202
Drag and Moment in Viscous Potential Flow
D. D. Joseph, T. Y. Liao and H. H. Hu
Department of Aerospace Engineering and Mechanics
University of Minnesota
Minneapolis, Minnesota 55455
May 1992
We consider solutions of the Navier-Stokes equations in which the velocity is
given by the gradient of a potential. We show that the drag on bodies and bubbles is the
same in viscous and inviscid potential flow. The lift on two-dimensional bodies is given
by the usual Kutta condition but the moment about the origin of the stresses acting on the
body is given by MI + 2 where is the viscosity, is the circulation and MI is the
usual moment for an inviscid fluid.
211
1
Viscous and Viscoelastic Potential Flow
Daniel D. Joseph and Terrence Y. Liao
Department of Aerospace Engineering and Mechanics
University of Minnesota, Minneapolis, MN 55455
January 12, 1993
Abstract
Potential flows of incompressible fluids admit a pressure (Bernoulli) equation
when the divergence of the stress is a gradient as in inviscid fluids, viscous fluids, linear
viscoelastic fluids and second-order fluids. We show that the equation balancing drag
and acceleration is the same for all these fluids independent of the viscosity or any
viscoelastic parameter and that the drag is zero in steady flow. The unsteady drag on
bubbles in a viscous (and possibly in a viscoelastic) fluid may be approximated by
evaluating the dissipation integral of the approximating potential flow because the
neglected dissipation in the vorticity layer at the traction-free boundary of the bubble gets
smaller as the Reynolds number is increased. Using the potential flow approximation,
the drag D on a spherical gas bubble of radius a rising with velocity U(t) in a linear
viscoelastic liquid of density and shear modules G(s) is given by
D 
2
3
a3 Ý U 12 a G(t )U( )d 

t
_
and in a second-order fluid by
D a 2
3
a2 12 1
_
_
_
_
Ý U 12 a U
where 1 < 0 is the coefficient of the first normal stress and is the viscosity of the fluid.
Because 1 is negative, we see from this formula that the unsteady normal stresses
oppose inertia; that is, oppose the acceleration reaction. When U(t) is slowly varying, the
two formulas coincide. For steady flow, we obtain D 12 a U for both viscous and
viscoelastic fluids. In the case where the dynamic contribution of the interior flow of the
bubble cannot be ignored as in the case of liquid bubbles, the dissipation method gives an
estimation of the rate of total kinetic energy of the flows instead of the drag. When the
dynamic effect of the interior flow is negligible but the density is important, this formula
for the rate of total kinetic energy leads to D ( a )VBg ex aVB
Ý U where a is the
density of the fluid (or air) inside the bubble and VB is the volume of the bubble.
Classical theorems of vorticity for potential flow of ideal fluids hold equally for
viscous and viscoelastic fluids. The drag and lift on two-dimensional bodies of arbitrary
cross section in viscoelastic potential flow are the same as in potential flow of an inviscid
fluid but the moment M in a linear viscoelastic fluid is given by
M MI 2 G(t ) ( ) 

t
_ d 
where MI is the inviscid moment and (t) is the circulation, and
M MI 2 2 1

t
in a second-order fluid. When (t) is slowly varying, the two formulas for M coincide.
For steady flow, they reduce to
M MI 2 
which is also the expression for M in both steady and unsteady potential flow of a viscous
fluid.
Potential flows of models of a viscoelastic fluid like Maxwell's are studied. These
models do not admit potential flows unless the curl of the divergence of the extra-stress
vanishes. This leads to an over-determined system of equations for the components of the
stress. Special potential flow solutions like uniform flow and simple extension satisfy
these extra conditions automatically but other special solutions like the potential vortex
can satisfy the equations for some models and not for others.
270
Breakup of a liquid drop suddenly exposed to a high-speed airstream
by
Daniel D. Joseph, J. Belanger & G.S. Beavers
University of Minnesota, Minneapolis, MN 55455
This paper is dedicated to Gad Hetsroni,
on the occasion of his 65th birthday,
to honor his many contributions
to the understanding of multiphase flows.
Abstract
The breakup of viscous and viscoelastic drops in the high speed airstream behind a shock
wave in a shock tube was photographed with a rotating drum camera giving one photograph
every 5s. From these photographs we created movies of the fragmentation history of viscous
drops of widely varying viscosity, and viscoelastic drops, at very high Weber and Reynolds
numbers. Drops of the order of one millimeter are reduced to droplet clouds and possibly to vapor
in times less than 500 s. The movies may be viewed at http://www.aem.umn.edu
/research/Aerodynamic_Breakup. They reveal sequences of breakup events which were
previously unavailable for study. Bag and bag-and-stamen breakup can be seen at very high
Weber numbers, in the regime of breakup previously called “catastrophic.” The movies allow us
to generate precise displacement-time graphs from which accurate values of acceleration (of
orders 104 to 105 times the acceleration of gravity) are computed. These large accelerations from
gas to liquid put the flattened drops at high risk to Rayleigh-Taylor instabilities. The most
BREAKUP OF A LIQUID DROP SUDDENLY EXPOSED TO A HIGH-SPEED AIRSTREAM
unstable Rayleigh-Taylor wave fits nearly perfectly with waves measured on enhanced images of
drops from the movies, but the effects of viscosity cannot be neglected. Other features of drop
breakup under extreme conditions, not treated here, are available on our Web site.
284
Accepted for publication in J. Fluid Mech.
1
Viscous potential flow analysis of Kelvin-Helmholtz
instability in a channel
By T. FUNADA1 AND D. D. JOSEPH2
1Department
2Department
of Digital Engineering, Numazu College of Technology, Ooka 3600, Numazu, Shizuoka, Japan
of Aerospace Engineering and Mechanics, University of Minnesota, Minneapolis, MN 55455, USA
(Received ?? and in revised form ??)
We study the stability of strati.ed gas-liquid .ow in a horizontal rectangular channel using viscous potential
.ow. The analysis leads to an explicit dispersion relation in which the e.ects of surface tension and viscosity
on
the normal stress are not neglected but the e.ect of shear stresses are neglected. Formulas for the growth
rates,
wave speeds and neutral stability curve are given in general and applied to experiments in air-water .ows.
The
e.ects of surface tension are always important and actually determine the stability limits for the cases in
which
the volume fraction of gas is not too small. The stability criterion for viscous potential .ow is expressed by
a critical value of the relative velocity. The maximum critical value is when the viscosity ratio is equal to
the
density ratio; surprisingly the neutral curve for this viscous .uid is the same as the neutral curve for inviscid
.uids. The maximum critical value of the velocity of all viscous .uids is given by inviscid .uids. For air at
20.C
and liquids with density ρ = 1 g/cm3 the liquid viscosity for the critical conditions is 15 cp; the critical
velocity
for liquids with viscosities larger than 15 cp are only slightly smaller but the critical velocity for liquids
with
viscosities smaller than 15 cp, like water, can be much lower. The viscosity of the liquid has a strong a.ect
on
the growth rate. The viscous potential .ow theory .ts the experimental data for air and water well when the
gas fraction is greater than about 70%.
294
Rayleigh-Taylor Instability
of Viscoelastic Drops at High Weber Numbers
D.D. Joseph*, G.S. Beavers*, T. Funada**
*University of Minnesota, Minneapolis, MN 55455
** Numazu College of Technology, Ooka 3600,
Numazu, Shizuoka, Japan 410-8501
Abstract
Movies of the breakup of viscous and viscoelastic drops in the high speed airstream behind a
shock wave in a shock tube have been reported by Joseph, Belanger and Beavers [1999]. A
Rayleigh-Taylor stability analysis for the initial breakup of a drop of Newtonian liquid was
presented in that paper. The movies, which may be viewed at http://www.aem.umn.edu/
research/Aerodynamic_Breakup, show that for the conditions under which the experiments
were carried out the drops were subjected to initial accelerations of orders 104 to 105 times the
acceleration of gravity. In the Newtonian analysis of Joseph, Belanger and Beavers the most
unstable Rayleigh-Taylor wave fits nearly perfectly with waves measured on enhanced images of
drops from the movies, but the effects of viscosity cannot be neglected. Here we construct a
Rayleigh-Taylor stability analysis for an Oldroyd B fluid using measured data for acceleration,
density, viscosity and relaxation time 1. The most unstable wave is a sensitive function of the
retardation time 2 which fits experiments when 2/1= O(10-3). The growth rates for the most
unstable wave are much larger than for the comparable viscous drop, which agrees with the
surprising fact that the breakup times for viscoelastic drops are shorter. We construct an
approximate analysis of Rayleigh-Taylor instability based on viscoelastic potential flow which
gives rise to nearly the same dispersion relation as the unapproximated analysis.
300
Capillary/C-InstabSh03-27.tex 1
Viscous Potential Flow Analysis
of Capillary Instability
T. Funada and D.D. Joseph
University of Minnesota
Aug 2001
Draft printed March 28, 2002
This paper is dedicated to Klaus Kirchg¨assner on the occasion of his 70th birthday.
Contents
1 Introduction 1
2 Governing equations and dimensionless parameters 3
2.1 Linearized disturbance equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.2 Dispersion relation for fully viscous flow (FVF) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.3 More viscous fluid outside . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.4 Dispersion relation for viscous potential flow (VPF) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
3 Growth rate curves, vs. 6
4 Maximum growth rates and wavenumbers, and vs. 11
5
vs. for IPF 16
6 Conclusions and discussion 16
_
k
_m
km
pJ
_
mI
km
Abstract
Capillary instability of a viscous fluid cylinder of diameter
surrounded by another fluid is determined by a
Reynolds number , a viscosity ratio
and a density ratio . Here
is the capillary collapse velocity based on the more viscous
liquid which may be inside or outside the fluid cylinder.
Results of linearized analysis based on potential flow
of a viscous and inviscid fluid are compared with the
unapproximated normal mode analysis of the linearized
Navier-Stokes equations. The growth rates for the inviscid
fluid are largest, the growth rates of the fully viscous problem
are smallest and those of viscous potential flow are between.
We find that the results from all three theories converge
when is large with reasonable agreement between
viscous potential and fully viscous flow with .
The convergence results apply to two liquids as well as to
D
J = V D_`= _`
_a= _`
` = _a=_`
m=
V = = _`
J
J > O( 10)
liquid and gas.
1 Introduction
Capillary instability of a liquid cylinder of mean radius
leading to capillary collapse can be described as a neckdown
due to surface tension in which fluid is ejected
from the throat of the neck, leading to a smaller neck and
greater neckdown capillary force as seen in the diagram in
figure 1.1.
The dynamical theory of instability of a long cylindrical
column of liquid of radius under the action of capillary
force was given by Rayleigh (1879) following earlier
work by Plateau (1873) who showed that a long cylinder
of liquid is unstable to disturbances with wavelengths
greater than . Rayleigh showed that the effect of inertia
is such that the wavelength corresponding to the
mode of maximum instability is exceeding
very considerably the circumference of the cylinder.
The idea that the wave length associated with fastest growing
growth rate would become dominant and be observed
in practice was first put forward by Rayleigh (1879). The
analysis of Rayleigh is based on potential flow of an inviscid
R
R
2_R
_
_ = 4:51 _ 2R;
305
J. Fluid Mech. (2004), vol. 505, pp. 365–377. c _ 2004 Cambridge University Press
DOI: 10.1017/S0022112004008602 Printed in the United Kingdom
365
The dissipation approximation and viscous
potential .ow
By D. D. JOSEPH AND J. WANG
Department of Aerospace Engineering and Mechanics, University of Minnesota,
Minneapolis, MN 55455, USA
(Received 18 June 2003 and in revised form 1 February 2004)
Dissipation approximations have been used to calculate the drag on bubbles and
drops and the decay rate of free gravity waves on water. In these approximations,
viscous e.ects are calculated by evaluating the viscous stresses on irrotational .ows.
The pressure is not involved in the dissipation integral, but it enters into the power of
traction integral, which equals the dissipation. A viscous correction of the irrotational
pressure is needed to resolve the discrepancy between the zero-shear-stress boundary
condition at a free surface and the non-zero irrotational shear stress. Here we show
that the power of the pressure correction is equal to the power of the irrotational
shear stress. The viscous pressure correction on the interface can be expressed by a
harmonic series. The principal mode of this series is matched to the velocity potential
and its coe.cient is explicitly determined. The other modes do not enter into the
expression for the drag on bubbles and drops. They vanish in the case of free gravity
waves.
315
J. Fluid Mech. (2003), vol. 479, pp. 191–197. c _ 2003 Cambridge University Press
DOI: 10.1017/S0022112002003634 Printed in the United Kingdom
191
Viscous potential .ow
By D. D. JOSEPH
Department of Aerospace Engineering and Mechanics, University of Minnesota, MN 55455, USA
joseph@aem.umn.edu
(Received 6 September 2002 and in revised form 10 December 2002)
Potential .ows u = 佫φ are solutions of the Navier–Stokes equations for viscous
incompressible .uids for which the vorticity is identically zero. The viscous term
μ佫2u = μ佫佫2φ vanishes, but the viscous contribution to the stress in an incompressible
.uid (Stokes 1850) does not vanish in general. Here, we show how the viscosity
of a viscous .uid in potential .ow away from the boundary layers enters Prandtl’s
boundary layer equations. Potential .ow equations for viscous compressible .uids are
derived for sound waves which perturb the Navier–Stokes equations linearized on
a state of rest. These linearized equations support a potential .ow with the novel
features that the Bernoulli equation and the potential as well as the stress depend on
the viscosity. The e.ect of viscosity is to produce decay in time of spatially periodic
waves or decay and growth in space of time-periodic waves.
In all cases in which potential .ows satisfy the Navier–Stokes equations, which
includes all potential .ows of incompressible .uids as well as potential .ows in the
acoustic approximation derived here, it is neither necessary nor useful to put the
viscosity to zero.
319
J. Fluid Mech. (2003), vol. 488, pp. 213–223. c _ 2003 Cambridge University Press
DOI: 10.1017/S0022112003004968 Printed in the United Kingdom
213
Rise velocity of a spherical cap bubble
By DANIEL D. JOSEPH
University of Minnesota, Aerospace Engineering and Mechanics, 110 Union St. SE, Minneapolis,
MN 55455, USA
(Received 23 October 2002 and in revised form 26 February 2003)
The theory of viscous potential .ow is applied to the problem of .nding the rise
velocity U of a spherical cap bubble (see Davies & Taylor 1950; Batchelor 1967). The
rise velocity is given by
U
√gD
=.
8
3
ν(1 + 8s)
_gD3
+
√2
3 _1 . 2s .
16sσ
ρgD2 +
32v2
gD3 (1 + 8s)2_1/2
,
where R = D/2 is the radius of the cap, ρ and ν are the density and kinematic
viscosity of the liquid, σ is surface tension, r(θ) = R(1 + sθ2) and s = r__(0)/D is
the deviation of the free surface from perfect sphericity r(θ) = R near the stagnation
point θ = 0. The bubble nose is more pointed when s < 0 and blunted when s > 0. A
more pointed bubble increases the rise velocity; the blunter bubble rises slower. The
Davies & Taylor (1950) result arises when s and ν vanish; if s alone is zero,
U
√gD
=.
8
3
ν
_gD3
+
√2
3 _1 +
32ν2
gD3 _1/2
,
showing that viscosity slows the rise velocity. This equation gives rise to a hyperbolic
drag law
CD = 6+32/Re,
which agrees with data on the rise velocity of spherical cap bubbles given by Bhaga
& Weber (1981).
324
J. Fluid Mech. (2004), vol. 511, pp. 201–215. c _ 2004 Cambridge University Press
DOI: 10.1017/S0022112004009541 Printed in the United Kingdom
201
Potential .ow of a second-order .uid over
a sphere or an ellipse
By J. WANG AND D. D. JOSEPH
Department of Aerospace Engineering and Mechanics, University of Minnesota,
Minneapolis, MN 55455, USA
(Received 9 May 2003 and in revised form 10 March 2004)
We study the potential .ow of a second-order .uid over a sphere or an ellipse. The
normal stress at the surface of the body is calculated and has contributions from the
inertia, viscous and viscoelastic e.ects. We investigate the e.ects of Reynolds number
and body size on the normal stress; for the ellipse, various angles of attack and
aspect ratios are also studied. The e.ect of the viscoelastic terms is opposite to that
of inertia; the normal stress at a point of stagnation can change from compression to
tension. This causes long bodies to turn into the stream and causes spherical bodies
to chain. For a rising gas bubble, the e.ect of the viscoelastic and viscous terms in
the normal stress is to extend the rear end so that it tends to the cusped trailing edge
observed in experiments.