Under consideration for publication in J. Fluid Mech. 1

The motion of a spherical gas bubble inviscous potential flow

By D. D. JOSEPH1, J. WANG1 AND T. FUNADA21Department of Aerospace Engineering and Mechanics, University of Minnesota, 110 Union St.

SE, Minneapolis, MN 55455, USA2Department of Digital Engineering, Numazu College of Technology, 3600 Ooka, Numazu,

Shizuoka, 410-8501, Japan

(Received Nov 10, 2004)

A spherical gas bubble accelerates to steady motion in an irrotational flow of a viscousliquid induced by a balance of the acceleration of the added mass of the liquid withthe Levich drag. The equation of rectilinear motion is linear and may be integratedgiving rise to exponential decay with a decay constant 18t/a2 where is the kinematicviscosity of the liquid and a is the bubble radius. The problem of decay to rest of a bubblemoving initially when the forces maintaining motion are inactivated and the accelerationof a bubble initially at rest to terminal velocity are considered. The equation of motionfollows from the assumption that the motion of the viscous liquid is irrotational. It is anelementary example of how potential flows can be used to study the unsteady motions ofa viscous liquid suitable for the instruction of undergraduate students. Another example,considered here, is the purely radial irrotational motion of a viscous liquid associatedwith the motions of a spherical gas bubble. This gives rise to an exact potential flowsolution of the Navier-Stokes equations in which the jump of the viscous component ofthe normal stress is evaluated on the potential flow. The equation of motion for theliquid is almost always called the Reyleigh-Plesset equation but the viscous terms wereintroduced by Poritsky (1951) and not by Plesset (1949). We show that when the normalstress equation is converted into energy equation in the conventional way used for inviscidfluid, the viscous normal stress term is converted into the viscous dissipation in the liquidevaluated on potential flow.

We consider a body moving with the velocity U in an unbounded viscous potentialflow. Let M be the mass of the body and M be the added mass, then the total kineticenergy of the fluid and body is

T =12(M +M )U2. (1)

Let D be the drag and F be the external force in the direction of motion, then the powerof D and F should be equal to the rate of the total kinetic energy,

(F +D)U =dTdt

= (M +M )UdUdt

. (2)

We next consider a spherical gas bubble, for which M = 0 and M =23pia3f . The drag

can be obtained by direct integration using the irrotational viscous normal stress and aviscous pressure correction: D = 12piaU . Suppose the external force just balances thedrag, then the bubble moves with a constant velocity U = U0. Imagine that the external

2 D. D. Joseph, J. Wang and T. Funada

force suddenly disappears, then (2) gives rise to

12piaU = 23pia3f

dUdt

. (3)

The solution is

U = U0e18a2

t, (4)

which shows that the velocity of the bubble approaches zero exponentially.

If gravity is considered, then F =43pia3fg. Suppose the bubble is at rest at t = 0 and

starts to move due to the buoyant force. Equation (2) can be written as

43pia3fg 12piaU = 23pia

3fdUdt

. (5)

The solution is

U =a2g

9

(1 e 18a2 t

), (6)

which indicates the bubble velocity approaches the steady state velocity

U =a2g

9(7)

exponentially.Another way to obtain the equation of motion is to argue following Lamb (1932) and

Levich (1949) that the work done by the external force F is equal to the rate of the totalkinetic energy and the dissipation:

FU = (M +M )UdUdt

+D. (8)

Since D = DU , (8) is the same as (2).Equations (8) or (2) can be used to consider the rectilinear motion of a bubble of

non-spherical shape, e.g. the oblate ellipsoidal bubble considered by Moore (1965) andJoseph & Wang (2004). The added mass M and the expression for the drag need to bechanged. The relation D = DU still holds and the drag depends linearly on U , thusthe equation of motion is still a linear equation of U and the solutions would be similarto (3) and (6).It is of interest to see how the viscosity alters the analysis of the force on a bubble in

translational motion. Batchelor (1967) has presented an analysis of the force on a body intranslational motion in an inviscid fluid. The same analysis applies to a bubble when itsshape is given. He writes (p. 404) We consider the total force F exerted instantaneouslyby the surrounding fluid on a body moving without rotation. This force arises from thepressure at the body surface, and with the aid of (6.2.5) we have

F = pndA =

tndA+

12

q2ndA

g xndA, (6.4.20)

the integrals being taken over the fixed surface A that coincides instantaneously withthe body surface. Batchelor (1967) considers the motion of a sphere of mass M movingwith velocity U through infinite fluid under the action of an applied force X (p. 453).He writes the equation of motion as

MU = X 12M0U+Mg M0g, (6.8.19)

The motion of a spherical gas bubble in viscous potential flow 3

where M0 =43pia3f is the mass of fluid displaced by the sphere. In the case of a gas

sphere moving under gravity alone, the above equation is equivalent to (5) without theviscous drag term. Batchelor (1967) gives the acceleration of the gas sphere U 2g andwrote Thus a spherical gas bubble moves from rest in water with an upward accelerationof 2g, and, since in this case boundary-layer separation seems not to occur (in a liquid freefrom impurities), continues to have this acceleration until either the bubble is deformedor the velocity becomes comparable with the terminal velocity considered in 5.14.The terminal velocity referred to here is obtained by equating the buoyancy force to theviscous drag 12piaU obtained from the dissipation calculation. Batchelor (1967) givesthe terminal velocity

a2g

9in his (5.14.12), which is the same as our (7). Without the

viscous drag in the equation of motion (6.8.19), Batchelor (1967) cannot show how thegas sphere approaches the terminal velocity.If the motion of the bubble irrotational and the liquid viscous, the irrotational shear

stress vanishes and a viscous contribution to the pressure pv is induced in a thin boundarylayer. In this case, drag terms arising from the normal stress

(pv + n)ndA (9)where

n = n () n (10)and the drag component of pv is known (see Joseph & Wang 2004) for spherical andoblate ellipsoidal bubbles. Following then the analysis given by Batchelor through to theacceleration reaction on page 407, we find equations of bubble motion like (3) and (5) inwhich the retarding effect of viscosity on the irrotational motion of the bubble is madeexplicit.The motion of a single spherical gas bubble in a viscous liquid has been considered by

some authors. Typically, these authors assemble terms arising in various situations, likeStokes flow (Hadamard-Rybczynski drag, Basset memory integral) and high Reynoldsnumber flow (Levich drag, boundary layer drag, induced mass) and other terms into asingle equation. Such general equations have been presented by Yang & Leal (1991) andby Park, Klausner and Mei (1995) and they have been discussed in the review paper ofMagnaudet & Eams (2000, see their section 4). Yang & Leals equation has Stokes dragand no Levich drag. Our equation is not embedded in their equation. Park et al. listedfive terms for the force on a gas bubble; our equation may be obtained from theirs if thefree stream velocity U is put to zero, the memory term is dropped, and the boundarylayer contribution to the drag given by Moore is neglected. Park et al. did not write downthe same equation as our equation (1) and did not obtain the exponential decay.It is generally believed that the added mass contribution, derived for potential flow

is independent of viscosity. Magnaudet and Eames say that ... results all indicate thatthe added mass coefficient is independent of the Reynolds, strength of acceleration and... boundary conditions. This independence of added mass on viscosity follows from theassumption that the motion of viscous fluids can be irrotational. The results cited byMagnaudet & Eams seem to suggest that induced mass is also independent of vorticity.Chen (1974) did a boundary layer analysis of the impulsive motion of a spherical

gas bubble which shows that the Levich drag 48/Re at short times evolves to the drag48Re

(1 2.21

Re

)obtained in a boundary layer analysis by Moore (1963). The Moore drag

4 D. D. Joseph, J. Wang and T. Funada

cannot be distinguished from the Levich drag when Re is large. The boundary layercontribution is vortical and is neglected in our potential flow analysis.Another problem of irrotational motion of a spherical gas bubble in a viscous liquid is

the expanding or contracting gas bubble first studied by Rayleigh 1917. The problem isalso framed by Batchelor 1967 (p.479) but, as in Rayleighs work, with viscosity and sur-face tension neglected. Vicosity and surface tension effects can be readily introducedinto this problem without approximation because the motion is purely radial and irro-tational; shear stresses do not arise. Though Plesset 1949 introduced a variable externaldriving pressure and surface tension, the effects of surface tension were also introducedand the effects of viscosity were first introduced by Poritsky 1951. His understanding ofirrotational viscous stresses is exemplary, unique for his time and not usual even in ours.The equation

2R

= pb p RR 32 R2 4R

R(11)

for the bubble radius R(t), is always called the Rayleigh-Plesset equation but Plesset didnot present or discuss this equation which is given as [62] in the 1951 of paper of Poritsky.It is well known when and are neglected, that equation (11) can be formulated as anenergy equation

ddtKE = (pb p) V

where

KE =12

R

(

r

)24pir2dr

and

V =ddt

(43piR3

)The equation

(pb p) V = dKEdt +D + 2V

R(12)

where the dissipation

D = 2 R

2

xixj

2

xixj4pir2dr

= 16pi2RR2

follows from (11) after multiplication by V . In this problem we demonstrate a directconnection between the irrotational viscous normal stress and the dissipation integral Dcomputed on potential flow.

This work was supported in part by the NSF under grants from Chemical TransportSystems.

REFERENCES

Batchelor, G. K. 1967 Introduction to fluid dynamics, Cambridge University Press.Joseph, D. D. & Wang, J. 2004 The dissipation approximation and viscous potential flow, J.

Fluid Mech. 505, 365-377.Lamb, H. 1932 Hydrodynamics, 6th edn. Cambridge University Press. (Reprinted by Dover,

1945)

The motion of a spherical gas bubble in viscous potential flow 5

Levich, V.G. 1949 The motion of bubbles at high Reynolds numbers, Zh. Eksperim. & Teor.Fiz. 19, 18.

Moore, D.W. 1965 The velocity of rise of distorted gas bubbles in a liquid of small viscosity.J. Fluid Mech. 23, 749766.

Plesset, M. 1949 The dynamics of cavitation bubbles ASME J. Appl. Mech. 16, 228-231.Poritsky, M. 1951 The collapse on growth of a spherical bubble on cavity in a viscous fluid.

Proceedings of the first U.S National Congress of Applied Mechanics held at the IllinoisInstitute of Technology, June 11-16, 1951, 813-821, ASME.

Rayleigh, Lord 1917 On the pressure developed in a liquid during the collapse of a sphericalcavity. Phil. Mag. 34, 94-98.