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Force on a small inclusion in a standing acoustic wave
J. Sound Vib. (1969)
10(2), 331-339
FORCE ON A SMALL INCLUSION IN A STANDING ACOUSTIC WAVE K. B. Dvs,rB@ Department of Applied Mathematics, University of Bergen, Norway (Received 18 October 1968)
The mean acoustic force on a small inclusion in a standing acoustic field is investigated. For a spherical inclusion, the force is always parallel to the wave vector of the applied field. For a non-spherical body, however, the acoustic force depends on the orientation of the body relative to the applied field. The forces on such a body have a mean moment, which is zero only for three orientations of the body. Only one of these equilibrium orientations is stable. When the body has attained the stable orientation, the mean force is parallel to the applied field.
1. INTRODUCI’ION
It is well known that bubbles can be trapped in a liquid by a standing acoustic field (see Eller [I] and the literature cited there). If, for example, a standing plane sound wave of sufficient intensity is applied vertically in a liquid, gas bubbles may be trapped in horizontal planes lying a distance X/2 apart (A is the wavelength). In these planes there exists an equilibrium between the buoyancy force and a force arising from the sound field. It is also observed [2] that small rigid bodies can be trapped in a liquid by a standing acoustic field. Theoretical expressions for the force on a gas bubble were derived by Dysthe [3] and by Eller [I] independently (the expressions agree except for an error in the sign in [3]). The purpose of the present paper is to extend the theory so that it is valid for any small inclusion in a standing sound field. In section 2 the properties of the scattered sound field are briefly discussed, and it is indicated that the first two terms of the expansion in l/r (r is the radial distance from the inclusion), are related to the dynamics of the inclusion-its volumetric pulsation, and oscillation relative to the fluid. In section 3, the dynamics of a rigid body in the sound field is studied. We obtain more information about one of the terms in the expansion of the scattered field. It is also shown that the sound field exerts a mean moment on a non-spherical inclusion. This mean moment vanishes for three different orientations of the body relative to the sound field, only one of which is stable. In section 4 an expression for the mean force on an inclusion is derived. For a nonspherical body, this force is shown to depend (both in magnitude and direction) on the orientation of the body relative to the sound field. When the inclusion has attained a stable orientation (in the sense of section 3), the mean force is parallel to the wave-vector of the applied sound field. This agrees with observations [2] which indicate that the inclusion has a stable orientation, and that there are no components of the mean force perpendicular to the wave vector of the applied field. t Present address: Institute for Plasma Research, Stanford University, Stanford, Calif., U.S.A. 331
332
K. B. DYSTHE
In section 5 the general formula for the mean force on an inclusion is applied to some special cases. 2. THE SCATTERED FIELD FROM A SMALL INCLUSION A small body of dimension I will be considered, immersed in an acoustic field of wavelength h 9 1. It will also be assumed that the dimension of the body considerably exceeds the thickness of the boundary layer surrounding it. The thickness of the boundary layer is of the order of magnitude Jvlw, where v is the kinematic viscosity and w is the angular frequency of the wave. Thus it is assumed Jv/w < 1. Further it is assumed that the amplitude of oscillations, which is of order of magnitude Mh (M is the Mach number), is much less than 1. Interest now centres on the scattered velocity field arising from the presence of the body in the sound field. With the assumptions above the scattered field is a potential field. Near the body (at distances small compared to h, but outside the boundary layer) the flow will satisfy Laplace’s equation. Denoting the velocity potential of the scattered field &, we have v=+,=o.
(1)
Assuming that the motion of the body is a pure translation with velocity u, and neglecting the boundary layer, the boundary condition is (v+s - (v - u))*u = v,,
(2)
where v is the unperturbed velocity field, n is a vector perpendicular to the boundary, and v, is the velocity of the boundary perpendicular to itself, with respect to the centre of gravity. For non-spherical bodies it is assumed that there exists a region at a distance r from the body such that I-er-eX.
In such a region, (1) is still valid and the solution can be written as a power series in I/r. Taking into account only the monopole and dipole contributions yields
where a and A are independent of position. The quantity a can be determined from the conservation of mass to be
where v is again the unperturbed velocity of the sound field and V .v is evaluated at the position of the body. V, V0 and Pare the instantaneous value, equilibrium value and rate of change of the volume of the body, respectively. The vector A depends on the actual shape of the body and its velocity relative to the fluid. Thus, in the approximate solution (3) the first term is due to the volumetric pulsation of the inclusion, induced by the sound field, and the difference in the compressibility of the inclusion from that of the fluid. The second term, however, is due solely to the motion of the inclusion relative to the fluid. This term would not change if the inclusion were replaced by a rigid body of the same form and which performed the same motion as the inclusion. To obtain a complete solution for A, (1) and (2) must be solved to the lowest order in an expansion in 1/r, with v,, = 0. From the linearity of equation (1) and the fact that the boundary condition is now linear
FORCE ON AN INCLUSION
IN AN ACOUSTIC
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both in $s and v - a, it follows that & (and hence A) is some linear function of v - u. For a spherical inclusion, A = +R3(v - u), (5) where R is the radius. The corresponding solution (3) is also valid near the spherical body. 3. DYNAMICS OF A RIGID INCLUSION IN A SOUND FIELD To analyse the quantity A of the scattered field any further, we must investigate thedynamics of the body in its motion relative to the fluid. For a spherical inclusion this is an easy task, on observing that the oscillating force acting on it is -
I p,,do=-p. body
$ ($r+~)d.=p,v,e+~pOA, body
where v = -V&, and pa is the total acoustic pressure. The surface integral is evaluated over the surface of the body. Integrating the corresponding equation of motion and inserting (5) yields “2!!!&, PO+
2P,
where p. and pi are the densities of the liquid and the inclusion, respectively. To find a similar relation for a non-spherical rigid body the existence of a region must again be assumed such that 1 e r -e A. It is now possible to simplify the model to that of a rigid particle immersed in an oscillating fluid whose velocity at infinity may be given by v = v. sin wt. It can readily be shown (e.g. [43) that, in a co-ordinate system moving with the velocity v, the kinetic energy E of the liquid is given by E=F(47rA.I-
V,,i),
(6)
where i = a - v is the velocity of the body with respect to the co-ordinate system chosen. As already mentioned, A is some linear function of u - v. Consequently, E becomes a quadratic function of i, which is written E= +t.M+. (7) The tensor M, which is taken to be symmetrical, is called the induced mass tensor. The Lagrangian of the system consisting of the fluid and body is L=fi.M.i+~i2-(m-poYo)4.r,
(8)
where I is the position of the centre of gravity of the body and m is the mass of the body. The last term is the potential energy arising from the “force of gravity”, rnq, and the “buoyancy force”, p. V,+. The viscous drag force is of the order of magnitude povZ(u - v). The ratio of this force to the rate of change of momentum of the body is of the order of magnitude v/w12. Consequently, the viscous drag force should be neglected according to the previous assumptions. The equation of motion is now $oMi+mf)=-(m-p,V,)v.
(9)
K. B. DYSTHE
334
Recalling that i = u - v the desired relation emerges between u and v: (M+mz)*u=(M+p~V,z)~v,
(10)
where Zis the unit tensor. From (10) it is seen that u and v are parallel only when the orientation of the body is such that v is parallel to one of the three principal axes of M. Next it will be demonstrated that the forces acting on the body have a moment K. Using (9) and choosing the origin of the co-ordinate system to be the mean position of the centre of gravity of the body, the Lagrangian (8) takes the form L = - J&n - po v,y vo *(M + ml)-’ vo COGcot,
(11)
where (M + mZ)-’ is the inverse tensor of (M + ml). Let us proceed to find the moment K of the forces with respect to the chosen origin. The change 8L in the Lagrangian due to a small rotation 68 of a rigid body, can be shown to be -K-68, where K is the moment of the forces with respect to some point on the axis of rotation. To obtain K by this procedure the variation in the Lagrangian (11) should be calculated by a small rotation of the body. This, however, is inconvenient because a rotation of the body will change the components of M. If instead the fluid is rotated through an angle -68, the moment K is easily obtained K = (m - p. I’o)2v. *(M + ml)-’ x v, cos2 wt.
(12)
It is seen from (12) that, except for the case where v. is parallel to one of the three principal axes of M, K is different from zero. The oscillating part of K will not have any great effect because it is a small quantity (the ratio between the rotational and translational velocities will be of the order M*h/l, which is assumed small). As can be seen from (12), however, the moment K has a mean value (9) given by (K) = +(m - p. Vo)2v.(M + ml)-’ x v,,
(13)
which is always in the plane perpendicular to v. (i.e. in the tangential plane of the wave surface going through the body). This seems at first to be incompatible with the assumption that the motion of the body is purely translational. It turns out, however, that one of the three “equilibrium” orientations where K = 0, is a stable one. This fact, together with some experimental evidence which will be considered in the next section, strongly suggests that the rigid body will rapidly find the stable equilibrium orientation, where it will remain. To investigate the stability of the equilibrium orientations, the work 6 W is calculated in order to rotate the body through some small angle 60 from an equilibrium orientation: 6W=-
j (K)*dO.
After some calculations this becomes (14) Here the co-ordinate axes are chosen along the principal directions of M with the x,-axis along vo. The eigenvalues of M, denoted by pr(i = 1,2,3), are all positive according to (7). The condition for stability is that 6 W shall be positive for any small rotation 68 = {M,, Se,, M3} from the equilibrium orientation. This is satisfied if, and only if,
Because M is invariant to rotations around a symmetry axis, the stability condition can be relaxed to PI L CL2 I4 2 p2.
FORCE ON AN INCLUSION
IN AN ACOUSTIC
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The conclusion is reached that the only stable orientation of a rigid body in an oscillating fluid is when the principal axis of M corresponding to the largest eigenvalue is parallel to vo. The same result is obtained using the expression for the mean total energy H of the fluid and body with respect to the co-ordinate system chosen,
It is easy to show that H has a minimum as a function of orientation ( Iv01constant) when v. is along the axis corresponding to the largest eigenvalue of M. Considering the energies corresponding to the three equilibrium orientations of the body, we denote by AH the difference between the smallest two of them. AH, which is proportional to the difference between the largest and the next largest of the eigenvalues pI (i = 1,2,3), can be taken to be a measure of the “strength” of stability. When the stability is weak it is to be expected that small perturbations of the fluid will bring the body away from the stable orientation. This is the case when the surface of the body is nearly a surface of revolution with the “axis of symmetry” perpendicular to the stable axis, or nearly a sphere. It has been observed that bodies of the first category may have a certain rotation around the “axis of symmetry”. In this section the inclusion has been assumed to be a rigid body, and the term a/r in (3) and v,, in (2) have thus been neglected in the calculation of the Lagrangian (8). By including the compressibility of the body, the Lagrangian will contain additional terms to account for the new degree of freedom (the volumetric pulsation). If riis assumed to be of the same order of magnitude as V. V*v, the relative order of magnitude of the additional terms compared to the terms given in (8), is 13/rhZ which is very much smaller than unity. The influence of the compressibility of the inclusion on its motion relative to the fluid, must therefore be negligible. Thus the assumption of a rigid inclusion is not essential for the conclusions of this section. 4. THE MEAN FORCE ON A SMALL BODY IMMERSED IN A STANDING ACOUSTIC FIELD To obtain an expression for the mean force on a small object immersed in an acoustic field we shall partly apply a method used by Westervelt [5]. The equation of momentum conservation for a fluid can be written, in the absence of external forces and viscosity, $J.w)=-V.{pz+pvv}r-V.P
(15)
where p, p and v are the pressure, density and velocity of the fluid, respectively. In the linear approximation for an acoustic field, v=-v+,
P=Po+po,,,
a+
P=Po+~jp
POa4
where c is the velocity of sound, and the velocity potential 4 satisfies a wave equation. Taking the mean value of equation (15), and neglecting terms of order three and higher in the field variable 4, V*(P)
= V*(- (L>Zf
where L=!f@-@)2)
is the Lagrangian density of the wave field.
po(V$V#))
= 0,
(16)
336
K. B. DYSTHE
Integrating (16) over the volume bounded by the surface of the body and some surface S enclosing the body, and applying the divergence theorem, we have for the mean force F F=$(P).da.
(17)
S
Thus the mean force acting on the body is equal to the flux of the mean stress tensor through some surface enclosing the body. Now taking 4 = c&,+ &, where +,, and $s are the velocity potentials of the standing and the scattered wave, respectively, (P) can be broken up into three terms: (P,), (P,)and (PO,). The meaning of the first two terms is evident and (Pas) represents the coupling term
From (17) it is evident that the scattered field only needs to’be known in some limited domain of space (near S) in order to calculate F. For a spherical inclusion we take S to be the surface of the body. For a non-spherical rigid body we take S to be a sphere centred in the body, of radius R large compared to the dimension I of the body and still small compared to the wavelength A. Near S, the velocity potential +s is given by (3) for both cases (for the spherical body we must of course assume 14 X). It is seen that fs (PO) *da = 0, because V +(PO) = 0 everywhere inside S. It can also be shown that the ratio of the terms corresponding to (P,) and (Pas) in (17) is of the order of magnitude 13/Rh2. The term corresponding to (P,) should consequently be neglected in our approximation. Inserting (3) into (5) and integrating we have, after some calculations, .
(19)
The first term in this expression is due to the pulsation of the inclusion, induced by the sound field, and the difference in compressibility between the body and the fluid. The second term is due to the oscillating motion of the inclusion relative to the fluid. For a we have the expression (4), and A is found from (6), (7) and (10) to be (20)
A=-(m-po~o)u/4~po=~~V~o where m-pPoVo
B=(M+mz)-‘~(M+p,V,z)
3vopo
.
Inserting these expressions into (19) we have F = po( I%$,) - Vo(Z+ B)*V(U) where u=po
(21)
(a+,12 at
/2c2
is the potential energy density of the standing acoustic field. It is seen from (21) that, for a non-spherical rigid body, the force F will depend on the orientation of the body. Generally F will have a component perpendicular to V$,. It has been observed [2] that small rigid bodies can be trapped in the neighbourhood of some horizontal planes when a standing plane sound wave is applied vertically in a liquid. No horizontal translation is observed for the trapped
FORCE ON AN INCLUSION
IN AN ACOUSTIC
WAVE
337
objects. This strongly suggests that the body will rapidly attain the stable equilibrium orientation. For this orientation namely, the component of F perpendicular to V+, is zero, and F can be written
F=poU%o>- VorV
(22)
where r=l+Pr-Po.PfPo 3Po
CL+ Pi’
Here pi is the density of the body (inclusion) and p=J-maxp.. Vo n For a rigid body of course dV/dt = 0, and the only contribution comes from the second term. 5. SOME APPLICATIONS
In this section we shall apply the general formula for the mean force on an inclusion to some special cases. We limit ourselves to spherical inclusions. For a sphere we have M=+p,
V,I,
B=
=Z, 2P* + PO
and consequently we have for the force
3pi
F = po( VV$,) - v, ____ V(U).
(23)
2P‘+ PO
We first look at two examples with compressible inclusions. 5.1.
A GAS BUBBLE IN A LIQUID
For a gas bubble in a liquid (V - Vo) satisfies the following differential equation:
(
$+s$+w:
1
(V-
vo)=-47rRo~,
WO
where R. is the equilibrium radius of the sphere, S accounts for the damping due to radiation, thermal conduction, and viscosity [6]. This equation is valid only when (V - Vo)/Vo Q 1, which can also be written
where pA is the acoustical pressure amplitude and p. is the equilibrium pressure. The eigenfrequency of the bubble o. is given by wo =
J
3~;
I
Ro,
where 7 is a polytropic exponent whose value, lying in the interval from one to y (y is the adiabatic constant), depends upon the radius and the wave frequency o (see [6]). For the force F, taking into account that pi Q po, F = 4rRo c’(&, - w2) - v,3p’ V(U> * p. I (w; - w2)2+ w262
(24)
K. B. DYSTHE
338
Because (a/~)~ Q 1 the damping only affects F very near the resonance, where the first term in the bracket becomes small. When not near a resonance, (24) can be simplified to F = 4rRo c2 __V(U). 0$-J For an infinite plane wave (i.e. $,, = u&coskxcosot)
(25)
F(=\FI) becomes
~RQ POk F= ___ M2 sin 2kx, d-C&j
where M is the Mach number (M = u,,/c). If the wave field is applied vertically the force F will be parallel-or anti-parallel-to the buoyancy force. If the amplitude of F exceeds the buoyancy force, it is easily seen that a family of planes exists (a distance h/2 apart) where a gas bubble has a stable equilibrium. The condition that F shall exceed the buoyancy force is M> 2R0(lw2 - w;I g/3k)“2. The formula (25) was found by Dysthe [3] and Eller [l] independently, and seems to be verified by experimental observations [ 1,2,7]. 5.2.
A LIQUID DROP
For a liquid drop in a gas or in another liquid, we have for v I?=_- I,+!!L~2~ pi c: at2 3 where c1is the velocity of sound in the liquid of the drop. The force becomes F=-Vb
For a liquid drop in a gas we have pI 9 p. and (26) reduces to F=-3V,/2 5.3.
V(U).
(27)
A RIGID SPHERE
For a rigid sphere we have, of course, that Ti= 0 and the expression for the force F is consequently F= _ v,, -----V(U) 3pf 2Pi + PO
(28)
which for the case pi s p. reduces to (27). As already mentioned, it has been observed [2] that small rigid bodies can be suspended against gravity by a standing plane wave applied vertically. “Stable horizontal planes” similar to the effect just described for a gas bubble, are observed. The condition that the force (28) shall exceed the force of gravity is M > (2gl Pi - Pal @PI + PO)/PlPO4”2. Take as a numerical example the values w = lo6 set-‘, p0 = 1 gr/cm3, pi = 2 gr/cm3 and c = 1.5 - lo5 cm/set. The above condition is then satisfied if M > 1.7 +10m4. REFERENCES 1. A. ELLBR 1968J.
2. H.
acoust Sot. Am. 43,170. Force on a bubble in a standing acoustic wave.
HOBABKunpublished communications.
FORCE ON AN INCLUSION
IN AN ACOUSTIC
WAVE
339
3. K. B. DYSTHE 1968 J. Sound Vib. 6,464. The force on a small gas bubble in a standing sound wave in water. 4. L. D. LANDAU and E. M. LIFSHITZ 1959 Fluid Mechanics. London: Pergamon Press. 5. P. J. WESTERVJZLT 1957 J. acoust. Sot. Am, 29,26. Acoustic radiation pressure. 6. C. DEMN JR. 1959 J. acoust. Sot. Am. 31,1654. Survey of thermal, radiation and viscous damping of pulsating air bubbles in water. 7. D. J. DUNN, M. KULJIS and V. G. WELSBY 1965 J. Sound Vib. 2, 471. Non-linear effects in a focused underwater standing wave acoustic system.
22
10(2), 331-339
FORCE ON A SMALL INCLUSION IN A STANDING ACOUSTIC WAVE K. B. Dvs,rB@ Department of Applied Mathematics, University of Bergen, Norway (Received 18 October 1968)
The mean acoustic force on a small inclusion in a standing acoustic field is investigated. For a spherical inclusion, the force is always parallel to the wave vector of the applied field. For a non-spherical body, however, the acoustic force depends on the orientation of the body relative to the applied field. The forces on such a body have a mean moment, which is zero only for three orientations of the body. Only one of these equilibrium orientations is stable. When the body has attained the stable orientation, the mean force is parallel to the applied field.
1. INTRODUCI’ION
It is well known that bubbles can be trapped in a liquid by a standing acoustic field (see Eller [I] and the literature cited there). If, for example, a standing plane sound wave of sufficient intensity is applied vertically in a liquid, gas bubbles may be trapped in horizontal planes lying a distance X/2 apart (A is the wavelength). In these planes there exists an equilibrium between the buoyancy force and a force arising from the sound field. It is also observed [2] that small rigid bodies can be trapped in a liquid by a standing acoustic field. Theoretical expressions for the force on a gas bubble were derived by Dysthe [3] and by Eller [I] independently (the expressions agree except for an error in the sign in [3]). The purpose of the present paper is to extend the theory so that it is valid for any small inclusion in a standing sound field. In section 2 the properties of the scattered sound field are briefly discussed, and it is indicated that the first two terms of the expansion in l/r (r is the radial distance from the inclusion), are related to the dynamics of the inclusion-its volumetric pulsation, and oscillation relative to the fluid. In section 3, the dynamics of a rigid body in the sound field is studied. We obtain more information about one of the terms in the expansion of the scattered field. It is also shown that the sound field exerts a mean moment on a non-spherical inclusion. This mean moment vanishes for three different orientations of the body relative to the sound field, only one of which is stable. In section 4 an expression for the mean force on an inclusion is derived. For a nonspherical body, this force is shown to depend (both in magnitude and direction) on the orientation of the body relative to the sound field. When the inclusion has attained a stable orientation (in the sense of section 3), the mean force is parallel to the wave-vector of the applied sound field. This agrees with observations [2] which indicate that the inclusion has a stable orientation, and that there are no components of the mean force perpendicular to the wave vector of the applied field. t Present address: Institute for Plasma Research, Stanford University, Stanford, Calif., U.S.A. 331
332
K. B. DYSTHE
In section 5 the general formula for the mean force on an inclusion is applied to some special cases. 2. THE SCATTERED FIELD FROM A SMALL INCLUSION A small body of dimension I will be considered, immersed in an acoustic field of wavelength h 9 1. It will also be assumed that the dimension of the body considerably exceeds the thickness of the boundary layer surrounding it. The thickness of the boundary layer is of the order of magnitude Jvlw, where v is the kinematic viscosity and w is the angular frequency of the wave. Thus it is assumed Jv/w < 1. Further it is assumed that the amplitude of oscillations, which is of order of magnitude Mh (M is the Mach number), is much less than 1. Interest now centres on the scattered velocity field arising from the presence of the body in the sound field. With the assumptions above the scattered field is a potential field. Near the body (at distances small compared to h, but outside the boundary layer) the flow will satisfy Laplace’s equation. Denoting the velocity potential of the scattered field &, we have v=+,=o.
(1)
Assuming that the motion of the body is a pure translation with velocity u, and neglecting the boundary layer, the boundary condition is (v+s - (v - u))*u = v,,
(2)
where v is the unperturbed velocity field, n is a vector perpendicular to the boundary, and v, is the velocity of the boundary perpendicular to itself, with respect to the centre of gravity. For non-spherical bodies it is assumed that there exists a region at a distance r from the body such that I-er-eX.
In such a region, (1) is still valid and the solution can be written as a power series in I/r. Taking into account only the monopole and dipole contributions yields
where a and A are independent of position. The quantity a can be determined from the conservation of mass to be
where v is again the unperturbed velocity of the sound field and V .v is evaluated at the position of the body. V, V0 and Pare the instantaneous value, equilibrium value and rate of change of the volume of the body, respectively. The vector A depends on the actual shape of the body and its velocity relative to the fluid. Thus, in the approximate solution (3) the first term is due to the volumetric pulsation of the inclusion, induced by the sound field, and the difference in the compressibility of the inclusion from that of the fluid. The second term, however, is due solely to the motion of the inclusion relative to the fluid. This term would not change if the inclusion were replaced by a rigid body of the same form and which performed the same motion as the inclusion. To obtain a complete solution for A, (1) and (2) must be solved to the lowest order in an expansion in 1/r, with v,, = 0. From the linearity of equation (1) and the fact that the boundary condition is now linear
FORCE ON AN INCLUSION
IN AN ACOUSTIC
WAVE
333
both in $s and v - a, it follows that & (and hence A) is some linear function of v - u. For a spherical inclusion, A = +R3(v - u), (5) where R is the radius. The corresponding solution (3) is also valid near the spherical body. 3. DYNAMICS OF A RIGID INCLUSION IN A SOUND FIELD To analyse the quantity A of the scattered field any further, we must investigate thedynamics of the body in its motion relative to the fluid. For a spherical inclusion this is an easy task, on observing that the oscillating force acting on it is -
I p,,do=-p. body
$ ($r+~)d.=p,v,e+~pOA, body
where v = -V&, and pa is the total acoustic pressure. The surface integral is evaluated over the surface of the body. Integrating the corresponding equation of motion and inserting (5) yields “2!!!&, PO+
2P,
where p. and pi are the densities of the liquid and the inclusion, respectively. To find a similar relation for a non-spherical rigid body the existence of a region must again be assumed such that 1 e r -e A. It is now possible to simplify the model to that of a rigid particle immersed in an oscillating fluid whose velocity at infinity may be given by v = v. sin wt. It can readily be shown (e.g. [43) that, in a co-ordinate system moving with the velocity v, the kinetic energy E of the liquid is given by E=F(47rA.I-
V,,i),
(6)
where i = a - v is the velocity of the body with respect to the co-ordinate system chosen. As already mentioned, A is some linear function of u - v. Consequently, E becomes a quadratic function of i, which is written E= +t.M+. (7) The tensor M, which is taken to be symmetrical, is called the induced mass tensor. The Lagrangian of the system consisting of the fluid and body is L=fi.M.i+~i2-(m-poYo)4.r,
(8)
where I is the position of the centre of gravity of the body and m is the mass of the body. The last term is the potential energy arising from the “force of gravity”, rnq, and the “buoyancy force”, p. V,+. The viscous drag force is of the order of magnitude povZ(u - v). The ratio of this force to the rate of change of momentum of the body is of the order of magnitude v/w12. Consequently, the viscous drag force should be neglected according to the previous assumptions. The equation of motion is now $oMi+mf)=-(m-p,V,)v.
(9)
K. B. DYSTHE
334
Recalling that i = u - v the desired relation emerges between u and v: (M+mz)*u=(M+p~V,z)~v,
(10)
where Zis the unit tensor. From (10) it is seen that u and v are parallel only when the orientation of the body is such that v is parallel to one of the three principal axes of M. Next it will be demonstrated that the forces acting on the body have a moment K. Using (9) and choosing the origin of the co-ordinate system to be the mean position of the centre of gravity of the body, the Lagrangian (8) takes the form L = - J&n - po v,y vo *(M + ml)-’ vo COGcot,
(11)
where (M + mZ)-’ is the inverse tensor of (M + ml). Let us proceed to find the moment K of the forces with respect to the chosen origin. The change 8L in the Lagrangian due to a small rotation 68 of a rigid body, can be shown to be -K-68, where K is the moment of the forces with respect to some point on the axis of rotation. To obtain K by this procedure the variation in the Lagrangian (11) should be calculated by a small rotation of the body. This, however, is inconvenient because a rotation of the body will change the components of M. If instead the fluid is rotated through an angle -68, the moment K is easily obtained K = (m - p. I’o)2v. *(M + ml)-’ x v, cos2 wt.
(12)
It is seen from (12) that, except for the case where v. is parallel to one of the three principal axes of M, K is different from zero. The oscillating part of K will not have any great effect because it is a small quantity (the ratio between the rotational and translational velocities will be of the order M*h/l, which is assumed small). As can be seen from (12), however, the moment K has a mean value (9) given by (K) = +(m - p. Vo)2v.(M + ml)-’ x v,,
(13)
which is always in the plane perpendicular to v. (i.e. in the tangential plane of the wave surface going through the body). This seems at first to be incompatible with the assumption that the motion of the body is purely translational. It turns out, however, that one of the three “equilibrium” orientations where K = 0, is a stable one. This fact, together with some experimental evidence which will be considered in the next section, strongly suggests that the rigid body will rapidly find the stable equilibrium orientation, where it will remain. To investigate the stability of the equilibrium orientations, the work 6 W is calculated in order to rotate the body through some small angle 60 from an equilibrium orientation: 6W=-
j (K)*dO.
After some calculations this becomes (14) Here the co-ordinate axes are chosen along the principal directions of M with the x,-axis along vo. The eigenvalues of M, denoted by pr(i = 1,2,3), are all positive according to (7). The condition for stability is that 6 W shall be positive for any small rotation 68 = {M,, Se,, M3} from the equilibrium orientation. This is satisfied if, and only if,
Because M is invariant to rotations around a symmetry axis, the stability condition can be relaxed to PI L CL2 I4 2 p2.
FORCE ON AN INCLUSION
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The conclusion is reached that the only stable orientation of a rigid body in an oscillating fluid is when the principal axis of M corresponding to the largest eigenvalue is parallel to vo. The same result is obtained using the expression for the mean total energy H of the fluid and body with respect to the co-ordinate system chosen,
It is easy to show that H has a minimum as a function of orientation ( Iv01constant) when v. is along the axis corresponding to the largest eigenvalue of M. Considering the energies corresponding to the three equilibrium orientations of the body, we denote by AH the difference between the smallest two of them. AH, which is proportional to the difference between the largest and the next largest of the eigenvalues pI (i = 1,2,3), can be taken to be a measure of the “strength” of stability. When the stability is weak it is to be expected that small perturbations of the fluid will bring the body away from the stable orientation. This is the case when the surface of the body is nearly a surface of revolution with the “axis of symmetry” perpendicular to the stable axis, or nearly a sphere. It has been observed that bodies of the first category may have a certain rotation around the “axis of symmetry”. In this section the inclusion has been assumed to be a rigid body, and the term a/r in (3) and v,, in (2) have thus been neglected in the calculation of the Lagrangian (8). By including the compressibility of the body, the Lagrangian will contain additional terms to account for the new degree of freedom (the volumetric pulsation). If riis assumed to be of the same order of magnitude as V. V*v, the relative order of magnitude of the additional terms compared to the terms given in (8), is 13/rhZ which is very much smaller than unity. The influence of the compressibility of the inclusion on its motion relative to the fluid, must therefore be negligible. Thus the assumption of a rigid inclusion is not essential for the conclusions of this section. 4. THE MEAN FORCE ON A SMALL BODY IMMERSED IN A STANDING ACOUSTIC FIELD To obtain an expression for the mean force on a small object immersed in an acoustic field we shall partly apply a method used by Westervelt [5]. The equation of momentum conservation for a fluid can be written, in the absence of external forces and viscosity, $J.w)=-V.{pz+pvv}r-V.P
(15)
where p, p and v are the pressure, density and velocity of the fluid, respectively. In the linear approximation for an acoustic field, v=-v+,
P=Po+po,,,
a+
P=Po+~jp
POa4
where c is the velocity of sound, and the velocity potential 4 satisfies a wave equation. Taking the mean value of equation (15), and neglecting terms of order three and higher in the field variable 4, V*(P)
= V*(- (L>Zf
where L=!f@-@)2)
is the Lagrangian density of the wave field.
po(V$V#))
= 0,
(16)
336
K. B. DYSTHE
Integrating (16) over the volume bounded by the surface of the body and some surface S enclosing the body, and applying the divergence theorem, we have for the mean force F F=$(P).da.
(17)
S
Thus the mean force acting on the body is equal to the flux of the mean stress tensor through some surface enclosing the body. Now taking 4 = c&,+ &, where +,, and $s are the velocity potentials of the standing and the scattered wave, respectively, (P) can be broken up into three terms: (P,), (P,)and (PO,). The meaning of the first two terms is evident and (Pas) represents the coupling term
From (17) it is evident that the scattered field only needs to’be known in some limited domain of space (near S) in order to calculate F. For a spherical inclusion we take S to be the surface of the body. For a non-spherical rigid body we take S to be a sphere centred in the body, of radius R large compared to the dimension I of the body and still small compared to the wavelength A. Near S, the velocity potential +s is given by (3) for both cases (for the spherical body we must of course assume 14 X). It is seen that fs (PO) *da = 0, because V +(PO) = 0 everywhere inside S. It can also be shown that the ratio of the terms corresponding to (P,) and (Pas) in (17) is of the order of magnitude 13/Rh2. The term corresponding to (P,) should consequently be neglected in our approximation. Inserting (3) into (5) and integrating we have, after some calculations, .
(19)
The first term in this expression is due to the pulsation of the inclusion, induced by the sound field, and the difference in compressibility between the body and the fluid. The second term is due to the oscillating motion of the inclusion relative to the fluid. For a we have the expression (4), and A is found from (6), (7) and (10) to be (20)
A=-(m-po~o)u/4~po=~~V~o where m-pPoVo
B=(M+mz)-‘~(M+p,V,z)
3vopo
.
Inserting these expressions into (19) we have F = po( I%$,) - Vo(Z+ B)*V(U) where u=po
(21)
(a+,12 at
/2c2
is the potential energy density of the standing acoustic field. It is seen from (21) that, for a non-spherical rigid body, the force F will depend on the orientation of the body. Generally F will have a component perpendicular to V$,. It has been observed [2] that small rigid bodies can be trapped in the neighbourhood of some horizontal planes when a standing plane sound wave is applied vertically in a liquid. No horizontal translation is observed for the trapped
FORCE ON AN INCLUSION
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337
objects. This strongly suggests that the body will rapidly attain the stable equilibrium orientation. For this orientation namely, the component of F perpendicular to V+, is zero, and F can be written
F=poU%o>- VorV
(22)
where r=l+Pr-Po.PfPo 3Po
CL+ Pi’
Here pi is the density of the body (inclusion) and p=J-maxp.. Vo n For a rigid body of course dV/dt = 0, and the only contribution comes from the second term. 5. SOME APPLICATIONS
In this section we shall apply the general formula for the mean force on an inclusion to some special cases. We limit ourselves to spherical inclusions. For a sphere we have M=+p,
V,I,
B=
=Z, 2P* + PO
and consequently we have for the force
3pi
F = po( VV$,) - v, ____ V(U).
(23)
2P‘+ PO
We first look at two examples with compressible inclusions. 5.1.
A GAS BUBBLE IN A LIQUID
For a gas bubble in a liquid (V - Vo) satisfies the following differential equation:
(
$+s$+w:
1
(V-
vo)=-47rRo~,
WO
where R. is the equilibrium radius of the sphere, S accounts for the damping due to radiation, thermal conduction, and viscosity [6]. This equation is valid only when (V - Vo)/Vo Q 1, which can also be written
where pA is the acoustical pressure amplitude and p. is the equilibrium pressure. The eigenfrequency of the bubble o. is given by wo =
J
3~;
I
Ro,
where 7 is a polytropic exponent whose value, lying in the interval from one to y (y is the adiabatic constant), depends upon the radius and the wave frequency o (see [6]). For the force F, taking into account that pi Q po, F = 4rRo c’(&, - w2) - v,3p’ V(U> * p. I (w; - w2)2+ w262
(24)
K. B. DYSTHE
338
Because (a/~)~ Q 1 the damping only affects F very near the resonance, where the first term in the bracket becomes small. When not near a resonance, (24) can be simplified to F = 4rRo c2 __V(U). 0$-J For an infinite plane wave (i.e. $,, = u&coskxcosot)
(25)
F(=\FI) becomes
~RQ POk F= ___ M2 sin 2kx, d-C&j
where M is the Mach number (M = u,,/c). If the wave field is applied vertically the force F will be parallel-or anti-parallel-to the buoyancy force. If the amplitude of F exceeds the buoyancy force, it is easily seen that a family of planes exists (a distance h/2 apart) where a gas bubble has a stable equilibrium. The condition that F shall exceed the buoyancy force is M> 2R0(lw2 - w;I g/3k)“2. The formula (25) was found by Dysthe [3] and Eller [l] independently, and seems to be verified by experimental observations [ 1,2,7]. 5.2.
A LIQUID DROP
For a liquid drop in a gas or in another liquid, we have for v I?=_- I,+!!L~2~ pi c: at2 3 where c1is the velocity of sound in the liquid of the drop. The force becomes F=-Vb
For a liquid drop in a gas we have pI 9 p. and (26) reduces to F=-3V,/2 5.3.
V(U).
(27)
A RIGID SPHERE
For a rigid sphere we have, of course, that Ti= 0 and the expression for the force F is consequently F= _ v,, -----V(U) 3pf 2Pi + PO
(28)
which for the case pi s p. reduces to (27). As already mentioned, it has been observed [2] that small rigid bodies can be suspended against gravity by a standing plane wave applied vertically. “Stable horizontal planes” similar to the effect just described for a gas bubble, are observed. The condition that the force (28) shall exceed the force of gravity is M > (2gl Pi - Pal @PI + PO)/PlPO4”2. Take as a numerical example the values w = lo6 set-‘, p0 = 1 gr/cm3, pi = 2 gr/cm3 and c = 1.5 - lo5 cm/set. The above condition is then satisfied if M > 1.7 +10m4. REFERENCES 1. A. ELLBR 1968J.
2. H.
acoust Sot. Am. 43,170. Force on a bubble in a standing acoustic wave.
HOBABKunpublished communications.
FORCE ON AN INCLUSION
IN AN ACOUSTIC
WAVE
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3. K. B. DYSTHE 1968 J. Sound Vib. 6,464. The force on a small gas bubble in a standing sound wave in water. 4. L. D. LANDAU and E. M. LIFSHITZ 1959 Fluid Mechanics. London: Pergamon Press. 5. P. J. WESTERVJZLT 1957 J. acoust. Sot. Am, 29,26. Acoustic radiation pressure. 6. C. DEMN JR. 1959 J. acoust. Sot. Am. 31,1654. Survey of thermal, radiation and viscous damping of pulsating air bubbles in water. 7. D. J. DUNN, M. KULJIS and V. G. WELSBY 1965 J. Sound Vib. 2, 471. Non-linear effects in a focused underwater standing wave acoustic system.
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