202 - University of Minnesota



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 5μs. 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 μs. The movies may be viewed at

/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 of Digital Engineering, Numazu College of Technology, Ooka 3600, Numazu, Shizuoka, Japan

2Department 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

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. k 6

4 Maximum growth rates and wavenumbers, _m and km vs. pJ 11

5 _

mI vs. km for IPF 16

6 Conclusions and discussion 16

Abstract

Capillary instability of a viscous fluid cylinder of diameter

D surrounded by another fluid is determined by a

Reynolds number J = V D_`=_`, a viscosity ratio m =

_a=_` and a density ratio ` = _a=_`. Here V = =_`

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 J is large with reasonable agreement between

viscous potential and fully viscous flow with J > O(10).

The convergence results apply to two liquids as well as to

liquid and gas.

1 Introduction

Capillary instability of a liquid cylinder of mean radius R

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 R 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 2_R. Rayleigh showed that the effect of inertia

is such that the wavelength _ corresponding to the

mode of maximum instability is _ = 4:51 _ 2R; 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

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.

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