Capillary Breakup of DiscontinuouslyI Rate
Thickening Suspensions
MASSACHUSETTS INSTffTE
by
Pawel J. Zimoch
B.S., Mechanical and Materials Science and Engi
2010, Harvard University (Cambridge, MA)
OF TECHNOLOGY
JUN 2 8 2012
UBRARIES
e
ARCH
Submitted to the Department of Mechanical Engineering
in partial fulfillment of the requirements for the degree of
Master of Science in Mechanical Engineering
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
June 2012
@ Massachusetts Institute of Technology 2012. All rights reserved.
A uthor .................
.......
Department of Mechanical Engineering
May 24, 2011
Certified by.....................
Anette Hosoi
Professor, Mechanical Engineering
Thesis Supervisor
Accepted by ...................
David E. Hardt
Graduate Officer, Department Committee on Graduate Students
Capillary Breakup of Discontinuously Rate Thickening
Suspensions
by
Pawel J. Zimoch
Submitted to the Department of Mechanical Engineering
on May 24, 2011, in partial fulfillment of the
requirements for the degree of
Master of Science in Mechanical Engineering
Abstract
In this study, we investigated the behavior of Discontinuously Rate Thickening Suspensions (DRTS) in capillary breakup, where a thin suspension filament breaks up
under the action of surface tension forces.
We performed experiments with 55% by weight suspension of cornstarch in glycerol. To minimize the effect of gravity on the experiments, we developed a new
experimental method, where the filament is supported in a horizontal position at the
surface of an immiscible oil bath by the interfacial tension of the oil-air interface. It
was found that after a brief transition period, the radius of the filament decreases
at an exponentially decaying rate, which is half the deformation rate at which the
apparent viscosity of DRTS appreciably increases beyond it's low-deformation rate
value. Late in the filament's evolution, a bead forms in its center, leading to formation of morphologically complex, high aspect ratio structures. It was found that the
formation of these structures is caused by the viscous drag exerted on the filament
by the oil bath.
The behavior of DRTS filaments in capillary breakup was modeled with 1- dimensional approximations to momentum and mass balance equations, which are valid in
the limit of slender geometry of the filament. The rheology of the suspension was modeled with a simple function diverging at the deformation rate at which the increase in
viscosity becomes appreciable. The governing nonlinear coupled partial differential
equations were solved numerically with a finite volume scheme using the Newton's
method. It was found that this simple model reproduces the observed behavior well.
It was found that in contrast to Newtonian filaments, the viscous stress in the
DRTS filaments reaches a plateau and does not increase indefinitely. This is a result
of a coupling between the nonlinear rheology of the suspension and the nonlinearity
associated with evolving shape of the filament. It was found that the evolution
of DRTS filaments with no external viscous drag depends on the value of a single
parameter, i/Wi, which is a function of the Weissenberg number Wi associated with
the flow, and the aspect ratio of the filament . When i/Wi < 1/3, the viscous stress
at the center of the filament scales as (- , and when i/Wi > 1/3, the viscous stress
scales as Wi- 1 . These findings are supported by analytical arguments based on the
governing equations in the regime where i/Wi < 1/3.
The formation of the beaded structures was investigated, focusing on the appearance of the first bead at the center of the filament. It was found that the viscous
drag from the environment plays a central role in formation of the beads. Numerical
solutions, theoretical arguments and experiments were found to be in agreement.
Thesis Supervisor: Anette Hosoi
Title: Professor, Mechanical Engineering
Acknowledgments
First and foremost, I would like to express my gratitude to my family and my girlfriend
Jacqueline Nkuebe, who supported me every day during my work on this project, and
whose presence in my life made this work enjoyable.
I would like to thank my advisor, Professor Anette (Peko) Hosoi, for her insightful
and constructive critique of my work, and for her unwavering support and trust in
me. I would also like to thank Professor Gareth H. McKinley for his willingness to
share with me his knowledge and experience.
Finally, I would like to thank all members of the Hatsopoulos Microfluids Laboratory for creating a fantastic work environment, and for sharing their experience with
me.
5
6
Contents
1
Introduction
9
2
Experiments
13
3
Mathematical Model
17
4
Results and Discussion
21
5
Conclusion
27
A Experimental methods
29
B Mathematical Model
33
B.1 The governing equations ........................
. 35
B.2 Viscosity function and its impact on filament evolution . . . . . . . .
42
B.3 Derivation of governing equation by Control Volume analysis . . . . .
44
B.4 Nondimensionalization of the governing equations . . . . . . . . . . .
47
B.5 Parameter range covered by this study and range of valdity of equations 51
53
C Numerical simulation
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
53
C .2 Source Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
54
C.1 Equations solved
D Model equations solutions and Analysis
D.1 Effect of thickening on evolution of the filament . . . . . . . . . . . .
7
71
72
D.2 No drag behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
73
D.3
78
Analytical arguments . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
Chapter 1
Introduction
Complex, high aspect ratio structures, such as strings with embedded functionalized
elements ("beads") (Figure 1.1), can find applications in optics [16], self-assembly,
bioengineering, and custom material design [7]. It was recently shown that such structures can be created by polymerizing a liquid filament undergoing capillary breakup
[16]. Exploiting surface tension driven instabilities in liquids enables generation of
such complex geometries at small scales [24] and allows use of efficient microfluidic
techniques for processing [22].
Currently, formation of Beads-on-a-String (BOAS) structures is achieved by exploiting either electrohydrodynamic (EHD) pressure [16] or complex properties of
polymers [3, 8, 21]. In both cases, the factors affecting the properties of the formed
structure, such as size and relative placement of beads, are not well understood
[3, 16, 24].
We show that BOAS structures can be formed in a controlled way by utilizing
the known effect of viscous drag on capillary breakup [26, 27, 28]. Placing a filament
undergoing breakup in a bath of immiscible viscous fluid results in creation of a rich
variety of structures, depending on the properties of both fluids [27, 26]. Here, we
couple this effect with Non-Newtonian rheology of Discontinuously Rate Thickening
Suspensions (DRTS) 1 to extend the lifetime of the filament and enable formation of
'The more commonly used name, Discontinuously Shear-Thickening Suspensions, is made more
9
Figure 1.1: Typical Beads-on-a-String (BOAS) structures. (a) Solution of PolyEthyleneOxide solution (PEO) in water, surrounded by air. (b) Solution of PEO
solution in water, in corn oil bath. (c) Solution of polystyrene in styrene oil,
in glycerol bath. (d) Suspensions of cornstarch in glycerol, in corn oil bath.
(e) Suspension of silica in water, in corn oil bath. Scale bars: 1mm.
high-aspect ratio structures and subsequent polymerization.
The defining feature of DRTS is a sharp increase in viscosity at a critical deformation rate
crit
(Figure 1.2) [1]. The behavior of these materials in a given flow can be
pt( ) [Pa-s]
200
-
100
-
50 20
0.01
0.1 0.5
5.0
Figure 1.2: Shear rheology of 55% wt. suspension of cornstarch in glycerol, measured in
a parallel plate geometry.
characterized by a Weissenberg number Wi = if1/it,
where Af9 is the characteristic
deformation rate of the flow. The suspensions are nearly Newtonian for Wi < 1, and
thickening significantly affects the flow when Wi ~ 1.
general here to account for the fact that the main deformation mode in capillary breakup is extension.
10
M
hmin/R
1.0
0.5
0.3
0.2
OSSOS
experiment
-
simulation
-
--
\
0.1
10
30
20
40
t [s]
50
Figure 1.3: (c) Dimensionless radius of a typical filament as a function of time. The shear
rheology of the suspension is shown in Fig. 1.2. ep = 0.1 s-1.
In capillary breakup, a liquid filament's diameter decays at the slowest rate at
which surface tension driving the flow is balanced by another force [10, 12]. In DRTS,
this leads to decay at a fixed deformation rate
/crit
at late times, due to a sharp
increase in viscosity at that rate. Therefore, the deformation rate of the filament
is limited to
/crit.
This is similar to capillary breakup of viscoelastic liquids, but
the underlying mechanism for the limitation of deformation rate is different [19, 13,
14, 6, 17].
This limitation in deformation rate leads to development of a thread of
axially uniform radius which decreases at an exponentially decaying rate (Figure 1.3)
[4, 25]. Most importantly for this study, DRTS exhibit no memory effects [19, 1]. This
significantly simplifies the problem of bead formation, as the state of stress depends
only on the instantaneous deformation rate.
11
12
Chapter 2
Experiments
We performed experiments with a suspension of cornstarch (ARGO) in glycerol
(SIGMA, p = 1.261, y 11.4-). The particle mass fraction was fixed at 55%. The shear
rheology of the suspension was measured in a parallel plate geometry (Figure 1.2).
The lowest viscosity of the suspension was yto
viscosity to Pma.
360Pa-s occurred between
15Pa- s and the sharp increase in
r11
_ 0.1-0.5s-.
The yield stress was
negligible in the conducted experiments - the suspension filaments always fully coalesced into droplets, such that the shape of drops after coalescence was determined
by surface tension only. The dependence of DRTS rheology on the deformation mode
is still not well understood [4, 25]. Here, we use shear rheology with
,crt ? 0.1 as a
useful proxy for behavior of these suspensions in capillary breakup, where extension is
the dominant deformation mode. Corn starch particles (p = 1.59-1.68g/cm 3 , average
diameter d.8
14pm) [5, 20] are heavier than glycerol, but their settling distance over
the duration of a typical experiment
(a
2 min.) was negligible (~ 20pm).
A small amount of the suspension was placed between two rods which were then
slowly separated until the suspension formed an axially uniform catenary. Due to high
viscosity of the suspension, this process was slow enough to allow careful handling.
The catenary was then gently placed on the surface of a bath of an immiscible oil,
and anchored at two glass microscope slides, placed a distance 2L apart, level with
13
W
cross-sectional
view
suspension
filament
glass slide
/
--
-
2L
side view
Figure 2.1: Experimental setup showing a DRTS catenary supported by two rods, and a
floating filament attached to glass slides at both ends. The filament is held at
the surface of the oil bath by surface tension.
the surface of the oil (Figure 2.1). The interfacial tension between suspension and
oil (o = 13.7 mN/m) supported the filament at the surface of the bath, in the same
fashion a water-air interface can support a steel needle [29]. The details of the transfer
process did not affect the subsequent evolution of the thread. The viscosity of the
oil used was pil = 2.84Pa-s (Poly-Alpha-Olefin, Cannon Instrument Company).
The initial radius of the filament R was between 1-1.5mm. Capillary breakup of the
filament proceeded, entirely at the surface of the oil bath. The main advantage of
this method compared to typical capillary breakup instruments is that the filament
remained horizontal throughout the experiment, minimizing the influence of gravity
[4, 25, 23]. This allows the suspensions to form filaments with aspect ratios L/R of
up to 50. The breakup process was imaged from above with a video camera.
After an initial transition period, the filament's minimum radius decreased at an
exp(-yexpt/2) with tex, = 0.1s-,
exponentially decaying rate hmin ~..-
which is consis-
tent with the onset of thickening in shear (Figure 1.2). After some time, a small bulge
appeared in the center of the filament, and grew to become a small droplet ("bead")
(Fig. 2.2).
The radius in the thinnest section of the filament continued decaying
exponentially, with the same rate as before bead formation, until it reached the size
of about 15-20 particle diameters (~ 300 pm), at which point the filament ruptured.
For long filaments, second, third and fourth generation of beads appeared on the
'Measured using the pendant drop method.
14
(a)
(b)
experiment
simulation
OS16s
22s
37s
44s
49s
Figure 2.2: Experimental observations of capillary breakup of a DRTS filament (left) and
numerical solutions of (3.2)-(3.6) with ( = 11, Wi = 10.99 and ( - 0.078
(right). The shear rheology and capillary thinning dynamics are shown in
Fig. 1.2 (a) and (b), respectively. The times indicated on the left correspond
to times in Fig. 1.2.
2hb
1.0
A
A
02
0.122b
3#L
ZA
k
1
3
4
4
5
6
7
8
9
Figure 2.3: Minium radius of the filament (hb) at bead formation as a function of the
length of the filament (1b) for external oil viscosity pi = 2.84Pa - s (A).
filament in an arrangement symmetrical about the central primary bead (Figure 2.2).
While the results reported in this letter are for 55% wt. suspension of cornstarch
in glycerol, similar behavior was observed in suspensions of cornstarch or silica in
various water-glycerol mixtures.
The geometry of the filament at the time of bead formation, that is when dhmin/dt
0, can be characterized by the smallest radius of the filament hb at that time, and
its length
21
b,
measured between the two bounding drops. For experiments with fil-
aments of various lengths in an oil of viscosity pi = 2.8Pa-s, hb was found to be
related to
lb
through a power-law with exponent 3/4 (Figure 2.3).
15
16
Chapter 3
Mathematical Model
To study formation of the bead structures, we consider a long, axisymmetric filament
with initial radius R, made of fluid of constant density p and rate-dependent viscosity
p(j), where ' = V/1/2 _y:,
and
'
is the rate-of-strain tensor (Figure 3.1).
The
filament is submerged in a bath of immiscible Newtonian fluid of density pen, and
viscosity pen. The interfacial tension between the two fluids is o-. We use a cylindrical
coordinate system, with origin at the center of the filament. The radius of the filament
is denoted by h(z, t). The initial shape of the filament is h = R(1 - f cos(gr/Lz)),
with E < 1 and R < L. Under these conditions, the filament is unstable under the
action of surface tension, and the initial disturbance of wavelength 2L and magnitude
c grows at a characteristic rate
/inst =
o/(poR). To describe the evolution of the
slender filament, we use 1-dimensional approximations to the momentum and mass
balance equations, derived for Newtonian filaments by Eggers and DuPont [111, and
successfully used for Non-Newtonian filaments in the past [2].
17
The dimensionless
versions of the governing equations are
Dil~
Oh
Dt
0 =
T(E)
=
1 a
~
[T(z)] - (~ 2
h2 0z
(3.1)
h
Oh2
Oh2
~+
~(3.2)
at
o
(
Ih2i
(3.3)
+ 3P ()4
1= ~
~+
I1 + h2
h 11
~(3.4)
(1 + h'2 )3!2
(34
1-a(3.5)
( )0= 1
k (1 - Wi' -) 1
where tilde signifies a dimensionless quantity and prime represents differentiation with
respect to j 1. Details on the derivation and nondimensionalization of these equations are presented in Appendix B. The dimensionless viscosity ft
p=/po diverges as
y Wi-1. Parameters k and a control the steepness of the divergence. Although
simplistic, this constitutive model captures the most important aspect of DRTS behavior in capillary breakup, namely the limitation of the dimensionless deformation
rate to Wi-1 . In solutions reported here a = 1 and k = 10. The qualitative behavior
of solutions was the same for other values of these parameters.
In nondimensionalizing equations (3.1)-(3.5) the characteristic scales were: length
R, time A =is
and velocity V = R/A. For convenience, from now on we consider
dimensionless quantities only, and drop the tilde.
T(z) is the total tension in the filament cross-section [8].
Oh
=
po/ /po-R is
the Ohnesorge number, which is a measure of relative importance of viscous and
inertial effects in surface tension driven flows. In service of simplicity and to relate
simulations to experiments, where Oh ~ 100, we consider the case Oh
>>
1, where
'Note that /C/2 is not the mean curvature. While h" is asymptotically negligible in the slender
filament limit, it is often retained in capillary breakup problems
18
2L
a7
h(z, t)
I2R
Penv~ IPenv
-------------------------------------------------------------------
6
I
t
z
1
V v(z, t)
Figure 3.1: Initial (dashed) and late-time (solid) filament shapes and velocity fields from
a numerical solution of (3.2)-(3.6) with C = 0 and i/Wi < 1/3. The solution
domain is enclosed in the dashed rectangle in the upper inset. Relative size of
the initial disturbance E is exaggerated for clarity. The velocity field reaches
a steady state, where the velocity gradient Ov/&z is limited to ±Wi'1 in the
region (0, (). The steady state velocity field results in a characteristic shape of
the filament, whose radius increases before merging with the bounding drop
(lower inset: experimental image).
inertia is negligible. This simplifies equation (3.1) to
0 = -- [T(z)] - (v.
az
(3.6)
Under these conditions, the evolution equations contain 3 dimensionless groups: Wi,
C, and (.
Wi = ignt/lc,t. For Wi > 1 thickening dominates early evolution of
the filament, while for Wi -+
0 Newtonian behavior is recovered.
Since p(
<
Wi- 1 ) ~ 1, Wi- 1 is the dimensionless stress at which increase in viscosity due to
thickening becomes appreciable. C = 2penv/(po ln( )) is the coefficient of drag exerted
on the filament by the outer fluid. Assuming po > penv (which corresponds to our
experiments), the filament acts like a rigid rod translating along its axis, and the
drag coefficient can be derived from slender body theory [9, 26]. In this regime, the
viscosity of the external fluid has negligible effect on the initial instability of the
filament, but becomes significant when h < 1. L is the dimensionless initial length
19
of the filament. ( = L - (3/2L) 1 / 3 is the length of the filament in late stages of its
evolution, once semi-spherical drops develop on either end (Figure 3.1).
20
Chapter 4
Results and Discussion
We solved equations (3.2)-(3.6) for an initially stationary filament, subject to symmetry and impermeability boundary conditions in the domain (0, L) with an upwinded
finite volume scheme using Newton's method. The evolution of the filament's shape
closely followed that observed in experiments (Figure 2.2).
Details regarding the
numerical method used are presented in Appendix C.
When ( = 0, there is no external viscous drag, and the filament evolution depends
on Wi and ( only. As expected, the filament initially evolves similarly to a Newtonian fluid, but eventually the radius of the filament in its thinnest section hmin begins
decreasing at an exponentially decaying rate hmin ~ exp(t/(2Wi)) (Figure 1.3). Deformation rate cannot exceed Wi-1, and therefore the velocity field in the filament
approaches a steady state, where Ov/Oz = -Wi-1 at 0 < z < /2 and &v/&z = Wi
at (/2 < z <
1
(Figure 3.1).
The deviation from a Newtonian evolution occurs when the viscous stress rT
p()
in the filament reaches Wi-1 (Figure 4.1). Curiously, the viscous stress inside
the filament then reaches a steady value
Tmax,
in stark contrast to Newtonian fluids
and polymer solutions, where the stress inside the filament grows as
-
h2
The parameter space (Wi, ) can be divided into two regions with respect to
At large Wi and small
, Tma, depends on ( only and scales as (~'.
21
[cite].
Tmax.
At small Wi and
M
100 t10
1.0
0.1
0.01
hmin
0.001
0.05
0.02
0.1
0.5
0.2
1.0
Figure 4.1: Viscous stress r = p(7- ) at the center of the filament as a function of the
minimum filament radius hmin for Newtonian (black solid line) and thicken-
ing filaments (( = 10, Wi = 0.04 (yellow), 0.1 (green), 0.4 (red), 1.0 (blue)),
calculated from numerical solutions of (3.2)-(3.6). The area shaded in blue
corresponds to filaments which cannot be characterized as slender, and therefore equations (3.2)-(3.6) are not fully applicable in this region.
large (, Trm,, depends on Wi only and scales as Wi-
1
(Figure 4.2).
Therefore, the rescaled value of the maximum viscous stress
Tmax(
can be rep-
resented as a function of a single parameter c/Wi, as shown in Figure 4.2.
~ 1 and at
-max(
(/Wi < 0.3 we find
s/Wi > 0.3 we find
At
-maxc
Wi- 1 .
This behavior of the filament at ( = 0 can be explained by considering the total
tension in the filament and it's relation to the filament's shape. Once the steady state
velocity field is reached, v and Ov/&z are known functions of z only, and the volume
conservation equation (3.2) can be easily solved for h(z, t), yielding
h(z, t) = H exp(-Wi t/2)
h(z, t)
=
H exp(-Wi t/2)
0 < z < (/2
(4.1)
(/2
/2 < z <,
(4.2)
where H is the smallest radius of the filament at the time 89v/&z first reaches Wi- 1 .
Details regarding deriving these expressions are presented in Appendix D. Equations
(4.1)-(4.2) are not valid close to z = ( and z = (/2, as boundary layers exist there to
enforce smoothness of the filament's surface. From dimensional considerations, these
boundary layers are of width
-
hmin.
22
3
Tmax(,
100
3
Tbeadd
/
0
0
15
10
-013
0
0
0
0E3
El
0
0
o3
7
5
El
El
El
El
0l
El
0l
El
C3
a
o
El
al
O
El
El
El
Dl~
El
El
El
C 0/
00~ El/ 0
00
0
0/e
o l El.4
El
El 0
00o
000
El
o
o0l
'S
Se
0o
S
eS
e0
l El
El
0l
e
0
S
*0
/**
10
0.3Wi
S
S
0
S
S
e
3
=
*Wi
100
10
-
/
1
simulations, C $ 0
1.0
A
experiments, (
03
simulations,
I
0.01
(
#
0
= 0
I
I
0.10
1.0
i
II/W
10
Figure 4.2: Comparison of simulation and experimental results. Filled circles: simulations
with ( = 0 and c/Wi < 1/3. Open squares: simulations with ( = 0 and
Q/Wi > 1/3 (see inset). Color represents the magnitude of c/Wi, with warm
colors representing larger values. Black crosses: simulations with ( 5 0.
Triangles: experimental results; each mark represents formation of the first,
central bead. The value of the ordinate is 3Tmaxz( for simulations with ( = 0,
and 3 Tbeadg for simulations with ( 5 0 and for experiments.
23
Equation (3.6) shows that when (
=
0 tension in the filament is spatially uniform.
Specifically, at late times the existence of a steady state velocity field dictates that
Ov/Dz = 0 at z = (/2. At the center of the filament (z = 0) h' = h
=
0. Equating
tension at z = 0 and at z = (/2, while recalling that h' < 1 due to slenderness of the
filament, leads to
3Tmax z=O
VhIz-
3pfI)lz-o = h"|I-O/
=
/2 can be estimated by evaluating h' from equations
(4.3)
.
(4.1)-(4.2) on either side
of the matching region of thickness hmin around z = (/2.
h"|z-/2
and therefore Tmax ~ (3 )
~
'
l
1
-
2hmin
(4.4)
. More details on these equations are presented in Ap-
pendix D.
This expression is not valid for Wi < 3(, as at
T
=
(3 )-
< Wi-1 the filament
would not have been influenced by thickening, while in deriving equation (4.3) we
assumed that the velocity field has reached steady state. If a filament with Wi < 3(
was to reach a steady state described by equations (4.1)-(4.2), the filament's geometry
would require
-max~
(3)-1, resulting in a contradiction. Therefore, filaments with
Wi < 3( never reach steady state described by equations (4.1)-(4.2), and in this
case Tmax ~ Wi'.
The predicted threshold value of c/Wi = 1/3 between these
two regimes is consistent with simulation results, which indicate the threshold value
(/Wi ~0 0.3 (Figure 4.2).
Now we consider the filament's evolution with ( f 0. At i/Wi < 1/3, the velocity
inside the filament eventually reaches steady state where the velocity for 0 < z < (/2
24
is v = Wi-'z. Substituting this expression into equation (3.6) and integrating yields
Oz
Tz-g/2 -Tz-o
=Tz
(V =
j
CW i-I z
(4.5 )
2Wi- .
Wi-lzdx
(4.6)
Since h = hmin in this region, equation (4.6) can be simplified to
1
3p()|=
/2 - 3p(')j|=o =
CWi-1
(
\2
.(47)
A bead begins to form when the deformation rate at the center of the filament
becomes smaller than the deformation rate in other parts of the filament. When
this happens, a small bulge appears in the middle of the filament due to volume
conservation. This leads to a reversal in the direction of the surface tension force,
which starts driving flow towards the initially small bulge. As a result, the bulge
grows, and a bead eventually forms.
When
= Wi-1 at the center of the filament, a bead cannot form as this is
the highest deformation rate allowed by the suspension, and the entire filament must
deform at that rate. When the viscous stress at the center of the filament decreases
below Wi- 1 , however, the center of the filament begins deforming slower than the
rest of the filament, and a small bulge forms there, leading to formation of the fist
bead.
Therefore, the first bead begins to form at the center of the filament when
3p( )lz=
/2 = 3Tbead
=
IjWiI
(2hmi
+ 3Wi
1.
(4.8)
For z > (/2, the filament's radius increases significantly. Since the influence of drag
on stress is inversely proportional to h2 , there is little decrease in tension beyond
z = (/2. Therefore, at z = (/2 the stress in the filament is unaffected by viscous
25
drag, and hence the first bead forms when
rbead
' Tmax
(4.9)
-
~(3
For c/Wi > 1/3, steady state velocity field is not reached. In the region 0 < z <
(/2, &v/az ~~Wi-1, and equation (4.7) is still expected to' hold. However, since the
shape of the filament is not described by equations (4.1)-(4.2), it is expected that
Tbead
due to drag, where
f is smaller than
(4.10)
~maxf(()
1 and depends on the exact shape of the filament.
Equations (4.9) and (4.10) indicate that the rescaled viscous stress at bead formation Tbead should collapse on the curve formed by rmax in Figure 4.2 for i/Wi < 1/3,
with an upwards deviation for c/Wi > 1/3 due to the correcting factor
indeed the case, and the numerical results show that
f(s)
~
f(
). This is
-1/4.
To validate the simulations, the rescaled bead onset stress from experiments
3rbeast
i
1en, 1nlbh)
o 'Yexp
Plenv In
ln
2n+
2hbpl
± 3 exp
lb
(4.11)
(4.11)
was overlapped on the master curve (Figure 4.2), showing satisfactory agreement with
the numerical results.
26
Chapter 5
Conclusion
In conclusion, we demonstrated that viscous drag from an external fluid in capillary
breakup can be used to generate complex high-aspect-ratio structures, such as Beadson-a-String, in a controlled manner.
To demonstrate this process, we conducted
experiments with Discontinuously Rate Thickening suspensions.
The behavior of
these suspensions in capillary breakup is controlled by the ratio of the characteristic
capillary instability growth rate to the deformation rate at which the thickening
becomes appreciable. The influence of drag eventually reduces the viscous stress at
the center of the filament below the thickening threshold Wi-'. At this point, a
small bulge develops in the filament, and later grows to become a bead. For longer
filaments, the bead formation process can be repeated several times in a symmetrical
arrangement about the first, central bead. By modulating the viscosity of the outer
fluid and the length of the filament, structures of varying complexity can be achieved.
The dependence of bead sizes and spacing on these factors remains to be investigated.
The ability to use external viscous drag to influence the process of bead formation
is not specific to DRTS. As Figure 1.1 shows, the same principle can be used to
provoke bead formation in polymer solutions. Understanding the effect of drag in the
context of polymer dynamics, however, requires analysis more complex than presented
here.
27
Despite recent progress, behavior of Discontinuously Rate Thickening Suspensions remains poorly understood in flows other than simple shear. The experimental
method developed in this study allows easy characterization of DRTS in flows dominated by extension, rising the possibility of investigating the thickening effect in
general flows.
28
Appendix A
Experimental methods
The experiments were conducted in the setup shown in Figure 2.1. The main advantage of this method is that during the process of capillary breakup the filament
remains horizontal, which minimizes the effect of gravity on the process, making it
possible to observe filaments with aspect ratios up to 50.
We used 55% wt. suspensions of cornstarch (ARGO) in glycerol (SIGMA) in experiments reported here, but qualitatively similar results are produced in any discontinuously rate thickening suspension. We tested suspensions of cornstarch in various
water/glycerol mixtures, as well as suspensions of rough silica particles (MinuSil) in
various water/glycerol mixtures. Typical behavior of such a silica suspension is shown
in Figure 1.1 (e).
Cornstarch is known to age with time in water, most likely due to swelling of
cornstarch particles. For this reason, the suspensions tested were prepared directly
before the experiments were conducted. Typically, the suspensions were prepared in
small batched of approximately 5.00 g. The ingredients in appropriate amounts were
places separately in a container, and then vigorously mixed by hand for approximately
2-3 minutes. The prepared suspensions were homogeneous under visual inspection,
and measuring the shear rheology of three samples prepared independently showed
that the uniformity of the response to deformation was satisfactory.
29
Pure glycerol, as used in the experiments reported here, is hygroscopic, and absorbs water moisture from the environment. As even a small water content might
change the viscosity of glycerol significantly, the suspensions were placed under oil
directly after preparation. The oil used for storing the suspensions while experiments
were under way (typically no longer than 1-2 hours) was the same as the oil used in
the experiments.
A small portion of the suspension was placed between two rods of 2 mm diameter,
and then the rods were separated slowly by about 2-3cm. Over about 15-20 seconds,
a thin, axially uniform catenary formed, as shown in Figure 2.1.
This catenary
was evolving slowly, allowing careful handling. Once the axially uniform catenary
developed, it was placed gently on the surface of the oil bath, between two glass
slides distance 2L apart, which were level with the surface of the bath (see Figure
2.1). The interfacial tension between the oil and air supported the suspension filament
at the surface of the bath, keeping it horizontal over the duration of the experiment.
Glass slides at both ends of the filament kept it fixed in place.
An alternative method of conducting the experiments, was to allow the filaments
to break the surface of the suspension, and settle close to the bottom of the container.
The containers used were made of polystyrene, which is hydrophobic. In the environment of the oils used, it was thermodynamically advantageous for the polystyrene
dish to be wetted by the surrounding oil than by the glycerol or water-based suspension. Therefore, presence of the oil in effect turned the bottom of the dish into
a perfectly hydrophobic surface, allowing the suspension to move along the bottom
of the dish with only minimum resistance, caused by the thin lubricating layer of
oil between suspension and the bottom of the dish. The magnitude of the resistive
force on the filament was difficult to estimate, mostly because the thickness of the
lubricating layer was difficult to measure with useful accuracy.
The method of "hovering" the suspension above the bottom of the container,
however, is very useful in testing the behavior of DRTS in flows other than capillary
breakup.
This is because the presence of the thin lubricating layer dramatically
30
reduces the drag on the suspension caused by the presence of a rigid boundary. DRTS
are very sensitive to high deformation rates, which are typically present close to any
rigid boundary. For this reason, boundary effects are very important in testing DRTS
in flows involving inhomogeneous deformation rates. Placing the tested suspension in
a bath of immiscible oil effectively isolates it from the effects of the rigid boundaries,
and allows generating flows with inhomogeneous deformation rates in a way which is
independent of the presence of rigid boundaries.
As an additional benefit, a relatively thin layer of the suspension (-
be formed on the container bottom.
2 mm) can
This layer can haver a large lateral extent,
effectively forming a layer in which quasi-2D flows can be established. This layer can
be visualized from above, making it possible to observe the dynamical response of
DRTS independently of force or stress measurements. This allows for investigations
of e.g. force propagation in DRTS where the force and state of the suspension can
be measured/observed in real time independently. This was not possible in earlier
investigations, where the state of the suspensions was typically inferred from the force
measurements or inferred after the experiments from another proxy, such as imprint
left in a clay surface [18].
The interfacial tension between the suspension and the oil was measured with the
pendant drop method [15].
31
32
Appendix B
Mathematical Model
This chapter describes the equations used to model the evolution of the filament of discontinuously thickening suspension in the experimental setup described in Appendix
A. We use a model that neglects the details of motion of the suspended particles and
instead describes the suspension dynamics with a constitutive relation embedded in a
continuum mechanics framework. The continuum model is based on a 1-dimensional
approximation to full 3-dimensional equations of momentum and mass balance. We
consider an axisymmetric filament in cylindrical coordinates, as shown in Figure 3.1.
The suspension dynamics are modeled with a deformation-rate dependent viscosity,
given in equation 3.5.
The dimensionless governing equations describing the evolution of the filament
are
2 (K:' + 3p_()
(-h2 + h 2 OZ [h
Ah2 Oh2V
0
oz
at
v(z
')]
0)
v(z = L) = 0
hh
z=O
02 (h2V)
OZ2
momentum balance
(B.1)
mass balance
(B.2)
(boundary conditions)
(B.3)
(B.4)
z=L
2
_
z=0
(h2v)
aZ2
(B.5)
z=L
33
where v (velocity) and h (filament shape) are dimensionless variables and (, Wi and
L (or () are the dimensionless groups that determine the evolution of the filament.
The limitations, derivation and physical interpretation of these equations are discussed in section B.1.
The constitutive model describing the dynamical behavior of discontinuously thickening suspensions and the relation between deformation rate and viscosity is described
in section B.2.
A simplified derivation of the governing equation using a simple control volume
analysis is presented in section B.3.
The nondimensionalization procedure and the pysical meaning and significance of
the dimensionless groups is presented in section B.4.
The range of the parameter values for which we seek solutions in Chapter 4 and
the reasoning behind it is discussed in section B.5.
34
B.1
The governing equations
We study our model system in capillary breakup with a 1-dimensional reduced-order
continuum model.
We consider an axisymmetric, infinitely long filament of constant density p and
deformation-rate dependent viscosity p()
suspended in an infinite bath of another,
immiscible fluid (outer fluid) of density Penv and constant viscosity penv, shown in
Figure 2.1. The interfacial tension between the fluids is o-. We neglect the gravitational forces. We use a cylindrical coordinate system with origin at the center of the
filament. The velocity of the fluid and pressure in the filament are given by v(x, t)
and p(x, t) and in the outer fluid by venv(;, t) and penv(1, t), where x is the location
of a point with respect to the origin x = [z, r, 6]. The spatial components of the
velocity field are v = [v2, v,, vo]. The cylindrical surface of the axisymmetric filament
is designated by S(t) and its generator is h(z, t). The vectors normal and tangential
(in the axial direction) to the surface is designated n and t, respectively.
The surface of the filament initially is a uniform cylinder of radius R. At time
t = 0+ a sinusoidal deformation of small amplitude e and wavelength 2L > 27rR is
imposed on the cylinder
h(z, t) = R - c cos(
L
z).1
(B.1)
Under these conditions, the filament surface becomes unstable due to interfacial tension, and initial variations in h(x, t) begin to grow (reference), preserving the periodic
nature and wavelength of the deformation. We consider the section of the filament in
the domain z E (, L).
In the limit of a slender filament and p/plenv > 1, the momentum and mass
'Note that this deformation does not conserve the volume of the filament.
35
balance equations can be simplified to a 1-dimensional form:
1 0
Dv
Dz
+ h23p
(v(OZ)
+
[env
V
ln(L) h 2
(momentum balance)
(B.2)
Oh2 Oh
0
at
2
i
(mass balance)
Oz
(B.3)
0
(boundary conditions)
v(z = 0) = v(z = L)
(B.4)
Oh
Oz
0
0
o
z=
Oh
(B.5)
Oz z=L
82 (h2 v)
OZ2 z=0
_
2
V
02(
(h2 v)
OZ2
z=L
(B.6)
where C' = IC'(h) describes the shape of the filament, v is the average axial velocity inside the filament and the last term in equation (B.2) describes the interaction
between the filament and the outer fluid.
B.1.1
Derivation of the equations
We consider the full 3-dimensional equations of momentum balance and mass balance
and simplify them by assuming slender geometry.
Momentum and mass balance
Navier-Stokes equations with deformation-rate-dependent viscosity and incompressible fluid continuity equations are used.
In the filament and outer fluid, respectively, the momentum balance is:
Dv
Dt
p-=
Penv
D"
Dt
V -T
(B.7)
V - Tenv,
(B.8)
=
36
where T and Ten, is the stress in the filament and outer fluid, respectively. Following
standard procedure, the stress is decomposed into an isotropic and deviatoric parts:
(B.9)
T = -pI + pM_i
Tenv = -Pen
+ pIenv'env.
(B.10)
Here, y and isn, are rate-of-strain tensors defined as
' = (V v + (Vv)T )
,:env = (_Vvenv
Additionally,
1
+ (V
(B.11)
B11
env)T) .
(B. 12)
is the scalar, frame-invariant rate-of-strain, defined as
1.
~y:y.
(B.13)
Both the fluid inside the filament and the outer fluid are considered incompressible,
and thus the mass balance reads
V -V= 0
(B.14)
_V - env = 0.
(B.15)
Boundary conditions
The outer fluid far from filament is quiescent and at zero pressure.
lim Penv = 0.
lim Wen = 0;
1X1-+00
(B.16)
1I-0o
Both pressure and velocity in the filament are symmetrical about the center of
37
the filament, and bounded:
Or r
;
-
0.
(B.17)
r=
Continuity of both fluids requires that their velocities are equal at the surface of
the filament:
v =venv
at S(t).
(B.18)
Stress must be balanced in the entire domain, requiring that the stresses at the
surface of the filament balance.
Most importantly, due to presence of interfacial
tension, there is a stress discontinuity across the surface of the filament. Therefore,
at S(t):
TTev
(B.19)
- 2o-/Cn,
where KC is the average curvature of S(t). For an axisymmetric filament with surface
generator h(z, t):
=
2
(
1 I(±h1
1±
hv/11 + h'2
h"
(1 + h'2)3/2
) .
(B.20)
The filament is considered infinite in length. Due to periodicity of the intial conditions (see next section) and lack of any symmetry-breaking agent, the filament's shape
will remain periodic with the initially prescribed period. Therefore, the condition of
38
infinite length can be replaced with conditions of periodicity:
0 = ez v(z = -L) = e -v(z = L)
0
= ez - v,(z
= ez -Ven_(z = -L)
0
Oz=
-L
aOh
Ozz=L
(no translation)
(B.21)
(B.22)
= L)
(periodic shape)
, h(z = -L) = h(z = L)
(B.23)
(B.24)
Initial conditions
A small amplitude sinusoidal deformation is imposed on an initially uniform, cylindrical filament of radius R.
h(z, t = 0+) = R (I - ecos
z
(B.25)
.
with L > rR to initiate capillary-driven instability.
Both fluids are initially at rest:
v(X,t = 0+) = 0;
env (X, t =
(B.26)
0+) = 0.
Slenderness of the filament
As shown in figure 2.2, the filaments are slender in shape, i.e. L
>
R. While the mass
balance must be valid at all points in the domain, this means that the variation of
velocity across the filament is much less than it's variation along the axial coordinate.
For this reason equations (B.7) and (B.14) can be simplified by expanding the pressure
and velocity fields in Taylor series about the center of the filament.
39
Taylor expansion of pressure and velocity fields
The Taylor expansions in r of the velocity and pressure fields about the center of the
filament are as follows:
v2(Z r, t) = vO(z, t) + v 1 (z, t)r + v2 (z, t)r 2 + p(z,r,t) = po(z,t) +p 1 (z,t)r +P 2 (z,t)r2 +
Due to boundary condition (B.17), pi = vi
l avo
Vr(z, r, t) =
2 Oz
vo = 0
(B.27)
.
(B.28)
0. Substituting (B.27) into (B.14) gives
1&8v 2
r -
4 Oz
_
r
-- - -
by symmetry.
and
(B.29)
(B.30)
Grouping governing equation terms by order
The governing equations are the zeroth order equations in the radial coordinate.
Equations (B.27) - (B.30) are substituted into equations (B.7) and (B.14). The
terms in the resulting equations are grouped by order of magnitude in r, taking into
account the fact that h'
-
r.
We find that the r-component of equation (B.7) is
satisfied identically at the lowest order, the 0-component is satisfied identically due
to symmetry, and the z-component reads
Dv
-
Dt
+ h2
Oz
(
OZ
(B.31)
Defining
K'/ =
1
h v1 +h'
2
+
h"
hi
(1 + h'2 )3 / 2 '
(B.32)
equation (B.31) can be rewritten as
Dvz
PDt =
(
+ 3py))
(o-K'
0Z
2
40
(B. 33)
The momentum balance equation satisfied at the lower order in r is
oh2
h
at
8h 2 v
= 0.+
OZ
(B.34)
Equations (B.33) and (B.34) are the basic equations of motion of the system.
B.1.2
Incorporation of the effect of environment drag
To model the influence of external oil, we model the filament as a slender rod and
the viscous drag stress on the surface of the filament is calculated from slender body
theory [9, 26].
As expressed in equation (B.19), the stress at the interface between the filament
and the outer fluid in the direction normal to the filament is
(B.35)
t - Tn = -t - Tenvn
Which in the limit yu> pen, implies that ' <
env.
Additionally, a typical Reynolds
number of the filament motion is
pvR
Ret ypical -io
1 4
(B.36)
Therefore, the filament can be considered as a rigid rod translating in the limit
of low Re in a viscous fluid, and thus the drag that the environment exerts on the
filament can be characterized with the slender body approximation as
Fd = 2p"env o
ln (
(B.37)
where ( is the aspect ratio of the filament. Note that beyond the weak dependence
on the aspect ratio this is independent of filament radius.
To incorporate this drag into the equation of motion, we note that the stress
associated with the drag is
Td
= Fd/h 2 , and so the momentum balance equation 41
equation (B.33) - becomes
Dve
D
Dt
B.2
2
108
h2
9Z2n
[h2 (o-iC' + 3p()-y)] --
env
V
h2
(B.38)
Viscosity function and its impact on filament
evolution
The most important impact of thickening on capillary breakup is that it limits the
deformation rate that the filament can achieve, because capillary force cannot drive
the filament past the point of thickening. Therefore, the highest value the deformation
rate can achieve is
jcrit.
In our model, we replicate this characteristic by modelling
the suspension viscosity with a function which diverges at
'
=
Acrit.
The function is
rescaled such that pL(0) = po and that the viscosity increases by a factor of two when
the deformation rate reaches 95% of the critical deformation rate, i.e. p(0.957crit) =
2[to. The viscosity function therefore is
1
pMi =
.I
10(1 - Wz 7
(B. 1)
It is important to note that this function neglects any yields stress or initial
shear thinning for
'
< icrit, which typically exist in shear thickening suspensions,
cf. Figure 1.2. In the suspensions used in our experiment, yield stress is very low even though the filament evolution and subsequence coalescence into droplets takes
several minutes, the suspensions never jam. Shear thinning is limited to very low
deformation rates, which are achieved only in the very stages of evolution of the
filament, and therefore do not affect its behavior significantly. Since the effect of
yield stress and shear thinning on the evolution of the filament are not large, for the
sake of simplicity our model neglects them.
It is important to note that in contrast to e.g. viscoelastic fluids, thickening
suspensions' viscosity increases both in extension and in compression. As noted in
42
chapter 4, this has important implications for the evolution of the filament. The
most important qualitative difference that appears in the shape of the filament is a
pronounced widening of the filament in the section where it merges with the bounding
drop. This is consistent with experiments, where the widening can be observed, as
shown in fig. 3.1.
B.2.1
Lack of memory or time dependence in the constitutive
equation
One of the most important difficulties in modeling flows of complex fluids, especially
polymer solutions and other viscoelastic fluids, is that their state of stress at a given
time depends not only on the instantaneous deformation rate at that time, but also
on the earlier deformation rates. This makes the dynamics of thinning filaments
made of such fluids very complex, and increases the importance of the constitutive
model used in solving the equations of motion. In particular, the phenomenon of
BOAS morphology formation is not well understood, in part because current evidence
suggests that the final stages of polymer stretching play an important role in creating
the beads [24]. As behavior of the polymer chains and its interaction with the flow is
highly complex, this makes predicting the properties of BOAS structures very difficult.
On the other hand, the state of stress of thickening suspensions depends exclusively
on the instantaneous deformation rate at the same time. This make it relatively easy
to couple deformation rate with the state of stress in the filament, which allow the
dynamics of filament thinning to be understood.
Our model captures this characteristic of thickening suspensions by assuming that
the viscous stress can be expressed as
r
(B.2)
-scous
where viscosity depends on the scalar deformation rate
43
V 1/2 j' : .
Treatment of the compression response in the literature
The response of thickening suspensions has mostly been considered in shear [1]. Studies focusing on extensional properties of thickening suspensions have most depended
on filament stretching experiments, where the entire filament is stretched [4, 25].
In capillary breakup, however, the fluid is first stretched, as it accelerates inside
the filament, but is later compressed, when it merges with the bounding drop. This
aspect of capillary breakup has not been considered so far, and in chapter 4 we showed
that it has important consequences for the evolution of the filament.
B.3
Derivation of governing equation by Control
Volume analysis
The governing equations given in equations 3.1-3.2 can be alternatively derived from
a Control Volume analysis.
We consider a thin section of an axially uniform filament between z and z + dz
Under the assumption of slender geometry, we find that the radial velocity is much
smaller than the axial velocity, due to the continuity equation. v is the average axial
velocity
v
=
v2 r dr.
The control volume is stationary.
44
(B.1)
B.3.1
Conservation of mass
The conservation of volume in the Control Volume is
j
+f
dt cv
pdV + j
dV + (v7rh 2 )zdz
-
pVdA
0
(B.2)
(v7rh 2 )z=
0.
(B.3)
cs
Letting dz -+ 0:
[(7rh 2 dz)t+dt - (7rh 2 dz)t] dz + [(v7wh 2)z+dz
-
(vwFh 2 ),] dt
Dividing both sides by dz dt and taking the limit as dz -+ 0 and dt -
0.
0, we arrive at
+ A 2 V= 0.
at
Oz
B.3.2
(B.4)
(B.5)
Conservation of Momentum
Consider the total force acting on the surface created by an intersection of the filament
with a plane perpendicular to the axis of the filament . This force is the tension in
the filament. There are two contributions to this force: the line force generated by
surface tension, and the area force generated by the stress Tzz acting on the surface.
The total force acting on the surface is
T =
irh
1- + 7h 2 Tzz.
1 + h'2
(B.6)
Where Tzz is the average stress acting on the surface, and is expressed with
Tzz = -p + 2p( ) &z
(9z
45
(B.7)
Pressure inside the filament can be obtained from the boundary stress condition,
equation (B.19):
Dv
p + PM1 Oz*=
-2K/C
p = 2o-C - p(
(B.8)
(B.9)
Dz
Substituting (B.9) into (B.7) yields
Dv
Tzz = -2o-K + 3pi(ai) 2.
(B.10)
The total force on the cross-section therefore is
27h
T
1 + h' 2
-2±K + 3 p(i) Dv
Oz
+ 7h 2
(B.11)
Recalling the expression for C- equation (B.20), substituting it into eqution (B.11),
and bringing the first term under the parenthesis, we get
T =7th2(
h
T = th12
T = irh 2
h'
2
1 +h'
h
0
(h1V + h'2
o-C + +3p
+
2
OVZ
+
h
+ 3p O)
(1 + h'2 )3 /2
+ 3p
(1 + h'2 ) 3 / 2 +
Bz
)
(B.12)
(B.13)
(B.14)
V)
Now, consider a material element of the suspension, placed between two planes at
z and z + dz. The total force acting on the material element is
F=Tz+dz -Tz
- 27rvdz,
(B.15)
where the contribution from the viscous environment drag acting on the external
surface of the filament was taken into account. This force is also proportional to the
46
acceleration of the filament
F =,7h2 dzp
Dv
(B.16)
.
Combining the two equation yields
Dv
-rh 2 dzp Dz =
Dv,
- Tz-
1 Tz+dz - Tz
dz
h2
Dz
27r(vdz
2
(v
h2
(B.17)
(B.18)
Taking the limit dz - 0, we arrive at
Dz
Dz
B.4
1 0
h2 9z
v
h2
(T(z)) - , z .
(B.19)
Nondimensionalization of the governing equations
The model is described with 4 dimensionless groups.
B.4.1
The Nondimensionalizat ion procedure and characteristic scales
Characteristic scales
Time
In capillary breakup problems, the characteristic time scale is the initial
growth rate of the capillary instability. This is the lowest deformation rate at which
the driving capillary forces are balanced by another force. Typically, there are two
possible forces that can balance surface tension: inertia and viscosity. If the balancing
force is inertia, the typical deformation rate is
Ainertia
=
47
pR3
R3
(B.1)
If the balancing force is the viscosity of the filament, the characteristic time is
Aviscou,
OR
(B.2)
In the filaments used in the experiments, the Ohnesorge number is approximately
Oh =
PO
pR
15
1
~ 100 >> 1
v/1500 x 0.001 x 0.0137
(B.3)
which means that viscosity dominates over inertia, and we choose the viscous time
scaling Aviscous.
The inverse of the characteristic time scale is the characteristic deformation rate
'typicaI
(B.4)
=A--
The characteristic length scale is R.
The characteristic velocity scale is simply
V
B.4.2
Aviscous
.
(B.5)
yo
The 4 dimensionless groups
The four dimensionless groups each describe a different physical ratio.
Oh =
-
the Ohnesorge number
VpRa
Ohnesorge number is the ratio of surface-tension-driven inertia to viscous resistance
in the filament. At Oh < 1 the main force balancing the surface tenion is inertia,
and at Oh
>>
1 surface tension force is balanced by viscosity.
Alternatively, one can consider Oh as a ratio of timescales. The inertial velocity
scale is vi =
',
which means that the inertial timescale - the characteristic time
scale of deformation is surface tension is balanced by inertia is t' =
48
' pR
3
The viscous velocity scale is v, = o/p, which means that the viscous time scale is
tv =A. The ratio of these two timescales is the Ohnesorge number
t
pRo
Since viscous and inertial resistance to motion grow monotonously with speed
and acceleration, the filament starting from rest will evolve according to the lowest
timescale. At Oh
>
1 the viscous timescale is much shorter than the inertial time
scale, and the filament will evolve according to the viscous velocity scaling, and the
surface tension is resisted by viscosity. At Oh
<
1, inertial timescale is much slower
than the viscous timescale, and the filament will evolve according to the inertial
velocity scale, and the surface tension will be resisted by inertia of the fluid.
Wi =
p/o-R
-
ratio of deformation rates
crit
Wi is the ratio of the characteristic deformation rate of the flow to the deformation
rate at which increase in viscosity due to the thickening effect becomes appreciable,
i.e.
Acrit.
When Wi <
1, the flow is nearly Newtonian, since p(
<
'crit)
- 1.
When Wi ~ 1, the increase in viscosity due to the thickening effect is substantial.
In the capillary breakup flow considered here, the characteristic deformation rate is
the initial growth rate of the capillary instability. Therefore if Wi < 1, the filament
will initially deform as if it was made of a Newtonian fluid, and the thickening sets
in late in its evolution. When Wi
>>
1, thickening restricts the initial growth rate of
the instability, and therefore significantly affects the evolution of the filament even in
the early stages of its evolution.
Since p(y <
cit)
4 1, and
T
=(7
)',
Wi'
is the approximate magnitude of
dimensionless viscous stress at which the thickening response become appreciable.
49
= L/R - the aspect ratio
The aspect ratio is determined by the initial shape of the filament. An alternative
measure of the filament's geometry is the ratio of the initial filament radius R to the
long-time length of the filament. Since the filament eventually entirely retracts into
the bounding droplets, the entire volume of the initial cylindrical filament will end
up in the semi-spherical droplets on either side of the filament. Therefore, just before
breakup the length of the filament is
3
IPenv
p 2ln
3/2
the drag coefficient
The drag coefficient determines the relative importance of the environment viscous
drag on the development of the filament. At ( = 0 there is no drag, and the filament
experiences no drag. Given that the overall drag term in equation (B.38) is inversely
proportional to h2 , for every non-zero ( the drag can reach arbitrarily large values,
for small enough h.
The functional form of the drag term derived in section B. 1.2 is only valid for the
case pienv/po < 1. However, given the comment above, even very small values of (
can have a significant effect on the dynamics of the system, in the limit of very small
h.
In practice, the limit of vanishing h is impossible to achieve because the suspension
is made up of finite-size particles. When the filament reaches the size of the particle,
its behavior must transition to that of a Newtonian fluid, and subsequently break up
in finite time.
50
B.5
Parameter range covered by this study and
range of valdity of equations
Analytically and numerically, we only consider filaments with negligible inertia, i.e.
the limit Oh -+ oc.
Because in our experiments we use very viscous suspensions, with Oh
>
1, in
this study we only consider the case of high Ohnesorge numbers. In this limit, the
momentum equation simplifies to
(h+2
h2 OZ
(IC' + 3pM
(h2±/
N
NN) ~~
0
(B. 1)
and the only dimensionless group affecting the dynamics of the system are (, ( and
Wi
As discussed in previous sections and directly above, there are 3 parameters that
influence the behavior of the system. It is important to be weary of the range of
parameters in which the model can be expected to give results that correspond to
the behavior of the physical system. The model only works for slender filaments.
That is, h/i < 1. This, however, is different from the parameter (, which describes
the geometry of the system at the beginning of its evolution. For filaments with
1 - 10, the model yields reliable results for late times only, once the geometry
of the filament becomes slender. The slenderness approximation is troublesome in
the region where the filament merges with the bounding droplets. There, h' is no
longer small, and the radial flow is not negligible. This is an issue shared by all
investigations of capillary breakup based on 1-D approximations to the momentum
and mass balance equations. A partial resolution to this problem is offered by the fact
that the asymptotically negligible terms h" are retained in the curvature expression
3.4. This ensures that the steady-state shapes of the bounding droplets are spherical,
in agreement with the experiments. As was already mentioned above, the functional
form of the drag term is only valid at penv/p'O < 1.
51
52
Appendix C
Numerical simulation
C.1
Equations solved
The governing dimensionless equations solved in this section are
0=1 a
f
=h2z ( h2
0=
2 +
Ah
ot
/ IC
v
3
'3
oaz)
Ah2 V
(C.2)
Oz
Where it was taken into account that Oh > 1 and
(pv-
= -
az
k
Ov\(.1
az)
(=
0. The viscosity function is
+-
1
2
(C.3)
+1
(1 - !LWi) ce+(L Wi - 1)c
k = 18.52;
(C.4)
a =1
This function has the property that p(0) = 1, as required by the nondimensionalization procedure. The factors k and a control the sharpness of the transition. The
value of a controls the asymptotic nature of divergence as ov/&z -+ Wi 1 , where the
smaller the value of a, the sharper the divergance. The value of parameter k was
chosen such that p(0.95Wi) = 2, to make sure that the results for curves with various
values of a can be compared with each other. The values of 0.95Wi and y
53
=
2 were
chosen arbitrarily to serve as measures of steepness of the transition. Simulations
were performed for various values of these parameters, but the most frequently used
ones were k = 18.512 and a = 1.
Description of numerical scheme
C.1.1
The numerical scheme employed is a volume-conservative fully implicit finite difference scheme.
The solution is obtained by discretizing the domain between z
=
0 and z
=
L, and
solving for h(z, t) and v(z, t). The soltution is obtained for half the initial instability
wavelength to limit computation time because of symmetry. The grid is uniform, but
the time step is adaptive. The scheme is fully implicit. The solutions to the nonlinear equations are obtained by Newton's method. Up to 5 iterations of the Newton's
method were used to obtain convergence. The maximum Error for all variables was
set at 10-3, and the time step was varied to match this limitation.
The equations are discretized using upwinded differences. This was necessary due
to a sharp increase in the value of viscosity as '
-+
Wi- 1 . While this is not a typical
shock feature that appears e.g. in the Burger's equation, the sharp increase causes
very sharp gradients in the values of curvature and viscous stress inside the filament.
The viscous component of the momentum equation and the mass conservation equation were upwinded following the direction of velocity. However, for stability, the
curvature component was upwinded in the opposite direction.
C.2
Source Code
Solve a set of non-linear equations F
(2)
= y, whereF
is a vector of non-linear equations
(* Clear all variable definitions *)
ClearAll[Evaluate[Context[
<> "*"]]
Clear[Subscript]
(* Set the right directory *)
Dir = NotebookDirectory[];
SetDirectory[Dir]
C:\\Users\\auv\\Desktop\\Pawel\\20120509\\data
54
Setting up the finite difference equations
H-equation
heqn = Bth[x, t] + 8 2,(vf[x,
VF[i-]:=
(
t]);
j
in+1X+1]+l]
V[i, n + 1];
i-11
dVFdxf = VF;i-F
dHdtf =- (H[i,+1])2_(H[i,n])2
d
Substitution rules for full grid points:
HErules = {Bth[x, t]->dHdtf, 8(vf[x, t])->dVFdxf};
V-equation
Viscosity equation
(*withtheseparameters, thedeviationofvisocsityfromit'szeroshearratevaluedecreasesby99%withinMl-yNDfromyND
*)
M1 = 18.5128;
M2 = 1.82139;
R1 = 1000.0;
R2 = 1000.0;
(*MyVisc[x_]:=R1 * Erf [(
+r
l(
MyVisc[x-]:=
+ R2 * Erf (- X -
1) *
-
1) *
;
+ (M+1))
If[
RI ==
O&&R2
== 0,
pA[x..]:=1,
p[x_]:=MyVisc
-
MyVisc[0] + 1];
Plot[{p[x], D[p[k], k]/.k -+ z}, {x, -2.12
*
1, 2.12 * 1}, PlotRange -+ {-10, 10}(*{-R1, 2.2R1}*)]
(* Table for export *)
ExpMu = Table[{i, p[i]}, {i, -2S, 2S, 4S/200}]; (*"Quiet" preventserrorsfromcomingup - atx =S,
indeterminateformpopsup*)
Curvature Pressure
press[x,
t]:=
hz,t]*(1+(h[z,t])21/2
[t
1
HfFunc[i-, n-]:=H[i,n + 1];
Hf2Func[i., n-]:=H[i-
1, n + 1];
HhFunc[i.-, n-:= H[i,n+1]X
1++H[i+1,n+1IX[i]
+1
dHdxfFunc[i-, n.]:= H
dHdxhFunc[i_, n.]:= H; i+1n+
HL2+1n+l-HliW+11u L _: i
d.5d
ddHddxfFunc[i,
n-]:
dd~dx~uc~.
ddHddxf2Func[Li,
dH
=
fFunc[i,
in+]H-1+]
;+
O
X +.5X
L-2
OXi-+5X
i
+_ X
0. X VHf=+1
HfHunc,+1, Hi,n+];-HI-n+1]
H i,
n4]::= d~x~n+1
.5 1+
-de
.X[i
dxf
Hne
X[i1i
,
H
2.21
-
-] =
Hf2 = HfFunc[i, n];
n];
dHdxf = dHdxfFunc[i,
n];
ddHddxf = ddHddxfFunc[i,
ddHddxf2 = ddHddxf2Func[i, n];
55
Hh = HhFunc[i, n];
Prules = {h[x, t]->Hf,
8x h[x,
t]->dHdxf, ix,x h[x, t]->ddHddxf};
Prules2 = {h[x, t]->Hf2, 8x h[x, t]->dHdxf, 8x,,h[x, t]->ddHddxf2};
PrulesH = {h[z, t]->Hh,B. h[z, t]->dHdxh, 8z,xh[x, t]->ddHddxh};
P[i.]:=Evaluate[press[x, t]/.Prules];
P2[Lj:=Evaluate[press[x, t]/.Prules2];
Ph[L] :=Evaluate[press[z, t]/.PrulesH];
Vequation proper
2
veqn = (*ADVterm-*)(*Oh *) - .9,(f[x, t]) + Dr(*Oh2*)
V[i,n+
dVdth =
-
Vi,n;
Vh = V[i - 1, n + 1];
visc[i-]:=yA
-1
FluxV[i-]:=H[i, n + 1]2
(3
*
Flux2V[i-]:=H[i - 1, n + 1]2
FluxP[i.]:=H[i, n + 1]2
;5X
visc[i] * (
(3
1-Vi-1,m+1
* visc[i] *
;
p[i;
2
Flux2P[Li]:=H [i - 1, n + 1] P2[i];
dFdxh =
1
((AlFIuxV[i+1]+(1-A1)FIux2Vli+1
H [i,n+1]2
((1-Al)FluxP[i+1]+AlFlux2P[i+1])-
)-((AlFluxV[i]+(1-AI)Flux2V[i])) +
i]
((1-A1)FluxPji]+AIFlux2P
i])
[ ]
ADV =
1]2Vii + 1, n + 1]2 -
Al ([+] i + 1, .+
H
(1 -
Al) (H[i, n + 1]2 Vi,
n + 1]2 - H[i -
H[i, n + 112 V[i, n + 1]2)/ (0.5(X[i + 1] + X[i]))+
2
1, n + 1] V[i _ 1, n + 112)/
(0.5(X[i] + X[i -
1])))
Substitution rules for V equation
Vrules =
{Btv[x, t]->dVdth, v[x,
t]->Vh, h[x,
t]->H[i,n
+ 1], Bx(f[x, t])->dFdxh, ADVterm -+ ADV};
Vrulesl = {Bctv[z, t]->dVdth, v[x, t]->Vh, h[z, t]->H[i,n + 1], 8i(f[x, t])->dFdxh, ADVterm -+ ADV};
Vrules2 = {Oitv[x, t]->dVdth, v[x, t]->Vh, h[x, t]->H[i, n + 1], Bz(f[z, t])->dFdxh, ADVterm -4 ADV};
Vrulesl = {atv[x, t]->dVdth,
v[x, t]->Vh,
h[x, t]->H[i, n + 1], ce (f[x,
t])->dFdxh, ADVterm
ADV};
-+
Vruleslm = {4tv[x, t)->dVdth, v[z, t]->Vh, h[z, t]->H[i, n + 1], 8x(f[x, t])->dFdxh, ADVterm -4 ADV};
VrulesB = {atv[z, t]->dVdth,
v[x,
t]->Vh, h[z, t]->H[i,n + 11, Ba(f[z, t])->dFdxhB, ADVterm -+ ADVB};
VrulesLR = {8tv[x, t]->dVdth, v[z, t]->Vh, h[z, t]->H[i,n + 1], Bz(f[z, t])->dFdxh, ADVterm
-
ADV};
VrulesC = {F3 v[z, t]->dVdth, v[z, t]->Vh, h[x, t]->H[i,n + 1], Bz (f[z, t])->dFdxh, ADVterm -+ ADV};
Boundary conditions and V- and H- function evaluation
Fv[L]:=Evaluate[veqn/.Vrules];
Fvl[i.]:=V[i, n + 1];
FvI[i-]:=V[i, n + 1];
Fv2[i-]:=Evaluate[Fv[i]/.
{V[i - 2, n + 1] -+ -V[i, n + 1],
H[i -
2, n
+ 1] -4 H[i, n + 1],
X[i - 2] -+ X[i]}];
56
FvIm[i_]:=Evaluate[Fv[i]/.
{V[i + 2, n + 1] H[i + 2, n + 1] -
-V[i, n + 1],
H[i, n + 1],
X[i}];
X[i + 2] --
2
VolF[i_]:=H[i,n + 1] V[i, n + 1];
Fh[i.]:=
+
Al
(1-
Al)
) dt +
FhI[i_]:=Evaluate[Fh[i]/.
-V[i - 1, n + 1],
{V[i + 1, n + 1] -
V[i + 2, n + 1] -+ -V[i
- 2, n + 1],
H[i + 1, n + 1] -1 H[i -
1, n + 1],
H[i + 2, n + 1] -+ H[i - 2, n + 1],
X[i
X[i
+
+
X [i - 1],
1] -
2] -+X[i - 2]}];
FhIm[i.]:=Fh[i];
Fhl[i_]:=Evaluate[Fh[i]/.
{V[i - 1, n + 1] -+ -V[i + 1, n + 1],
-V[i + 2, n + 1],
V[i -
2, n + 1] -
H[i -
1, n + 1] -> H[i + 1, n + 1],
H[i - 2, n + 1] -+ H[i + 2, n + 1],
X[i -- 1] -+X[i + 1],
X[i + 21}];
X[i - 21
Fh2[i_]:=Fh[i];
Set up the F-vector
Hrules =
{
H[i -
2, n + 1] -
H[i -
2, n] -+
V[i -
2, n
Himm,
Himm,
+ 1] -
Vimm,
V[i - 2, n] -) Vimm,
+ 1] -4 Him,
H[i -
1, n
H[i -
1, n] -4 Him,
V[i -
1, n
V[i -
1, n] -
+ 1] -
Vim,
Vim,
H[i, n + 1] -+ Hi,
H[i, n] -+ Hi,
V[i, n + 1] -+ Vi,
V[i, n] -+
Vi,
H[i + 1, n + 1] -+ Hip,
H[i + 1, n] -+ Hip,
V[i + 1, n
+
11 -+ Vip,
V[i + 1, n] -+ Vip,
H[i + 2, n + 1] -+ Hipp,
H[i + 2, n] -> Hipp,
57
H[i, n +1]2)
V[i + 2, n + 1] -*
Vipp,
V[i + 2, n] -4 Vipp,
H[i + 3, n + 1] -4 Hippp,
H[i + 3, n] --
Hippp,
X[i
-
2] -4 Ximm,
X[i
-
1] -4 Xim,
X[i] -+
Xi,
X[i + 1]
-
Xip,
X[i + 2]
-
Xipp,
X[i + 31 -+ Xippp,
Al -.
alpha
};
vcom = Compile[
{{Himm, .Real}, {Vimm, -Real}, {Him, -Real}, (Vim, -Real}, (Hi, -Real},
{Vi, -Real}, (Hip, -Real}, {Vip, -Real}, {Hipp, -Real}, {Vipp, -Real},
{Hippp, -Real}, {Ximm, -Real}, {Xim, -Real}, {Xi, -Real}, {Xip, -Real},
{Xipp, -Real}, {Xippp, -Real}, (count, -Real}, {alpha, -Real}},
Evaluate[
Which[
count == 1, Evaluate[Fv1[i]/.Hrules],
count==2, Evaluate[Fv2[i]/.Hrules],
count == I, Evaluate[FvI[i]/.Hrules],
count==I -
1, Evaluate(FvIm[i]/.Hrules],
True, Evaluate[Fv[i]/.Hrules]]],
{{If[-, -, -), .Real},
(a, -Real, 1},
(Al, -Real},
{With[-, -], -Real},
{U, -Real, 1},
{I, -Real}}];
vcomwrap = Compile[
{{L, -Real, 1}, (count, -Real, 1}, {alpha, -Real}},
vcom[L[[1]], L[2]], L[[3]], L1[4]], L[[5]], L[[6]], L[[7]], L][8]], L[[9]], L{10]], L{[11]], L[[12]},
L[[13]], L[[14]], L[[15]], L[[16]], L[[17]], count[[1]], alpha],
{{fvcom[-, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -], -Real},
(a, -Real, 1}}];
hcom = Compile[
((Himm, -Real}, (Vimm, _Real}, (Him, -Real}, {Vim, -Real}, (Hi, -Real}, {Vi, -Real},
(Hip, -Real}, {Vip, -Real}, (Hipp, -Real}, {Vipp, SReal}, {Ximm, -Real}, (Xim, -Real},
{Xi, -Real}, {Xip, -Real}, {Xipp, -Real}, (count, -Real}, {alpha, -Real}},
Evaluate[
Which[
count == 1, Evaluate[Fhl]i]/.Hrules],
count == 2, Evaluate[Fh2[i]/.Hrules],
count ==
I, Evaluate[FhI[i]/.Hrules],
count ==
I - 1, Evaluate[FhIm[i]/.Hrules],
True, Evaluate[Fh[i]/.Hrules]]],
((a,
-Real, 1},
(Al, -Real}}*)
{I,
-Real}}];
hcomwrap = Compile[
((L, SReal, 1}, (count, -Real, 1}, {alpha, -Real}},
hcom[L[[1]], L[[2]], L[[3]], L[[4]], L[[5]], L[[6]], L[[7]], L[[8]], L[[9]], L[[10]], L[[11], L[[12]],
58
L[[13]], L[f14]], L[[15]], count[[1]], alpha],
{{hcom[-, -, -, 1,- - , , -
- ,..
-- -, -, -, -], -Real},
{ac, -Real, 1}}];
F[U.]:=
With[
{UV = Join[{0, 0, 0,0}, U, {0, 0, 0, 0}],
dxV = Join[{0, 0), dx, {0, 0,
0)],
UH = Join[{0, 0, 0, 0}, U, {0, 0, 0,
0)],
dxH = Join[{0, 0), dx, {0, 0)]),
With[
{ParV = Join [Partition[UV, 11, 2], Partition[dxV, 6, 1], 2],
ParH = Join[Partition[UH, 10, 2], Partition[dxH, 5, 1], 2]},
Riffle[
Maplndexed[hcomwrap[#1, #2, a[[#2[[1]]]]]&, ParH],
Maplndexed[vcomwrap[#1, #2, a[[#2[[1]]]]1&, ParV]]]];
Construct the Jacobian
VFuncs =
{
H[i - 2, n + 1],
V[i - 2, n + 1],
H[i -
1, n + 1],
V[i - 1, n + 1],
H[i, n + 1],
V[i, n + 1],
H[i + 1, n + 1],
V[i + 1, n + 1],
H[i + 2, n + 1,
V[i + 2, n + 1],
H[i + 3, n + 1]
};
HFuncs =
H[i -
{
2, n + 1],
V[i - 2, n + 1],
H[i - 1, n + 1],
V[i - 1, n + 1],
H[i, n + 1],
V[i, n + 1],
H[i + 1, n + 1],
V[i + 1, n + 1],
H[i + 2, n + 1],
V[i + 2, n + 1]
};
vdcom = Compile[
{{Himm, -Real}, {Vimm, -Real}, {Him, -Real), {Vim, -Real}, {Hi, -Real}, {Vi, .Real},
{Hip, -Real}, {Vip, -Real}, {Hipp, -Real}, {Vipp, -Real}, {Hippp, .Real},
{Ximm, -Real}, {Xim, -Real}, {Xi, -Real}, {Xip, -Real}, {Xipp, .Real}, {Xippp, -Real},
{count, -Real}, {alpha, -Real)},
EvaluatefWith[{Fi = 1},
Which[
count == 1, Evaluate[D[Fv1[i], {VFuncs}]/.Hrules],
count==2, Evaluate[D[Fv2[i], {VFuncs}]/.Hrules],
count == I, Evaluate[D[FvI[i], {VFuncs)]/.Hrules],
count==I - 1, Evaluate[DfFvIm[i], {VFuncs}]/.Hrules],
59
True, Evaluate[D [Fv[i], {VFuncs}]/.Hrules]]]],
{{Map[..-,
-Real, 1},
{With[-, -,-Real}}I];
vdpadcom = Compile[
{{L, -Real, 1}, {d, Integer}, {count, _Real), {alpha,
Real)),
With[
{list = vdcom[L[[1]], L[[2]], L[[3]], L[[4]], L[[5]], L[[6]], L[[7]], L[[8]], L[[9]], L[[10]], L[[11]],
L[[12]], L[[13]], L[[14]], L[[15]], L[[16]], L[[17]], count, alpha],
multiply = {1, -1,
1, -1,
1, -1,
1, -1,
1, -1,
1}},
ArrayPad[
If[d - 6 < 0,
list + ArrayPad[Take[list * multiply, 6 -
d],
{(6 - d) + 1 - 1 + 2,11 - ((6 - d) + 1) - (6 - d) + 1 - 2)],
If[TG - (d + 5) < 0,
list + ArrayPad[Take(list * multiply, TG - (d + 5)],
{11 - ((d + 5) -
TG + 1) - ((d + 5) - TG)(*+1*), (d + 5) - TG + 1}],
list]],
{d - 6, TG - (d + 5))]],
{{ArrayPad[_, _], _Real, 1},
{vd com[-, -, -, -, -, -, -, -, .., .., -, -, -, -, _, -, _], _Real, 1},
{TG, _Real))];
hdcom = Compile[
{{Himm, -Real}, {Vimm, _Real}, {Him, .Real), {Vim, -Real), {Hi, _Real},
{Vi, SReal), {Hip, _Real}, {Vip, SReal), {Hipp, -Real), {Vipp, -Real),
{Ximm, Real), {Xim, _Real), {Xi, _Real), {Xip, Real), {Xipp, -Real),
{count, _Real), {alpha, -Real),
Evaluate[With[{Fi = 1),
Which[
count == 1, Evaluate[D[Fhl[i], {HFuncs}]/.Hrules],
count == 2, Evaluate[D[Fh2[i], {HFuncs}]/.Hrules],
count == I, Evaluate[D[FhI[i], {HFuncs}]/.Hrules],
count == I -
1, Evaluate[D[FhIm[i], {HFuncs}]/.Hrules],
True, Evaluate[D[Fh[i], {HFuncs}]/.Hrules]]]],
{{Map[-, -], SReal, 1))];
hdpadcom = Compile[
Real),
{{L, SReal, 1}, {d, Integer), {count,
{alpha, _Real)),
With[
{list = hdcom[L[[1]], L[[2]], L[[3]], L[[4]], L[[5]], L[[6]],
L[[7]],
L[[8]], L[[9]], L[[10]],
L[[11]],
L[[12]], L[[13]], L[[14]], L[[15]], count, alpha],
multiply = {1, -1,
1, -1,
1, -1,
1, -1,
1, -i)),
ArrayPad[
If[d - 5 < 0,
list + ArrayPad[Take[list * multiply, 5 - d],
{(5 - d) + 1 If[TG -
1, 10 -
((5 -
d) + 1) -
(5 - d) + 1)],
(d + 5) < 0,
list + ArrayPad[Take[list * multiply, TG - (d + 5)],
{10 -
((d + 5) - TG + 1) -
((d + 5) - TG)(*+1*), (d + 5) - TG + 1)],
list]],
{d - 5, TG -
(d + 5))]],
{{ArrayPad[-,
-Real, 1),
{hdcom[-, -, ., -, _,
_, -,_,
, , -, _,
,
-Real, 1},
{TG, _Real))];
J[U_]:=
60
With[
{UV = Join[{0, 0, 0, 0}, U, {0, 0, 0, 0, 0}],
dxV = Join[j{0, 0}, dx, {0, 0, 01],
UH = Join[{0, 0, 0, 01, U, {0, 0, 0, 0}],
dxH = Join[{0, 01, dx, {0, 0}]},
With[
{ParV = Join[Partition[UV, 11, 2], Partition[dxV, 6,11,2],
ParH = Join[Partition[UH, 10, 2], Partition[dxH, 5, 1], 2]},
Riffle[
MapIndexed[hdpadcom[#1, 2 * (First[#2] - 1) + 1, #21[l]], a[[#2[[1]]]]]&, ParH],
Maplndexed[vdpadcom[#1, 2 * (First[#2] -
1) + 2, #2[[1]], a[[#2[[1]]]]]&, ParV]
1]];
Define TimeStep function
(* Perform the timestep *)
TimeStepfPPrev_, Prev_, Dt_, Residue_]:=
*
EvaluateVelocitygiventheheightprofileatt
= to *
*
(* Formulate guess *)
U =
Prev;
(*Formulatetargetvector - frompreviousiteration*)
(*Y = Prev; *)
Hpr = Prev[[1;;TG - 1;;2]];
YO = Table[0, {i, 1, 1}];
Y = Riffle[Hpr, YO];
dt = Dt;
Resid = {}; (* reset the residuals value *)
(*LoopforNewton'smethodtosolvethenon -
linearfinitedifferenceequation*)
For[q = 1, q <; Newt, q++,
c = AbsoluteTiming]
Jac = SparseArray[J[U]];
Fval =F[U];
mid = Timing[U = U + LinearSolve[Jac, Y - Fval];];];
CurrResid = Abs[Y - F[U]]; (* must use the new solution to evaluate F *)
HRes = Total[Take[CurrResid, {1, TG - 1, 2}]];
VRes = Total[Take[CurrResid, {2, TG, 2}]];
(* Calculate residuals *)
If[Residue,
Resid = Append(Resid, {HRes, VRes}];,];
(* Stop if sufficient accuracy was reached *)
If[HRes < MaxResidue&&VRes < MaxResidue, Break[], ];];
If[Ni > 1, Print["Ni=", Ni], ];
Return[{U, Resid}];
Regridding functions
ViscF[H_, V_, dx-]:=Table
5
3p [3-
-4Jt,
0.5 dx
(d H
Kterm[H_, dx_]:=Table
i1-
{i, 2, 1 -
Hi
i
dr/2
,{,2il]
1+(IoHt+HA
V_, dx_]:=Table
23 ViscFH[H_,
{i, 2, I -
1}i
Afqi~~)h
i xi1
[]
/
H
1}];
61
I - 1});
RowFunc[x]:= 2[2]] {[1
-lJ]r[2]]
[2]'
+[[2]]x[[3]'
[[2]]+[[3]]
}
MyAccumulate[dx_]:=
Abs [Accumulate[A.dx] -
d 2 1])
Step[dx.., func_]:=(
xtry = MyAccumulate[dx];
dxgoal = Map[func, xtry];
NG = 0.5dxgoal + 0.5dx;
NG=Dom Total[NG]_ NG[[1]] NG[[I]]
Return[NG];);
MakeGrid2[dx_, Func.]:=(
(*Print["mygrid"]; *)
dx2 = dx;
dxtry = Step[dx2, Func];
xtry = Abs [Accumulate[A.dxtry] -
dxtry[1]]]
i = 1;
While[
Total[Abs[dxtry/Max[dxtry]
- Map[Func, xtryJ] > 10-
3
(*&&i < 1000*),
dxtry = Step[dxtry, Func];
xtry = MyAccumulate[dxtry];
i++; ]
Return[{dxtry, i}];);
DomSpan[x_]:=Total[x] -
[212
FindGrid[DensFunc_]:=(
XF = MakeGrid2[dx, DensFunc];
iterno = XF[[2]];
XF = XF[[1]];
Return[{XF, iterno}];
Regridding[U_, dx_]:=(
Hg = Take[U, {1, TG - 1, 2)];
Vg = Take[U, {2, TG, 2}];
xoh = Accumulate[A.dx] -
dx1] ; (* Old locations of grid points *)
Vi = DViscF[Hg, Vg, dx]; (* function according to which the regridding takes place *)
Vi2 = Abs Table
V,
{i, 2, Lengt[Vg]}
;
Ord = Ordering[Vi];
Ord2 = Ordering[Vi2];
xboundmin = 1.5;
.
xboundmax = 0.9xmax;
xmini = 1;
minxval = xoh[[Ord2[[xmini] + 1]];
While[minxval < xboundmin||minxval > xboundmax,
62
xmini = xmini + 1;
minxval = xoh[[Ord2[[xmini]] + 1]];
]; (* coordinate at which minimum in grid spacing is located *)
xmaxi = Length[Ord];
maxxval = xoh[[Ord[[xmaxi]] + 1]];
While[maxxval < xboundmin||maxxval > xboundmax,
xmaxi = xmaxi - 1;
maxxval = xoh[[Ord[(xmaxi]] + 1]]]; (* coordinate at which maximum in grid spacing is located *)
dxsmall = 0.1;
dxsmall2 = 0.3;
varmax = If[Max[Vi] > 0.05, 1, 0];
varmin = 1;
densityGrid[x.]:=
With I{va = 1 -
(1-
(1 - dxsmall)Exp
(*
2
(-m3inxval)
dxsmall)varminExp
-1xmax)2
(1 -
dxsmall)Exp
- (1-
_(-_0
2
dxsmall)varmaxExp [(-maxxval)2-
, If[va <
dxsmall, dxsmall, va]
density function for the new grid *)
dxtab = Map[densityGrid, xoh];
densityAdjust = Interpolation[Transpose[{xoh, dxtab}]];
Dom = xmax - xmin;
NewGrid = FindGrid[densityAdjust[#]&]; (*actualregriddinghappenshere.Outputisinrelativevalues*)
NewGrid = NewGrid[[1]];
XF = NewGrid;
dx2 = (xmax -
xmin) *
(* Mapping relative values of grid spacing onto actual values *)
F;
Total[XF]-
x2h = Accumulate[A.dx2] -
2
dx2i[111 ; (* New locations of grid points *)
(* Interpolating function for height *)
IntH = Interpolation[Pair[xoh, Hg, g], InterpolationOrder
-
3];
(* Interpolating function for velocity *)
IntV = Interpolation[Pair[xoh, Vg, g], InterpolationOrder -+ 3];
(* New values of H *)
Quiet[H2 = Map[IntH, x2h]];
(* New values of V *)
Quiet[V2 = Map[IntV, x2h]];
Return[{Riffle[H2, V2], dx2, U}];
Print["Regrid Complete"];
63
Solve
AR = 11;
001
gflow =
gsusp = 0.35;
gCR = gsusp/gflow
2.8
DragTest =
{
{AR, gCR, Drag}
};
For[serg = 1, serg <; Length[Test], serg++,
Print["xmax=", Test[[serg, 1]],
",
"gND=", Test[[serg, 2]],
",
"Dr=", Test[[serg, 3]]];
(* Define the spatial grid *)
xmin = 0.0;
xmax = Test[[serg, 1]];
dxval = 0.03;
I = Min[Round[(xmax - xmin)/dxval], 300];
Print["I=", I];
A = SparseArray[{Band[{1, 1}] -4 0.5, Band[{2, 1}] -
0.5}, {I, I}];
(* Matrix that gives relative spacings between grid points *)
Pair[x_, y_, L_]:=Map[{z[[#]], y[[#]]}&, L]; (* Make pairs of lists x and y *)
w = Table[1, {x, 1, I}];
dx =
w
* (xmax - xmin)/(Total[w] - 1/2(w[[1]] + w[[I]]));
varmin = 0;
varmax = 0;
afun[x-]:=If[x > 0, -1,
1];
dXFULL = Accumulate[A.dx] -
dxl2
afun2[list_]:=Join[{afun[list[[1]]]}, Map[afun, list], {afun[Last[list]]}];
a = Table[1, {i, 1, I}];
(* define initial conditions *)
Hboundary = 1;
Z = 0.00001;
(*Z = 0.4; *)
Hini = -ZSin
Accumulate[A.dx] -
jdx[[l1]]
* r-
+ (Hboundary - Z); (* IC for height *)
Vini = Table[0, {I}]; (* IC for velocity *)
TG
21; (*Totalnumberofgridpoints, fullandhalf*)
g = Range[I];
Resids = {}; (* reset residuals *)
64
(* set Maximum Error per grid point *)
MaxErrorGrid = 0.5 * 10-5; (*0.5 * 10-
5
isequivalenttooldMaxError = 10-
3
atI = 100*)
(*MaxresiduepergridpointwhenusingNewton'sMethod*)
(*residuesarecalculatedseparatelyforVandH.Errorsintimeintegrationarecalculatedjointly*)
MaxResidueGrid = 10-6; (*10-
6
4
isequivalenttooldresidueerrorof10 -atI
= 100*)
MaxResidue = MaxResidueGrid * I;
(* Run the simulation *)
TimeConstrained[
(*Quiet[*)
Parallelize[
SimDate = DateString[];
INI = Riffle[Hini, Vini]; (* set the initial conditions *)
(* Variable timestep counters *)
TimeCheck = 1;
TimeCheckLast = 0;
TimeCheckIndex = 1;
Newt = 5; (* Number of Newton iterations *)
(*Simulationparameters - physical*)
Oh = 1000.0; (* Ohnesorge number *)
Dr = Test[[serg, 3]];
-yND = Test[[serg, 2]]; (* critical deformation rate *)
Comms = "Corresponds to N1000 oil deep cs55glyc 5 crop";
(*Simulationparameters - numerical*)
Regrid = False;
TS = 0.0001;
dt = TS;
TSmin =10-8
TSmax = 100000.0;
T = 0;
(*MaxError = 1 * 10-3; *)
MaxError = TG * MaxErrorGrid;
Error = 0;
Nmax = 10000; (* maximum number of iterations *)
Res = True; (*Saveresidues?Yesif'True', noif'False'*)
If[Res,
Result = {{0, INI, dx, {0,
0}}},
Result = {{0, INI, dx, {}}}];
init = TimeStep[INI, INI, TS, Res]; (*Firstsolveriteration - useinitialcondition*)
Previ = init[[1]];
PPrevi = INI;
T = T + TS;
Result = Append[Result, {T + TS, init[[1]], dx, init[[2]]}];
65
report = 0;
(* Loop over time steps *)
Timing[
For[m = 1, m < Nmax&&TS > TSmin, m++,
If[(TimeChecklndex - TimeCheckLast) == TimeCheck,
(tTrue*)
sol = TimeStep[PPrevi, Previ, TS, Res];
soll = TimeStep[PPrevi, Previ, TS/2, Res];
sol2 = TimeStep[Previ, soil [[1]], TS/2, Res];
Error = Total[Abs[sol[[l]] - sol2[[1]]]];
If[Error > MaxError * 2,
TimeCheck = Round[TimeCheck/2
+
1];
sol = sol;
T = T + TS;
TS = TS/(3/2);
If[TS < TSmin,
Print["times step too small"],];,
(*TimingcheckatmostafterlOsteps - avoidsrunningtolongwithoutcheckingappropriateness
oftimestepmagnitude*)
TimeCheck = If[TimeCheck < 10, TimeCheck + 1, TimeCheck];];
If[Error < MaxError/2,
T = T
+
TS;
TS = TS * 3/2;
If[TS > TSmax,
TS = TSmax];
TimeCheck = Round[TimeCheck/2],
T = T + TS];
TimeCheckLast = m;,
(*False*)
sol = TimeStep[PPrevi, Previ, TS, Res];
TimeChecklndex = m;
T = T
+
TS; ];
report = report + 1;
(* Save the result of the simulation to the variable Result *)
If[IntegerQ[m/l],
Result = Append[Result, {T, sol[[l]], dx, sol[[2]]}];
report = 0; ];
PPreviH = Take[Previ, {1, TG - 1, 2}];
PPreviV = Take[Previ, {2, TG, 2}];
MVF = DViscF[PPreviH, PPreviV, dx];
(*****************Monitorsimulationoutput******************)
66
If[IntegerQ[m/10],
PPreviH = Take[Previ, {1, TO -
1, 2}];
PPreviV = Take[Previ, {2, TO, 2}];
MyViscFAct = ViscF[PPreviH, PPreviV, dx];
Kt = Kterm[PPreviH, dx];
(*MVF = Table[(MyViscFAct[[i]] - MyViscFAct[[i -
1]])/dx[[i + 1]], {i, 2, Length[MyViscFAct]}]; *)
MVF = DViscF[PPreviH, PPreviV, dx];
xoh = Accumulate[A-dx] -
Idx[[1]];
2
volfi = Previ[[1;;TG - 1;;2]] Previ[[2;;TG;;2]];
2
- 1;;2]] PPrevi[[2;;TG;;2]];
volfim = PPrevi[[1;;TG
erle
Table
[=((voli[i
-
}((volfl[[i +
1]] -
1]] +dx[[i+
1]];+
dxji]])
volfl[[i - 1]])/(0.5(dx[[i - 1]] + dx[[i + 1]]) + dtx[[i]])-
(volflm[[i + 1]] - volfim[[i -
1]])/(0.5(dx[[i -
1]] + dx[[i + 1]]) + dx[[i]])), {i, 2, Length[volfl] -
(*Print[ListPlot[Transpose[{xoh, a}], PlotStyle -+ Red, PlotRange -+ All]]; *)
Print[ListPlot[Pair[xoh, PPreviV, Range[I]], Joined -+ False, ImageSize -+ Small]];
Print[ListPlot[Transpose[{xoh[[2;;I -
1]], MyViscFAct}], PlotStyle -+ Green, PlotRange -+ All,
Joined -+ False, ImageSize -+ Small]];
Print[ListPlot[Transpose[{xoh[[2;;I -
1]], Kt}], PlotStyle -+ Purple, PlotRange -+ All, Joined -
False,
ImageSize -+ Small]];
(*Print[ListPlot[Transpose[{xoh, dx}], PlotStyle -+ Brown, PlotRange -+ All, Joined -+ True, ImageSize -+ Small]]; *)
(*Print[ListPlot[Transpose[{xoh[[2;;I - 1]], err}], PlotRange -+ All]]; *)
Print ListPlot Pair Accumulate[A.dx] -
j dx[[1]], Take[sol[[1]],
{1, TO, 2}], g , PlotRange -+ All
(*{{0, xmax}, {0, 3}}*), ImageSize -4 Small, AspectRatio -+ 3/xmax,
Epilog -+ Inset[ToString[Last[Result] [[2,21 -
1]]], Scaled[{0.8, 0.8}]]]];
Print[ListPlot[Transpose[{xoh[[2;;I - 2]], MVF}], PlotStyle -+ Orange, Joined -+ False, PlotRange -+ All]];
,;
*
Conditionsforstoppingthesimulation****** *
(* Stop the simulation is any part of the output becomes a complex number *)
If[MemberQ[Map[Head, sol[[1]]], Complex], Print[ "Complex"]; Break[], ];
(*Dbreak = 0.005;
(* Break if either bead is created or simulation goes past treshold of 0.1 *)
If[sol[[1, TO -
1]] > Previ[[TG -
1]],
Print["BEAD"];
Print[Previ[[TG -
1]]];
(*Result = Append[Result, "Bead"];
Break[],
If[sol[[1, TO -
1]] < Dbreak,
Print["NO BEAD"];
Print[sol[[1, TO - 1]]];
(*Result = Append[Result, "No Bead"]; *)
Break[],
(*If[m < 50000000, Regrid = False, Regrid = True]; *)
67
(***********************Regriddingprocedure*************************)
If [IntegerQ [i-] &&Regrid,
(*Print[ "regridding"]; *)
Reg = Regridding[sol[[1J, dx]; (* perform the regridding *)
(*Notonlythecurrentgrids, butalsotheonebeforethathastobeportedtonewgrid*)
xoh = Accumulate[A.dx] -
Idx[[1];
PPreviH = Take[Previ, {1, TO -
(* Old locations of grid points *)
1, 2}]; (* values of H at old grid points *)
PPreviV = Take[Previ, {2, TG, 2}]; (* value of V at old grid points *)
(* Interpolating function for height *)
PPreviHint = Interpolation[Pair[xoh, PPreviH, g], InterpolationOrder
-
3];
-
3];
(* Interpolating function for velocity *)
PPreviVint = Interpolation[Pair[xoh, PPreviV, gJ, InterpolationOrder
dx = Reg[[2]]; (*NEW GRID*)
xnh = Accumulate[A.dx] -
ldx[[1]]; (* NEW locations of grid points for H*)
PPreviHnew = Map[PPreviHint, xnh]; (* values of past H in new grid
5)
PPreviVnew = Map[PPreviVint, xnh]; (5 values of past V in new grid
5)
PPrevi = Riffle[PPreviHnew, PPreviVnew;
Previ = Reg[[1]]; (* Substitute new grid
(* update
5)
a *)
dXFULL = Accumulate[A.dx] -
jdx[[1]];
(*a = Map[afun, dXFULL]; *)
a = afun2[Take[PPreviV, {2, I -
(*Print["regrid complete"];
1}]];
5)
PPrevi = Previ;
Previ = sol[[1]];];
(* Print out simulation progress report *)
If [IntegerQ [M] , Print[{m, c, mid[[1]], Error, T, TS}],]
]]],
60 * 60];
SetDirectory[Dir];
(* Simulation parameters to be exported
5)
Params =
Join[
Map[
Flatten,
{{"IC", N[Riffle[Hini, Vini]]},
68
{"grid", N[c]},
{ "visc",
ExpMu},
{"Oh", N[Oh]},
{"MaxError", N[MaxError]},
{ "NewtIter", N[Newt]},
{ "DateRun", SimDate},
{ "ICSource", ICSource},
{ "ICFile", ProfileSource},
{ "PressTerm", PressTerm},
{"Drag", Dr},
{"NVT", NVTParam},
{"gcrit", -yND},
{"ViscParamMRS", {M1, M2, S, R1, R2}}}]];
ResultExp = Map[Flatten, Result[[i;;Length[Result]]]];
ToExport = Join[Params, ResultExp];
(* name for the file *)
Day = Block[{$DateStringFormat = {"Year", "Month", "Day"}}, DateString[]j;
Time = Block[{$DateStringFormat =
{ "Hour",
"Minute", "Second" }}, DateString[]];
Newt = If[R1 == 0, "Newt", "NonNewt"];
filename = StringJoin[Newt, " ", D, Day, " ", T, Time, "",
ToString[xmax/Hboundary], " ", "Dr.", ToString[Dr], "",
ToString[yND],"
" ",
"Oh.", ToString[Oh], " ", "AR.",
"RDr.", ToString[RDr], " ", "gND.",
", "Ml.", ToString[M1], " ", "M2.", ToString[M2], " ", "R1.", ToString[R1]," ", "R2.",
ToString[R2], " ", "I.", ToString[I],
" ",
"Z.", ToString[Z],
" ",
Comms];
file = Stringioin[filename, ".dat"];
(* Export the file *)
Timing[Export[file, ToExport]];
]
69
70
Appendix D
Model equations solutions and
Analysis
Solutions to the governing equations of our model reveal a coupling between material
and geometrical nonlinearities, resulting in two regimes - one of which is dependent
on the material properties, the other on the geometry of the filament.
The equations governing the filament's evolution, presented in Chapter 3 are
solved numerically using a finite volume method, described in Appendix C.
The solutions of the equations confirm the observations made in Chapter 2 that
the most important influence of the thickening effect in tested suspensions is to limit
the rate at which the filament thins to
ycrit,
the critical deformation rate at which
the thickening effect kicks in.
It is found that the evolution of the filament, specifically the viscous stress developed at the center of the filament, depends on the combination of the critical
deformation rate, icit, and the length of the filament (. Below a certain threshold in
the value of the combined parameter s/Wi, the evolution depends on ( only, whereas
above the threshold the evolution depends on Wi only. Section D.2 presents the
results of the simulations leading to this conclusion.
The results of the simulations are validated using simplified analytical arguments.
71
Using the fact that in the late stages of filament evolution deformation rate is constrained to
Acrrt,
the velocity field in the filament is known, which allows the volume
conservation equation to be solved for the shape of the filament. Using the fact that
in a highly viscous filament inertia is negligible and tension in the filament is uniform,
we use the information about the shape of the filament to show that below a threshold
value of the parameter c/Wi, the stress evolution is dominated by (, whereas above
the threshold, the stress evolution depends on Wi only. This corresponds well with
the results of numerical simulations.
D.1
Effect of thickening on evolution of the filament
As noted earlier, the thickening effect and its particular embodiment in the form of
the model viscosity function (3.5) mean that the velocity gradient inside the filament
cannot reach values beyond Wi-1. In contrast, n a Newtonian filament even for very
viscous filaments the velocity gradient increases monotonously in time, and its growth
is only stopped by filament fragmentation once the neck reaches diameter comparable with the liquid's molecular scale. Since the viscosity in the model describing the
thickening effect is relatively independent of
'
until it gets very close to Wi- 1 , it is
reasonable to expect that the filament will evolve according to a Newtonian filament
with the same low-deformation-rate viscosity. However, when the stress in the filament reaches ~ Wi- 1 , the thickening filament's behavior begins to deviated from
that of the Newtonian filament.
This is shown in Figure 4.1, where the value of the viscous stress p(y) is plotted
against the minimum radius of the diameter. For a Newtonian filament, the viscous
stress increases as 1/hmin, as expected. For the thickening filament, however, after the
stress reaches the value of approximately Wi- 1 , it plateaus, increasing much more
slowly. This nature of this increase depends on the exact shape of the thickening
72
function. For more steep thickening responses, the increase is very shallow, as shown
in Figure 4.1.
For much smaller values of Wi- 1 , the stress in the thickening filament also plateaus,
but the nature of deviation from the Newtonian curve is different.
This is the subject of the next section.
D.2
No drag behavior
Behavior of filament without drag can be understood in terms of a single variable,
which combines two relevant dimensionless groups.
There are two dimensionless groups governing the evolution of the thickening
filament in the case without drag: Wi and (.
The evolution of the filament can
be described in two ways: by describing the evolution of the shape in time, and by
describing the evolution of the stress with changes in the shape of the filament, which
is a measure of the coupling between the material properties and the surface tension
forces.
Shape evolution
As explained above, the shape of the filament initially evolves
similarly to a Newtonian fluid, but when the stress inside filament reaches Wi-1, the
deformation rate cannot increase beyond that point. therefore, the deformation rate
is fixed at Wi-1, and the volume conservation equation can be solved
&h2
at
2h-
at
Oh
Ot
= -v
Oh
V-z
+ 2hv
hOv
3
2
8z
73
Oz
= -
+
ah2v
+ h
Oh
Oz
Oz
5z
0
(D.1)
=0
(D.2)
=
h
W
2
1
(D.3)
This equation allows solutions with Oh/Oz = 0, leading to
h
-
at
hWi1
=-
h = Hexp
2
h
(D.4)
Wi-i t
(D.5)
2)
where H is the radius of the filament at the time the deformation rate first reaches
Wi1.
Therefore, after the stress inside filament reaches Wi'-, the filament becomes
axially uniform
(
Oh/Oz = 0) and its radius decreases at an exponentially decaying
rate. This is shown in Figure 1.3. In the temporal evolution of the minimum radius
of the filament, length of the filament does not play a role.
Stress evolution
The viscous stress that the filament has to withstand is an im-
portant indicator of the dynamics and evolution of the filament, and in the case of
Non-Newtonian fluids also an indicator of the state of the fluid. Since the stress is
governed by the momentum balance equation, which in the limit Oh
>>
1 includes no
inertia, it makes sense to consider the evolution of the viscous filament as a function
of the shape of the filament, for which a convenient proxy is the minimal radius of the
filament hmin. For highly viscous filaments, the minimum diameter occurs always in
the center of the filament, and when thickening sets in, the filament becomes axially
uniform in shape as explained above, so the minimum diameter is equivalent to the
diameter at the center of the filament.
There are two dimensionless groups affecting the behavior of the system: Wi
and (. To investigate how they affect the evolution of stress inside the filament, we
construct a 2-dimensional parameter space, shown in the inset of Figure 4.2. The
limit Wi
-±
0 corresponds to Newtonian behavior.
First, we select Wi'- = 0.004 and vary L between 5 and 19 (Figure D.1). As
the filament thins down, the stress increases, and eventually it plateaus. However,
we see that for large values of ( the curves collapse onto a single curve, indicating
74
p(Th'
100:10 E
0.1
0.01
0.00 1
''
0.02
0.05
'
' ' ' ''
0.10
0.20
'
'
' '
0.50
h mi
' ' '
1.00
Figure D.1: Evolution of the viscous stress p()y at the center of the evolving filaments
as a function of the minimum filament radius hmin. Derived from numerical
solutions to equations (3.2)-(3.6) with Wi- 1 - 0.04, C = 0 and L ranging
from 5 to 19. Arrow indicates the direction of increasing L.
100
10
0.1
0.01
0.001
'hI
0.02
0.05
0.20
0.10
0.50
1.00
Figure D.2: Evolution of the viscous stress p( )y at the center of the evolving filaments
as a function of the minimum filament radius hmin. Derived from numerical
solutions to equations (3.2)-(3.6) with Wi- 1 = 0.1, ( = 0 and L ranging from
5 to 19. Arrow indicates the direction of increasing L.
that after the length passes a certain threshold, the exact value of ( does not matter.
We conduct the same experiment, this time with Wi- 1 = 0.1 (Figure D.2)and see
the same behavior, i.e. the curves for largest values of ( collapse onto a single curve.
However, the threshold for when the collapse starts occurring is different for the two
values of Wi. Also, while the curve onto which the high ( curves collapse is different,
the curves for small ( collapse on each other for the two values of Wi. Therefore, one
can expect that there exist a threshold in the value of (, whose exact value depends
on Wi for which the evolution of the stress is as follows.
For small values of (, the evolution of the filament stress depends strongly on
75
100
10
0.1
0.01
0.001
0.02
0.20
0.10
0.05
0.50
1.00
hmn
at the center of the evolving filaments
Figure D.3: Evolution of the viscous stress p()
as a function of the minimum filament radius hmin. Derived from numerical
solutions to equations (3.2)-(3.6) with L = 6, ( = 0 and Wi- 1 ranging from
0.01 to 1. Arrow indicates the direction of increasing Wi-1.
but not on Wi. Above a certain threshold value of (, which depends on Wi, the
evolution of the stress does not depend on (, but becomes strongly dependent on Wi.
Next, we select ( = 6 and vary Wi between 0.01 and 1 (Figure D.3). Again, as
the filament thins down, the value of the stress increases until it reaches a plateau.
Here, we see that for small value of Wi the curves collapse onto a singe curve, while
for larger values of Wi the curves do not collapse.
Conducting a similar experiment at L = 15 (Figure D.4) we see a very similar
behavior. However, we notice that the stress value for the master curve is different
than for the other value of (, and the threshold value of Wi for which the collapse
occurs is also different. However, for the uncollapsed curves, the value of the plateau
stress is the same for both values of (. This strengthens the conclusions drawn from
the experiment described above.
This leads to a conclusion that there is a bounding region in the phase diagram,
which separates "large" ( from "small", and likewise "large" and "small" Wi. Since
the threshold value for Wi depends on (, and the threshold value of ( depends on
Wi, we can represent the threshold as a single curve, as shown in Figure 4.2.
This also suggests that the boundary between "large" and "small" values of
and
Wi depends on the combination of the two parameters. Indeed, it is found that the
controlling parameter is (Wi-
1
. For "large" (Wi- 1 , the value of the plateau stress
76
100
0.1
0.01
0.001
hmj,
0.02
0.05
0.50
0.20
0.10
1.00
at the center of the evolving filaments
Figure D.4: Evolution of the viscous stress p()
as a function of the minimum filament radius hmin. Derived from numerical
solutions to equations (3.2)-(3.6) with L = 15, ( = 0 and Wi- 1 ranging from
0.01 to 1. Arrow indicates the direction of increasing Wi-1.
is controlled exclusively by (, while for "large" values of (Wi- 1 , the plateau value of
stress is controlled exclusively by Wi. To verify this dependence, we take the limiting
value of stress at each combination of parameters, and call it Tmax.
As explained
above
Tmax = Tmax(
Tmax =
)
< 1
(D.6)
for (Wi- 1 > 1
(D.7)
for (Wi-
Tmax(Wi)
If Tmax is plotted as a function of Wi alone, a series of lines will be generated, each
for a different value of
. At very large Wi, all lines will collapse onto a single
curve, while at low Wi, they will plateau, with each plateau scaling in value like 1/i.
Therefore, both horizontal and verical axis can be rescales by (, and a plot of rmax
versus (Wi-1 can be generated, with the expectation that all curves should collapse
onto a single master curve, with a plateau of magnitude ~ 1 for (Wi
a linear increase rmax( ~ (Wi- 1 for (Wi-
1
1
<
1 and
> 1. indeed, as Fig 4.2 demonstrates,
such collapse is achieved, confirming the dependence of the stress evolution inside the
filament on the two parameters.
77
D.3
Analytical arguments
The behavior of the filament and emergence of the two regimes can be understood
by considering the limitation on velocity gradient imposed by the thickening and its
effect on volume conservation.
The behavior described in the previous section shows three unexpected behaviors
1. There is a clear threshold between i-dominated and Wi-dominated evolution
of the filament, depending on the value of the single parameter (Wi-1
2. When the stress evolution is dominated by the aspect ratio, the value of the
plateau stress scales as 1/i.
3. When the stress evolution is dominated by the dimensionless critical deformation rate, the stress scales as Wi-1.
The fundamental reason for the sharp transition and the relative scalings in the two
regimes is the influence that the limitation in deformation rate has on the shape of
the filament.
As mentioned above, the deformation rate of the filament cannot exceed Wi-',
both in extension and in compression. This means that a fluid element initially close
to the center of the filament will first accelerate and then decelerate, and the absolute
value of the velocity gradient never exceeds Wi-1. Therefore, there are two regions:
one where Ov/&z > 0, close to the center of the filament, and one where &v/&z < 0,
close to the bounding droplet - see Figure 3.1.
As shown in section D.2, in the region where Wi- 1 > 0 the volume conservation
equation yields the solution
h(z, t) = H exp
-
where H is the value of the radius at the time where '
78
(D.1)
Wi-1 for the first time.
In the region where Wi-1 < 0, the volume conservation equation is
Oh
Oh
O
2h Oh +2hvO +h2
0
at
z
x
Oh
Oh
2h- + 2hWi 1 ( - z)
- h2 Wi-1 =0.
ot
az
(D.2)
(D.3)
This equation admits solutions with
2h
Oh
= h 2 Wi- 1
at
(D.4)
which yields
2hWi
1(
- z)
hWi
(
- 2h 2 Wi-1
0
(D.5)
= h2 Wi- 1
az
Oh
(D.6)
Oz
- z)
=
_h
Oz
az
h-z
(D.7)
- z
which yields the solution
h(z, t) = H exp
where the fact that h(z, t)z- 1 /2-
-
2Wi
"
(D.8)
- z
h(z, t)z=/ 2+ was used as a boundary condition on
equation D.7.
According to this equation, the filament's radius diverges as z
unphysical. In reality, the assumption that Ov/Oz = Wi
-±
L, which is
is not valid in the region
where the filament merges with the drop and where the accelerating and decelerating
portions of the filament merge.
boundary layers exist around z =
To enforce smoothness of the filament's surface,
and z = /2. On dimensional grounds, thickness
of these boundary layers is hmin, that is the smallest radius of the filament.
79
In absence of inertia, the tension inside the filament is uniform, so
T(z) = h2 (C'+
3p(j)Ov/&z).
(D.9)
This tension is uniform in the filament, but might change in time. As mentioned
above, at one place inside the filament Bv/&z = 0, and the tension there is
T(z) = h 2 (Ck')
assuming that h' <
=
h2
1
(D.10)
+ h" )
1 due to slenderness of the filament.
At the same time, the
tension at the center of the filament, where h' =h"= 0, is
T(z) = h 2
(D.11)
+ 3p( )
Hence, we get
+ 3p( )
h2
=
h2
+ h")
(D.12)
(D.13)
under the assumption that h is the same at both points. Therefore, the stress at the
center of the filament is equal to the second derivative at the point where &v/&z = 0.
To estimate h" at this point, we compare the shapes of the filament in the regions
where av/&z > 0 and Ov/&z < 0. The second spatial derivative of h at this point is
approximately
h"
h~
h'|1 /2+-
-
h/
h
h
0
1
(D.14)
and therefore
3p( ) = (-'
80
(D. 15)
Hence, the viscous stress inside the filament is approximately equal to (3 )-1.
However, the reasoning outlined above is based on the assumption that the filament's
shape is defined by the limitation on the deformation rate caused by the thickening
response. This response sets in when the viscous stress reaches Wi-1, and therefore
the above analysis implicitly assumes that &- > Wi-'.
However, the above analysis also is independent of Wi, and only assumes that the
velocity profile is triangle-shaped and that the transition region between the regions
&v/8z < 0 and Ov/8z > 0 is of width of order h. Therefore, filaments with Wi < --'
never reach this state, as that would imply lowering the stress below Wi-1, leading
out of thickening, which is inconsistent.
Therefore, filaments with Wi
1
> (-
reach the viscous stress of Wi-1 and this
is the plateau value of stress at which they remain. Filaments with Wi-1 < (-', on
the other hand, reach thickening when stress reaches Wi- 1 , but then as the velocity
profile continues to develop, the viscous stress continues to increase until it reaches
(',
where it plateaus.
This explains the results of the simulations: for values of Wi-' < (-', the maximum viscous stress attained depends only on ( and is equal to (-' due to geometrical constraints associated with the limitation on the deformation rate. When
Wi- 1 > (-1,
the maximum viscous stress attained depends only on Wi and is equal
to Wi- . This explains the collapse of various numerical experiments on the master
curve in Figure 4.2.
81
82
Bibliography
[1] H. A. Barnes. Shear-thickening ("dilatancy") in suspensions of nonaggregating
solid particles dispersed in Newtonian liquids. Journal Of Rheology, 1989.
[2] Osman A. Basaran and Ozgur E. Yildirim. Deformation and breakup of stretching bridges of Newtonian and shear-thinning liquids: comparison of one-and
two-dimensional models. Chemical Engineering Science, 2001.
[3]
Pradeep P. Bhat, Santosh Appathurai, Michael T. Harris, Matteo Pasquali,
Gareth H. McKinley, and Osman A. Basaran. Formation of beads-on-a-string
structures during break-up of viscoelastic filaments. Nature Physics, 6(8):625631, June 2010.
[4]
Erica E. Bischoff White, Manoj Chellamuthu, and Jonathan P Rothstein. Extensional rheology of a shear-thickening cornstarch and water suspension. Rheologica
Acta, 49(2):119-129, December 2009.
[5]
Eric Brown, Nicole A Forman, Carlos S. Orellana, Hanjun Zhang, Benjamin W.
Maynor, Douglas E. Betts, Joseph M. DeSimone, and Heinrich M. Jaeger. Generality of shear thickening in dense suspensions. Nature Materials, 9(3):220-224,
January 2010.
[6] Eric Brown and Heinrich M. Jaeger. Dynamic Jamming Point for Shear Thickening Suspensions. Physical Review Letters, 103(8):086001, August 2009.
[7] Eric Brown, Alice Nasto, Athanasios G. Athanassiadis, and Heinrich M. Jaeger.
83
Strain Stiffening in Random Packings of Entangled Granular Chains. Physical
Review Letters, 108(10), March 2012.
[8] Christian Clasen, Jens Eggers, Marco A. Fontelos, Jie Li, and Gareth H. McKinley. The beads-on-string structure of viscoelastic threads. Journal of Fluid Mechanics, 556:283, May 2006.
[9] R. G. Cox. The motion of long slender bodies in a viscous fluid part 1. general
theory. Journal of Fluid Mechanics, 44(04):791-810, 1970.
[10] Jens Eggers. Nonlinear dynamics and breakup of free-surface flows. Reviews of
modern physics, 1997.
[11] Jens Eggers and Todd F. Dupont. Drop Formation in a One-Dimensional Approximation of the Navier-Stokes Equation. Journalof Fluid Mechanics, 262:205-221,
1994.
[12] Jens Eggers and Emmanuel Villermaux.
Physics of liquid jets.
Reports on
Progress in Physics, 71(3):036601, February 2008.
[13] V. M. Entov and E. J. Hinch. Effect of a spectrum of relaxation times on the
capillary thinning of a filament of elastic liquid. Journal Of Non-Newtonian Fluid
Mechanics, 72(1):31-53, September 1997.
[14] Abdoulaye Fall, N. Huang, Frangois Bertrand, Guillaume Ovarlez, and Daniel
Bonn. Shear Thickening of Cornstarch Suspensions as a Reentrant Jamming
Transition. Physical Review Letters, 100(1), January 2008.
[15] S FORDHAM.
On the Calculation of Surface Tension From Measurements
of Pendant Drops.
Proceedings of the Royal Society of London Series a-
Mathematical and Physical Sciences, 194(1036):1-16, 1948.
[16] S Grilli, Sara Coppola, Veronica Vespini, Francesco Merola, Andrea Finizio, and
84
Pietro Ferraro.
3D lithography by rapid curing of the liquid instabilities at
nanoscale. Proceedings of the National Academy of Sciences, 2011.
[17] Young Sil Lee and Norman J. Wagner. Dynamic properties of shear thickening
colloidal suspensions. Rheologica Acta, 42(3):199-208, 2003.
[18] Bin Liu, Michael Shelley, and Jun Zhang. Focused Force Transmission through
an Aqueous Suspension of Granules. Physical Review Letters, 105(18):188301,
October 2010.
[19] Gareth H. McKinley and Tamarapu Sridhar. Filament-stretching rheometry of
complex fluids. Annual Review Of Fluid Mechanics, 2002.
[20] Florian S. Merkt, Robert D. Deegan, Daniel I. Goldman, Erin C. Rericha,
and Harry L. Swinney. Persistent Holes in a Fluid. Physical Review Letters,
92(18):184501, May 2004.
[21] M6nica S N Oliveira, Roger Yeh, and Gareth H. McKinley. Iterated stretching,
extensional rheology and formation of beads-on-a-string structures in polymer
solutions. Journal Of Non-Newtonian Fluid Mechanics, 137(1-3):137-148, August 2006.
[22] S. Pennathur, C. D. Meinhart, and H. T. Soh. How to exploit the features of
microfluidics technology. Lab On A Chip, 8(1):20, 2007.
[23] Matthieu Roche, Hamid Kellay, and Howard A. Stone.
Heterogeneity and
the Role of Normal Stresses during the Extensional Thinning of Non-Brownian
Shear-Thickening Fluids. Physical Review Letters, 107(13), September 2011.
[24] R. Sattler, C. Wagner, and Jens Eggers. Blistering Pattern and Formation of
Nanofibers in Capillary Thinning of Polymer Solutions. Physical Review Letters,
100(16), April 2008.
85
[25] M. I. Smith, R. Besseling, Michael E. Cates, and V. Bertola. Dilatancy in the
flow and fracture of stretched colloidal suspensions. Nature Communications,
1(8):114-115, January 2010.
[26] Howard A. Stone and John R. Lister. Capillary breakup of a viscous thread
surrounded by another viscous fluid. Physics Of Fluids, 1998.
[27] M. Tjahjadi, Howard A. Stone, and J. M. Ottino. Satellite and Subsatellite
Formation in Capillary Breakup. Journalof Fluid Mechanics, 243:297-317, 1992.
[28] S. Tomotika. On the instability of a cylindrical thread of a viscous liquid surrounded by another viscous fluid. Proceedings of the Royal Society of London
Series a-Mathematical and Physical Sciences, 150(870):322-337, 1935.
[29] Dominic Vella and L. Mahadevan. The "Cheerios effect". American Journal Of
Physics, 73(9):817-825, 2005.
86