation. It was found that as in the case of the spherical drop the
migration velocity of the confined drop is always lower than that of an unbounded drop. Brady et
al. [145] presented numerical simulations of a droplet inside a rectangular box and showed that for
low Marangoni numbers the drop rapidly settles to a quasi steady state whereas for high Marangoni
numbers the initial conditions affect significantly the behaviour of the droplet. In the case of severe
confinement inside a tube, the drop can become quite long. Such a case was studied by Hasan &
Balasubramaniam [146] and Wilson [147] who focused their attention on the thin film region away
from the drop ends, and were able to derive a relation between the migration velocity and the film
thickness. Later, Mazouchi & Homsy [148, 149] used lubrication theory to determine the liquid film
thickness and migration velocity for the case of a cylindrical and polygonal tube, respectively.
It is well known that the surface tension of common fluids, such as air, water, and various oils,
decreases linearly with increasing temperature; all of the above mentioned studies have considered
such fluids. In this thesis, we are interested in the thermocapillary migration of a deformable bubble
inside a cylindrical tube filled with liquids that exhibit a non-monotonic dependence of the surface
tension on temperature. In particular, these so-called “self-rewetting” fluids [150–154], which are
9
non-azeotropic, high carbon alcohol solutions, have parabolic surface tension-temperature curves
with well-defined minima; the parabolicity of these curves increases with alcohol concentration.
These fluids were first studied by Vochten & petre [150] who observed the occurrence of the minimum
in surface tension with temperature in high carbon alcohol solutions. Petre & Azouni [151] carried
out experiments that involved imposing a temperature gradient on the surface of alcohol aqueous
solutions, and used talc particles to demonstrate the unusual behaviour of these fluids. Experimental
work on these fluids was also carried out under reduced-gravity conditions by Limbourgfontaine et
al. [152]. The term “self-rewetting” was coined by Abe et al. [155] who studied the thermophysical
properties of dilute aqueous solution of high carbon alcohols. Due to thermocapillary stresses,
and the shape of the surface tension-temperature curve, the fluids studied spread “self-rewet” by
spreading spontaneously towards the hot regions, thereby preventing dry-out of hot surfaces and
enhancing the rate of heat transfer.
Due to the abovementioned properties, “self-rewetting” fluids were shown to be associated with
substantially higher critical heat fluxes in heat pipes compared to water [156–158]. Savino et al. [153]
illustrated the anomalous behaviour of self-rewetting fluids by performing experiments to visualise
the behaviour of vapour slugs inside wickless heat pipes made of pyrex borosilicate glass capillaries.
They found that the size of the slugs was considerably smaller than that associated with fluids
such as water. More recently, work on self-rewetting fluids was extended to microgravity conditions
for space applications on the International Space Station. Savino et al. [154], and Hu et al. [159]
demonstrated that the use of these fluids within micro oscillating heat pipes led to an increase in
the efficiency of these devices. In a slightly different context, it was very recently shown that the
presence of a minimum in surface tension can also have a significant impact on the dynamics of the
flow giving rise to very interesting phenomena such as the thermally induced “superspreading” [160].
Phase change in falling drops
Multiphase flows with phase-change are ubiquitous and have several industrial applications, for
instance, energy generation, manufacturing, and combustion. Phase change in interfacial flows may
occur due to chemical reaction, evaporation, melting, etc. In the past, several researchers have
discussed chemically reacting flows [161, 162], which is not the subject of present discussion. In this
case, when the temperature and pressure are favourable for the reaction to occur, a change in phase
can take place and mass of a new species (product of chemical reaction) increases at the expense of
existing ones (reactants). In the present study, we discuss the dynamics of a blob of a heavier fluid
falling under the action of gravity inside a lighter fluid initially kept at a higher temperature, and
undergoing evaporation.
If the static pressure at the interface of both the fluids is less (more) than the saturation vapour
pressure at the given temperature, evaporation (condensation) can occur at the interface. Due
to the relevance in many industrial applications, such as spray combustion, film evaporation and
boiling, and naturally occurring phenomena in oceans and clouds, several investigations on evaporation/condensation have been conducted [163–165]. The phase change phenomena occurring during
evaporation/condensation depend on several factors, including the environmental conditions. Thus,
computationally, it is an extremely difficult problem, and most of the previous studies are experimental in nature.
Recently, with the advent of computational fluid dynamics due to the development of powerful
10
supercomputers, many researchers have tried to incorporate the phase-change models to their multiphase flow solvers [166–168]. Although accurate in describing the interface, the method of Esmaeeli
& Tryggvason [167] is computationally expensive because of the explicit interface tracking and usage
of linked lists for describing the elements on the interface. Workers in this area also have tried using
one-fluid approaches, like volume-of-fluid (VOF), diffuse-interface and level-set methods to compute
phase-change phenomena (see e.g Schlottke & Weigand [168]).
In spite of the above-mentioned work, the effect of viscosity and density ratios with temperature dependent fluid properties have not been investigated on falling drop undergoing evaporation.
Recently, the dynamics of bubbles (ρr < 1) and drops (ρr > 1) has been studied by Tripathi et al.
1.3
Outline of the thesis
The present work is an attempt to study some aspects of the abovementioned phenomena. Many
a times, no clear question was there to start with, but the questions emerged as we observed the
unexcpected and expected solutions of the Navier-Stokes equations, which are bubbles and drops.
The next chapter (Chapter 2) discusses the general formulation and numerical methods common
to the entire work. In most of the axisymmetric simulations, we have employed a bespoke finitevolume diffuse-interface code and compared it with the results obtained from the volume-of-fluid
code. Gerris alone has been used in all of the three-dimensional simulations because of its adaptive
mesh refinement feature. Moreover, extensive validations have been presented in this chapter.
In chapter 3, the similarities and differences between a rising bubble and a falling drop are
investigated theoretically and numerically. We also investigate the exclusive behaviour that bubbles
and drops exhibit. Scaling relationships and numerical simulations show a bubble-drop equivalence
for moderate inertia and surface tension, so long as the density ratio of the drop to its surroundings
is close to unity. When this ratio is far from unity, the drop and the bubble are very different. We
show that this is due to the tendency for vorticity to be concentrated in the lighter fluid, i.e. within
the bubble but outside the drop. As the Galilei and Bond numbers are increased, a bubble displays
under-damped shape oscillations, whereas beyond critical values of these numbers, over-damped
behaviour resulting in break-up takes place. The different circulation patterns result in thin and
cup-like drops but bubbles thick at their base. These shapes are then prone to break-up at the sides
and centre, respectively.
In chapter 4, we present the results of one of the largest numerical study of three-dimensional
rising bubbles and falling drops. Herein, we study bubbles rising due to buoyancy in a far denser
and more viscous fluid. We show that as the size of the bubble is increased, the dynamics goes
through three abrupt transitions from one known class of shapes to another. A small bubble will
attain an axially symmetric equilibrium shape dictated by gravity and surface tension, and travel
vertically upwards at a terminal velocity thereafter. A bubble larger than a first critical size loses
axial symmetry. We show that this can happen in two ways. Beyond the next critical size, it breaks
up into a spherical cap and many satellite bubbles, and remarkably, the cap regains axial symmetry.
Finally, a large bubble will prefer not to break up initially, but will change topologically to become
an annular doughnut-like structure, which is perfectly axisymmetric.
Chapter 5 presents a study of the buoyancy-driven rise of a bubble inside an infinite viscoplastic
medium, assuming axial symmetry. To account for the viscoplasticity, we consider the regularized
11
Herschel-Bulkley model. We employ the Volume-of-Fluid method to follow the deforming bubble
along the domain. Our results indicate that in the presence of inertia and in the case of weak
surface tension the bubble does not reach a steady state and the dynamics may become complex for
sufficiently high yield stress of the material.
Rising bubble in a self-rewetting fluid with a temperature gradient imposed on the container
walls has been studied in Chapter 6. To account for the non-monotonicity of surface tension we
consider a quadratic dependence on temperature. We examine the Stokes flow limit first and derive
conditions under which the motion of a spherical bubble can be arrested in self-rewetting fluids even
for positive temperature gradients. We then employ a diffuse-interface method [169] to follow the
deforming bubble along the domain in the presence of inertial contributions. Our results indicate
that for self-rewetting fluids, the bubble motion departs considerably from the behaviour of ordinary
fluids and the dynamics may become complex as the bubble crosses the position of minimum surface
tension. As will be shown below, under certain conditions, the motion of the bubble can be reversed,
and then arrested, or the bubble can become elongated significantly.
Finally, a preliminary study of the effect of volatility of liquids have been presented for the case of
falling evaporating drops in Chapter 7. To this end, the open-source code, gerris, created by Popinet
[170] is used and a phase-change model, similar to that employed by Schlottke & Weigand [168], is
incorporated to gerris in order to handle the complex phenomena occurring at the interface. We
found that the vapour is generated more on the regions of the interface with relatively high curvature,
and the vapour generation increases with breakup of the drop. Furthermore, a competition between
volatility and the dynamics governs the vapour generation in the wake region of the drop. This is
an ongoing work, and only few of the results are presented in this Chapter.
1.4
Future work
As mentioned before, a large number of studies have focused upon the dynamics of single drops
and bubbles from theoretical, experimental and numerical analyses. With the growth of computing
speed, researchers [66] have developed techniques to simulate hundreds of bubbles simultaneously in
a flow. Complex problems such as these are completely three dimensional and it is very difficult to
visualize the flow field experimentally. Moreover, flows involving thermal gradients, non-Newtonian
fluids, evaporation, moving contact lines, surfactants and other complexities change the dynamics
drastically. Such dynamics is often not possible or challenging to study theoretically or numerically,
while being of great importance to the industries and natural physics. Apart from this, the simple
coalescence and breakup of drops and bubbles is less understood and the studies are mostly experimental. To gain a better understanding of these processes, better numerical techniques have to
be devised. The present work acknowledges the three-dimensional nature of bubble and drop motion, and attempts to incorporate various complexities into the dynamics of a single bubble/drop.
This work could be naturally extended to include multiple bubbles/drops, contact lines and other
additional physics mentioned above.
12
Chapter 2
Formulation and numerical
methods
The bubble and drop dynamics has been investigated numerically and theoretically. In this chapter
we describe the numerical methods employed, and their validations. The flow is assumed to be
incompressible and in the case of air bubbles in liquids, the height of rise of air bubble is assumed
to be small enough to cause any change in the bubble volume. Axisymmetric and three-dimensional
simulations have been carried out in this work, the domains of computation for which are depicted
in Fig. 2.1. The formulation and validation present in this chapter are contained in our published
works [19, 22–24] .
2.1
Formulation
The equations of mass, momentum and energy conservation which govern the flow can be respectively
written as:
∇ · (ρu) = −ṁv ,
(2.1)
�
�
�
∂u
ρ
+ u · ∇u = −∇p + ∇ · µ(∇u + ∇uT ) + Fb + Fs ,
∂t
(2.2)
∂(ρcp T )
+ ∇ · (ρcp uT ) = ∇ · (λ∇T ) − Δhv ṁv ,
∂t
(2.3)
�
wherein u, p and T denote the velocity, pressure, and temperature fields of the fluid, respectively; t
represents time; Fb and Fs are the additional body and surface forces, respectively. Here, ṁv is the
mass source term per unit volume, non-zero only at the interface. The sign convention is such that
a positive ṁv is associated with evaporation and a negative ṁv with the condensation. Note that
ṁv is set to zero when there is no phase change, therefore the source term has been considered only
in Chapter 7. In addition to this, the advection-diffusion equation for the vapour volume fraction is
as follows
13
(a)
(b)
Figure 2.1: Schematic diagram of the simulation domains considered to solve (a) axisymmetric
(dot-dashed line represents the axis of symmetry), and (b) three-dimensional rising bubble problem.
Bubble size is not to scale. Similar domains are considered for falling drop problem with inverted
gravity. The domain is considered to have a square base of size, L in three-dimensions and a circular
base of diameter, L in cylindrical coordinates. The outer and inner fluids are designated by ‘o’ and
‘i’, respectively. The height of the domain, H is chosen according to the expected dynamics of the
bubble/drop.
∂cv
ṁv
+ ∇ · (cv u) = ∇ · (Dav ∇cv ) +
,
∂t
ρv
(2.4)
In Eqs. (2.1)–(2.4), ρ, µ and cv are the density, viscosity and the volume fraction of the vapour,
respectively; Dav represents binary diffusion coefficient of the gas mixture. The specific latent heat of
vaporization, specific heat and thermal conductivity are denoted by Δhv , cp and λ, respectively. In
addition to this, for the non-isothermal cases i.e. in Chapters 6 and 7, the hot and cold temperatures
are designated by Th and Tc , respectively.
The body and surface forces in Eq. (2.2), in the absence of electric, magnetic, or any other forces
except gravitational and surface tension forces, can be written as
Fb = −ρg�ez ,
(2.5)
Fs = δ [σκn̂ + ∇s σ] ,
(2.6)
and
wherein, δ(= |∇cb |) is the dirac distribution function, n̂ is the unit normal to the interface, κ is
14
the local curvature of the interface separating the two phases, �ez is the unit vector in the vertically
upward direction, and σ is the coefficient of surface tension. For non-isothermal systems (discussed
in Chapters 6 and 7), the surface tension coefficient is assumed to be a quadratic function [153] of
temperature (where a choice of β2 = 0 yields a linear dependence of surface tension on temperature)
σ = σ0 − β1 (T − Tc ) + β2 (T − Tc )2 ,
dσ
where β1 ≡ − dT
|Tc and β2 ≡
1 d2 σ
2 dT 2
(2.7)
| Tc .
The parameters β1 and β2 are set to zero for isothermal systems Furthermore, a one-fluid approach is followed and only 6 equations (Eqs. (2.1)–(2.4)) are solved for all the fluid phases. Therefore, the fluid properties are defined to be dependent of a colour function (volume fraction) (ca )
which varies from 0 to 1 in a thin region between the two phases for a given pair of fluids. The
advection equation for the colour function is discussed in the next section. Density (ρ), viscosity
(µ) and thermal conductivity (λ) are volume averaged between the two phases, whereas the specific
heat capacity (cp ) is mass averaged as follows
ρ = (1 − ca )ρi + (ca − cv )ρo + cv ρv ,
(2.8)
µ = Gi (T )(1 − ca )µi + Go (T ) [(ca − cv )µo + µv cv ] ,
(2.9)
cp =
λ = (1 − ca )λi + (ca − cv )λo + cv λv ,
�
�
(c −cv )cp,o ρo +cp,v cv ρv
ρo
(1 − ca )cp,i ρi + ca a(ca −c
v )cp,o ρo +ρv cv
ρ
(2.10)
,
(2.11)
wherein ρo , ρi and ρv are the densities of outer fluid (fluid ‘o’), inner fluid (fluid ‘i’), and pure water
vapour, respectively; µo , µi , and µv are the viscosities of these fluids, respectively. Similarly the
thermal conductivities and specific heat capacities for these fluids are denoted by λo , λi , and λv , and
cp,o , cp,i , and cp,v , respectively. The functions Gi (T ) and Go (T ) allow the viscosity to be temperature
dependent. If the inner and outer fluids are liquid and gas, respectively, the Reynolds’ [171] and
Sutherland’s [172] models are used to express the variation in viscosity of the two phases with
temperature
−
�
T −T c
Th −Tc
�
Gi (T ) = e
,
�
�3/2
T − Tc
Go (T ) = 1 +
.
Th − Tc
(2.12)
(2.13)
The following scalings are employed in order to render the governing equations dimensionless:
(x, y, z) = R (�
x, y�, z�) , t =
R�
t, (u, v, w) = V (�
u, v�, w),
� p = ρo V 2 p�,
V
� o, σ = σ
µ=µ
�µo , ρ = ρ�ρo , cp = �
cp cp,o , λ = λλ
�σ0 , T = T�(Th − Tc ) + Tc ,
β1 =
�
�
σ0 �
σ0
˙ v ρA g/R, δ = δ ,
�2 , ṁv = m
M1 , β 2 =
M
�
Th − Tc
(Th − Tc )2
R
where the tildes designate dimensionless quantities, the velocity scale is V =
√
(2.14)
gR, and σ0 is the
surface tension at the liquid gas interface at reference temperature Tc . After dropping tildes from
15
all dimensionless terms, the non-dimensional governing equations are given by
ρ
�
∇ · (ρu) = −ṁv ,
�
(2.15)
�
�
∂u
1
+ u · ∇u = −∇p +
∇ · µ(∇u + ∇uT ) − ρ�ez + Fs ,
∂t
Ga
∂(ρcp T )
1
ṁv
+ ∇ · (uρcp T ) =
∇ · (λ∇T ) −
,
∂t
GaP r
Ja
∂cv
+ ∇ · (cv u) = ∇ ·
∂t
�
1
∇cv
Pe
�
+
ṁv
,
ρrv
(2.16)
(2.17)
(2.18)
where Ga(≡ ρo V R/µo ), P e(≡ V R/Dav ), Ja(≡ cp,o (Th − Tc )/Δhv ), P r(≡ ρo cp,o /λo ) and Fs denote
the Galilei number, the Peclet number, the Jackob number, the Prandtl number and the dimensionless surface tension force, respectively. The dimensionless fluid properties can be written as,
ρ = (1 − ca )ρro + [(ca − cv ) + ρrv cv ] ,
�
�
µ = e−T (1 − ca )µro + 1 + T 3/2 [(ca − cv ) + µrv cv ] ,
λ = (1 − ca )λro + [(ca − cv ) + λrv cv ] ,
�
�
(c −cv )+cp,rv cv ρrv
(1 − ca )cp,ro ρro + ca a(ca −c
)+ρ
c
v
rv v
cp =
,
ρ
(2.19)
(2.20)
(2.21)
(2.22)
where ρro = ρi /ρo , ρrv = ρv /ρo , µro = µi /µo , µrv = µv /µo , λro = λi /λo , λrv = λv /λo , cp,ro =
cp,i /cp,o and cp,rv = cp,v /cp,o , respectively. The dimensionless surface tension force, Fs becomes
Fs =
δ
[σκ�n + ∇s σ] ,
Bo
(2.23)
where Bo(≡ ρo gR2 /σ0 ) is the Bond number. To close the problem, we define the initial and boundary
conditions now. The relevant initial and boundary conditions used in this work are discussed in each
chapter for the respective problems. A list of dimensionless groups frequently used in bubble and
drop studies relevant to our work are listed below. Some of the dimensionless numbers listed in the
table will be discussed later.
2.2
Numerical methods
In this work, we have used two numerical techniques, namely difuse interface method and volume
of fluid method to capture the interface separating the pairs of fluids. Both the methods and their
validation have been presented below. Volume of fluid and diffuse interface methods, both, belong
to the class of interface capturing methods, which means that the interface is not explicitly tracked,
but is reconstructed by means of a colour function. Thus the fluid properties are smeared in a region
containing a few computational cells. Volume of fluid method prevents the smearing of the interface
by reconstructing the interface at every time step, and thus providing a sharp interface. These two
methods are different in how the colour function is advected (or diffused), thus determining how the
interface is defined. These will be described briefly in the following text.
16
ρr (ρro )
µr (µro )
λr (λro )
cp,ro (cp,r )
ρrv
µrv
λrv
cp,rv
Ga
Bo
Mo
Pr
Pe
Re
We
Bn
m
M1
M2
ρi /ρo
µi /µo
λi /λo
cp,i /cp,o
ρv /ρo
µv /µo
λv /λo
cp,v /cp,o
ρo R3/2 g 1/2 /µo
ρo gR2 /σ
gµ4o /ρo σ 3
cp,o µo√
/λo
Dav /R gR
ρo U R/µo
ρo U 2 R/σ
√
τ0 R/µ0 gR
n−1
2
(g/R)
� dσ
�
− dT Tc ΔT /σ0
� 2 �
1 d σ
ΔT 2 /σ0
2 dT 2
Density ratio of inner to outer fluid
Viscosity ratio of inner to outer fluid
Thermal conductivity ratio of inner to outer fluid
Specific heat capacity ratio of inner to outer fluid
Density ratio of vapour to outer fluid
Viscosity ratio of vapour to outer fluid
Thermal conductivity ratio of vapour to outer fluid
Specific heat capacity ratio of vapour to outer fluid
Gallilei number
Bond number
Morton number
Prandtl number
Peclet number for diffusion of vapour in dry air
Reynolds number based on velocity, U
Weber number based on velocity, U
Bingham number
Fluid consistency for Bingham fluid
First derivative of surface tension coefficient
Tc
Second derivative of surface tension coefficient
Table 2.1: Frequently used dimensionless groups relevant to the present work.
2.2.1
Diffuse-interface method
In the diffuse-interface framework, the interface is captured by tracking the volume fraction (colour
function) of the outer fluid, ca . A thin region where the two fluid may mix is defined at the interface,
such that the advection-diffusion equation for the volume fraction of the outer fluid becomes
∂ca
1
+ ∇ · (uca ) −
∇ · (M ∇φ) = 0,
∂t
P ed
(2.24)
where P ed is a very large number of the order of �−1 , wherein � is of the order of grid size; M
is the mobility defined as ca (1 − ca ); φ is the chemical potential of the fluid system defined as the
change in free energy with respect to ca . Additionally, the dimensionless force per unit volume Fs
(excluding the tangential force) in the Navier-Stokes equation (Eq. (2.16)) is obtained as,
Fs =
φ∇ca
.
Bo
(2.25)
The pressure and the volume fraction of the outer fluid are defined at the cell-centres, and the velocity
components are defined at the cell faces, respectively. In our code a fifth order weighted-essentiallynon-oscillatory (WENO), and central difference schemes are used to discretize the advective and
diffusive terms appearing in the advection-diffusion equation for the volume-fraction, respectively.
In order to achieve second-order accuracy, the Adams-Bashforth and the Crank-Nicholson methods are used to discretize the advective and dissipation terms in Eq. (2.16), respectively. The
implementation is similar to that discussed in the work of Ding et al. [169].
17
2.2.2
Volume of fluid method: Gerris
In the volume of fluid (VOF) framework, the surface tension force is included as a force per unit
volume in the Navier-Stokes equation in the following manner
Fs =
δ
(κσn̂ + ∇s σ) .
Bo
(2.26)
The curvature, κ is calculated using a generalized height-function method implemented in gerris.
The volume fraction of the outer fluid (ca ) is advected with the local fluid velocity as follows:
∂ca
+ u · ∇ca = 0.
∂t
(2.27)
A piecewise linear interface calculation (PLIC) is employed to reconstruct a sharp interface at
every time-step from the volume fraction data. This avoids smearing of the interface due to the
numerical diffusion.
We have used an open-source flow solver, gerris, based on VOF framework. Gerris uses a generalized height-function method for calculating the curvature of the interface, thus improving the
accuracy of the surface tension force calculation for the VOF methods. Level set and front-tracking
methods used to be considered as the state-of-the-art for high fidelity interfacial flow simulations, but
with the implementation of height-functions [173], VOF methods are getting the recognition as stateof-the-art, again. Moreover, gerris uses a balanced force algorithm for inclusion of surface-tension
force in the Navier-Stokes equations, which combined with the height-function implementation reduces the amplitude of spurious velocity to machine error (i.e. lowest possible error achievable on
a computer calculation). Another feature of gerris, the dynamic adaptive mesh refinement, allows
one to cluster the grid more in the desired regions dynamically, thus saving the computation time
remarkably.
2.3
Validation
A number of tests have been performed to check the accuracy of gerris [174] and the diffuse-interface
method [169]. We present below a few validation cases relevant to the bubble and drop dynamics.
First, we check the domain and grid dependence of results for our numerical methods. The results
have been obtained for both, diffuse interface method and volume of fluid method, however only one
set of validation exercises (for gerris) are shown here. The results for diffuse-interface method have
been found to compare well with the results obtained using gerris, however the computation cost
was several times that for gerris.
2.3.1
Grid convergence test
We start with a test against the discretization errors, such that the results would be independent of
the grid size used. The shape of the bubble at t = 4 and 7 for two different grids (Δx = Δz = 0.029
and 0.015) in a computational domain of size 16 × 48 are shown in Fig. 2.2. It is found that a
square grid with Δx = Δz = 0.029 is enough to get results to within 0.1% accuracy. A similar test
is presented for three-dimensional bubble in Fig. 2.3. In Fig. 2.3 we show bubble shapes obtained
using two different grids. It reveals that grid convergence is achieved for simulations having the
18
smallest grid less than 0.029. Thus all the three-dimensional simulations have been conducted using
this grid size.
(a)
(b)
Figure 2.2: Effect of grid refinement on the shape of the bubble at (a) t = 4, and (b) t = 7 for
Ga = 3.09442, Bo = 29, ρr = 7.4734 × 10−4 and µr = 8.1536 × 10−6 . The solid and dot-dashed lines
correspond to the results obtained using Δx = Δz = 0.015 and 0.029, respectively.
(a)
(b)
Figure 2.3: Grid convergence test. The shapes of the bubble for two different grid sizes at t = 3 is
shown. The parameter values are Ga = 70.7, Bo = 200, ρr = 10−3 and µr = 10−2 . The smallest
grid sizes in panels (a) and (b) are about 0.029 and 0.015, respectively. The grid refinement criteria
used here are based on the vorticity magnitude and the gradient of volume fraction (ca ).
2.3.2
Effect of domain size
The effect of domain size is investigated in Fig. 2.4, where the axial rise velocity of the spiralling
bubbles is plotted versus time for two different values of dimensionless base width, L. It is found
that doubling the size of the lateral cross-section of the domain (i.e doubling L) has a negligible
effect (less than 0.2% on the rise velocity as shown in Fig. 2.4) on the flow dynamics. Although the
results are somewhat different for the two domain sizes at later times for these parameter values,
the results do not differ significantly so as to change the shape regime. Thus, a domain of L = 30 is
found to be satisfactory for a qualitative study of shape regimes of bubbles. We also found that the
19
shapes of the bubble for both the cases are close to each other. In addition to this, it is to be noted
here that [9] and [34] considered L to be about 30 in their experimental and numerical studies of the
path instability of rising bubble, and fragmentation process of a falling drop, respectively. Thus a
computational domain 30 × 30 × 120 is used for three-dimensional simulations of bubbles and drops.
All the dimensions are scaled with the radius of the bubble.
2.5
2
1.5
w
L
1
30
60
0.5
0
0
3
6
t
9
12
15
Figure 2.4: Effect of domain size on upward velocity, w of a bubble exhibiting a spiralling motion.
The dashed and solid lines represent domains of base width, L = 30 and 60, respectively. The
dimensionless parameters used for the simulations are: Ga = 100, Bo = 0.5, ρr = 10−3 and
µr = 10−2 .
For axisymmetric simulations, a half-domain size of 16 × 48 was found to be sufficient to simulate
bubbles and drops in an unconfined medium for the parameter values considered in the present work
i.e. for Ga > 2. Fig. 2.5 shows the results for two different domain sizes for the parameter values:
Ga = 3.09442, Bo = 29, ρr = 7.4734 × 10−4 and µr = 8.1536 × 10−6 . It is concluded that a domain
size of 16 × 48 can be used to simulate bubble rise in an unbounded surrounding fluid medium.
Figure 2.5: Effect of domain size on the bubble shape at t = 4, and t = 7 (left to right) for
Ga = 3.09442, Bo = 29, ρr = 7.4734 × 10−4 and µr = 8.1536 × 10−6 . The solid and dot-dashed lines
correspond to computational domains 8 × 24 and 16 × 48, respectively. The results are generated
using square grid of Δx = Δz = 0.015.
20
(a)
(b)
(c)
(d)
Figure 2.6: Comparison of the shape of the bubble obtained from our simulation (shown by solid
red line) with those from the level-set simulations of Sussman & Smereka [5] (dashed line) at various
times: (a) t = 0, (b) t = 0.8, (c) t = 1.6 and (d) t = 2.4. The parameter values are Ga = 100,
Bo = 200, ρr = 0.001 and µr = 0.01. The transition to toroidal bubble (topological change) is
observed at t = 1.6, which matches exactly with the result of Sussman & Smereka [5].
2.3.3
Comparison with numerical simulations
Comparison with Sussman & Smereka [5]
In order to validate our code, in Fig. 2.6 we compare our results obtained for Ga = 100, Bo = 200,
ρr = 0.001 and µr = 0.01 with those of Sussman & Smereka [5], who studied the fluid dynamics of
rising bubble with topology change in the framework of a level-set approach. The dashed lines on the
left hand side of each panel are the results from Sussman & Smereka [5], whereas the present results
are plotted by solid red lines on the right hand side of the panels. It can be seen that the topology
changes observed in our simulation agree excellently with the results of Sussman & Smereka [5].
Comparison with Han & Tryggvason [6]
The simulation domain was taken to be the same as that of Han & Tryggvason [6], i.e. 10 × 30 and
the grid size was taken to be Δx = Δz = 0.015 for the parameter values stated in Fig. 2.7. The
dimensionless time at which the drop breaks up (tbr ≈ 25.0) is in agreement with the that of [6].
The oscillations in velocity are well replicated too.
2.3.4
Comparison with the experimental result of Bhaga & Weber [1]
In Fig. 2.8, we compare the shape of the bubble obtained from our simulation (shown by the red
line) with the corresponding results given in the experiment of Bhaga & Weber [1] (shown by the
gray scale picture). The parameter values used to generate this figure are Ga = 3.09442, Bo = 29,
ρr = 7.4734 × 10−4 and µr = 8.1536 × 10−6 , which are the values at which the experimental shape
is presented by Bhaga & Weber [1], after suitable transformation, as follows:
Ga =
�
Bo3BW
64M oBW
� 14
,
and
21
Bo =
BoBW
,
4
(2.28)
Figure 2.7: Variation of upward velocity of center of gravity of the drop with time for Ga = 219.09,
Bo = 240, ρr = 1.15 and µr = 1.1506. The dashed line is the result due to Han & Tryggvason [6]
and the solid line is the result of the present simulation. The figure is plotted till breakup. It
could be seen that the results match to a very good accuracy, however small deviations can be seen
which may be attributed to the differences in the interface tracking/capturing methods in the two
simulations.
where M oBW = gµ4o /ρo σ 3 , BoBW = 4gR2 ρo /σ, where the subscript BW refers to Bhaga & Weber
[1]. It can be seen that the shape of the bubble obtained from our numerical simulation is in
qualitative agreement with the experimentally obtained bubble of Bhaga & Weber [1]. Note that
a dimple at the bottom of the bubble (if one exists) in the experiment will not be visible in this
photograph, and would appear as a horizontal edge at the bottom.
Figure 2.8: Comparison of the shape of the bubble obtained from the present diffuse interface
simulation (shown by red line) with that of Bhaga and Weber [1]. The parameter values are Ga =
3.09442, Bo = 29, ρr = 7.4734 × 10−4 and µr = 8.1536 × 10−6 . The dimple is not clearly visible
in the experimental result because it is hidden by the periphery of the bubble. Also, the apparent
dimple seen from the side view is different from the actual dimple because of the refraction due to
the curved bubble surface.
Next we validated the volume of fluid code (gerris) by comparing the results obtained using it
with the experimental results of Bhaga & Weber [1] in Fig. 2.9 for different values of Ga and Bo.
Furthermore, the streamline pattern in the wake of the bubble for different Ga and Bo is also plotted
22
in Fig. 2.10. It can be seen that the results are in good agreement.
Figure 2.9: Comparison of terminal shape of the bubble with Bhaga & Weber [1] (left to right):
Ga = 2.316, Bo = 29, Ga = 3.094, Bo = 29, Ga = 4.935, Bo = 29, and Ga = 10.901, Bo = 84.75.
The results in the bottom row are obtained from the present three-dimensional simulations. The
results from the corresponding axisymmetric simulations are shown by red lines in the top row.
It is also noted here that this solver has been extensively validated with theoretical results for
capillary instability of a cylindrical liquid coloumn [175], and linear instability theory for a two phase
mixing layer and a two-dimensional drop in a shear flow [174]. More test cases are available at the
webpage for gerris test-suite (http : //gf s.sourcef orge.net/tests/tests/).
2.3.5
Comparison with analytical results
The next two comparisons are not so much for validation, since the analytical results are for idealised
limits, but a demonstration that the analytical results are valid in a range of parameters lying near
the idealized limits.
In the Hadamard [7] limit
By balancing the drag force with the weight of the bubble/drop, and neglecting the inertial and
surface tension forces, Rybczinsky [54] and Hadamard [7] analytically derived an expression for
terminal velocity (famously known as Hadamard-Rybczinsky equation), which is given by
Vt =
2 R2 g(r − 1)ρ1
3
µ1
�
1+m
2 + 3m
�
.
(2.29)
Thus the dimensionless terminal velocity can be written as:
2Ga(1 − r)
V�t =
3
�
1+m
2 + 3m
�
.
(2.30)
In an example simulation with the parameters Ga = 0.1, Bo = 0.1, ρr = 10−3 and µr = 10−2 ,
we found that the terminal velocity is Vt = 0.0329 for a domain of size 16 × 48 and larger. The
corresponding dimensionless velocity obtained using Eq. (2.30) is 0.0331. In Fig. 2.11 and 2.12, we
plot streamlines obtained from the Hadamard solution and our numerical simulations for Ga = 0.1.
The qualitative similarity is apparent. The small discrepancies may be attributed to the fact that
at Re → 0 an infinitely large computational domain is required for accurate solutions, and also to
23
Figure 2.10: Comparison of streamline pattern in the wake of the bubble with Bhaga & Weber [1]
for the following dimensionless parameters: (a) Ga = 7.9, Bo = 17, (b) Ga = 9, Bo = 21, (c)
Ga = 12.6, Bo = 17, (d) Ga = 17.8, Bo = 27, (e) Ga = 21.9, Bo = 17, and (f) Ga = 33.2, Bo = 11.
The rest of the parameter values are ρr = 7.747 × 10−3 and µr = 10−2 . The results on the left hand
side and right hand side of each panel are from the present simulations and Bhaga & Weber’s [1]
experiments, respectively.
small deviations from a spherical shape at our finite surface tension values, whereas the analytical
result assumes a perfectly spherical bubble.
Comparison with potential flow solution
In Table 2.2 we compare the terminal velocities obtained from our numerical simulations for Ga = 50
and 100 with those obtained from the analytical solution of [14], who studied a rising spherical cap
bubble in the potential flow regime. The other parameter values are Bo = 10, ρr = 0.001 and
µr = 0.01. In this parameter range, the computationally obtained bubble resembles a spherical cap.
It can be seen that the potential flow assumption is able to predict the terminal velocity qualitatively.
Cases
a
b
Joseph [14]
0.864
0.882
Present work
0.883
0.906
Table 2.2: Comparison of the terminal velocities by Joseph [14] and the present work for the parameter values: (a) Ga = 50, and (b) Ga = 100. The rest of the parameter values are Bo = 10,
ρr = 0.001 and µr = 0.01.
24
Figure 2.11: Streamlines obtained from the analytical result for Hadamard flow (Re → 0) in a
spherical bubble (left hand side), and volume of fluid simulation with gerris (right hand side) for a
domain of half-width 16R. The dimensionless parameters are: Ga = 0.1, Bo = 0.1, ρr = 10−3 and
µr = 10−2 .
0.04
0.035
0.03
w 0.025
Present study
0.02
Theory (Hadamard, 1911)
0.015
0.01
0
2
4
t
6
8
10
Figure 2.12: Comparison of present numerical result with Hadamard-Rybczynski [7] theory. The
terminal velocity agrees well for a domain size of 30×30×120 and for the parameter values: Ga = 0.1,
Bo = 0.1, ρr = 10−3 and µr = 10−2 . The center of the vortex is well predicted by our numerical
simulation.
2.4
Effect of regularization parameter
Rising bubble in a viscoplastic medium has been studied in Chapter 5. The outer fluid is modelled
as a regularized Herschel-Bulkely fluid as follows:
µo =
τ0
n−1
+ µ0 (Π + �)
,
Π+�
(2.31)
where τ0 and n are the yield stress and flow index, respectively, � is a small regularization parameter,
and µ0 is the fluid consistency; Π = (Eij Eij )
wherein Eij ≡
1
2
1/2
is the second invariant of the strain rate tensor,
(∂ui /∂xj + ∂uj /∂xi ).
In the dimensionless form,
µo =
Bn
n−1
+ m (Π + �)
,
Π+�
25
(2.32)
where Bn ≡ τ0 R/µ0 V is the Bingham number, and m = (V /R)
n−1
, wherein V (=
√
gR) is the
velocity scale. The characteristic scales and non-dimensionalization is explained in Chapter 5. After
careful evaluation, we have chosen the value of � down to 10−3 in order to neither affect the yield
surface by overly increasing � nor produce numerical instabilities or stiff equations by decreasing
it further; similar values for � have been used earlier by Singh and Denn [111]. The effect of
regularization parameter is shown in Figs 2.13 and 2.14. The height and width of the bubble are
denoted by h and w, respectively (see Fig. 5.1). Also, a comparison with Papanastasiou’s model,
presented in Fig. 2.15, shows a good match with the regularized model. Hence the present regularized
model is employed to simulate a rising bubble inside a viscoplastic fluid in Chapter 5.
(a)
(b)
(c)
Figure 2.13: The unyielded region in the non-Newtonian fluid (shown in black) at time, t = 10 for
different values of the regularized parameter, �: (a) � = 0.01, (b) � = 0.001, (c) � = 0.0001. The rest
of the parameter values are Ga = 70.71, Bn = 14.213, µr = 0.01, ρr = 0.001, m = 1 and Bo = 30.
The unyielded regions for � = 0.001 and 0.0001 are visually indistinguishable.
(a)
(b)
5
1.6
ε
4
3
ε
0.01
0.001
0.0001
0.01
0.001
0.0001
1.4
1.2
zCG
h/w
2
1
1
0.8
0
0
5
10
t
15
20
0
5
10
t
15
20
Figure 2.14: (a) Temporal variation of the center of gravity (zCG ), (b) the aspect ratio (h/w) of the
bubble for different values of the regularization parameter, �. The rest of the parameter values are
the same as those used to generate Fig. 2.13.
26
(a)
(b)
Figure 2.15: The bubble shape (shown by red line) and unyielded region in the non-Newtonian fluid
(shown in black) at time, t = 2 for (a) regularised model, (b) Papanastasiou’s model. The rest of
the parameter values are � = 0.001, Ga = 70.71, Bn = 14.213, µr = 0.01, ρr = 0.001, m = 1 and
Bo = 30. The aspect ratios of the bubble in (a) and (b) are the same (h/w = 1.018).
27
Chapter 3
Bubbles and drops: Similarities
and differences
3.1
Introduction
Bubbles and drops are often studied separately. To our knowledge, no published work discusses the
reason for the differences between a bubble and a drop in their motion, deformation or the manner of
their breakup. In a fluid mechanics symposium, “Fluids Days 2014”, organized on the 80th birthday
of Prof. Roddam Narasimha, I presented a poster containing Fig. 3.1 which attracted attention
from other researchers. Some people looked confused and they had no clue as to why a bubble
should behave different from a drop if the density and viscosity were inverted. Prof. Garry Brown,
among others, mused about why should a drop be different from a bubble. Some of them, including
Prof. Roddam Narasimha, suggested that if I tried to adjust the viscosities or other dimensionless
numbers in some way, I could get an equivalence between the bubble and the drop motion. However,
experts on the subject (Prof. K. R. Sreenivasan) seemed to know that a bubble and a drop could
never behave the same because of the basic violation of the dynamical similarity inherent in the
problem.
A series of very natural questions arose in this regard: Can we derive a general theory for the
motion of both bubbles and drops? Can a rising bubble be designed to behave as the mirror image
of a falling drop? If yes, under what conditions? If not, what are the fundamental differences
between a bubble and a drop that differentiate them? This chapter deals with an analytical as well
as numerical analysis of bubbles and drops in a unified sense for the first part, and the differences
and some interesting dynamics is dealt with in the latter part.
3.2
In Hadamard flow regime
The motion of a spherical fluid bubble/drop in a different quiescent fluid in creeping flow regime
is termed as Hadamard flow, or Hadamard-Rybczynski flow in the honour of Hadamard [7] and
Rybczynski [54] who determined the flow field in this kind of flow for the first time in the year
1911, independently. A spherical vortex solution which has the similar streamfunction solution The
analytical derivation of the stream function for such flow can be found in standard textbooks such
28
(a)
(b)
Figure 3.1: Figure taken from the poster presented on Prof. Roddam Narasimha’s 80th birthday.
Rising bubble and falling drop for parameter values: (a) Ga = 50, Bo = 29, ρr = 7.47 × 10−4 and
µr = 8.15 × 10−6 , and (b) Ga = 30, Bo = 29, ρr = 10 and µr = 10.
as those authored by Batchelor [176] or Leal [177]. The theory assumes that the shape of the
bubble/drop is maintained to be spherical as a result of a balancing surface tension force, which
could be possible for a very high surface tension force as compared to inertial or viscous forces. This
implies that in the dimensionless parameter space, we are concerned with very low Gallilei and Bond
numbers i.e. Ga << 1 and Bo << 1. The dimensionless stream function for Hadamard’s flow inside
the fluid sphere is given by:
ψiH = −
x2 (1 − cos2 θ)(1 − x2 )
,
4(1 + m)
(3.1)
and that for the surroundings is given by
ψoH =
1 − cos2 θ
4
�
2x2 −
3m + 2
m
x+
m+1
x(1 + m)
�
,
(3.2)
where x and θ are the radial distance and the polar angle in a spherical coordinate system having
its origin at the center of the spherical bubble/drop.
The bubble/drop reaches a terminal velocity when the drag force is balanced by the buoyant
weight of the bubble/drop. Also known as Hadamard-Rybczinsky equation, the terminal rise velocity
is given by
�
1 + µr
,
(3.3)
2 + 3µr
√
or, in the dimensionless form, where velocity is scaled with gR, the terminal velocity can be written
as
�
�
V∗
2
1 + µr
Vt = √ t = Ga(ρr − 1)
.
(3.4)
3
2 + 3µr
gR
Vt∗ =
2 ρo R2 g(ρr − 1)
3
µo
29
�
Figure 3.2: Theoretical streamlines in a spherical bubble for the Hadamard
flow(Re << 1). The
√
stagnation ring (center of the spherical vortex) lies at a distance of 1/ 2 from the axis of symmetry.
From Eq. (3.4), it seems possible to design a bubble which is a mirror image of a drop, only by
modifying the viscosity ratio (µr ). It should be noted again that a bubble is defined as a fluid blob
for which ρr < 1 and a drop is a fluid blob for which ρr > 1. This implies that for a given bubble,
the only unknown is the viscosity ratio of the drop, which can be determined from the following
relation:
(rd − 1)
�
1 + md
2 + 3md
�
= (1 − rb )
�
1 + mb
2 + 3mb
�
.
(3.5)
Strictly in the zero Reynolds number limit, Ga can also be different for a bubble and a drop,
thus yielding infinite solutions for the equivalence of a bubble and a drop. But for a finite value of it,
the Gallilei number has to remain fixed in order to have the same flow dynamics for the outer fluid
in both the systems i.e. bubble and the corresponding drop). Thus, only a part of the solution has
been arrived at with this analysis. Moreover, we have not found the range of relevant parameters
for which a rising bubble may behave similar to a falling drop. Next, it is investigated if a similar
acceleration phase can be obtained for a bubble and a drop and the conditions required to do so.
Figure 3.3: A sketch showing a spherical body falling under gravity and the forces acting on it,
where z represents the vertical coordinate, and Fb , Fg and FD denote the gravitational, buoyancy
and drag forces, respectively.
Consider a spherical immiscible mass (solid or fluid) in a quiescent fluid falling under the action
30
of gravity as shown in Fig. 3.3. According to the Newton’s second law of motion,
FB − Fg + FD = md ad ,
(3.6)
wherein, FB , Fg and FD are the forces exerted on the body due to buoyancy, gravity and viscous
drag. The mass of the falling body i.e. drop is denoted by md . Here, we have neglected the
added mass and Basset history forces to keep the solution simple. More accurate solutions can be
obtained by including these terms. Additionally, the acceleration of the center of gravity of the drop
is assumed to be ad . Eq. (3.6) can be written in terms of flow and material properties as
4 3
4
πR ρo g − πR3 ρi g − f (m)µo Rv + O(v 2 ) =
3
3
�
�
4 3
πR ρi ad .
3
(3.7)
In creeping flow regime, the higher order velocity terms can be neglected and the drag force can
be approximated as being proportional to velocity of the center of gravity of the drop. Note that
the drop is translating in the negative z direction, hence the drag force is taken to be proportional
to −v to make it positive. Also, for a general drop material (Newtonian fluid or solid) the drag force
may depend on the viscosity ratio (m). Eq. (3.7) can be simplified as
3
f (m)µo Rv
4πR3
3
1−r−
f (m)µo v
4πR2 ρo g
3
v
√ f (m)µo √
1−r−
4πρo R gR
gR
g(m) ∗
1−r−
v
Ga
ρo g − ρ i g −
= ρ i ad ,
ad
,
g
ad
=r ,
g
=r
= ra∗d
(3.8)
(3.9)
(3.10)
(3.11)
where the superscript ∗ represents dimensionless quantities. Also, a function g(m)(= 3f (m)/4π)
has been introduced to keep the expression succinct. Rewriting Eq. (3.11) in terms of only vertical
velocity and dropping the superscript from dimensionless quantities, we get
dv
g(µr )
1 − ρr
+
v=
.
dt
ρr Ga
ρr
(3.12)
Eq. (3.12) is a first order ordinary differential equation which can be solved using �a standard
� g(µr ) �
method involving the integrating factor. The integrating factor can be calculated as exp
ρr Ga dt
�
�
g(µr )t
which is equal to exp ρr Ga . Thus the solution of Eq. (3.12) can be written as
�
g(µr )t
1 − ρr
e ρr Ga dt + k1 ,
ρr
−g(µr )t
Ga(1 − ρr )
v=
+ k1 e rGa .
g(µr )
g(µr )t
ve ρr Ga =
� �
(3.13)
(3.14)
Initially the drop is at rest i.e. v = 0 at t = 0. This implies
k1 = −
Ga(1 − ρr )
.
g(µr )
31
(3.15)
Therefore, the vertical velocity of the drop can be described by the following equation
v=
�
−g(µr )t
Ga(1 − ρr ) �
1 − e ρr Ga .
g(µr )
(3.16)
We note, from Eq. (3.16) that the velocity of center of gravity of the drop is always negative as
ρr > 1 for a drop (by definition). Thus, we can write two separate equations for a bubble and a
drop as follows
Gab (1 − rb )
vb =
g(mb )
�
Gad (1 − rd )
vd =
g(md )
�
and,
1−e
1−e
−g(mb )t
Gab rb
−g(md )t
Gad rd
�
,
(3.17)
�
.
(3.18)
It is evident from Eqs (3.17) and (3.18), that for a bubble to have similar motion as that for a
drop, the following conditions must be satisfied:
and
Gab rb
Gad rd
=
,
g(mb )
g(md )
(3.19)
1 − rb
rd − 1
=
.
rb
rd
(3.20)
Additionally, the Gallilei numbers for the bubble and the drop systems must be the same for
the dynamical similarity of the flow in the respective outer fluids, i.e. Gab = Gad . Therefore, a
bubble can, in principle, be designed to behave like a drop in the creeping flow regime according
to Eqs (3.19) and (3.20). An interesting result that is apparent from these conditions is that an
equivalent solid drop cannot be designed for a solid bubble even in the creeping flow limit. This
is true because g(mb ) = g(md ) = 4.5 for solid spheres translating in Stokes regime. This implies,
from Eq. (3.19), that the density ratios for bubble and drop systems are identical and only a trivial
solution is obtained. However, for a fluid bubble it is possible to design an equivalent drop which
has a similar motion in the vertically downward direction as that of the bubble in the upward
direction. From the Hadamard-Rybczynski solution, the function g(µr ) comes out to be equal to
(6 + 9µr )/(2 + 2µr ). Rewriting the equivalence conditions, Eqs (3.19) and (3.20), in terms of the
density and viscosity ratios
rb (1 + mb )
rd (1 + md )
=
,
2 + 3mb
2 + 3md
and
rd =
rb
.
2rb − 1
(3.21)
(3.22)
Enforcing the condition that the drop density ratio is always positive, we arrive at a limit to the
bubble density ratio i.e. rb > 0.5. On substituting the value of rd from Eq. (3.22) in Eq. (3.21), we
get
1 + mb
=
2 + 3mb
�
1
2rb − 1
32
�
1 + md
.
2 + 3md
(3.23)
Let us define q(mb ) and q(md ) such that
1 + mb
,
2 + 3mb
1 + md
q(md ) =
.
2 + 3md
q(mb ) =
(3.24)
(3.25)
We note that q(mb ) and q(md ) lie between 1/3 and 1/2 for all positive real values of mb and md .
Therefore, the maximum and minimum bounds on the bubble density ratio, rb can be calculated as
1.25 and 0.83, respectively. This condition requires that the bubble density ratio, rb should always
be greater than 0.834, and by definition rb < 1. Furthermore, it is noted that not all viscosity ratios
are available to choose from, for rb even slightly different from 1. Thus, it is concluded that a fluid
drop can be designed to behave similar to a bubble in the creeping flow limit for a bubble density
ratio greater than 0.834, however not all viscosity ratios are feasible even in this regime, and one
needs to satisfy Eqs (3.21) and (3.22) to choose the density and viscosity ratios.
Generally, Boussinesq approximation is applied in this limit, wherein the effect of density difference is lumped into a body force equal to the buoyant weight per unit volume of the bubble/drop.
Boussinesq approximation is an approximate way of accounting for the density difference and in
this regard Han & Tryggvason [6] have shown that the approximation is reasonable for drop density
ratios in the range 1 < r < 1.6. Boussinesq approximation, thus allows one to choose the bubble
density ratio for a particular drop density ratio using the relation, rb = 2−rd , such that a drop would
behave similar to the bubble. In Fig. 3.4 we compare the Boussinesq criterion for the equivalence
of a bubble and a drop to our criteria derived from the consideration of the Hadamard-Rybczynski
solution.
Figure 3.4: Vertical location of the center of gravity as a function of time for a drop (ρr = 1.214 and
µr = 76), an equivalent bubble based on Boussinesq approximation (ρr = 0.786 and µr = 76) and
an equivalent bubble based on conditions (3.21) and (3.22) (ρr = 0.85 and µr = 0.1). The rest of
the parameters are Ga = 6 and Bo = 5 × 10−4 . The bubble designed using the Hadamard’s solution
is shown to be better than the one derived using the often employed Boussinesq condition.
33
3.3
Bigger bubbles and drops
In the previous section, the bubbles and drops in the Hadamard regime are investigated. For the
given gravitational acceleration, bigger bubbles and drops in the same kind of outer fluids have
higher Ga and Bo, and therefore have inertia dominated dynamics which often involves deviation
from spherical shape and even breakup/topological changes. In the inertia dominated regime, the
shapes of bubbles and drops are visibly different. Typically, bigger bubbles that rise up in a straight
path often develop a dimple at their rear end [1, 5] while drops for a bag like structure which thins
down away from the thick, core [6, 34]. For very high inertia and low surface tension, bubbles tend
to change their topology into a doughnut like shape and drops tend to break from their periphery commonly known as bag break-up. Although there are few other breakup modes that are exhibited
by bubbles and drops like shear breakup and catastrophic breakup, the important feature of bubbles
is to remain in blobs (except for an exception of skirted bubbles at low Ga and high Bo) and for
the drops to thin out into sheet like structures.
The motion of bubbles and drops may be expressed as the continuity and momentum equations
as follows:
and
∇ · u = 0,
(3.26)
Du
∇p
1
∇ · n̂
=−
+
∇ · (µ(∇u + ∇uT )) − ĵ +
δ(x − xs )n̂,
Dt
ρ
ρGa
ρBo
(3.27)
wherein ρ and µ are the dimensionless density and viscosity of the entire fluid system, respectively,
having a sharp change in their values at the bubble/drop interface; ĵ is the unit vector in the
vertically upward direction; δ(x − xs ) is the Dirac-delta function being unity at the interface (which
is defined by the position vector x = xs ) and zero elsewhere; n̂ is the unit normal to the interface,
and D(≡ ∂/∂t + u · ∇) is the material derivative. Here the surface tension force is written as a
volume source term as proposed by Brackbill et al. [178].
Thus the surrounding fluid obeys
Duo
1 2
= −∇p +
∇ uo − ĵ,
Dt
Ga
(3.28)
and the bubble/drop fluid is governed by
Dui
∇p
m 2
=−
+
∇ ui − ĵ,
Dt
r
rGa
(3.29)
with the interfacial conditions being the continuity of velocity and stress components at the interface
(x = xs ). The surface tension term appears in the normal stress balance at the interface, which is
pi |s − po |s = κ/Bo where κ(= ∇ · n̂) is the local curvature of the interface.
Returning to the question of equivalence between a bubble and a drop, we observe the Eqs (3.28)
and (3.29). It is noted that the viscous diffusion and surface tension terms can be matched by
making the kinematic viscosity ratio (m/r), Ga and Bo to be the same for the bubble and the drop
systems, but the pressure gradient term cannot be made to match in both the inner and outer fluids
simultaneously for the two systems. Thus we conclude that for higher Ga and Bo, the equivalence
is theoretically impossible for all bubbles and drops.
After failing to find any exact equivalence between a bubble and a drop at higher Gallilei and
34
Bond numbers, let us follow an order of magnitude analysis to find if a bubble can be designed to,
atleast, approximately behave as a drop for this regime. The force balance in terms of the order of
magnitudes may be expressed as
� 2�
�
�
� σ �
Δ�
p
v�
v�
∼ O ρ�
+O µ
�o 2 + O[g�
ρ] + O
,
R
R
R
R2
(3.30)
wherein, the different terms on the right hand side represent the inertial, viscous, gravitational, and
surface tension contribution to the total force per unit volume. Here, the text decoration, tilde is to
represent the dimensional quantities. For pressure gradients to be the similar in both, bubble and
drop, the terms involving density and viscosity. i.e.
v�2
� 1,
gR
(3.31)
or, the dimensionless velocity, v (which is also the Froude number in this case) should be several
order of magnitudes less than 1. When the bubble/drop reaches a terminal velocity and shape, the
drag force is comparable its buoyant weight. Therefore,
µ
�o
v�
∼ gΔ�
ρ,
R2
(3.32)
Non-dimensionalization of this equation yields,
v ∼ Ga(r − 1).
(3.33)
Additionally, the condition that surface tension force is very large as compared to the inertial forces,
we have
or, in the dimensionless form,
ρ�o
v�2
σ
� 2,
R
R
1
v�√ .
Bo
(3.34)
(3.35)
From relations (3.33) and (3.35), the condition for equivalence between a bubble and a drop, in
dimensionless form can be expressed as
Ga2 (r − 1)2 �
1
.
Bo
(3.36)
According to the inequality (3.36), the equivalence can be obtained in the limit, ρr → 1 even if
Ga and Bo are not small. This analysis gives the condition for equivalence of the forces in bubble
and drop system. A different route can be adopted to equate the pressure distribution on the bubble
and the drop for any Ga and Bo, thus providing a condition for similarity of shapes.
Shape equivalence - pressure arguement
Assuming the surface tension forces to be large compared to inertia and viscous forces, and using
subscripts b and d for the bubble and the drop and unscripted variables for the continuous phase,
35
we write the pressure difference at the tip of the fluid blob as
pb − p = σ b k b ,
pd − p = σ d k d ,
pd = p + k d σ b
rd
,
rb
or,
pd /rd = p/rd + kd σb /rb ,
(3.37)
pb /rb = p/rb + kb σb /rb .
(3.38)
and
Although these equations are for the tip of the bubble, they could also be thought to be valid for
the average pressure (p) and curvature (κ) over the whole surface of the bubble in an approximate
sense. Subtracting equation (3.38) from (3.37), we get
�
�
1
1
σb
p
−
=
(kd − kb ) .
rd
rb
rb
(3.39)
Replacing rb by 2 − rd and σb /rb by σd /rd , we get
p
�
�
1
2rd − 1
σd
−
=
(kd − kb ) ,
rd
rd
rd
or,
2p(1 − rd ) = σd (kd − kb ) .
(3.40)
For the shape of the bubble and the drop to be almost the same, i.e. mathematically kb − kd ∼ �,
where � << 1. Eq. (3.40) suggests that
rd ≈ 1 + �σd /p ∼ 1 + �
, or
rd = 1 + δ,
(3.41)
where δ << 1. Hence rb = (1 + δ)/(1 + 2δ) ≈ (1 + δ)(1 − 2δ), or
rb ≈ 1 − δ.
(3.42)
Eqs (3.41) and (3.42) show that the equivalence in pressure at the surface of the bubble and
drop is possible only when the bubble and drop densities are very close to 1. Therefore the analyses
presented above suggest that the similarity between a bubble and drop is not possible for bigger and
faster moving bubbles/drops.
36
(a)
(b)
Figure 3.5: (a) Evolution of bubble shape with time for ρr = 0.9, µr = 0.5, Ga = 50, Bo = 50; (b)
evolution of drop shape with time for ρr = 1.125, µr = 0.625, Ga = 50, Bo = 50. The direction of
gravity has been inverted for drop to compare the respective shapes with those of the bubble. Even
for high Ga and Bo, the dynamics can be made similar if density ratios are close to unity.
3.4
Differences in bubble and drop dynamics
At small inertia and moderate surface tension, bubbles and drops do not remain spherical, but show
similar behaviour for density ratios close to 1, as shown in Fig. 3.5. We remove the effect of gravity
(which causes a pressure gradient to appear in the fluid) to observe if any differences arise between
the dynamics of a bubble and drop. We replace the Gallilei number with a Reynolds number defined
as Re ≡ U0 R/ν based on the initial drop velocity U0 . For a density ratio close to unity (Fig. 3.6), a
bubble and a drop behave similarly and stop after some time due to viscous dissipation. However,
for density ratios far from unity (Fig. 3.7), the bubble oscillates and comes to rest, while a drop
breaks up after forming a bag-like structure. This suggests that a bubble and a drop cannot behave
similarly even when gravity is not present.
The numerical results presented in this chapter have been obtained using the diffuse interface
solver as well as gerris (see Chapter 2). A bubble is initialized as a sphere with stagnant conditions
at a height of 8R from the bottom of the domain (Fig. 2.1(a)). A drop is initialized in a similar
fashion, with the opposite sign of gravitational acceleration. The axis of symmetry passes through the
diameter of bubble/drop and a Neumann condition on scalars (p, and ca ) and vertical component of
velocity, and a zero dirichlet condition on radial velocity component is imposed. Neumann boundary
conditions are imposed for all variables on the remaining boundaries.
Bubbles and drops of higher inertia where inequalities (3.31) and (3.36) are not followed are
shown in Fig. 3.8. Drop shapes obtained numerically for a density ratio close to unity are shown for
this case in Fig. 3.8(b). The Galilei and Bond numbers are the same for the bubble and the drop,
37
Figure 3.6: Dynamics in the absence of gravity: (a)evolution of bubble shape with time for ρr = 0.9,
µr = 0.5, Re = 50, Bo = 50, (b) evolution of drop shape with time for ρr = 1.125, µr = 0.625,
Re = 50, Bo = 50. The initial shape of both drop and bubble was kept spherical and the initial
velocity given to the fluid blobs is U0 = 1 for both. The shapes of bubble and drop tend to be similar
for density ratios close to unity.
38
Figure 3.7: Dynamics in the absence of gravity: (a) evolution of bubble shape with time for ρr = 0.52,
µr = 0.05, Re = 50, Bo = 50, (b) evolution of drop shape with time for ρr = 13, µr = 1.25, Re = 50,
Bo = 50. The initial shape of both drop and bubble was kept spherical and the same initial velocity
U0 given to both fluid blobs. The bubble regains a spherical shape, whereas the drop breaks up in
the bag-breakup mode.
and the densities are related by Eq. (3.22). The Reynolds number based on the terminal velocity of
the bubble and the drop is about 16. The kinematic viscosity ratio m/r for the bubble is kept the
same as the drop in Fig. 3.8(a) and related by Eq. (3.21) in Fig. 3.8(c). It is apparent that a drop
and its equivalent bubble behave qualitatively the same. The velocities of the drop and bubble are
closer to each other when the viscosity relation (3.21) is used whereas the shapes are closer together
when they have the same kinematic viscosity ratio.
When the density ratio is far from unity, no equivalence is possible. Eq. (3.21) is no longer valid,
nor possible to satisfy. We therefore compare drops and bubbles of the same m/r. Fig. 3.9 makes
it evident that neither the shape nor the velocity of the drop and bubble are similar to each other.
Shown in color in this figure is the residual vorticity Ω [8], which is a good measure of the
rotation in a flow. A detailed discussion of what is the best way to estimate rotation within a drop
is available in [179]. In axisymmetric flow, the velocity-gradient tensor may be spilt into a symmetric
part and an anti-symmetric part. The anti-symmetric part is the vorticity, of magnitude ω, oriented
azimuthally. The eigenvalues of the symmetric part are given by ±s/2, where s = (4u2 +(u+w)2 )1/2 .
The vorticity in turn can be decomposed into shear part and a pure rotational part. The latter is
39
Figure 3.8: Evolution of (a) bubble shape with time for ρr = 0.9, µr /ρr = 0.56. (b) drop shape with
time for ρr = 1.125, µr /ρr = 0.56. (c) drop shape with time for ρr = 1.125, with viscosity obtained
from Eq. (3.21). The direction of gravity has been inverted for the drop in order to compare the
respective shapes with those of the bubble. In all three simulations, Ga = 50, Bo = 50, and the
initial shape was spherical.
termed the residual vorticity, defined [8] as
ωres
=
0 for |s| > |ω|
=
sgn(ω)(|ω| − |s|) for |s| ≤ |ω|
(3.43)
where sgn(ω) is the signum function. The more standard Okubo-Weiss parameter
W = s2n + s2s − ω 2 ,
(3.44)
wherein sn (≡ ∂x u − ∂z w) and ss (≡ ∂x w − ∂z u) are the normal and the shear components of the
strain rate tensor respectively, is another measure of rotation in the flow. Both measures give similar
images of the vortex cores in our simulations, but since the residual vorticity takes care to remove
the shear part of the vorticity, we present results using this quantity.
At later times in Fig. 3.9, it is evident that residual vorticity is concentrated within the bubble
but outside the drop. This is the primary difference between a bubble and a drop. The region of low
pressure and high vorticity tends to lie in the lighter fluid. In the case of the bubble, this causes an
azimuthally oriented circulation in the lower reaches, which then leads to a fatter base and aids in a
pinch-off at the top of the bubble. In the case of a drop, the vorticity being outside means that the
lower portion of the drop is stretched into a thin cylindrical sheet, and an overall bag-like structure
is more likely. Also a pinch off in this sheet region is indicated rather than a central pinch-off. We
present in Figs 3.10 and 3.11 streamlines at various stages of evolution in this simulation. Closed
40
(a)
(b)
Figure 3.9: Evolution of (a) bubble and (b) drop shapes with time, when densities of outer and
inner fluid are significantly different. As before, for the drop (b), the direction of gravity has been
inverted. In both simulations Ga = 50 and Bo = 10. The other parameters for the bubble system
are ρr = 0.5263 and µr = 0.01, while for the drop system ρr = 10 and µr = 0.19. Note the shear
breakup of the drop at a later time. Shown in color is the residual vorticity [8].
41
Figure 3.10: Streamlines in the vicinity of a bubble for t = 1, 2, 3 and 4 for parameter values
Ga = 50, Bo = 10, ρr = 0.5263 and µr = 0.01. The bubble is shown in grey and a red outline. The
circulation can be seen lying inside the bubble, which does not allow the bubble to thin out at its
base.
Figure 3.11: Streamlines in the vicinity of a drop for t = 1, 2, 3 and 4 for parameter values Ga = 50,
Bo = 10, ρr = 10 and µr = 0.19. The bubble is shown in grey and a red outline. The direction of
gravity has been inverted to compare the shapes with those in Fig. 3.10. The circulation is seen to
move out of the drop, making the drop to thin out at its trailing end.
streamlines are visible in the region of lower density, indicative of regions of maximum vorticity
being located in the lighter fluid.
The fact that regions of low pressure and high vorticity would concentrate in the less dense fluid
follows directly from stability arguments. A region of vorticity involves a centrifugal force oriented
radially outwards, i.e., pressure increases as one moves radially outwards from a vortex. There is a
direct analogy between density stratification in the vicinity of a vortex and in a standard Rayleigh
Bénard flow [180]. In the latter, we have a stable stratification when density increases downwards.
In the former, we have a stable stratification when density increases radially outwards, i.e., when
the vortex is located in the less dense region.
Fig. 3.12 is a demonstration that for the same outer fluid even if the viscosity of the bubble and
the drop were kept the same, and only the densities of the two were different, the behavior discussed
above is still displayed.
Our results thus indicate that density is the dominant factor rather than viscosity in determining
the shapes of inertial drops and bubbles. In particular, the vorticity maximum tends to migrate to
the region of lower density, and this has a determining role in the shape of the structure. Since large
density differences bring about this difference, these are effectively non-Boussinesq effects.
We note that given the large number of parameters in the problem, including initial conditions,
which we have kept fixed, the location of maximum vorticity in the less dense region may not be
universally observed in all bubble and drop dynamics. For example, the Widnall instability [181]
in drops resembles the central break-up we have discussed. In the usual set-up of the Widnall
instability, the densities of the inner and outer fluid are close to each other, so we may expect the
42
Figure 3.12: Evolution of (a) bubble and (b) drop (gravity reversed) shapes with time. Parameters
for both bubble and drop systems are: Ga = 100, Bo = 50 and µr = 10. The density ratio for the
bubble and drop are ρr = 0.52 and ρr = 13 respectively, based on Eq. (3.22). The figure shows
that the density, rather than viscosity, decides the location of vortical structures, which results in
altogether different deformation in bubbles and drops.
43
drop to behave similar to a bubble, and the initial conditions are not stationary.
Our arguments above on vorticity migrating to the lighter fluid do not depend on gravity being
present. We also confirm this in simulations which obtain the motion of a bubble and a drop started
with a particular initial velocity in a zero-gravity environment. These are shown in Fig. 3.7.
3.5
Before breakup
Break-up of drops and bubbles is typically a three-dimensional phenomenon on which much has been
said, see e.g. the experimental studies of Elzinga & Banchero [182], Blanchard [183] and Hsiang &
Faeth [184], the theoretical work of Kitscha & Kocamustafaogullari [185] and Cohen [186], and
numerical studies of Jing & Xu [187] and Jalaal & Mehravaran [34]. The transient behavior of liquid
drops has been discussed extensively [188–191], especially in the context of internal combustion
engines, emulsification, froth-formation and rain drops. However, the transient behavior of bubbles
has not been commented upon as much, and we make a few observations, regarding large-scale
oscillations, that are not available in the literature to our knowledge. We note that since our
simulations are restricted to axisymmetric break-up they may not always capture the correct breakup location or shape.
Various parameter ranges are covered in numerous papers in the past 100 years, and it is known
that drops and bubbles break up at higher Bond numbers. The Bond number below which a bubble
of very low density and viscosity ratio does not break, but forms a stable spherical cap, is about
8 [192] which is similar to that found in our simulations. We begin by associating a time scale ratio
with the Bond number. It may be said that surface tension would act to keep the blob together
whereas gravity, imparting an inertia to the blob, would act to set it asunder. The respective
�
time-scales over which each would act may be written as Ts = ρR3 /σ for surface tension and
�
Tg = R/g for gravity. The ratio
Tg
1
=√ ,
(3.45)
Ts
Bo
is a measure of the relative dominance. At Bo >> 1, surface tension is ineffective in preventing
break-up, and we may expect a break-up at a time of O(1), since we use gravitational scales. For
Bo ∼ 1, it is reasonable to imagine a tug-of-war to be played out between surface tension and inertia
in terms of shape oscillations, with a frequency of O(1). Since a bubble usually breaks up at the
centre by creating a dimple, the vertical distance Dd of the top of the dimple from the top of the
bubble is a useful measure to observe oscillations. Fig. 3.13 shows the dimple distance as a function
of time for various Bond numbers, and our expectations are borne out.
Figs 3.14 and 3.15 are typical streamline patterns in the vicinity of bubbles in the break-up
and recovery cases respectively. Instantaneous streamlines are plotted by taking the velocity of the
foremost point of the bubble as the reference, but the picture is qualitatively unchanged when the
velocity of the centre of gravity of the drop is chosen instead. Both cases are characterized by a
large azimuthal vortex developing within the bubble initially. In the break-up case (Fig. 3.14), this
vortex is sufficient to cause the bubble surface to rupture and obtain a topological change, from a
spherical-like bubble into a toroidal one. In all the cases of bubble recovery we have simulated, of
which a typical one is shown (Fig. 3.15), there develop at later times several overlaying regions of
closed streamlines, which act to counter the effect of rupture by the primary vortex, and to bring
44
Figure 3.13: Variation of dimple distance versus time for different Bond numbers for Ga = 50,
ρr = 7.4734 × 10−4 , µr = 8.5136 × 10−6 . The tendency of a bubble to break from the center is
evident. However, a bubble may form a skirt for intermediate Bond numbers (Bo = 15), which may
lead to breakup or shape oscillations in certain cases.
Figure 3.14: Streamlines in and around the bubble at time, t = 1, 1.5, 2.0 and 2.5 respectively, for
Ga = 50, Bo = 29, ρr = 7.4734 × 10−4 and µr = 8.5136 × 10−6 . The shape of the bubble is plotted
in red. The toroidal vortex inside the bubble maintains the thickness of its base as the liquid jet
penetrates the remaining air film at the top.
Figure 3.15: Streamlines in and around the bubble at time, t = 2.5, 5, 7, 9 and 11 respectively, for
Ga = 50, Bo = 15, ρr = 7.4734 × 10−4 and µr = 8.5136 × 10−6 . Three toroidal vortices form inside
and outside the bubble which compete with the surface tension force to make the bubble shape
oscillate.
45
back the drop to a shape that is thicker at the centre. During each oscillation, we see the upper and
lower vortices form and disappear cyclically.
3.5.1
Effects of viscosity
In this section we study the effects of viscosity on the tendency to break-up. That decreasing external
fluid viscosity (increasing Gallilei number) will increase the tendency to break-up is demonstrated
in Figs 3.16 and 3.17 for two Bond numbers. Also oscillations become more prominent at higher
Ga. If the viscosity ratio is low enough, the outer fluid is able to shear-break the drop. It is already
known [186] that a higher Weber number is required to break a drop when the surrounding fluid is
more viscous. This indicates that the more the viscous drag, i.e. the less the inertia of the blob, the
less willing it is to break. If the surrounding fluid is more viscous, we would need to increase gravity
or reduce surface tension to break a blob. Thus, in effect, a higher Bond number is needed to break
a blob. Fig. 3.17 shows that when all the physical properties are kept the same while the kinematic
viscosities are reduced in the same proportions for inner and outer fluids, the bubble tends to break
up.
Figure 3.16: Variation of dimple distance versus time for different Gallilei numbers for Bo = 8,
ρr = 7.4734 × 10−4 , µr = 8.5136 × 10−6 . The bubble shapes are shown at corresponding times
for Ga = 5 (top) and 125 (bottom). The shape oscillations ensue after a threshold in outer fluid’s
viscosity i.e. Ga.
3.5.2
Drop breakup
A typical breaking drop, with its associated streamlines is shown in Fig. 3.18. As discussed above,
the breakup is very different from that of a bubble, since the primary vortical action is outside, and
causes a thinly stretched cylindrical, rather than toroidal shape.
The vortex in the wake of the drop tends to stretch the interface (and surface tension is not high
enough to resist the stretching) which leads to an almost uniformly elongated backward bag. New
eddies are formed due to flow separation at the edge to the “bag” and a toroidal rim is detached
46
Figure 3.17: Variation of dimple distance Dd versus time for Bo = 29, ρr = 7.4734 × 10−4 , µr =
8.5136 × 10−6 . Bubble shapes are shown for non-oscillating (top, black), oscillating (blue) and
breaking (bottom, black) bubbles.
Figure 3.18: Streamlines in and around the drop at time, t = 4.5, 6 and 7.5, respectively (from left
to right), for Ga = 50, Bo = 5, ρr = 10 and µr = 10. The circulation zones form outside the drop,
as observed in Fig. 3.9.
after some time from this bag. A drop too responds to Bond number, but the response is shown in
terms of an early break-up at high Bond numbers, as seen in Fig. 3.19. The shape at break-up too
evolves with the Bond number, as shown.
3.6
Summary
A bubble and a drop, starting from rest and moving under gravity in a surrounding fluid, cannot in
general be designed to behave as one another’s mirror images (one rising where the other falls). We
have shown that the underlying mechanism which differentiates the dynamics is that the vorticity
tends to concentrate in the lighter fluid, and this affects the entire dynamics, causing in general a
thicker bubble and a thinner drop. However, if inertia is small, surface tension is large, and a drop
is only slightly heavier than its surrounding fluid, a suitably chosen bubble can display dynamics
similar to it. In this limit, the Hadamard solution can be exploited to design a bubble with its
47
Figure 3.19: Variation of break-up time with Bond number for Ga = 50, ρr = 10 and µr = 10. A
typical bag breakup mode is shown in this figure. Shapes of the drop just before breakup are shown
for various Bond numbers.
density and viscosity suitably chosen to yield the same acceleration at any time as a given drop.
We are left with an interesting result: while a solid ‘bubble’ can never display a flow history which
is the same as a solid ‘drop’, a Hadamard bubble can. Also, although density differences are small,
the Boussinesq approximation cannot lead us to the closest bubble for a given drop.
We find numerically that a similarity in bubble and drop dynamics and shape is displayed up
to moderate values of surface tension and inertia, so long as the density ratio is close to unity. In
axisymmetric flow, the vorticity concentrates near the base of the bubble, which results in a pinch-off
at the centre whereas the cup-like shape displayed by a drop, and the subsequent distortions of this
shape due to the vorticity in the surrounding fluid, encourage a pinch-off at the sides. Bubbles of
Ga higher than a critical value for a given Bo will break up at an inertial time between 2 and 3. For
Ga or Bo just below the critical value, oscillations in shape of the same time scale occur before the
steady state is achieved.
48
Chapter 4
Three dimensional bubble and
drop motion
4.1
Introduction
In the preceding chapter, we have investigated the motion of rising bubbles and falling drops under
the assumption that the dynamics is axisymmetric. This assumption breaks down for large Gallilei
and Bond numbers, as observed in several experimental as well as numerical studies [1, 11, 32, 83].
Interestingly, the fact that this dynamics is three-dimensional was first documented by Leonardo Da
Vinci in the 1500s, in his book Codex leicester which was recently discovered by Prosperetti [18].
Leonardo Da Vinci found that the bubbles rise in zigzagging and spiralling trajectory even if released
axisymmetrically under a coloumn of water. This is currently known as path instability and it has
been the subject of a host of experimental [9, 42, 78, 79], numerical [10, 80] and analytical [81, 82]
studies. Most of the workers embarking on this study find it satisfactory to investigate the effect
of initial bubble diameter on the rising dynamics, and no experimental investigations are available
to our knowledge which study the effects of just the surface tension or viscosity of liquid on the
bubble rise. However, a recent study discusses the effect of viscosity ratio on the drop dynamics and
breakup for immiscible liquid-liquid systems [193].
A vast majority of the earlier experimental and theoretical studies have had one of the following
goals (i) to obtain the rise velocity (ii) to evaluate the path instability (iii) to understand bubbly
flows, (iv) to make quantitative estimates for particular industrial applications, and (v) to derive
models for estimating different bubble parameters. Most of these restrict themselves to only a few
Ga or Bo. Our study, in contrast, is focussed on the dynamics of a single bubble/drop. Starting
from the initial condition of a spherical stationary blob, we are interested in delineating the physics
that can happen. We cover a range of several decades in the relevant parameters.
In the first part of this chapter, a study of bubbles rising due to buoyancy in a far denser and
more viscous fluid is presented. Therefore in this part of the work, ρr and µr are fixed at 10−3
and 10−2 , respectively. We show that as the size of the bubble is increased, the dynamics goes
through three abrupt transitions from one known class of shapes to another. A small bubble will
attain an axially symmetric equilibrium shape dictated by gravity and surface tension, and travel
vertically upwards at a terminal velocity thereafter. A bubble larger than a first critical size loses
49
axial symmetry. We show that this can happen in two ways. Beyond the next critical size, it breaks
up into a spherical cap and many satellite bubbles, and remarkably, the cap regains axial symmetry.
Finally, a large bubble will prefer not to break up initially, but will change topologically to become an
annular doughnut-like structure, which is perfectly axisymmetric. In the latter part of this chapter,
we present a three-dimensional study of the effect of viscosity and density ratios on drop dynamics.
It is shown that the effect of density ratio is to increase the inertia of the drop and thus the drop
tends to breakup with an increase in the density ratio. The effect of viscosity ratio is shown to delay
the breakup. Also, it is confirmed by three-dimensional simulations that a drop tends to break up
from periphery rather than its center, whereas a bubble often breaks up at its center, at high inertia
and low surface tension forces. A large portion of this chapter is contained in one of our published
works [19].
4.2
Bubbles
The results of more than 130 three-dimensional simulations of single bubbles rising due to buoyancy
are presented in this section. The open-source volume-of-fluid solver, gerris has been used due to
its dynamic adaptive grid refinement feature and one of the best algorithms for inclusion of surface
tension force in Navie-Stokes equation [175]. The simulation domain is shown in Fig. 2.1(b). The
boundary conditions on all sides of the domain is symmetry i.e. Neumann condition for scalars
(p and ca ) and velocity components tangential to the given boundary, and zero dirichlet condition
for velocity components normal to the given boundary. The bubble is initialized as a sphere of
unit radius in the dimensionless terms. The dimensionless governing equations and the constitutive
relations (Eqs 2.15-2.27) can be simplified to:
ρ
�
�
∇ · u = 0,
�
�
∂u
1
δ
+ u · ∇u = −∇p +
∇ · µ(∇u + ∇uT ) − ρ�ez +
κ�n,
∂t
Ga
Bo
(4.1)
(4.2)
∂ca
+ u · ∇ca = 0,
∂t
(4.3)
ρ = (1 − ca )ρr + ca ,
(4.4)
µ = (1 − ca )µr + ca ,
(4.5)
along with,
The results obtained from the simulations are discussed next.
4.2.1
Regimes of different behaviours
Fig. 4.1 represents a summary of what happens to an initially spherical bubble rising under gravity
in a liquid. A range of ratios of gravitational, viscous and surface tension forces have been simulated
(in about 130 simulations). Several features emerge from this phase plot, which is divided into five
regions. Region I, at low Bond and Galilei numbers, is shown in pink. In this region, surface tension
is high and gravity is low, so it is understandable that the bubble retains its integrity. It attains a
constant ellipsoidal shape, of which a typical example is shown in the figure in that region, and takes
50
Figure 4.1: Different regimes of bubble shape and behaviour. The different regions are: axisymmetric
(circle), asymmetric (solid triangle) and breakup (square). The axisymmetric regime is called region
I. The two colors within the asymmetric regime represent non-oscillatory region II (shown in green),
and oscillatory region III (blue) dynamics. The two colors within the breakup regime represent
the peripheral breakup region IV (light yellow), and the central breakup region V (darker yellow).
The red dash-dotted line is the M o = 10−3 line, above which oscillatory motion is not observed in
experiments [1,9]. Typical bubble shapes in each region are shown. In this and similar figures below,
the bubble shapes have been made translucent to enable the reader to get a view of the internal
shape.
on a terminal velocity going straight upwards. The bubble is axisymmetric in this region. Region
II, at high Bond numbers and low Galilei numbers, is demarcated in green color. The bubble here
has two distinct features, an axisymmetric cap with a thin skirt trailing the main body of bubble.
The skirt displays small departures from axisymmetry in the form of waves, e.g., a wavenumber 4
mode is barely discernible in the typical shape shown. Bubbles in this region travel upwards in a
vertical line as well, and practically attain a terminal velocity after the initial transients and display
shape changes only in the skirt region. The extreme thinness, in parts, of the skirt presents a great
challenge for numerical analysis, and a detailed study of this region is left for the future. Region III,
depicted in blue colour, occupies lower Bond and higher Galilei numbers. Here surface tension and
inertial forces are both significant, and of the same order. Bubbles display strong deviations from
axisymmetry in this region, at relatively early times, and rise in a zigzag or a spiral manner. Bubbles
remain integral but their shapes change with time. Region IV is shown in light yellow colour, and
region V is in dark yellow. The bubble, faced with higher gravity and relatively weak surface tension,
breaks up or undergoes a change of topology in these regions. Remarkably, the dynamics may be
described well as axisymmetric up almost to the break-up. Region IV is a narrow region which may
be described roughly as having a moderate value of the product GaBo. At low Ga and high Bo (i.e
high Morton number) the bubble in this regime breaks into a large axisymmetric spherical cap and
several small satellite bubbles in the cap’s wake. We term this a peripheral break-up, since it involves
51
a pinch-off of a skirt region of the kind seen in region II. For high Ga and low Bo (i.e lower M o)
a new breakup dynamics is observed, not hitherto described to our knowledge, which is discussed
below with Fig. 4.9. Significant among the results is the fact that in region IV, after break-up
axisymmetry is regained and the final spherical cap bubble attains a constant shape and terminal
velocity. Finally, the bubbles shown is region V are under the action of high inertial force and low
surface tension force. A qualitatively different kind of dynamics is seen here. A dimple formation
in the bottom centre leads to a change of topology: to a doughnut-like or toroidal shape as seen
in the figure. Close to the boundary of region IV, the change of topology may be accompanied
by an ejection of small satellite bubbles. As Ga and Bo are increased further in this region, a
perfectly axisymmetric change of topology of the whole bubble is observed. Unlike in the other
regions, this new shape is not permanent. It eventually loses symmetry, and evolves into multiple
bubble fragments. The boundaries between the five regions are easy to distinguish because the time
evolution is qualitatively different on either side. Details of how a bubble is assigned to a particular
region are provided in the below. Moreover the sum of the kinetic and surface energies usually goes
to a maximum at the transition between two regions, and falls on either side. This is discussed in
more detail below. We had mentioned the Morton number above, defined as M o = Bo3 /Ga4 . This
Figure 4.2: Dynamics expected for bubbles in different liquids. Constant Morton number lines,
each corresponding to a different liquid, are overlayed on the phase-plot to demonstrate that our
transitions can be easily encountered and tested in commonly found liquids. The initial radius of
the air bubble increases from left to right on a given line. Circles, triangles and squares represent
air bubbles of 1 mm, 5 mm and 20 mm radii, respectively.
combination deserves a separate name because it depends only on the fluid properties and not on
the bubble size. Air bubbles in a particular fluid at a particular temperature will lie on constant
Morton number lines, which are straight lines in the log-log phase plot. The red dashed line in Fig.
4.1 corresponds to a Morton number of 10−3 , which is the Morton number mentioned in numerous
experiments, see e.g. [1, 9] below which spiralling and zigzagging trajectories are seen. Note that
52
the boundary between regions II and III, i.e. between straight and zigzagging trajectories, in our
simulations lies very close to this. The lines of constant Morton number corresponding to some
common liquids at different temperatures are shown in Fig. 4.2. Since we have used very small
viscosity and density ratios, our results apply to various air-liquid systems. In the examples given,
the liquid densities are not far from water, and we know from [52,194] that the dynamics is insensitive
to viscosity ratio for small µr . Moving upwards and to the right on a given line, the bubble size
increases, and typical bubble sizes are indicated in the figure. We see that our results apply to a
range of liquids in which an estimate of bubble motion may be desired, for instance crude oil, water
at different temperatures and cooking oils. A 1 mm radius bubble in water at room temperature
will execute spiral or zigzag motion whereas a 20 mm bubble in honey will develop a skirt but move
upwards in a straight line.
We had recently shown [22] that a bubble is more likely to lose its original topology to attain
a doughnut shape at high inertia and low surface tension, whereas a drop under the same Bond
and Galilee numbers would tend to break into several drops. We predicted that non-Boussinesq
effects are qualitatively different in drops and bubbles, since highly vortical regions are stable when
situated within the lighter fluid. The present three dimensional simulations are a confirmation of
this physics.
4.2.2
Path instability and shape asymmetry
Figure 4.3: Agreement and contrast between present and previous results for different flow regimes.
Comparison between the onset of asymmetric bubble motion obtained in the numerical stability
analysis of Cano-Lozano et al. [10] (solid black line), and the present boundary between regions I
and II. Also given in this figure are five different conditions (diamond symbols) studied by Baltussen
et al. [11]. The dynamics they obtain are as follows: A - Spherical, B - Ellipsoidal, C - Boundary
between skirted and ellipsoidal, D - Wobbling and E - Peripheral breakup. The correspondence
between present results and [11] is excellent. Grace et al. [12] obtained spherical bubbles below the
solid blue line shown.
53
Two portions of our phase space have received particular attention earlier. The first, which we
have spoken of earlier, is the onset of zigzagging motion, famously referred to as the path instability.
Ryskin & Leal [195] and many other studies believed the path instability to occur due to vortex
shedding from the bubble. Indeed in the motion of solid objects through fluid this is the only way
in which one can get a path which is not unidirectional. Magnaudet & Mougin [80] assumed the
bubble to be ellipsoidal in shape, and studied a constant velocity flow past such a bubble to obtain
the instability of its wake. The bubble shape and position were held fixed during the simulation.
An asymmetric wake was taken to be indicative of the onset of zigzagging motion. Cano-Lozano
et al. [10] repeated a similar analysis, but on a realistic bubble shape, which they obtained from
axisymmetric numerical simulations. The bubble was held in a constant velocity inlet flow equal to
the terminal velocity obtained in their simulations for the axisymmetric shape. Wake instabilities
were the investigated from a three-dimensional simulation of this fixed bubble. We find that this
simplified method yields a good qualitative estimate of the onset of zig-zagging motion at low
inertia. A comparison with our more exact three-dimensional simulations is shown in Fig. 4.3,
where quantitative discrepancies are noticed, especially for large inertia, i.e., Ga > 50. Also given
in this figure is a comparison with the very recent results of Baltussen et al. [11]. For five different
pairs of Ga and Bo, the dynamics predicted by these authors may be seen to be confirmed by
present results. While we did not distinguish our shapes into spherical and ellipsoidal, we note
that the boundary provided by Grace et al. [12], also shown in this figure, between spherical and
non-spherical shapes, is consistent with our findings. The line falls well within our regions I and II
where we have ellipsoidal drops, and in region II lies close to the minimum Bo of our computations.
(a)
(b)
(c)
Figure 4.4: Dynamics and shapes of region III bubbles: trajectory of the bubble centroid for (a)
Ga = 70.7, Bo = 10, and (b) Ga = 100, Bo = 4, and (c) shape evolution of bubble corresponding to
the latter case. In panel (c), the radial distance of the center of gravity (rs ) of the bubble measured
in the horizontal plane from the original location is shown below the shapes at each time.
A point to note is that unlike solid spheres, departures from vertical motion in a bubble can be
caused either by shape asymmetries, or unsteady vortex shedding, or both. The stability analyses
discussed above take account only of the latter, whereas experiments, e.g. those of De Vries et
al. [196] in clean water found a regime of path instability where no vortex shedding was expected.
54
(a)
(b)
(c)
Figure 4.5: Differences between two dimensional and three dimensional bubble shapes: (a) A region
III bubble at t = 20 for Ga = 100 and Bo = 0.5, (b) at t = 30 for Ga = 100 and Bo = 4, again in
reign III, and (c) a region IV bubble at t = 5 for Ga = 70.71 and Bo = 20. The second row shows
the side view of the three-dimensional shapes of bubbles rotated by 90 degrees about the x = 0 axis
with respect to the top row.
(a)
(b)
Figure 4.6: Characteristics of a region III bubble of Ga = 100 and Bo = 0.5. (a) Oscillating upward
velocity, with different behaviour at early and late times, (b) trajectory of the bubble centroid. The
two regions corresponding to two different behaviours in the rise velocity correspond to the inline
oscillations and zig-zagging motion.
In fact a recent analytical study [82] attempts to explain that vortex shedding is the effect, rather
than a cause of the path instability in rising bubbles. Without a statement as to cause and effect,
we expect an intimate connection between loss of symmetry and loss of a straight trajectory. Any
asymmetry in the plane perpendicular to gravity should result in an imbalance of planar forces.
Similarly any asymmetry in the planar forces, due to vortex shedding or otherwise, should result in
shape asymmetry. In accordance with these expectations, we find that path instability and shape
asymmetry go hand in hand, so the onset of path instability is just the boundary between regions
I and III. Not just the onset, but the entire region III, where the bubble shape is strongly nonaxisymmetric, coincides with the regime where path instability is displayed. Fig. 4.4 shows the
trajectory, and the shape of a typical bubble in this region at different times. A helical-like motion
is executed in the cases shown, while the shape is continuously changing. The bubble does not
adopt a standard geometry. Incidentally, in several of the simulations, the centre of the helix does
not coincide with the original location in the horizontal plane. Nor are the windings of the helix
55
(a)
(b)
Figure 4.7: Region III bubble corresponding to that shown in Fig. 4.6 (Ga = 100 and Bo = 0.5).
(a) Iso-surfaces of the vorticity component in the z direction at time t = 15 (ωz = ±0.0007) and
26 (ωz = ±0.006), (b) The evolution of the shape of the bubble. The radial distance of the center
of gravity (rs ) of the bubble measured in the horizontal plane from the original location is shown
below the shapes at each time.
periodic or regular. Most trajectories in this regime are indicative of chaotic dynamics. We also
obtain trajectories resembling widening spirals, or those which execute a zig-zag motion with the
centroid lying close to some vertical plane and there seems to be no particular region in the Ga − Bo
plane where one or the other dominates. Zig-zag and helical motion is accompanied by oscillations
in the vertical velocity as well, so the bubble alternately speeds up and slows down on its way. An
example is seen in the vertical velocity plot of Fig. 4.6a. Two kinds of oscillatory behaviour in the
velocity are clearly visible in the figure, one with increasing oscillations at early times, and one with
a different character at later times. At later times the dynamics is more erratic, but amplitudes of
variation are lower. In the first part vorticity is generated in the wake but remains vertically aligned
and attached to the bubble. At time t > 14 the drop begins to display zig-zag motion (see Fig.
4.6b). The wake now consists of a pair of counter-rotating two-threaded vortices, often considered
to be a first sign of path instability [80]. This is soon followed by shedding of the vortices, which
begins at t > 20. We find that the onset of the second type of unsteadiness may be attributed to
the start of the vortex shedding off the bubble surface. The vertical component of vorticity in this
regime is shown in Fig. 4.7a. In some cases we find resemblences to the hairpin vortices of Gaudlitz
& Adams [83]. The manner in which the shape of the bubble evolves during this process is shown
in Fig. 4.7b. The correspondence between asymmetry in shape and the path instability is obvious.
A few animations are available in http://www.iith.ac.in/∼ksahu/bubble.html.
We bring out the importance of three-dimensional simulations in Fig. 4.5 in regions III and IV.
We saw that the path instability is deeply connected to shape asymmetries, so region III dynamics
are inherently three-dimensional. In region IV the break-up is not axisymmetric. We note that
region I can be well obtained from axisymmetric simulations.
56
4.2.3
Breakup regimes
We now examine the dynamics of bubbles destined for break-up, of regions IV and V. The contrast
in bubble behaviour between these two regions is evident in Fig. 4.8. At early times both bubbles
are axisymmetric. The region IV bubble develops a skirt, in this case similar to the one seen in
region II, with the difference that this skirt then breaks off in the form of satellite bubbles, leaving
an axisymmetric spherical cap. The region V bubble was seen to first undergo a change in topology
into a doughnut or toroid shape. Beyond time t = 5, the toroid is subject to further instability,
and breaks into a number of droplets. Pedley [197] had predicted that a perfectly toroidal bubble
of circular cross section will undergo instability beyond a time tc . In our scales, the instability time
of Pedley may be written as tc ∼ GaBo1/2 f 3/2 , where f is the ratio of the inner radius of the toroid
to the initial radius R. Given that our toroidal bubble has a cross section very far from circular,
we expect instability to set in much sooner, and find break-up at times an order of magnitude lower
than tc . In addition the history of the flow, including the vortex patterns, contribute to hastening
instability.
(a)
(b)
Figure 4.8: Time evolution of bubbles exhibiting a peripheral and a central breakup. Threedimensional and cross-sectional views of the bubble at various times (from bottom to top the
dimensionless time is 1, 2, 4 and 5). (a) region IV, a bubble breaking into a spherical cap and
several small satellite bubbles, Ga = 70.7 and Bo = 20, and (b) region V, a bubble changing in
topology from dimpled ellipsoidal to toroidal, Ga = 70.7 and Bo = 200.
Figure 4.9: A new breakup mode in region IV for Ga = 500 and Bo = 1. Bubble shapes are shown
at dimensionless times (from left to right) t = 2, 4, 6, 7, 8, 9 and 9.1).
Region IV bubbles show different breakup dynamics at higher inertia and surface tension (low
57
M o). For large M o, a wide skirt was seen to form which then broke off into small bubbles, whereas
for lower M o values small bubbles are ejected from the rim of the bubble while it recovers from an
initially elongated shape to the spherical cap shape. Bubbles of even lower M o values, i.e., at high
inertia and surface tension, are subjected to strong vertical stretching giving rise to a far narrower
skirt, which results in an ellipsoidal rather than a cap-like bubble, and a small tail of satellite bubbles,
as seen in Fig. 4.9. This type of break-up has not been reported before, to our knowledge.
Figure 4.10: Comparison of our 3D results with those of Bonometti & Magnaudet [13] for bubble
breakup. The light yellow and dark yellow colours represent the regions for peripheral and central
breakup. The corresponding data points from the present numerical simulation are shown as blue
and black squares, respectively.
Before break-up, departures from symmetry are small in region IV bubbles. Similarly region V
bubbles are symmetric up to toroid formation. We may thus ask whether cap or toroid formation
requires three-dimensionality. The transition from a spherical cap to a toroidal shape, as obtained
by Bonometti & Magnaudet [13] by means of axisymmetric computations are compared in Fig. 4.10
to our region IV – region V boundary, showing that the two trends agree qualitatively. The first
difference between the axisymmetric and 3D simulations was seen in region IV in Fig. 4.5. While the
2D simulations can only obtain break-up in the form of a ring that detaches from the spherical cap,
our simulations enable the ejection of satellite bubbles. Another feature which the axisymmetric
simulations will miss is the fact that the centre of gravity moves in the horizontal plane. Thirdly,
just below the lowest point given by Bonometti & Magnaudet [13], we obtain a protrusion of region
V (seen in deep yellow in Fig. 4.10) pointing to the left and downwards in the Ga − Bo plane.
The dynamics in this protrusion region is asymmetric, and seems to have been missed by other
axisymmetric simulations.
We have now seen that a bubble which is initially spherical with a Ga and Bo corresponding to
regions IV and V will break up eventually. Does this mean that no single bubble can display a Ga
and Bo corresponding to this region? The answer is a no. Large single bubbles have been created
58
experimentally by many [33, 198]. It has been found in all of these studies that the stable shape
for large-sized bubbles is a spherical cap. The initial conditions are extremely important for large
bubbles, and experimenters take great care to generate an initial bubble which itself is in the form
of a spherical cap. This is done by specially designed dumping cups. In fact Landel et al. [33] noted
that only with a cup whose shape was very close to the final spherical cap bubble shape could they
generate a stable bubble. Not just the curvature but particular care had to be taken to match the
angle subtended by the cup shape at the centre of curvature to that of the final bubble shape, and to
minimize external perturbations. In summary it was very difficult to create a single large spherical
cap bubble since if these conditions were not enforced, the bubble would break up and satellite
bubbles were inevitably present in the wake. Additionally, Wegener & Parlange [28] observes that
in general spherical cap bubbles undergo tilting and wrinkling of their bottom, which results in the
occasional peel off of satellite bubbles.
The largest spherical cap bubbles that have been thus observed, to our knowledge, have Ga ∼ 104
and Bo ∼ 102 [33], which are well beyond the regime we have investigated. Batchelor [199] conducted
a stability analysis of a steady rising spherical cap bubble to obtain an estimate of the largest stable
bubble. This size is far smaller, and lies in regime V of our phase plot. These studies, and the
computations of Ohta et al. [200], underline the importance of initial conditions in this problem. In
addition to spherical cap bubbles, toroidal bubbles too of much larger size have been experimentally
observed by Landel et al. [33] for different initial conditions and parameters. Our results show that
a bubble which starts from a spherical shape has a vastly different fate, and can stay integral only
when much smaller.
4.2.4
Upward motion
The vertical velocities of bubbles in the different regions is characterised in Fig. 4.11. In region I,
the vertical velocity monotonically increases and saturates at a terminal value. In region II, some
minor oscillations are displayed initially owing to the skirt formation, but again a terminal velocity is
reached. Region III displays oscillations of amplitude ∼ 25% of the average velocity, but these were
seen to quieten down somewhat once vortex shedding begins. Regions IV and V display irregular
but large oscillations in the velocity. In both regions the oscillations are small at later times, but
while in region IV, the final velocity is close to its maximum, in region V the upward movement of
the centre of gravity of the dispersed phase has slowed down to about half its original velocity. This
is because the bubble has disintegrated considerably in the latter case.
The variation of dimensionless terminal velocity, wT versus Bo for different values of Ga is plotted
in Fig. 4.12. It can be seen that decreasing the value of Bo results in an increase in the terminal
velocity for all values of Ga; however, as expected the rate of increase of the terminal velocity is
higher for higher values of Ga. The bubbles which exhibit peripheral breakup (i.e. bubbles lying in
region IV in our phase-plot, Fig. 4.1) tend to have an increase in their average rise velocity because
of the presence of satellite bubbles [33].
59
(a)
(b)
(c)
2.5
2.5
2.5
2
2
2
1.5
1.5
w
1.5
w
w
1
1
1
0.5
0.5
0.5
00
2
4
t
6
00
8
2
4
t
8
6
00
10
(d)
w
20
t
30
40
(e)
2.5
2.5
2
2
1.5
1.5
w
1
1
0.5
0.5
00
10
2
4
t
6
8
00
10
5
t
10
15
Figure 4.11: Rise velocity for bubbles having markedly different dynamics. (a) region I: axisymmetric
(Ga = 10, Bo = 1) (b) region II: skirted (Ga = 10, Bo = 200), (c) region III: zigzagging (Ga = 70.7,
Bo = 1), (d) region IV: offset breaking up (Ga = 70.7, Bo = 20) and (e) region V: centrally breaking
up bubble (Ga = 70.7, Bo = 200). In addition to the upward velocity, the in-plane components are
unsteady too in regions III to V.
4.3
4.3.1
Determination of the behaviour type
Shape analysis
Assignment of a given bubble dynamics to a region is straightforward given that behaviour is so
different on either side of each boundary. Bubbles which break up and those which do not are clearly
evident in visual examination of the time evolution of the shape. Similarly the difference between
the two kinds of break-up (region IV-V) is very evident. The boundary between regions II and III
is again evident by visual examination, since (a) the shapes are very different on either side of the
boundary (b) the path in region II is oscillatory whereas region III bubbles move up in a straight
line.
The green and blue colour in the phase plot (Fig. 4.1) combine to give the region in the Ga-Bo
plane where the bubble assumes an asymmetric shape. The asymmetry is computed as follows. The
bubble is cut with 8 vertical planes in order to get 8 cross sections, each successive plane separated
by an angle of π/8 radians. The area of a vertical face of each cross-section of the bubble is calculated
and the percentage difference in the area with respect to a reference cross-section (lying in the y − z
plane) is obtained. The root-mean-squared value of this data at each time step represents the degree
of asymmetry, δa .
Because of the O(Δx2 ) scheme used in finite volume discretization, the error in calculation of
area of cross-section(A) may be estimated as
ΔA
ΔLv
ΔLh
≈
+
,
A
Lv
Lh
60
(4.6)
Figure 4.12: Variation of dimensionless terminal velocity with Bo for different Ga. The terminal
velocity tends to decrease with decreasing surface tension because of the increased drag on the
bubble.
where Lv and Lh are the bubble dimensions in the vertical and horizontal directions, in the crosssectional plane. The errors in the bubble dimensions, ΔLv and ΔLh are of the order of the square
of the smallest grid size, i.e. 0.0292 for a simulation with the coarsest mesh used in our study i.e.
Δx = 0.029. Thus we obtain ΔA/A ≈ 2 × 0.0292 , or ≈ 0.0017 which is about 0.2%. The root-mean-
squared error percentage is calculated for all the cross-sections, which is denoted by δa in this text.
To be conservative, any variation within 0.5% in δa is considered to represent a symmetric bubble
whereas, the bubble is considered to be asymmetric when δa exceeds 0.5%.
As described in the main manuscript, shape asymmetry can be seen with or without an accompanying asymmetrical motion in the horizontal plane. The motion of the bubble is obtained
by tracking the center of gravity (centroid of the bubble) of the bubble with time. Our measure
of deviations from azimuthal symmetry, δa is for both kinds of asymmetry i.e. oscillatory as well
as non-oscillatory, whereas the centroid motion gives information about the deviation from vertical
motion, i.e. the path instability.
4.3.2
Energy analysis
We found that the sum of the kinetic and surface energies usually goes to a maximum at the transition
between any two of the five regions identified in the phase plot (Fig. 4.1), showing minima on either
side (Fig. 4.13). The kinetic and surface energies have been computed at the steady state/quasisteady state for bubbles lying in all the regions of the phase plot except for the region V.
4.4
Drops
Three-dimensional study of drops falling under gravity has been presented in this section. This is an
ongoing work, therefore only some of the preliminary work has been discussed here. Effect of inertia
on the drop dynamics is shown in Fig. 4.14. For low density ratios the drop remains almost spherical
61
(a)
(b)
250
60
200
150
40
TE
TE
100
20
50
0
10
0
0.1
100
1
10
100
Bo
Ga
Figure 4.13: Variation of the sum of kinetic and surface energies (T E) for (a) Bo = 20, and (b)
Ga = 100. The peak in energy corresponds to the boundaries of the regions shown in Fig. 4.1.
and deforms into a dimpled ellipsoidal shape for slightly higher values of the parameter. The drop
tends to take an upward opening cup-like structure for higher values of density ratio, which is also
observed in axisymmetric simulations [6,22]. For the values of ρr approximately greater than 20, the
surrounding medium tends to shear off a thin portion of the drop leading to a thin skirt-like structure
emanating from the periphery of the drop. This shearing may occur at multiple locations at drop
surface, resembling a Kelvin-Helmholtz like instability which is more pronounced for larger values of
ρr (for instance see the 6th and 7th row of Fig. 4.14). For density ratios of the order of 100 or more,
a violent breakup may occur, leading to multiple fragments of the drop. These regimes may change
depending on the other parameters i.e. Ga, Bo and µr . A computational study of fragmentation
of falling drops has been conducted by Jalaal and Mehravaran [34]. For higher density ratios, the
dynamics may be more chaotic and would need further attention. Drop fragmentation results due
to Villermaux & Bossa [191] show a reverse bag breakup mode which are not observed for density
ratios upto 100 in the present results. Although this regime(very low Bond number, high Gallilei
number and high density ratio) is very difficult to simulate and the computational cost is very high,
the dynamics need to be understood as it is an essential part in understanding rain. One such result
is shown for ρr = 1000 in Fig. 4.15. The drop has a very high inertia, which causes the KelvinHelmholtz instabilities to grow (Fig. 4.15(a)) on the surface of the drop and shear away fragments
of it(Fig. 4.15(b)) violently.
The effect of viscosity ratio on drop dynamics is depicted in Fig. 4.16. It is observed that the
breakup is delayed as the viscosity ratio is increased. The breakup occurs from the periphery of
the drop after it forms a skirt-like structure. The instability grows on the skirt which ultimately
separates ring-like structures from the drop, in its wake. It can be noticed that the phenomenon
is largely axisymmetric before breakup, thus can be understood with axisymmetric simulations in
these cases. However for low Bond and Gallilei numbers the drop starts to deviate from axisymmetry
and may execute zigzagging or spiralling motion, which is a part of the ongoing work.
62
Figure 4.14: Time evolution of drops for different values of density ratios (ρr ) for parameter values:
Ga = 40, Bo = 5 and µr = 10.
4.5
Conclusions
In the first part of this work, we studied the rise under gravity of an initially static and spherical
bubble whose density and viscosity are fixed to be much smaller than that of the surrounding fluid.
The parameters that govern the dynamics are the Galilee and the Bond numbers. Our extensive
fully three-dimensional study, with Ga and Bo ranging from 7 to 500 and 0.1 to 200, respectively,
brings to light a number of features. We find five distinct regions in the phase plot, with sharply
defined boundaries. The bubble is axisymmetric in region I, non-axisymmetric in regions II and
III, and breaks in regions IV and V. Region II, where the bubble consists of an axisymmetric
spherical cap and a skirt with minor asymmetries, is distinguished by the M o ∼ 10−3 line from the
dramatically asymmetrical bubbles of region III. This Morton number has been found in experiments
to be the highest at which path instabilities are seen. Region II bubbles are non-oscillatory whereas
all bubbles of region III display path instabilities, in the form of spirals or zig-zags. This shows an
intimate connection between shape and path asymmetries. In regions IV and V the bubble motion is
unsteady and shows two different kinds of topology change: peripheral break-up and toroid formation
respectively, the latter is followed by break-up. Moving along lines of constant Morton number on
63
(a)
(b)
Figure 4.15: A large liquid drop violently breaking up while falling in the air at times t = 4 and 5
(from left to right) for parameter values: Ga = 40, Bo = 5, ρr = 1000 and m = 10 .
this plot, i.e., for bubbles of increasing radius placed in a given surrounding liquid, there are thus
up to three transitions which take place. Some older experiments [1] have given crude boundaries
between different shapes of bubbles in regions I to III, with very good agreement with present
simulations in the transition from axisymmetric to wobbly. At low Morton number in region IV,
we show a new kind of bubble break-up, into a bulb-shaped bubble and a few satellite drops. Each
transition is clearly distinguishable in terms of the completely different behaviour on either side. A
maximum in kinetic plus surface energy occurs on the transition boundaries as shown in Fig. 4.13.
The importance of studying this problem in three-dimensions is brought out at many places in this
chapter. Other three-dimensional studies have obtained the path instability, but not the transition
to other regimes. We hope that this work will motivate experiments on initially spherical bubbles
to check our phase plot.
In the latter part of this work, we studied the dynamics of falling drops. As opposed to bubbles,
falling water drops in air are challenging to study due to high inertia and high surface tension
forces. To our knowledge no numerical work exists on extensive simulations of high density ratio
drops falling in air, whereas a vast literature exists on simulation of bubbles. Most of the numerical
work on fragmentation and atomization of sprays has been done for low density ratios [34, 174].
A preliminary study of effect of density and viscosity ratio has been carried out in this work. A
thorough study of the effect of Ga and Bo is being carried out currently.
64
Figure 4.16: Time evolution of drops for different values of viscosity ratios (µr ) for parameter values:
Ga = 40, Bo = 5 and ρr = 10.
65
Chapter 5
Bubble rise in a Bingham plastic
5.1
Introduction
The motion of droplets in fluids that exhibit yield stress is important in many engineering applications, including food processing, oil extraction, waste processing and biochemical reactors. Yield
stress fluids or viscoplastic materials flow like liquids when subjected to stress beyond some critical
value, the so-called yield stress, but behave as a solid below this critical level of stress (see Chapter 1
for a brief review). As a result the gravity-driven bubble rise in a viscoplastic material is not always
possible as in the case of Newtonian fluids but occurs only if buoyancy is sufficient to overcome the
material’s yield stress [86, 87]; the situation is also similar for the case of a settling drop or solid
particle [88].
The first constitutive law proposed to describe this material behavior is the Bingham model [89]
which was later extended by Herschel & Bulkley [90] to take into account the effects of shear-thinning
(or thickening). According to this model the material can be in two possible states; it can be either
yielded or unyielded, depending on the level of stress it experiences. As the common boundary of
the two distinct regions the so-called yield surface is approached, the exact Bingham model becomes
singular. In simple flows this singularity does not generate a problem, but, in more complex flows
the discontinuous behaviour of the Bingham model may pose significant difficulties due to the fact
that in most cases the yield surface is not known a priori but must be determined as part of the
solution. Nevertheless, there are examples of successful analysis of two-dimensional flows using this
model at the expense of relatively complicated numerical algorithms [88, 91–93]. A simpler way to
overcome these difficulties is to modify the Bingham constitutive equation in order to produce a
non-singular constitutive law, by introducing a ‘regularization’ parameter [94]. This method has
been used with success by several researchers in the past [87, 95–98] and when used with caution
can give significant insight in the behaviour of viscoplastic materials. As mentioned in the literature
review, several authors have investigated the creeping flow [110, 111] and steady state dynamics [87]
of bubbles in viscoplastic media. In the numerical simulations of [87], even in the cases where a
steady solution could be obtained it is not certain that this solution is stable. Therefore a question
that arises is whether for some parameter values it is possible to get a time-dependent solution and
what would be the dynamics of the bubble flow in this case. This is the question that our study
attempts to address.
66
In this part of the work, we assume axial symmetry and study the buoyancy-driven rise of a bubble
inside an infinite viscoplastic medium. To account for the viscoplacity we consider the regularized
Herschel-Bulkley model. We employ the Volume-of-Fluid (see Chapter 2 for more details) method
to follow the deforming bubble along the domain. Most of the work presented here is contained in
one of our papers in press [24].
5.2
Formulation
We consider the rise of a bubble (Newtonian fluid ‘B’) in a viscoplastic material (fluid ‘A’) under
the action of buoyancy within a cylindrical domain of diameter H and height L, as shown in Fig.
5.1. We use an axisymmetric, cylindrical coordinate system, (x, z), to model the flow dynamics, in
which x and z denote the radial and axial coordinates, respectively, the latter being aligned in the
opposite direction to gravity. The bubble is initially present at a distance zi above the bottom of
the domain located at z = 0. The governing equations of the problem correspond to those of mass
and momentum conservation (Eq. (2.15)-(2.16)) as described in Chapter 2. After dropping off the
energy and vapour advection-diffusion equations, the dimensionless governing equations relevant to
this problem are,
ρ
�
∇ · u = 0,
�
(5.1)
�
�
∂u
1
δ
+ u · ∇u = −∇p +
∇ · µ(∇u + ∇uT ) − ρ�ez +
κ�n,
∂t
Ga
Bo
∂ca
+ u · ∇ca = 0,
∂t
(5.2)
(5.3)
along with the following dependence of density on the volume fraction of the outer fluid:
ρ = (1 − ca )ρr + ca .
(5.4)
The outer fluid viscosity (dimensional), µo is given by the regularized Herschel-Bulkley model
µo =
τ0
n−1
+ µ0 (Π + �)
,
Π+�
(5.5)
where τ0 and n are the yield stress and flow index, respectively, � is a small regularization parameter, and µ0 is the fluid consistency; Π = (Eij Eij )
tensor, wherein Eij ≡
1
2
1/2
is the second invariant of the strain rate
(∂ui /∂xj + ∂uj /∂xi ). The effect of the regularization parameter � has been
presented in Figs 2.13 and 2.14 of Chapter 2. Finally, we set n = 1 henceforth so that our nonNewtonian fluid corresponds to a Bingham plastic and the effect of a shear-dependent viscosity will
be ignored for the purposes of the present study. An important ingredient of every study that concerns the flow of a viscoplastic material is the determination of the position of the yield surface and
when using a regularized model this can be achieved a posteriori by using the following criterion:
yielded material: T > τ0 ,
(5.6)
unyielded material: T ≤ τ0 ,
(5.7)
67
Figure 5.1: Schematic diagram of a bubble of fluid ‘B’ rising inside a Bingham fluid ‘A’ under the
action of buoyancy. The bubble is placed at z = zi ; the value of H, L and zi are taken to be 20R,
48R, and 10.5R, respectively. Initially the aspect ratio of the bubble, h/w is 1, wherein h and w are
the maximum height and width of the bubble.
where T denotes the second invariant of the stress tensor in material ‘A’,
�
1
T =
τij τji
2
�1/2
,
(5.8)
and τij is given by
τij = µo Eij .
(5.9)
In addition to the non-dimensionalization procedure presented in Chapter 2, the viscosity, µ, is
non-dimensionalized as:
µ=
�
�
Bn
n−1
+ m (Π + �)
c + (1 − c)µr ,
Π+�
(5.10)
where the definitions of Bn, m and µr are given in Chapter 2. The position of the yield surface is
determined by evaluating the dimensionless second invariant of stress tensor, T , inside fluid ’A’ and
finding the locus of points for which T = Bn. Rest of the dimensionless parameters are same as
those discussed in Chapter 2.
5.3
Results
We numerically solve the governing equations in a finite-volume framework using gerris as well
as a bespoke diffuse-interface solver. The results from the open-source solver, gerris have been
cross-checked with our diffuse-interface code results for accuracy. Note that in the framework of
the diffuse-interface method, the advection equation of the colour function is modified to contain
68
a diffusion term with a very low dimensionless diffusion coefficient, of the order of grid size (see
Chapter 2 for details).
We assume that the flow is symmetrical about the axis x = 0. Stress free boundary conditions
are imposed at the rest of the boundaries. The domain width is chosen such that the yielded region
is well within the boundaries. A dimensionless domain of H = 32 and L = 48 has been found to be
a reasonable choice for the set of parameter values considered in the present study. We compare the
shape of the bubble along with the unyielded region obtained using the simple regularized viscosity
model (Eq. (5.10)) with the Papanastasiou’s model [109] (Eq. (5.11)) in Fig. 2.15 of Chapter 2. It
can be seen that the results agree qualitatively. Thus in this part of the work, the rest of the results
are generated using the simple regularized viscosity model (Eq. (5.10)).
µo = Bn
�
1 − e−N Π
Π
�
+ m (Π + �)
n−1
.
(5.11)
In Fig. 5.2, we present an illustration of the convergence of the numerical solutions upon mesh
refinement. The parameters chosen for this case are Re = 70.71, Bn = 14.213, µr = 0.01, ρr = 0.001,
m = 1 and Bo = 30. Other validations for this code can be found in Chapter 2.
(a)
(b)
Figure 5.2: The shape of the bubble along with the mesh at t = 1.5 are shown for (a) finer and (b)
coarser grids. Adaptive grid refinement has been used in the interfacial and yielded regions. The
smallest mesh size in the finer and coarser grids are 0.015 and 0.0625, respectively. Note that the
finer grid has been used to generate the results presented in the subsequent figures. The parameter
values are Ga = 70.71, Bn = 14.213, µr = 0.01, ρr = 0.001, m = 1 and Bo = 30. The aspect ratios
of the bubble obtained using the finer and courser grids are 1.002 and 1.003, respectively.
69
5.3.1
Discussion
We begin the discussion of our results by examining the dependence of the results on the regularisation parameter, �, used in the viscosity model for fluid ‘o’, given by Eq. (5.10), for Ga = 70.71,
Bn = 14.213, µr = 0.01, ρr = 0.001, m = 1 and Bo = 30. In Fig. 2.13, we show that the bubble
rise is accompanied by its deformation and the development of a yielded region which surrounds the
bubble at t = 10, in which the stress generated from the bubble motion is sufficient to exceed the
yield stress in fluid ‘o’; this region is itself surrounded by unyielded fluid. Also shown in Fig. 2.13 is
the formation of three small unyielded regions: two at the bubble equator, and one near the dimple
located at the bubble base; similar predictions have been presented by Tsamopoulos et al. [87] using
the Papanastasiou model [109].
It is seen that the dependence of the shapes of the bubble and the yielded region surrounding it, as
well as the extent of the unyielded regions immediately adjacent to the bubble becomes progressively
weaker with increasing �. At this point, it should be noted that decreasing the value of � the system
of equations becomes stiffer and more difficult to handle numerically. This may also result in the
appearance of numerical noise and therefore very small values of � should actually be avoided. A more
accurate evaluation of the yield surface position is possible, as was shown recently by Dimakopoulos
et al. [93] using the augmented Langrangian method at the expense of a significantly more complex
numerical algorithm. Nevertheless, for the purposes of this study, the calculated yield surfaces are
considered to be reasonably accurate. We have also found that the time evolution of the bubble
aspect ratio (h/w), which is defined as the ratio of instantaneous maximum height of the bubble to
its maximum width, and its centre of gravity, zCG , exhibit a similar dependence on � and become
virtually indistinguishable with decreasing �, as shown in Fig. 2.14. Thus, the rest of the results
discussed in this chapter have been generated using � = 0.001.
Next, we study the bubble rise dynamics by examining the temporal evolution of the bubble
aspect ratio and centre of gravity for varying Bingham number, Bn, with Ga = 7.07, Bo = 10,
µr = 0.01, ρr = 0.001, and m = 1. It is seen in Fig. 5.3 that for low Bn values, which reflect the
presence of a weak yield stress, the bubble undergoes severe deformation at relatively early times
before assuming a constant aspect ratio. More specifically, for Bn = 0.071 the aspect ratio is found
to be approximately equal to 0.48 in good agreement with the predictions given by Tsamopoulos et
al. [87]. We also found that, as expected, the rise velocity of the bubble decreases with Bn due to
the increased resistance associated with the larger yield stresses (see Fig. 5.3).
In the low Bn range, the bubble achieves a constant rise speed rapidly, as shown by the linear
dependence of zCG on time. In particular for Bn = 0.071 the calculated terminal velocity is approximately equal to 0.765 in agreement with the predicted value of 0.75 given in Tsamopoulos et
al. [87]. The extent of bubble deformation and rise speed decrease with increasing Bn for Bn less
than unity for the parameters used to generate the results shown in Fig. 5.3; the same trend was
also found in Tsamopoulos et al. [87]. For higher Bn values, e.g. Bn = 0.99, we notice that the
bubble aspect ratio (1.05) and terminal velocity (0.226) differ significantly from the predictions of
Tsamopoulos et al. [87], i.e. 1.25 and 0.07, respectively. The difference cannot be attributed to the
finite viscosity of the fluid since, as shown in Fig. 5.4, increasing the viscosity ratio, µr , leads to the
decrease of the rise velocity. We notice though that even at late times the deformation of the bubble
has not reached a steady state (see Fig. 5.4b) and continues to change. As shown in Fig. 5.3b, the
latter effect is more prominent for even higher values of the Bn number where we see clearly that
70
(a)
(b)
1.2
16
Bn
0
0.071
0.354
0.99
1.343
14
zCG
Bn
1
0
0.071
0.354
0.99
1.343
h/w
0.8
12
0.6
10
0
10
20
t
0.4
0
30
10
20
t
30
Figure 5.3: (a) Temporal variation of the center of gravity, (b) the aspect ratio of the bubble for
different values of Bn. The parameter values are Ga = 7.071, µr = 0.01, ρr = 0.001, m = 1 and
Bo = 10.
(a)
(b)
3
1.06
2
1.04
zCG
µr
1
0
0
3
6
t
9
12
µ
h/w
0.001
0.005
0.01
0.05
0.001
0.005
0.01
0.05
1.02
1
0
15
3
6
t
9
12
15
Figure 5.4: (a) Temporal variation of the center of gravity, (b) the aspect ratio of the bubble for
different values of µr . The parameter values are Ga = 7.071, Bn = 0.99, ρr = 0.001, m = 1 and
Bo = 10.
the flow does not reach a steady state and that the bubble aspect ratio exhibits finite amplitude
oscillations. These oscillations in the bubble deformation may lead to yielding of the surrounding
material and thereby could be responsible for the enhancement of the bubble motion.
Fig. 5.5 depicts the spatio-temporal evolution of the shape of the bubble and its surrounding
unyielded region as a function of Bn for the same parameters used to generate Fig. 5.3. Inspection
of this figure shows that the extent of the unyielded region increases with Bn, as expected, and for
Bn < 1, the bubble widens as it rises, which is consistent with the results shown in Fig. 5.3(b) for
the same range of parameter values and in accordance with the findings of Tsamopoulos et al. [87].
These shapes become steady with increasing time. For Bn = 1.34, it is evident that the bubble
aspect ratio exceeds unity, which is also consistent with Fig. 5.3(b), likely brought about by the
confinement due to the smaller yielded region associated with this value of Bn; it is also evident
that the shapes of the bubble and unyielded regions do not achieve a steady-state in this case.
In Fig. 5.6 we show contour plots of the radial and axial components of the velocity field for
Bn = 0, 0.35, 0.99, 1.34, and the rest of the parameters remain unchanged from those of Fig. 5.3.
The case with Bn = 0 corresponds to the Newtonian case. It is clearly seen that the radial and
71
axial velocity components exhibit stagnation contours that separate regions of outward and inward,
and upwards and downwards motion, respectively; at regions where the radial motion of the fluid is
negligible, unyielded zones are likely to occur. The stagnation contour associated with the vertical
component moves progressively closer to the interface with increasing Bn; for the largest Bn studied,
it is evident that the regions nearest the top and bottom of the bubble move upwards, while the
remaining regions move downwards leading to bubble elongation. The stagnation contour associated
with the radial component emanates from rightmost bubble edge at a negative angle to the horizontal
in the Newtonian case. This contour becomes essentially horizontal and the bubble, whose bottom
is dimpled in the Newtonian case, becomes well-rounded with inceasing Bn as the bubble becomes
flatter at the equatorial plane.
It is also important to study the effect of bubble deformability on its dynamics; this is done by
varying the Bond number, Bo, which reflects the relative significance of surface tension to gravtiational forces. In Fig. 5.7, it is seen clearly that for low Bo, for which surface tension forces are
dominant, bubble deformation is small and its rise speed is constant, increasing with Bo. For larger
Bo, however, the bubble dynamics gain in complexity. The bubble appears to undergo sudden acceleration between periods of constant rise speed; these periods become shorter and the magnitude
of the acceleration increases with Bn, as shown in Fig. 5.7a. This zCG dynamics is associated
with large bubble deformation as can be ascertained upon inspection of Fig. 5.7b: the aspect ratio
undergoes nonlinear oscillations about unity as the bubble ‘swims’ upwards, whose wavelength and
amplitude increase with Bo.
In order to rationalise the results presented in Fig. 5.7 and further highlight the role of bubble
deformation in the ‘swimming’ motion discussed, we show in Fig. 5.8 the spatio-temporal evolution
of the shape of the bubble and the unyielded regions for Bo = 1 and Bo = 30 while the rest of
the parameters remain unchanged from those of Fig. 5.7. It is seen that for Bo = 30, at relatively
early times unyielded regions are situated in the equatorial region of the bubble, and the bubble
aspect ratio is close to unity. With increasing time, the extent of the unyielded regions decreases
due to the shear stress associated with the bubble acceleration and the bubble elongates as it rises
through a yielded region of increasing size. The bubble then decelerates to a constant rise speed, its
aspect ratio decreases, and the decrease in shear stress in the vicinity of the interface leads to the
development of unyielded zones in the equatorial and south pole regions; the former become more
pronounced with increasing time, and the bubble aspect ratio decreases below unity as the bubble
decelerates. The process is then repeated. In contrast, no such process is evident in the case of
Bo = 1 for which the bubble appears to suffer negligible deformation and the size of the unyielded
regions remains largely unaltered.
In Fig. 5.9, we show the effect of Bo on the contour plots of the radial and axial velocity
components for Bo = 1 and Bo = 30; these plots are shown for t = 6 and t = 8.5 that correspond
to the times at which the bubble achieves its maximal and minimal aspect ratio for Bo = 30,
respectively. As can be seen from this figure, the magnitude of both components remains essentially
unchanged for the Bo = 1 case, while, for the same times in the Bo = 30 case, the axial and radial
velocity components dominate at t = 6 and t = 8.5, resulting in bubble elongation, and flattening
and dimpling, respectively.
72
Figure 5.5: The evolution of the shape of the bubble (shown by red lines) and the unyielded region
in the non-Newtonian fluid (shown in black) for different values of Bingham number. The results
of the Newtonian case are shown for the comparison purpose. The rest of the parameter values are
the same as those used to generate Fig. 5.3.
73
(a)
(b)
(c)
(d)
Figure 5.6: Contour plots for the radial (right) and axial (left) velocity components for (a) Bn = 0
at t = 6 (Newtonian case), (b) Bn = 0.354 at t = 6, (c) Bn = 0.99 at t = 20 and (d) Bn = 1.34 at
t = 20. In each panel the shape of the bubble is shown by red line. The rest of the parameter values
are the same as those used to generate Fig. 5.3.
5.4
Conclusions
In this chapter, we have examined the axisymmetric dynamics of bubble rise in Bingham fluids.
We have used an open-source finite-volume flow solver, gerris based on volume-of-fluid methodology
to study the flow, which involves the numerical solution of the equations of mass and momentum
conservation, and an equation of the volume fraction of the Bingham fluid. The momentum equation
accounts for surface tension and gravitational effects, while the density and viscosity are volume
fraction-weighted with respect to the corresponding quantities of the two fluids. For the Bingham
fluid, the formula for the viscosity contains a regularisation parameter; convergence of our results
was achieved upon mesh-refinement and reduction of the magnitude of this parameter to sufficiently
small values.
Our numerical results indicate that in the presence of weak yield stress the bubble achieves
a constant rise speed relatively rapidly, whilst its aspect ratio, defined as the ratio of its height
to its width asymptotes to a value less than unity; unyielded zones are confined to regions that
surround but are not immediately adjacent to the bubble. With increasing yield stress, the bubble
74
(a)
15
14
(b)
Bo
0.1
1
5
30
Bo
0.1
1
5
30
1.4
1.2
13
zCG
h/w
12
1
11
0.8
10
0
5
10
t
15
20
0
5
10
t
15
20
Figure 5.7: (a) Temporal variation of the center of gravity, (b) the aspect ratio of the bubble for
different values of Bo. The rest of the parameter values are Re = 70.71, Bn = 14.213, µr = 0.01,
ρr = 0.001, and m = 1.
rise is unsteady, and the bubble aspect ratio exhibits oscillations above a value that exceeds unity.
Unyielded zones near the equatorial and south pole regions of the bubble have also been observed
to form for sufficiently large yield stress in agreement with earlier studies in the literature [87, 93].
We have also shown that bubble deformation has a profound impact on the dynamics. In the
case of strong surface tension, the rise is steady and the bubble suffers negligible deformation. For
weak surface tension, however, the rise is unsteady, periods of approximately constant rise speed are
separated by rapid acceleration stages that coincide with oscillations in the bubble aspect ratio about
unity whose amplitude increases with decreasing surface tension. These oscillations also coincide
with the formation and destruction of unyielded zones in the equatorial regions. The motion executed
by the bubble for this range of parameters resembles ‘swimming’ as the bubble appears to grab hold
of the unyielded zones to propel itself upwards.
75
Figure 5.8: The evolution of the shape of the bubble (shown by red lines) and the unyielded regions
in the Bingham fluid (shown in black) for different values of Bo. The rest of the parameter values
are the same as those used to generate Fig. 5.7.
76
(a)
(b)
(c)
(d)
Figure 5.9: Contour plots for the radial (right) and axial (left) velocity components for (a) Bo = 1
at t = 6, (b) Bo = 1 at t = 8.5, (c) Bo = 30 at t = 6 and (d) Bo = 30 at t = 8.5. In each panel the
shape of the bubble is shown by red line. The rest of the parameter values are the same as those
used to generate Fig. 5.7.
77
Chapter 6
Non-isothermal bubble rise
6.1
Effect of temperature gradients
Interfacial flows with temperature gradients in the surrounding medium invariably create interfacial
tension gradient along the interface separating fluid pairs. A gradient in interfacial tension cause the
fluid to flow towards the regions of high surface tension so as to minimize the surface energy of the
system. A typical problem where Marangoni stresses play a significant role is the thermocapillary
migration of bubbles and drops. Much of the work in this field has been reviewed by [112] and [113].
A brief literature review is presented in Chapter 1. In this chapter, we present the buoyancy-driven
rise of a bubble inside a tube imposing a constant temperature gradient along the wall. To account
for the non-monotonicity of surface tension we consider a quadratic dependence on temperature.
We examine the Stokes flow limit first and derive conditions under which the motion of a spherical
bubble can be arrested in self-rewetting fluids even for positive temperature gradients. We then
employ a diffuse-interface method [169] to follow the deforming bubble along the domain in the
presence of inertial contributions. Our results indicate that for self-rewetting fluids, the bubble
motion departs considerably from the behaviour of ordinary fluids and the dynamics may become
complex as the bubble crosses the position of minimum surface tension. As will be shown below,
under certain conditions, the motion of the bubble can be reversed, and then arrested, or the bubble
can become elongated significantly. A large portion of this chapter appears in one of our published
works [23].
6.2
Formulation
Apart from the continuity, Navier-Stokes and Cahn-Hilliard equation/advection equation within the
diffuse interface/volume of fluid framework, we solve the temperature equation to allow for the
conduction and convection within the fluid. The set of equations to be solved are the same as
discussed in Chapter 2. An axisymmetric domain is considered to solve the governing equations.
It is to be noted that only in this work, a bounded domain has been considered. The geometry
is shown in Fig. 6.1 for clarity, where a no-slip boundary condition is imposed on the wall of the
cylinder. Other boundary conditions are same as those employed in Chapter 3.
The viscosity (dimensional) is assumed to depend on the temperature and the volume fraction
78
Figure 6.1: Schematic diagram of a bubble moving inside a Newtonian fluid under the action of
buoyancy. The initial location of the bubble is at z = zi ; unless specified, the value of H, L and
zi are 6R, 48R, and 10.5R, respectively. The acceleration due to gravity, g, acts in the negative z
direction.
as follows (from Eqs 2.12 and 2.13):
µ = cµo e
c
−( TTm−T
−Tc )
+ (1 − c)µi
�
1+
�
T − Tc
Tm − Tc
�3/2 �
,
(6.1)
where Tc and Tm are the temperature at the bottom of the tube (z = 0) and temperature at z = zm ,
wherein zm corresponds to the vertical location where surface tension reaches its minimum; µA and
µB are the viscosity of the liquid and gas at temperature Tc , respectively. This viscosity dependence
on temperature for the liquid and gaseous phases are taken from [201]. The density (ρ) and thermal
conductivity (λ) are calculated as linear functions of the volume-fraction of the outer fluid [128].
In order to model the behaviour of a self-rewetting fluid, we use the following relationship for
the functional dependence of the surface tension on temperature (from Eq. 2.7):
σ = σ0 − β1 (T − Tc ) + β2 (T − Tc )2 ,
dσ
where β1 ≡ − dT
|Tc and β2 ≡
1 d2 σ
2 dT 2
(6.2)
|Tc . A linear temperature variation is imposed in the vertical
direction with a constant gradient γ, and Fig. 6.2 shows the corresponding variation in σ with z
for different values of β1 and β2 . As can be seen from this figure, the parabolic dependence of σ
on z becomes more pronounced, with a deeper minimum, located at z = zm , for increasing β1 and
β2 ; this is expected to alter the type of Marangoni flow observed in case fluids that exhibit a simple
linear variation of σ with T which we will refer to in this chapter as ‘linear’ fluids. Below, we will
explore the dynamics of the bubble as it rises starting from zi , which may be either below or above
z = zm , for both linear and self-rewetting fluids. The dependence of this dynamics on β1 and β2 ,
which parameterise the behaviour of various self-rewetting fluids, will also be studied.
The governing equations are non-dimensionalized with the characteristic scales as described in
79
1.2
M1=0.4, M2=0.2
1.12
M1=0.2, M2=0.1
M1=0.1, M2=0.05
1.04
σ 0.96
0.88
0.8
0
10
5
z
15
20
25
Figure 6.2: Variation of the liquid-gas surface tension along the wall of the tube for Γ = 0.1 and
various values of M1 and M2 .
Chapter 2, however the following variables are non-dimensionalized differently:
T = T�(Tm − Tc ) + Tc , β1 =
where the velocity scale is V =
σ0
σ0
�1 , β2 =
�2 , γ = (Tm − Tc ) Γ,
M
M
Tm − Tc
(Tm − Tc )2
R
√
(6.3)
gR, σ0 is the surface tension at Tc , and the tildes designate
dimensionless quantities. After dropping tildes from all non-dimensional terms, the governing dimensionless equations are given by
∇ · u = 0,
�
�
∂u
1
+ u · ∇u = −∇p +
∇ · µ(∇u + ∇uT ) − ρ�ez + Fs ,
∂t
Ga
(6.4)
(6.5)
∂(ρcp T )
1
+ ∇ · (uρcp T ) =
∇ · (λ∇T ),
∂t
GaPr
(6.6)
∂c
+ u · ∇c = 0,
∂t
(6.7)
where Ga ≡ ρo V R/µo denotes the Gallilei number; Pr ≡ cp µo (Tc )/λo is the Prandtl number, wherein
cp is the specific heat capacity at constant pressure, and λo is the thermal conductivity of the liquid.
The dimensionless viscosity, µ, has the following dependence on T and ca :
�
�
µ = ca e−T + (1 − ca )µr 1 + T 3/2 .
(6.8)
The dimensionless density and thermal diffusivity are the same as those described in Chapter 2. In
Eq. (6.5), the surface tension force Fs is given by (continuum surface force formulation [178])
Fs =
�
κδ �
1 − M1 T + M2 T 2 �n.
Bo
(6.9)
In this relation, the dependence of σ on T , using Eq. (6.2) has been included. As the bubble is
80
assumed to reach a terminal location for some values of M1 and M2 , the Stokes flow assumption is
valid when the bubble slows down and tends to stop the flow. Therefore, we can solve the equations
of motion in creeping flow regime and find the terminal location of the bubble without fully solving
the Navier-Stokes equation.
6.3
Analytical results: Stokes flow limit
In this section, we provide a discussion of our analytical results. We derive expressions for the
terminal velocity of a spherical bubble rising vertically through a quiescent liquid in the Stokes flow
limit in which thermocapillary stresses arise due to a temperature gradient imposed on the liquid.
For the purpose of this calculation we will ignore the presence of walls and consider the case of
unconfined flow. We show how the dependence of the surface tension on temperature, represented
by Eq. (6.2), affects the terminal bubble speed, and the magnitude and sign of the temperature
gradient required to arrest bubble motion; this is also contrasted with the case of a linear fluid.
We adopt a spherically-symmetric coordinate system, (r, θ), with the polar angle, θ, measured
from the bottom of the bubble (θ = 0) to the top of the bubble (θ = π); u = ur ir + uθ iθ is the
velocity field in which ur and uθ represent its radial and azimuthal components, and ir and iθ denote
the unit vectors in the r and θ directions, respectively. The z-axis originates at the bubble centre
and is oriented vertically upwards so that it coincides with the axis of symmetry of the bubble. The
unit vector in the z-direction is expressed by iz = −ηir + (1 − η 2 )1/2 iθ in which η ≡ cos θ. Note that
we have adopted a different coordinate system from that in Section ?? temporarily for the purpose
of this calculation and redefined the origin of the z-axis. At the end of the present subsection, we
shall revert to the use of cylindrical coordinates. We also adopt a frame of reference that moves with
the centre of the bubble, which is scaled on the steady translational speed of the bubble, U ; this
speed will be determined as part of the solution. Note that all quantities presented in this subsection
are in dimensional terms.
We assume that heat transfer is dominated by conduction so that the temperature field in the
liquid, T , is governed by
∇2 T = 0.
(6.10)
We impose a linear temperature distribution in the liquid, so that at large distances from the bubble
we have
T∞ (z) = T∞ (0) + γz � .
(6.11)
Here, T∞ (0) denotes the temperature at z � = 0, the position of the centre of the bubble. At the
bubble surface, r = R, we demand continuity of the thermal flux:
∂T
=
∂r
�
λB
λA
�
∂Tg
,
∂r
(6.12)
where λB denotes the thermal conductivity of the gas. We assume that λA � λB so that Eq. (6.12)
reduces to
∂T
= 0.
∂r
81
(6.13)
The general solution of Eq. (6.10) is given by
T =
∞ �
�
� r �n
An
R
n=0
+ Bn
� r �−(n+1) �
R
Pn (η),
(6.14)
where Pn (η) are Legendre polynomials of the first kind of degree n. We apply the no-flux condition
given by Eq. (6.13) at r = R:
Bn =
�
n
n+1
�
An .
(6.15)
�
r �n
n � r �−(n+1)
+
Pn (η).
R
n+1 R
(6.16)
Substitution of Eq. (6.15) into Eq. (6.14) gives
T =
∞
�
An
n=0
��
To match to the far field condition, we set z � = r cos(π − θ) = −r cos θ = −rη in Eq. (6.11) so that
T∞ (z) = T∞ (0) − γrη. Matching this equation to Eq. (6.16) yields
A0 = T∞ (0),
A1 = −γR,
An = 0 for n ≥ 2.
(6.17)
Substitution of these values into Eq. (6.16) gives
�
�
1 � r �−3
T = T∞ (0) − γr 1 +
η.
2 R
(6.18)
The solution for the velocity field in the fluid, u = (ur , uθ ), is subject to the following boundary
conditions:
u → −iz
as
|r| → ∞,
(6.19)
ur = 0
at
r = R,
(6.20)
τrθ +
∂σ
=0
∂θ
at
r = R,
(6.21)
where τrθ is the tangential stress.
As noted above, the flow is axisymmetric about the z-axis, hence the solution to the Stokes flow
problem can be expressed in terms of the streamfunction, ψ [202]:
ψ = UR
2
�
−
� r �2
R
Q1 (η) +
∞ �
�
Cn
n=1
� r �2−n
R
+ Dn
� r �−n �
R
�
Qn (η) .
(6.22)
This is the general solution for flow past an axisymmetric body of arbitrary shape in the Stokes
flow limit. Here, Qn are integrals of Pn (η), and closely related to the Gegenbauer polynomials. The
polynomials relevant to the present work are
Q1 (η) =
1 2
(η − 1),
2
Q2 (η) =
η 2
(η − 1).
2
(6.23)
The solution expressed by Eq. (6.22) is chosen to satisfy the following equation
E 4 ψ = 0,
82
(6.24)
where E 2 is given by
E2 ≡
∂2
(1 − η 2 ) ∂ 2
+
,
∂r2
r2
∂η 2
(6.25)
as well as the far field condition given by Eq. (6.19). The streamfunction ψ is related to ur and uθ
by
ur = −
1 ∂ψ
1
∂ψ
, uθ = −
.
r2 ∂η
r(1 − η 2 )1/2 ∂r
(6.26)
It can be shown that for flow problems that have ψ expressed as in Eq. (6.22), the component
of the dimensional force along the axis of symmetry, Fz , exerted by the surrounding fluid on an
axisymmetric body of arbitrary shape with its centre of mass at |x| = 0 is given by the following
general formula
Fz = 4πµA U RC1 .
(6.27)
At steady-state, this drag force balances the buoyancy force:
4πµA U RC1 =
4
4
π(ρA − ρB )R3 g ≈ πρA R3 g,
3
3
(6.28)
where we have assumed that ρA � ρB . Equation (6.28) suggests that C1 is the only coefficient that
we need to compute in order to determine the terminal velocity of the bubble.
The no-penetration condition given by Eq. (6.20) can be re-expressed as ∂ψ/∂η = 0 at r = R.
It follows from this condition that ψ is constant at r = R. Since ψ = 0 at θ = 0 and θ = π,
corresponding to η = ±1, for all r because of symmetry, then Eq. (6.20) can be re-written as
ψ=0
at
r = R.
(6.29)
Application of this condition yields
0 = −Q1 (η) +
∞
�
(Cn + Dn ) Qn (η).
(6.30)
n=1
The tangential stress balance given by Eq. (6.21) can be re-expressed as
−µA r
∂
∂r
�
1
∂ψ
2
2
1/2
∂r
r (1 − η )
�
+
∂σ
= 0 at r = R.
∂θ
(6.31)
Using Eq. (6.2), the surface tension gradient, ∂σ/∂θ is then given by
∂σ
∂θ
=
=
×
∂σ ∂T
�∂T ∂θ
�
�
� ��
1 � r �−3
−β1 + 2β2 T∞ (0) − Tc − γr 1 +
η
2 R
� �
�
�
1 � r �−3
γr 1 +
(1 − η 2 )1/2 ,
2 R
83
(6.32)
where Tc is a reference temperature. Substitution of Eqs. (6.22) and (6.32) into Eq. (6.31) yields
−2Q1 (η)
+
=
=
2C1 Q1 (η) − 4D1 Q1 (η) +
∞
�
n=2
((2 − n)(1 + n)Cn − n(n + 3)Dn ) Qn (η)
�
�
��
−3γR
3
−β1 + 2β2 T∞ (0) − Tc − γRη (1 − η 2 )
2µA U
2
�
�
−3γRβ1
2β2
9γ 2 R2 β2
1−
(T∞ (0) − Tc ) Q1 (η) −
Q2 (η).
µA U
β1
µA U
(6.33)
From Eq. (6.30) we get
−Q1 + (C1 + D1 )Q1 +
∞
�
(Cn + Dn ) Qn = 0,
(6.34)
n=2
whence
C1 + D1 = 1,
Cn = −Dn for n ≥ 2.
and
(6.35)
For β2 = 0, i.e. for a linear fluid, then from Eq. (6.33) we get
−2Q1 + 2C1 Q1 − 4D1 Q1 = −
3γRβ1
n(n + 3)
Q1 , and Cn =
Dn , for n ≥ 2.
µA U
(n + 1)(2 − n)
Thus, we deduce that
C1 = 1 −
For n ≥ 2,
γRβ1
,
2µA U
D1 =
n(n + 3)
Cn =
Dn = −Dn ⇒
(n + 1)(2 − n)
�
γRβ1
.
2µA U
�
n(n + 3)
+ 1 Dn = 0.
(n + 1)(2 − n)
(6.36)
(6.37)
(6.38)
However, the coefficient of Dn is not zero, so Dn = Cn = 0 for n ≥ 2. From Eq. (6.28), we arrive at
the following expression for U , after making use of C1 from Eq. (6.37):
U=
ρA R2 g
3µA
�
1+
3 γβ1
2 ρA Rg
�
.
(6.39)
Thus, if U = 0 the bubble rise is arrested provided
γ = γc = −
2ρA Rg
.
3β1
(6.40)
This equation implies that for a linear fluid, the temperature gradient must be negative in order for
the bubble to come to rest [202].
For a self-rewetting fluid with β2 �= 0, and following a similar procedure to that discussed above,
the relevant coefficients are
84
C1
=
C2
=
D1
=
D2
=
�
�
γRβ1
2β2
1−
(T∞ (0) − Tc ) ,
2µA U
β1
9γ 2 R2 β2
−
,
10µA U
�
�
γRβ1
2β2
1−
(T∞ (0) − Tc ) ,
2µA U
β1
9γ 2 R2 β2
,
10µA U
1−
(6.41)
(6.42)
(6.43)
(6.44)
and Cn = Dn = 0 for n ≥ 3. Substitution of C1 into Eq. (6.28) yields the following expression for
the terminal velocity U :
ρA R 2 g
U=
3µA
�
�
��
3 γβ1
2β2
1+
1−
(T∞ (0) − Tc ) .
2 ρA Rg
β1
(6.45)
The expression for γc that leads to bubble arrest and U = 0 is given by:
γc = −
2
ρA Rg
�
�.
3 β 1 − 2 β2 (T (0) − T )
1
∞
c
β1
(6.46)
For β2 = 0, this equation reduces to Eq. (6.40). Equation (6.46) suggests that γc is positive
(negative) if 2β2 (T∞ (0) − Tc ) /β1 > 1 (2β2 (T∞ (0) − Tc ) /β1 < 1) in the case of a self-rewetting
fluid, in contrast to the case of a linear fluid in which γc < 0.
We note that if we had assumed that the temperature distribution given by Eq. (6.11) applied
everywhere, including at the bubble surface, r = R, then the formula for the terminal velocity would
have been expressed by
U=
ρA R 2 g
3µA
�
1+
�
��
γβ1
2β2
1−
(T∞ (0) − Tc ) ,
ρA Rg
β1
(6.47)
and the expression for γc by
γc = −
�
ρA Rg
β1 1 −
2 ββ21
(T∞ (0) − Tc )
�.
(6.48)
Reverting back to the coordinate system of the previous section the position of the bubble centre
is given by
z=
T∞ (0) − Tc
γ
(6.49)
and using Eq. (6.46) it is possible to derive an expression for the terminal vertical position of the
bubble, zc :
zc =
�
β 1 γc
ρA Rg
+
2
3
�
1
.
β2 γc2
(6.50)
In the following section, we compare the predictions of Eq. (6.50) with those obtained from the
numerical simulations. We turn our attention now to the numerical results.
85
15
Isothermal
Linear
Self-rewetting
14
13
zCG
12
11
10
0
2
4
t
6
8
10
Figure 6.3: Temporal variation of the center of gravity of the bubble for the parameter values
Ga = 10, Bo = 10−2 , ρr = 10−3 , µr = 10−2 , Γ = 0.1 and αr = 0.04. The plots for the isothermal
(M1 = 0 and M2 = 0), linear (M1 = 0.4 and M2 = 0) and self-rewetting (M1 = 0.4 and M2 = 0.2)
cases are shown in the figure. The horizontal dotted line indicates the prediction of Eq. (6.51) for
the self-rewetting case.
6.4
Numerical results
In this section, we present a discussion of our numerical results starting with a presentation of the
numerical procedures used to carry out the computations. To account for the effects of inertia and
confinement we solve the governing equations numerically. For this part of the study we have made
use of the cylindrical coordinates. We use a bespoke finite-volume flow solver as well as gerris (more
details are provided in Chapter 2) to simulate the bubble rise in a non-isothermal medium.
Below, we present a discussion of our results for the following set of ‘base’ parameters: Bo = 10−2 ,
Ga = 10, H = 6, µr = 10−2 , M1 = 0.4, ρr = 10−3 , zi = 10.5, and Γ = 0.1, which are consistent
with the case of a small air bubble rising in water due to buoyancy, in the presence of strong mean
surface tension and Marangoni effects, and appreciable inertial contributions. We will contrast the
difference in behaviour between bubble motion in linear and self-rewetting fluids by studying the
effect of parameter M2 on the dynamics.
We begin the discussion of our results by showing in Fig. 6.3 the temporal variation of the centre
of gravity, zCG , of a rising bubble for three different cases: the isothermal case, and the cases of a
simple linear fluid, and a self-rewetting one rising in a tube whose walls are heated with a linear
temperature profile of constant gradient Γ > 0. It can be seen from this figure that following an
initial, relatively short, acceleration period, the bubble reaches a constant, terminal speed for both
the isothermal case, and the linear fluid in the non-isothermal case. The terminal velocity is higher
for the non-isothermal case due to the presence of Marangoni stresses driving liquid towards the cold
region of the tube and thereby enhancing the upward motion of the bubble. For the self-rewetting
fluid in the non-isothermal case, zCG also reaches a constant speed for a certain time duration; this
is, however, followed by a drop in zCG before a terminal zCG value is reached. Thus, the motion of
a bubble rising initially in a self-rewetting fluid, whose temperature is essentially increasing linearly,
86
is first reversed and then arrested. The fact that the bubble motion comes to a halt in the selfrewetting and not the linear fluid in a positive temperature gradient was also suggested by Eq. (6.45)
in the Stokes flow limit. This property can be used to manipulate bubbles by simply shifting the
temperature gradient along the wall appropriately. This might be of interest to researchers working
in microfluidics and multiphase microreactors. The predictions of the dimensionless version of Eq.
(6.50), given by
�
�
Bo
1
ΓM1
zc =
+
,
(6.51)
2
3
M2 Γ2
are also shown in Fig. 6.3. For the parameters used to generate the results presented in this figure,
zc ∼ 11.67, which is in good agreement with the numerical predictions, despite the fact that strong
inertial contributions are present in the flow as represented by Ga = 10. We will examine the
mechanism underlying this behaviour below.
In Fig. 6.4a, we examine the dependence of the terminal velocity Vt of a bubble rising in a linear
fluid on the parameter M1 ; the latter governs the strength of the linear variation of the surface
tension with temperature. As explained above, for positive values of Γ the induced Marangoni
stresses increase the rise velocity of the bubble. The terminal velocity, though, appears to reach
a plateau with M1 indicating that the strength of Marangoni stresses saturates at large M1 , and
the dynamics are dominated by the remaining parameters. In the case of self-rewetting fluids, one
parameter that we need to take into account is the position of minimum surface tension with respect
to the bubble because it will affect the action of induced Marangoni stresses; this effect will be
studied in detail below. For the time being, we have positioned the center of gravity of the bubble
above the position of minimum surface tension (zi = 10.5, zm = 10) at t = 0. In this case, as the
bubble rises, it comes into contact with liquid of increasingly lower surface tension. The induced
Marangoni stresses drive liquid upwards, towards the hot region of the tube, and inhibit the upward
motion of the bubble. This is evident at early times in Fig. 6.3 where it is shown that the rise
velocity for the self-rewetting fluid is lower than for the isothermal case.
In order to study the effect of Marangoni stress in more detail, we examine in Fig. 6.4b the
terminal distance reached by bubbles moving in a self-rewetting fluid as a function of M1 with
M2 = M1 /2; the latter restriction is imposed in order to keep the position where the minimum
surface tension arises constant. As shown in Fig. 6.4b, this distance increases with decreasing M2 ,
which indicates that an increase in the self-rewetting character of the fluid leads to a larger degree
of bubble retardation: in the limit M2 → 0, a steady, terminal speed is reached for Γ > 0. The
numerical predictions for the terminal distance shown in Fig. 6.4b are also in good agreement with
those obtained from Eq. (6.51): for the parameters used here, zc ∼ (16.67, 13.33, 12.22, 11.67, 11.33)
for M1 = (0.1, 0.2, 0.3, 0.4, 0.5). Also, for Bo ∼ 0, and M2 = M1 /2, Eq. (6.51) reduces to zc ∼ Γ−1 ,
which for the parameters in Fig. 6.4b leads to zc ∼ 10; this appears to be the value to which the
terminal distance limits with increasing M1 .
Interestingly, the onset time for motion reversal, treversal , has a non-monotonic dependence on
M1 : starting from a global maximum at small M1 , treversal exhibits a shallow minimum, followed
by a local maximum, before undergoing a sharp decrease with increasing M1 . This is probably due
to the effect of inertia and the interplay of buoyancy and Marangoni stresses which act in opposite
directions. For low M1 values, the Marangoni stresses are initially relatively weak and take a long
time before they grow to change the direction of motion of the bubble. As M1 increases, Marangoni
87
(a)
5
1.18
1.12
1.07
4
1.03
Vt 3
2
Ar = 1.00
1
0
0.1
0.2
0.3
0.4
M1
(b)
25
M1 = 0.1
M1 = 0.2
M1 = 0.3
20
M1 = 0.4
M1 = 0.5
zCG
z c=
16.67
15
13.33
12.22
11.67
11.33
10
0
2
4
t
8
6
10
(c)
7
treversal
6
5
4
3
0.1
0.2
0.3
0.4
0.5
M1
Figure 6.4: (a) The terminal velocity of the center of gravity of the bubble along with the aspect
ratio for different values of M1 for M2 = 0; (b) temporal variation of the center of gravity of the
bubble for M2 = M1 /2; (c) variation of the time at which zCG reaches its maximum for different
values of M1 . The rest of the parameter values are Ga = 10, Bo = 10−2 , ρr = 10−3 , µr = 10−2 ,
Γ = 0.1 and αr = 0.04. The numerical predictions of Eq. (6.51) are shown by the filled square
symbols on the right vertical axis.
88
25
20
Ga
1
5
10
20
zCG
15
10
0
2
4
t
6
8
10
Figure 6.5: Effect of Ga on the temporal evolution of the bubble centre of gravity for Bo = 10−2 ,
ρr = 10−3 , µr = 10−2 , M1 = 0.2, M2 = 0.1, Γ = 0.1 and αr = 0.04. The prediction of Eq. (6.51) is
shown by the dotted line.
stresses become stronger initially, so the bubble decelerates faster and less time is needed for the
motion reversal. However, as Marangoni stresses gain in relative significance, the initial acceleration
of the bubble becomes considerably smaller and this results in small rise velocities initially and
therefore the bubble now has to move for longer times before it reaches the position of motion
reversal. Finally for even higher values of M1 , the Marangoni stresses are so strong that they
outweigh buoyancy, and the bubble very soon starts moving in the opposite direction.
The effect of inertia on the bubble motion in a self-rewetting fluid, parameterised by the Galileo
number, Ga, is also of interest, and is shown in Fig. 6.5. As can be seen from this figure, at low
values of Ga, the bubble centre of gravity, zCG , increases monotonically with time before reaching
a terminal value. With increasing Ga, however, the bubble exhibits flow reversal; the maximal zCG
values reached increase progressively with Ga prior to flow reversal, which then culminates in the
bubble motion being arrested. The onset of flow reversal also appears to be an increasing function
of Ga. For the parameters used in Fig. 6.5, we find using Eq. (6.51) that zc ∼ 13.33. As it is shown
in Fig. 6.5 this value is quite close to our calculations for the terminal position of the bubble even
for high values of Ga and in the presence wall confinement despite the fact that Eq. (6.51) was
derived an unconfined bubble moving in the Stokes flow limit. This is explained by the fact that
at the latter stages of the flow, for all values of Ga, the migration velocity of the bubble decreases
significantly, entering into the creeping flow regime.
Next we examine the effect of mean surface tension, characterised by the Bond number, Bo, on
the bubble dynamics in a self-rewetting fluid; this is shown in Fig. 6.6a,b for Ga = 5 and Ga = 10,
respectively. It is seen clearly in Fig. 6.6a,b that there exists a critical value of Bo above which flow
reversal is no longer possible and zCG undergoes a monotonic rise with time whose rate decreases,
and eventually saturates, with increasing Bo. These results highlight the role of bubble deformation
in the dynamics: minimising deformation, which is promoted by small values of Bo, accelerates flow
reversal, leading to lower terminal zCG values. For Ga = 5, measures of interfacial deformation are
provided by the bubble length, lB and aspect ratio of the bubble, Ar whose temporal variation are
89
(a)
30
(b)
25
Bo
0.01
0.02
0.03
0.05
1
5
10
25
zCG 20
20
zCG
15
15
10
0
2
4
t
(c)
6
8
10
0
10
2.1
2
4
t
(d)
6
8
2
Bo
1
5
10
1.8
1.6
2
lB
lB
1.9
1.8
0
10
Bo
0.01
0.02
0.03
0.05
2
1.4
1.2
1
0.8
4
t
(e)
6
8
0.6
0
10
2
4
t
(f)
6
8
10
4
1.08
1.04
Bo
0.01
0.02
0.03
0.05
3
Ar
Ar
1
2
Bo
1
5
10
0.96
0
2
4
t
6
8
10
1
0
2
4
t
6
8
10
Figure 6.6: Effect of Bo on bubble motion for (a) Ga = 10 and (b) Ga = 5; effect of Bo on the (c,d)
length of the bubble, lB , (e,f) aspect ratio of the bubble, Ar for Ga = 5. The rest of the parameters
values ρr = 10−3 , µr = 10−2 , M1 = 0.4, M2 = 0.2, Γ = 0.1 and αr = 0.04.
90
shown in Fig. 6.6c,d and Fig. 6.6(e), (f), respectively for small and large Bo values. Inspection of
these panels reveals that the extent of deformation increases with Bo, as expected.
In order to elucidate the reasons underlying the behaviour depicted in Fig. 6.6, we show in
Fig. 6.7 the evolution of the bubble shape and that of the temperature distribution in the fluid
surrounding the bubble for two values of Bo; the rest of the parameters remain fixed at their ’base’
values. Also shown in Fig. 6.7 are streamlines which represent the structure of the flow within the
bubble, in the surrounding fluid flowing past it, as well as in its wake region. It is seen that the bubble
in the Bo = 10 case (shown in Fig. 6.7(a)), which rises starting from zi = 10 that coincides with
the surface tension minimum in Fig. 6.2, undergoes significant deformation; this begins at relatively
early times, and culminates in the formation of a cap bubble. This deformation is accompanied by
the formation of a pair of counter-rotating vortices within the bubble, and another pair in the wake
region; the lateral and vertical extent of the latter increase with time, as the bubble rises towards
the warmer regions of the tube. No evidence of flow reversal is observed, which is consistent with
the results presented in Fig. 6.6.
For the Bo = 10−2 case, it is seen from Fig. 6.7b that the bubble suffers negligible deformation,
and its rise is accompanied by the formation of a pair of counter-rotating vortices form inside the
bubble at early times, as was also observed in the Bo = 10 case (shown in Fig. 6.7(a)). This
flow structure persists until t = 5 at which the bubble is seen to develop a wake region, and two
more vortices are formed within the bubble; this coincides with the onset of flow reversal, as can be
ascertained upon inspection of Fig. 6.6(a). At later times, the direction of the flow is reversed as
indicated by the direction of the streamlines associated with the t = 10 panel for Bo = 10−2 in Fig.
6.7(b), which points upwards since the liquid flows past a descending bubble. This is brought about
by the fact that the vertical temperature gradient across the bubble is positive which gives rise to
a positive surface tension gradient since z > zm (viz. Fig. 6.2). This, then, sets up a Marangoni
stress, which acts in the opposite direction to the flow past the rising bubble, retarding its motion.
This stress becomes increasingly dominant, counterbalances, and then exceeds the magnitude of the
buoyancy force, leading to the reversal of the bubble motion and its eventual arrest.
We have also studied the effect of varying the initial location of the bubble on the dynamics of the
centre of gravity of bubble, zCG . In Fig. 6.8, we show the temporal evolution of zCG as a parametric
function of zi with the rest of the parameters fixed at their ‘base’ values. For situations in which
the initial location of the bubble is lower than that associated with the surface tension minimum
in Fig. 6.2, z = zm , the surface tension gradient across the bubble re-inforces the buoyancy-driven
bubble rise. This results in an increase in zCG with time until the bubble reaches elevations such
that z > zm for which the sign of the surface tension gradient across the bubble is reversed, which
drives Marangoni flow that acts to retard and eventually reverse the direction of bubble motion. The
time associated with the onset of flow reversal decreases with increasing zi . For sufficiently large
values of zi , the bubble moves in the negative z-direction under the action of Marangoni stresses
whose magnitude exceeds that of the buoyancy force. The terminal value of zCG appears to be
weakly-dependent on zi for large zi values. Also, zc ∼ 13.33 from Eq. (6.51), which is in good
agreement with the numerical predictions.
As was mentioned in the introduction, self-rewetting fluids have been used in heat pipes associated
with substantially higher heat fluxes than normal liquids. In these applications, the bubbles are
very confined, usually forming slugs. It therefore seems appropriate to investigate the effect of
91
(a)
(b)
Figure 6.7: Evolution of bubble shape (blue line), streamlines (lines with arrows), and temperature
contours (shown in color) with time for (a) Bo = 10 and (b) Bo = 10−2 . The initial location of the
bubble, zi = 10. The inset at the bottom represents the colormap for the temperature contours.
The rest of the parameter values are Ga = 10, ρr = 10−3 , µr = 10−2 , M1 = 0.4, M2 = 0.2, Γ = 0.1
and αr = 0.04.
92
24
zi
5
8
10.5
12
15
20
16
zCG
12
8
4
0
2
4
t
8
6
10
Figure 6.8: The effect of initial location of the bubble on the temporal evolution of the center of
gravity, zCG . The rest of the parameter values are Ga = 10, Bo = 10−2 , ρr = 10−3 , µr = 10−2 ,
M1 = 0.2, M2 = 0.1, Γ = 0.1 and αr = 0.04. The prediction of Eq. (6.51) is shown by the dotted
line.
(a)
(b)
2.8
Bo
10
100
2.6
3.2
2.8
lB
2.4
zi
5
8
10.5
12
15
lB
2.4
2.2
2
2
0
2
4
6
t
8
10
12
0
2
4
t
6
8
10
Figure 6.9: (a) Evolution of the length of the bubble, lB for two values of Bo when he initial location
of the bubble zi = 8. (b) The effects of initial location of the bubble on elongation of the bubble for
Bo = 100. The radius of the tube, H = 2.5. The rest of the parameters are Ga = 10, ρr = 10−3 ,
µr = 10−2 , M1 = 0.4, M2 = 0.2, Γ = 0.1 and αr = 0.04.
93
(a)
(b)
Figure 6.10: Evolution of bubble shape (blue line), streamlines (lines with arrows), and temperature
contours (shown in color) with time for (a) Bo = 10 and (b) Bo = 100, and H = 2.5. The initial
location of the bubble zi = 8. The inset at the bottom represents the colormap for the temperature
contours. The rest of the parameters are Ga = 10, ρr = 10−3 , µr = 10−2 , M1 = 0.4, M2 = 0.2,
Γ = 0.1 and αr = 0.04.
94
(a)
(b)
4
3.5
11
M1=1.8, M2=0.9
M1=1.0, M2=0.5
M1=0.4, M2=0.2
10
Isothermal
lB
3
zCG
9
2.5
8
2
0
2
4
t
6
8
10
0
2
4
t
6
8
10
Figure 6.11: Evolution of (a) the length of the bubble, lB , (b) the location of center of gravity, in
a tube having H = 2.1. The initial location of the bubble zi = 8. The rest of the parameters are
Ga = 5, ρr = 10−3 , µr = 10−2 . The non-isothermal curve is plotted for Γ = 0.1 and αr = 0.04.
confinement. We have done this by varying the value of the dimensionless radius of the tube, H and
plotted in Fig. 6.9(a) the temporal evolution of the bubble length, lB , for Bo = 10 and Bo = 100,
and H = 2.5. For this set of simulations we place the bubble below the position of minimum surface
tension (zi = 8, zm = 10). As seen in this figure, the bubble undergoes a contraction at early times,
which is followed by rapid expansion for both values of Bo. This is then followed by a sustained
increase (decrease) in lB with time for Bo = 100 (Bo = 10). The effect of the initial location of
the bubble, zi on the elongation of the bubble is investigated in Fig. 6.9(b). It can be seen that
the length of the bubble, lB increases as we move the initial location of the bubble in the positive z
direction.
The evolution of the bubble shape, temperature distribution, and flow structure for the dynamics
associated with Bo = 10 and Bo = 100 for the parameter values the same as those used in Fig.
6.9(a) are shown in Fig. 6.10. Inspection of this figure reveals that the bubble remains essentially
bullet-shaped for Bo = 10, which is in contrast to the cap-like shape adopted by the bubble for
the same Bo and larger H value. For Bo = 100, the bubble develops filaments in its wake region,
driven by the formation of a pair of counter-rotating vortices in this region, which leads to bubble
elongation. This elongation is sustained by the action of the vortices whose size grows with time
and they cause the stretching of the filaments from the main body of the bubble towards the wake
region.
Next, we study the effect of self-rewetting character of the liquid surrounding the bubble on its
elongation in the presence of confinement effects. The results are shown in Fig. 6.11 for Bo = 10,
Ga = 5 while the tube radius has been reduced to H = 2.1 to intensify the effect of confinement; the
rest of the parameters remain fixed at their ‘base’ values. It is seen clearly from Fig. 6.11(a) that
the bubble elongation rate increases with M2 (= M1 /2); the maximal lB is reached at an earlier time
with increasing M2 . For the largest M2 values studied, the bubble undergoes a weak contraction
to an essentially terminal lB value. The bubble rise speed also increases with M2 , as shown in Fig.
6.11(b) which depicts the temporal evolution of the bubble centre of gravity, zCG .
We contrast in Fig. 6.12 the flow dynamics associated with the isothermal and (M1 = 1.8, M2 =
0.9) cases shown in Fig. 6.11. It is seen that in contrast to the isothermal case in which the bubble
95
(a)
(b)
Figure 6.12: Evolution of bubble shape with time for (a) isothermal case, and (b) M1 = 1.8, M2 = 0.9
(temperature contours shown in color). The inset at the bottom represents the colormap for the
temperature contours. The rest of the parameters are the same as those used to generate Fig. 6.11.
96
shape remains approximately spherical, the bubble rising in a non-isothermal, self-rewetting fluid
deforms significantly in the presence of confinement, and assumes the shape of a Taylor bubble. It
is difficult to compare this with the behaviour of a bubble in a linear fluid since there is no obvious
basis for such a comparison.
6.5
Concluding remarks
We have carried out an analytical and numerical investigation of a gas bubble rising in a nonisothermal liquid in a cylindrical tube whose walls have a linearly-increasing temperature. Two
types of liquids were considered: a ‘linear’ liquid whose surface tension decreases linearly with
temperature; and a so-called ‘self-rewetting’ liquid which exhibits a parabolic dependence of the
surface tension on temperature, with a well-defined minimum. Attention was focused on how the
latter can affect the development of thermocapillary Marangoni stresses and, in turn, the bubble
dynamics.
We have shown that in the Stokes flow limit, the motion of a spherical bubble can be arrested
in a self-rewetting liquid, and derived a formula for the terminal distance in this case, even if the
temperature gradient in this liquid, which surrounds the bubble, is positive. This is in contrast to the
case of a linear liquid in which a negative gradient is necessary to bring the bubble motion to a halt.
We have also studied the bubble motion numerically to account for the presence of thermocapillarity,
buoyancy, inertia, interfacial deformation, and confinement effects. Our results have demonstrated
that the motion of the bubble in a self-rewetting fluid can be reversed and then arrested in the limit
of weak bubble deformation. In this limit, good agreement between the numerical and analytical
predictions for the terminal distance was found, even for appreciable inertial contributions; this
is due to the fact that during the latter stages of the flow, the bubble enters the creeping flow
regime prior to reaching its terminal location. The flow reversal becomes accentuated for strongly
self-rewetting liquids in the presence of significant inertia. These phenomena are absent in the case
of linear liquids and are attributed to the thermocapillary Marangoni stresses which oppose the
direction of the buoyancy-driven bubble rise when the bubble crosses the vertical location associated
with the surface tension minimum. These stresses gain in significance during the course of the flow
and eventually become dominant leading to reversal and arrest of the bubble motion.
We have also shown that a bubble in a self-rewetting fluid undergoes considerable elongation for
significant confinement, forming a Taylor bubble; this is absent in the case of isothermal flows in
which the bubble remains essentially spherical.
Non isothermal bubbles may appear in a variety of situations and this study is but a glimpse of
what may happen when a bubble rises in a fluid with temperature gradients. At higher temperatures,the liquid may boil and increase the size of the bubble, or generate vapour bubbles on walls
of the container. But even at lower temperatures, liquid surfaces undergo phase change in an open
atmosphere under certain conditions. Evaporation can occur at room temperature also and cause
lakes and creeks to dry. A preliminary study was carried out to understand evaporation of liquid
drops falling under gravity.
97
Chapter 7
Evaporating falling drop
7.1
Introduction
Phase-change is commonly observed in unconfined air-liquid flows, for instance gasoline droplets
evaporating in an internal combustion engine, solidification of alloys, evaporation of ocean water
during wave-breaking and condensation of snow on falling snow crystals. Many industrial processes
require melting, evaporation and solidification of certain materials to either manufacture a desired
product or as coolants and other supporting components. Therefore understanding phase-change is
very important for industries as well as for natural phenomena. Phase-change is difficult to measure
experimentally and hence numerical techniques would be a great contribution to the research in
this area. As discussed in Chapter 1, there have been a significant amount of work on modelling
the boiling phenomenon. However the evaporation of drops below their boiling point is difficult to
simulate because of the dependence of evaporation rate on the local pressure at the drop surface
and the change in temperature as the evaporation proceeds. To model the process of evaporation
accurately, pressure at the liquid surface should be calculated with good accuracy. Due to the
presence of high density ratio and surface tension for a liquid drop in air, most of the numerical
techniques generate spurious currents, which are errors in the velocity field in the vicinity of the
interface. Therefore, a good interface capturing scheme is necessary to simulate evaporation below
boiling point of the liquid. Below, we discus the numerical scheme and incorporation of a phasechange model in a state-of-the-art flow solver - gerris, created by Stephane Popinet [170].
7.2
Formulation
We conducted three-dimensional numerical simulation of a drop (at temperature Tc ) of radius R
falling under the action of gravity, g inside another fluid initially kept at a higher temperature
(temperature Th ). The schematic showing the initial configuration of the drop is presented in Fig.
2.1 (b). The inner and outer fluids are designated by ‘i’ and ‘o’, respectively. At time, t = 0 a
spherical drop is placed at a height, H = 42R from the bottom of the computational domain of
size 30R × 30R × 60R. The governing equations are the same as those described in Chapter 2 (Eqs
(2.15)-(2.18)). The vapour volume fraction cv is initialized as zero at time t = 0. The boundary
conditions on all sides of the domain is imposed as follows: Neumann condition for scalars (p, ca , T ,
98
and cv ) and the velocity components tangential to the given boundary, and zero dirichlet condition
for velocity components normal to the given boundary. The evaporation model is discussed below.
7.2.1
Evaporation model
The interfacial mass source per unit volume, ṁv is given by
ṁv = A
�
Dav ρg
1−ω
�
∇ω · n̂,
(7.1)
where A is the area of the interface per unit volume, and ω is the mass fraction of vapour inside
the gas phase, given by
ω=
cv ρv
.
c a ρg
(7.2)
The gas-liquid interface is assumed to be at the saturation condition, such that the gradient of
vapour mass fraction across the interface can be estimated as,
n̂ · ∇ω ≈
ωn − ωsat
,
xn − xsat
(7.3)
where xn and xsat are the locations of a neighbouring point lying in the gas phase and the
location of the interface, respectively. Correspondingly, ωn and ωsat are the vapour mass fractions
at xn and xsat , respectively.
The saturation vapour mass fraction is calculated, using Dalton’s law of partial pressure, as
ωsat = Mgv
psat
,
p
(7.4)
where Mgv is the molar mass ratio of vapour to gas-mixture, p is the pressure field, and psat is the
saturation vapour pressure depending on the local temperature as follows (employing the Wagner
equation)
ln
�
psat
pcr
�
=
a1 τ + a2 τ 1.5 + a3 τ 3 + a4 τ 6
,
1−τ
(7.5)
where pcr is the critical pressure, and τ = 1 − T /Tcr , wherein Tcr is the critical temperature. For
water, the coefficients ai (i = 1 to 4) in the above equation are, a1 = −7.76451, a2 = 1.45838,
a3 = −2.77580 and a4 = −1.23303. The critical pressure (pcr ) and temperature (Tcr ) for water
are 220.584 kPa and 647 K, respectively. The values of Th and Tc are fixed at 343 K and 293 K,
respectively.
The dimensionless interfacial source term for mass transfer (ṁv ) is given by
ṁv = A
where ρrg =
cv
ca ρrv
�
+ 1−
cv
ca
�
, P e(≡
√
�
gRR
Dav )
ρrg
P e(1 − ω)
�
∇ω · n̂,
(7.6)
is the peclet number for the diffusion of the vapour in
dry air, and A is the area of the interface per unit volume. In Eq. (6.5), surface tension of the liquid
gas interface is given by the following constitutive equation
σ = 1 − MT T,
99
(7.7)
Figure 7.1: Vapour mass source calculated only in the interfacial cells. Normal to the interface
(yellow, dashed line) and its components (yellow, solid lines) are shown.
where MT = γT T1 /σ0 .
7.2.2
Model implementation in gerris
The model implementation is similar to that of Hardt & Wondra [203], in that the mass source is
smeared about the interface. A positive and a negative source term is added to the vapour and
liquid side of the interface without having to add any source term right at the interface. This
approach is simple to implement as it does not require one to modify the advection scheme of the
VOF volume-fraction variable, ca . The pressure field thus generated, automatically drives the fluid
from the interface towards the vapour phase. As shown in Fig. 7.1, the mass source is computed in
the cells containing the interface. The geometric reconstruction of a sharp interface is exploited to
accurately calculate the vapour mass fraction gradient in a cell given by Eq. (7.3). The geometric
location of the interface is calculated as xsat , which is not possible for diffuse interface or other
methods without a sharp interface reconstruction or tracking. The mass source is thus calculated
using Eq. (7.6), where A is derived from the geometry of the interface. The mass source thus
calculated is smeared in 3-4 cells about the interface and the mass source in the interfacial cells is
made zero. The mass source thus obtained of either side of the interface is weighted such that the
total mass flux per unit time remains same as that for the interfacial mass source. Next, the mass
source is made positive and negative on the gas and liquid side of the interface, respectively. This
is a simple model which doesn’t need any extensive modification of the existing code and is easy to
implement. This method is faster than that of Hardt & Wondra [203] in that a diffusion equation
is not solved to smear the mass source about the interface, instead a corner averaging is performed.
Some of the preliminary results are presented in the next section.
100
7.3
Results: Evaporating falling drops
We present the three-dimensional simulations of evaporating drops of three different volatilities. The
properties of the three cases considered are as follows:
(a)
(b)
(c)
(d)
Figure 7.2: Drop shape and vapour volume fraction contours with minimum and maximum levels
as 0 and 10−3 , for a water drop falling in air at time, t = 1, 3, 4 and 5 (from left to right). The
other parameters are: Ga = 500, Bo = 0.025, ρrb = 1000, ρrv = 0.9, µrb = 55, µrv = 0.7, P e = 200,
λrb = 26, λrv = 1.0, cp,rb = 4, cp,rv = 2, MT = 0.2, Tc = 293K, and Th = 343K.
Water drop in air
As mentioned earlier, the coefficients for Wagner equation for water drop in dry air are, a1 =
−7.76451, a2 = 1.45838, a3 = −2.77580 and a4 = −1.23303. The critical pressure (pcr ) and
temperature (Tcr ) for water are 22.06 MPa and 647 K, respectively. The values of Th and Tc are
fixed at 343 K and 293 K, respectively. The drop shape at different times is shown in Fig. 7.2 along
with the contours of vapour volume fraction. Other parameters are mentioned in the figure caption.
Due to evaporation, a spherical envelope of water vapour is formed at an initial time (Fig. 7.2(a)),
which is then convected to the wake of the drop. The amount of vapour keeps on increasing as the
time progresses. A lower pressure in the wake region promotes evaporation at the rear side of the
drop which reaches a saturation as the wake becomes saturated with the vapour. Also, a drop in
temperature in the wake region causes the evaporation to slow down after some time. Deformation
and breakup increase the rate of evaporation which is evident from Fig. 7.2(d).
Chloroform drop in dry air
The coefficients for Wagner equation for chloroform drop in dry air are, a1 = −6.50419, a2 =
0.010117, a3 = −0.37359 and a4 = −2.2322. The critical pressure (pcr ) and temperature (Tcr ) for
water are 5.33 MPa and 537 K, respectively. The values of Th and Tc are fixed at 343 K and 293
K, respectively. A chloroform drop falling in dry air is shown at different times in Fig. 7.3 for the
parameter values mentioned in the caption. It is noted that the amount of vapour generated is more
as compared to a falling water drop. The spherical envelope of vapour at an initial time, t = 1 (Fig.
7.3(a)) convects to the wake of the drop as the drop accelerates downwards. A vapour trail is seen
101
(a)
(b)
(c)
(d)
Figure 7.3: Drop shape and vapour volume fraction contours with minimum and maximum levels
as 0 and 3 × 10−3 , for a chloroform drop falling in air at time, t = 1, 3, 5 and 7 (from left to right).
The other parameters are: Ga = 100, Bo = 0.1, ρrb = 1480, ρrv = 0.9, µrb = 281.2, µrv = 0.7,
P e = 230, λrb = 6, λrv = 1.0, cp,rb = 1.05, cp,rv = 2, MT = 0.2, Tc = 293K, and Th = 343K.
behind the drop as it falls. The drop elongates and breaks up into several fragments (Fig. 7.3(d))
which promotes the evaporation further.
(a)
(b)
(c)
(d)
Figure 7.4: Drop shape and vapour volume fraction contours with minimum and maximum levels
as 0 and 3 × 10−3 , for a chloroform drop falling in air at time, t = 1, 3, 5 and 7 (from left to right).
The other parameters are: Ga = 100, Bo = 0.1, ρrb = 1480, ρrv = 0.9, µrb = 281.2, µrv = 0.7,
P e = 230, λrb = 6, λrv = 1.0, cp,rb = 1.05, cp,rv = 2, MT = 0.2, Tc = 293K, and Th = 343K.
Octane drop in dry air
The coefficients for Wagner equation for an octane drop in dry air are, a1 = −8.1622, a2 = 2.1052,
a3 = −5.4164 and a4 = −0.1583. The critical pressure (pcr ) and temperature (Tcr ) for water are 24.9
MPa and 568.5 K, respectively. The values of Th and Tc are fixed at 343 K and 293 K, respectively.
An octane drop falling in dry air is shown at different times in Fig. 7.4 for the parameter values
102
mentioned in its caption. The octane drop behaves similar to a chloroform drop, with a difference
in the breakup dynamics (Fig. 7.4) , which alters the vapour generation. This would lead to an
altogether different vapour concentration for the octane drop even when the dynamics is slightly
different from that of the chloroform drop.
7.4
Future work
We have recently started the study of evaporation and it would be our future task to study this
parametrically vast problem and identify the variables which play crucial role to govern this phenomenon. The dynamics of the drop can change the evaporation rate and vapour distribution in
air. It would be important to study the effect of inertia and volatility on evaporation. It would also
be interesting to study the effect of breakup on the vapour field generated. This would require a
more accurate calculation of mass source on the drop, which is an ongoing work and has not been
included in this thesis. A better numerical method is needed to develop where the source term does
not have to be smeared and can be treated sharply at the interface.
103
Chapter 8
Conclusions
We started off to understand canonical bubble and drop motion under gravity. Some time ago, Rama
madam saw a big bubble at Visvesvaraya industrial and technological museum, Bangalaore (India)
and was fascinated by it. When she came to hyderabad, she asked something to the effect of - how
big can a bubble get? Soon after we started investigating bubble dynamics, I had the opportunity
to present my preliminary work at the Fluids Days 2013 meeting organized on the birthday of Prof.
Roddam Narasimha. As mentioned in the introduction (Chapter 1), Prof. Garry Brown asked us the
question - ”why should a bubble and drop behave differently?” - which led us to ponder over it and
come up with a vorticity argument as mentioned in Chapter 3. We analysed bubble and drop motion
under axisymmetric assumption, and later on showed regions of axisymmetry and asymmetry by
performing extensive three-dimensional simulations.
The fully three-dimensional nature of rising bubbles and falling drops was observed and was
attempted to quantify. Bubbles and drops may not always be in an isothermal or a Newtonian fluid.
A few complexities were added to the system and the assumptions of constant temperature and
Newtonian nature of the fluid were relaxed.
Hence, in the next part of the work, we considered the problem of a rising bubble inside a
viscoplastic material. This problem had been studied either assuming the flow to be steady or
in Stokes flow regime, previously. All of the previous studies reported steady shapes of bubble
or a at least a terminal shape. By computer simulations, we showed that for a particular range of
parameters, the bubble motion may become unsteady and presented a mechanism for the “crawling”
motion of the bubble.
In another study, inspired from an experiment done by Khellil Sefiane, we tried to investigate the
bubble rise dynamics in a confined tube with a “self-rewetting” fluid as the surrounding medium (see
Chapter 6 in detail). It was observed that a positive temperature gradient can also act to reverse
the flow when a “self-rewetting” fluid is used instead of a fluid for which surface-tension depends
linearly on temperature. A theoretical expression was derived to estimate the terminal location of
the bubble which agreed very well with the computational result.
In real-life drops like rain drops, fuel droplets in an internal combustion engine, paint droplets
through a nozzle, evaporation and condensation have to be accounted for, to capture the essential
physics. The final part of this work was to develop a nuemrical technique to simulate evaporation of
falling drops under the action of gravity. This was done by modifying the source-code of an already
existing finite volume intefacial flow solver - gerris [170] to include phase change. A preliminary
104
study of falling drops of different volatility were studied to obtain the vapour field and drop shapes.
In conclusion, this work has developed from a basic study of bubbles and drops and went on to
add complexities such as non-Newtonian behaviour, Marangoni forces, and evaporation, keeping the
focus on their motion and deformation.
105
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