**by RE CAFLISCH · Cited by 334 — We derive a system of effective equations for wave propagation in a bubbly liquid. Starting from a microscopic description, we obtain the effective equations by **

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J. FZud Mech. (1985), voZ. 153, pp. 25S273 Printed in Great Britain 259 Effective equations for wave propagation in bubbly liquids By RUSSEL E. CAFLISCH, MICHAEL J. MIKSISt, GEORGE C. PAPANICOLAOU AND LU TING Courant Institute of Mathematical Sciences, New York University, New York 10012 (Received 16 November 1983 and in revised form 19 October 1984) We derive a system of effective equations for wave propagation in a bubbly liquid. Starting from a microscopic description, we obtain the effective equations by using Foldy™s approximation in a nonlinear setting. We discuss in detail the range of validity of the effective equations as well as some of their properties. 1. Introduction Wave propagation in a liquid containing gas bubbles is a complex phenomenon that has been studied theoretically both in the linear small-amplitude regime and in the weakly nonlinear regime, including effects of temperature, surface tension, viscosity, etc. (d™Agostino & Brennen 1983; Batchelor 1969; Drew & Cheng 1982; Drumheller & Bedford 1979; Hsieh 1982; Van Wijngaarden 1968,1972; Wallis 1969; and additional references therein). A good deal of the theory of waves in a bubbly liquid can be deduced from a set of nonlinear differential equations that were proposed by Van Wijngaarden (1968, 1972). These equations were written down on the basis of physical reasoning. It is not clear how they arise from the equations that describe the microscopic motion of the liquid and the gas bubbles. The purpose of this paper is to show that the equations of Van Wijngaarden can be obtained from the microscopic equations in a specific asymptotic limit that we describe in detail. From this analysis one gets a clear idea of the range of validity of Van Wijngaarden™s equations. Let us review briefly Van Wijngaarden™s equations. The macroscopic state of the gas-bubbleliquid mixture is described by its density p(t, x), pressure p(t, x), velocity u(t, x), gas volume fraction B(t, x) and bubble radius R(t, x) for some time t > 0 and x in three-dimensional space RS. The bubble radius field R(t,x) is a continuum variable and specifies some average bubble radius for bubbles in the neighbourhood of a point x. The equations of Van Wijngaarden (1972) are pt+V*(pu) = 0, (1.1) p(u,+u.Vu)+Vp = 0, (1 *2) P = PA1 -11, (1.3) M. = constant, 1-B t Present address: Department of Mathematics, Duke University, Durham, North Carolina, 27706.

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260 R. E. CaJEisch, M. J. Miksis, 0. C. Papanicolaou and L. Ting Initial conditions are given, for example, for p, u, p, R and R,. The volume fraction /3 and the gas pressure pg are determined by (1.4) and (1.5) once the constants on the right are prescribed. The liquid density pc is related to p by an equation of state for the liquid, or is simply taken as constant if the compressibility of the liquid is negligible. We find in $524 that (l.lF(1.6) (in the somewhat simpified form ((5.1)-(5.4) given below) can be derived by an adaptation of Foldy™s method (Foldy 1945; Carstensen 6 Foldy 1947). The essential point in this method is to argue that the pressure and velocity fields felt by each bubble are the macroscopic ones; each bubble does not feel the local fields of the other bubbles. Obviously this requires small gas-bubble volume fraction. Let n be the number of bubbles per unit volume, R, a typical bubble radius and h a typical wavelength of a disturbance in the mixture, with h 9 R,. Then we will show that (1.1 )-( 1.6) are valid if nh2Ro is of order one. Note that the volume fraction $rR,3n = $(R,/A)2 (nA2Ro) is then small. It is useful to note why (1.1)-( 1.6) are reasonably good for sound propagation in a bubbly liquid (Van Wijngaarden 1972). Equations (1.1) and (1.2) are the usual conservation laws for mass and momentum of the mixture. Equation (1.3) defines the macroscopic density as the liquid density pc times the liquid volume fraction. The term pg,!3 could be added to account for the mass density of the gas, but the ratio pg/pc is negligibly small for typical values of pg and pc. Equation (1 -4) is a consequence of the assumption that the mass of the gas per unit mass of the liquid is the constant pg/3/pc( 1 -p). This is valid when the gas and the liquid move with the same velocity. The isothermal equation of state in the gas (1.5) can be used to eliminate pg in the mass ratio, and this gives (1.4). Equation (1.6) is Rayleigh™s equation (Plesset & Prosperetti 1977 ; Keller & Miksis 1980 ; Prosperetti 1983) for radial bubble oscillations of a single bubble, with p being the pressure far away from the bubble. Note that in the macroscopic description the bubble radius is a field variable R(t,x), but (1.6) involves only time derivatives. In $4 we explain how (1.6) arises from a Foldy approximation in the continuum limit. The presence of (1.6) in the system (1.1 )-( 1.6) indicates that typical interbubble distances must be large compared with typical bubble radii, i.e. small gas-bubble volume fraction. The plan of this paper is as follows. In $2 we introduce the microscopic equations of motion, including a description of the bubble geometry, and a number of assumptions about the physical conditions. The appropriate scaling of the microscopic equation is introduced in $3, and under this scaling the non-dimensionalized macroscopic equations (4.1)-(4.4) are derived in $4. In dimensional form the equations are (5.1)-(5.4). The effective sound speed, the resonant frequency and an energy function for (4.1)-(4.4) are derived and analysed in $5. In $6 the addition of surface tension, viscosity and heat conduction is outlined. 2. The microscopic problem We consider wave propagation through a liquid with gas bubbles dispersed in it, for example water with air bubbles. Let p, u, p denote the fluid density, velocity and pressure. Whenever necessary the subscript G or g will be added to denote the property of liquid or gas respectively. We make a number of assumptions about the physical characteristics of the fluid motion. First, since we are interested in wave propagation rather than in bulk motion, we assume that the bubble centres do not move. Secondly, we assume that the bubbles are spherical with a uniform internal pressure distribution. The first assumption is

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Effective equations for wave propagation in bubbly liquids 261 consistent with the scaling in $3. The drift velocity of the bubble centre is of the order of the velocity far away from the bubble (as in (3.7)) and of higher order than the velocity of the bubble surface (as in (3.12)). The pressure will be uniform because the inertia of the gas is negligible. The sphericity assumption is self-consistent with the approximate solution found in $4, since in that solution the wavelength is much larger than the bubble radius and the bubble feels only a uniform pressure fluctuation. The sphericity could also be justified on the basis of surface tension. However, as a third simplifying assumption we do not explicitly include surface tension, viscosity or heat conduction. This assumption is removed in $6. The fourth assumption is that the liquid is nearly incompressible with constant density and sound speed and that the flow is irrotational. Scaling assumptions will be introduced at the end of this section and in $3. Suppose there are N bubbles with centres x,, . . ., xN and radii R,(t), . . . , RN(t) in the region Q. The corresponding equations of motion in the liquid region {x:lx-x,l > Rj for all j} are 1 – (pt+u*Vp)+V.u = 0, Pc C? P,(Ut + U*VU) + vp = 0, (2.2) with p, and c, taken to be constant and with V x u = 0. The boundary conditions on the bubble surfaces { I x – xj I = Rj} are continuity of pressure and normal velocity, i.e. a Rj = u.A, P = P, (2.4) at I x – x, I = Rj for j = 1, . . . , N, in which R is the normal to the bubble surface. The equation af state for the gas in the jth bubble is xy, j=l, , N, pg = ($nRJ in which MI is the mass of the jth bubble (which is constant in time) and the parenthetical term in (2.5) is the gas density. Equations (2.1)-(2.5) should be complemented by initial conditions for p, u and R,, as well as specification of the constantsp,, c,, xj, M,, K, y. We are interested in the limit of an infinite number of bubbles; so we make one further assumption that the bubble configuration tends to a continuum. This is formulated as follows. For each N let the bubble-centre configuration be {xy, . . . , xz} and define @'(A) number of points xy in a set A (2.6) — – N N Then there is a function O(x), the continuum bubble-centre density, which is positive in Q and zero outside, such that as N tends to infinity I P

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262 R. E. Cajisch, M. J. Miksis, G. C. Papaniwlaou and L. Ting for all subsets A in space. Another equivalent but more useful way of stating (2.7) is that as N tends to infinity for each continuous function 4. 3. Scaling Let A denote the wavelength of a disturbance propagating in the bubbly liquid, let V be the volume of the region Q containing the bubbles and let R, be a typical bubble radius. The dimensionless inter-bubble-centre distance E, dimensionless bubble radius S and gas volume fraction b are defined by An additional parameter, which enters in $4, is The first scaling assumption is that E and S are very small and that x is of order one (relative to E and 6). Then (3.4) implies that s = 0(•3) ; (3.5) this in turn implies that the volume fraction t? is very small, i.e. /3 = O(S2). The density will be scaled relative to the liquid density pc. The pressure will be scaled relative to some reference equilibrium pressure p,, such as the atmospheric pressure. The typical bubble mass M,, is related to p, and R, by We scale velocities relative to a reference speed E, which should be thought of as the effective sound speed of the bubbly liquid. Its value will be derived a posteriori in $5. Velocities are taken to be small compared with E, so that convective effects are of higher order. All this leads to the following scaling in which the dimensionless variables are primed : p =popf, I( = E6W = R, fad, 8 = V-W. J Here f is a reference frequency such that Af = E. The function 8′ is the ratio of the local number density of bubbles (N6) to the global number density of bubbles (N/ V), and is one for a uniform mixture. The definitions (3.7) implicitly assert a second scaling assumption of this paper: the dimensionless variables in (3.7) are assumed to have order-one magnitude in the

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EfSective equations for wave propagation in bubbly liquids 263 continuum limit of small 6. However, the scaling is non-uniform, since u™ turns out to be of order 8-™ in the neighbourhood of the bubble surfaces (cf. (3.12)). This boundary-layer behaviour cannot be anticipated by the general scaling (3.7). Next define dimensionless parameters f and C by The parameters c and C may be quite large in practice. However, as a further scaling assumption we require f and C to be of order one relative to 8. Using (3.7) we obtain the following dimensionless form of the microscopic equations (2.1)-(2.6). In the region {x™: Ix™-xiNI > RiN for allj} we have ~c-yp;.+SaU‚-V‚p‚)+V‚*u‚ = 0, (3.10) u;*+~u™*vfu™+g7™p™ = 0. (3.11) On the jth bubble surface I x™ – xiN I = RiN there are boundary conditions (3.12) a at‚ 1 ™ u™-A = 8-1 – R™N p™ = FY(RiN), and the gas pressure and bubble configuration are described by (3.13) (3.14) (number of xiN in a set A™)+ (3.15) v1 A3 N — These equations are complemented by initial conditions p™(0,x™) = @(x™), U™(0,X™) = ii(x™), RiN(0) = a(x;N), R;N(O) = R,,(X;N), M;N = B(XjN), (3.16) (3.17) (3.18) in which V x ii = 0 and and & are smooth functions defined in Sa. Equation (3.18) and V x u™ = 0 hold for all time. The initial data @, G are for the ambient fluid away from the bubbles. The actual spatial dependence of the initial data will be assumed to be consistent with the expansion described in $4. In particular initially u™ must satisfy (3.12) with radial velocity given by (3.17). Following (2.8), (3.15) may be written as (3.19) for any continuous q5. Before continuing, we make several remarks about the parameter 6. Using the expressions (5.9) and (5.11) for the effective sound speed E and resonant frequency wo, which will be derived in $5, we find that <=-(A) 4na w a 9 p= ( I--- 4n p)-™ = [ 1-2- (y-l. 3Y w 3y •3 na3 wa (3.20) (3.21)
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264 R. E. Caflisch, M. J. Mikeis, G. C. Papanicolaou and L. Ting Thus 5 is essentially the square of the ratio of the resonant frequency to the reference frequency, The relations in (3.21) show that upper limits on 5-l or w/oo are implicit in our scaling assumptions. The assumption that 5 be of order one relative to S may be understood in two different ways. First it can be thought of as a relation Lk = O(S2) Pd (3.22) between the geometric parameter S and the physical parameters pg and pd. To derive (3.22), note that pg? x R, so that Alternatively the order-one size of &' could be a statement about the size of the pressure variations in the liquid. If these variations are of size Ap, rather than p,, then we should change the pressure scaling in (3.7) top = p,+ Apop'. The dimension- less parameter would then be replaced by g= (1/S2)Apo/pdcZ. If pc and E are fixed, this is the statement that Ap, = O(62). (3.23) On the other hand, after taking the limit S+O with y fixed, as will be done in $4, we may consider 6 to be large. The effective equations (4.1)-(4.4) for average quantities p, ii, R imply that, for 5 large, pg(R) x p, u is size 5, and jj, P obey the acoustic equations for the liquid alone. Thus the effective equations are uniformly valid in this limit of pure liquid. 4. Derivation of effective equations The main result of this paper is a systematic derivation of effective equations ((4.1)-(4.7) below) from the scaled equations (3.10)-(3.18), in the limit Sgoing to zero. In this section primes are dropped from (3.10)-(3.18). We shall show formally the following proposition. Suppose that the u, p and RY satisfy (3.10)-(3.18), Let S+O and N-tao, with x, g and C held constant. Then u and p converge to ii and p, and RY converges to the continuum radius field R (in the sense that RY-R(xY)+O for allj). Furthermore, the limits ii, p and R satisfy equations cC2Pt + V-ii- ($tR'tl~)~ = 0, iit+pp = 0, RRtt +:RZ; = C(F(R) -PIl with the equation of state and initial conditions P(0, x) = W), a(o,x) = 17(x), R(O,x) = &), Rt(O, x) = &x), and with 8 and M = i@ defined by (3.15) and (3.18) (with primes dropped).
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266 R. E. CafEisch, M. J. Miksis, G. C. Papanicolaou and L. Ting the wave equation $,,-C2 A$ = 0 without boundary conditions. N is the nonlinear operator corresponding to the first integral on the right of (4.18) and FY is the nonlinear operator associated with the sum over j on the right of (4.18). The operators FY are nonlinear because the radii RY(t) of the spheres HY(t) depend on the solution via (4.11) and (4.12). In Foldy™s method, expressed somewhat differently here, (4.19) is solved approx- imately as follows. For 6 small and N large with x = NV-™h36 = 0(1), we shall show that the field 9 will tend to a limit $, the continuum limit. It is then sufficient to calculate FY($) for each sphere separately and to evaluate the limit of the sum over Nin (4.19). This is what is meant by saying that each bubble feels only the average pressure and velocity fields around it and not the local fields of the other bubbles. We proceed to implement this. First it is clear that the nonlinear term N(4) will make no contribution in the limit because it is formally of order cY2. Now consider a single bubble which may be centred at the origin and let z = x/S, so that the sphere has radius R(t) on the z-scale. To leading order in S the potential $ outside the bubble satisfies A& = 0 (lzl > R), (4.20) along with the boundary conditions (4.21) and ~(Vz$)8+$t = -@‚(R) on IzI = R. (4.22) Far away from the bubble, for (zI large, we expect that $ behaves like 3, the continuum potentiad field. Clearly the solution is (4.23) which satisfies (4.20), (4.21) and the large-)z( condition. Condition (4.22) leads to Rayleigh™s equation (4.3). With the 9 determined locally about a bubble in the above way, we look at a typical integrand in FY, ZY(x,t,7) = dS[G,(t-~,x-y) ~ aJS(T™Y)+JSr(~,y)A.VG(t-7,x-~)~, (4.24) I an where the integral is over the surface of the jth bubble, and x is away from the other bubbles. Note that FY = 1: d7. We may replace aJS/an in (4.24) by 6-™ aRY/at in view of (4.1 1). We may also evaluate Q at y = xiﬂ, making an error of order 6. Since $, by (4.22) is of order one, we see that (4.25) a Z~ﬂ™(X, t,7) = 47~4Rjﬂ™(~))~G~(t-~, X-X~) t RY(7) plus terms of order S2. sense that Let R(t, x) satisfy (4.3). Then Ry(t) is close to the continuum field R(t, xY) in the rN (4.26)

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Effective equations for wave propagation in bubbly liquids 267 The precise nature of this convergence will not be checked in detail. Using (4.26) in (4.25) allows us to replace Ry(7) by R(7, xy). If we insert this expression for Iy(x, t, 7) in the sum in (4.18) and ignore terms that are of order a2, we obtain $,(t,x) = -(4nNS)cZ lim d~G~(t-~,x-xy)R~(~,xy)R~(~,xy) Ntm I-1 (4.27) Using (3.4) and (3.19) on the right-hand side of (4.27), we arrive at the following integral equation for the continuum field $: The integral equation (4.28) is equivalent to (4.29) (4.30) which is equivalent to (4.1), (4.2), (4.5) and (4.6), with V$ = u and R solving (4.3), (4.4) and (4.7). This finishes the demonstration of the main proposition. 5. Properties of the effective equations The system of effective equations (4.1 )-(4.4) has in dimensional variables the form 2 1 pt + V-u- (ax 4 R3n), = 0, Pc cc MY Pg=+s) 1 Pd RRtt+$RZ; = – (p g – P). (5.3) (5.4) Here we assume that pc, c,, M, K and n = NO (the number of bubble centres per unit volume) are constants. Initial values are given for p, u, R and R,. In this section we shall compare (5.1)-(5.4) with Van Wijngaarden™s equations, compute the effective sound speed and resonant frequency, and derive an energy function for (5.1)-(5.4). The expression for sound speed and resonant frequency are well known; the energy function is new. Van Wijngaarden™s equations (l.lF(1.6) reduce to (5.1)-(5.4) if (i) the volume fraction is assumed to be small, (ii) the velocity fluctuations are assumed to be small and (iii) the liquid is assumed to be nearly incompressible, which is the familiar approximation used in acoustics. The last condition is implemented by replacing dp,/dt by c;~ dpldt and then treating pc and the liquid sound speed c, as constants. In addition we replace the isothermal equation of state (1.5) by the more general equation (5.3) and identify the volume fraction as B = $nR3n.

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268 about u = 0, p = p, and R = R,, such that R. E. CaJiech, M. J. Miksis, G. C. Papanicolaou and L. Ting To obtain the effective sound speed and resonant frequency, we linearize (5.1)-(5.4) 1 Pc cc to obtain 7pt+V*u-4nRinR, = 0, (5.6) Here 1 Rtt+wtR=– Ro Pr p. wo = ($)i (5.9) is the natural frequency of radial bubble oscillations. The dispersion relation for (5.6)-(5.8) is (5.10) with Po = ipRtn. In (5.10) C is the effective phase velocity of infinitesimal disturbances. In our derivation of the effective equations (5.1)-(5.4) there is no condition that the disturbance frequency w be less than the resonance frequency wo. However, the derivation is valid only if the solution of (5.1)-(5.4) is bounded. Note that if dissipative effects were included, as in 56, the solution would be bounded through resonance. When w is small compared with w,, i.e. the low-frequency case, (5.10) simplifies to (5.11) which is the well-known formula for the effective sound speed C (Van Wijngaarden 1972). With the values p, = los dyn/cm2 (atmospheric pressure), pe = 1 g/cm3 for water, R, = lo-‘ cm and y = 1, the resonant bubble frequency wo = 1.7 x lo4 rad/s (2750 Hz). If w is less than lo00 Hz, say, and Po is of order then cc2 is negligible (cc = 1400 m/s) in (5.11) and C is about 100 m/s. This is even smaller than the sound speed in the gas (cg – 330 m/s). Only for very small bubble volume fraction, say of order 0.01 % (Po = or smaller, does E of (5.11) differ from that of the simpler formula c 3- — YPO (5.12) Po Pc ‘ at low frequencies. This striking behaviour of the effective sound speed of the mixture as a function of bubble volume fraction has been confirmed experimentally by Silberman (1957), and makes bubbly liquids an interesting medium. For example, layers of bubbly liquids can be used to reflect or isolate sound fields (Domenico 1982). This behaviour is also interesting mathematically, because the bubbly liquid is a two-component composite medium with singular behaviour at small volume fraction: a very small volume of bubbles changes the effective properties of the mixture drastically.

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Effective equations for wave propagation in bubbly liquids 269 The energy density for the system (5.1)-(5.4) is given by 11 n E = +pL~B+-7pB+2xnpd R3R;+-p,($zR8). 2 PLCC Y-1 (5.13) With this definition, we have the energy-conservation equation (5.14) a at – E+V.@U) = 0. For suitable conditions at infinity, this equation implies the conservation of energy : I E(t, x) dx = 0. dt (5.15) The various terms in the energy density (5.3) have the following physical interpretation. The first two terms +pd u2+f(pd cj)-l pa are the energy density for a linearized, isentropic, compressible flow without bubbles. The last term in (5.13), n(y- 1)-‘pgtnR3, is the energy of the gas bubbles with equation of state (5.3). This is seen by noting that dn d dty-1 13 dt p,($R3) = -p – (n$zR3), — (5.16) which has the form dE = -p, d V, with V = n3xR3 being the total gas-bubble volume. The term n 2xpd R3R; is the number of bubbles per unit volume times the kinetic energy in an incompressible flow outside a radially oscillating sphere of radius R(t). The flow outside the sphere has the form (5.17) so the kinetic energy of the fluid induced by the radial oscillations of the bubble is r rm = 2xpC RSR;. (5.18) Equations (5.1)-(5.4) can be put in variational form by introducing the potential (5.19) 1 PC q3 defined by u = vq3, 6, = –p. The Lagrangian density is given by (5.20) It is also interesting to note that (5.1)-(5.4) is a, Hamiltonian system. Since the energy E is strictly positive and SEdx is conserved, finite-energy solutions of (5.1)-(5.4) are stable. In the case of plane waves, when all quantities in (5.1)-(5.4) depend on one space coordinate z1 and the velocity is u = (ul,O,O), one can show much more. If, at t = 0, p, u1 and R are smooth (with continuous first derivatives) and R is positive for all zl, then there is a smooth bounded solution of (5.1)-(5.4) with R > 0 for all t. This shows that there is no bubble cavitation or shock-wave formation.

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