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Symmetry in bubble dynamics
Communications in Nonlinear Science and Numerical Simulation 9 (2004) 83–92 www.elsevier.com/locate/cnsns
Symmetry in bubble dynamics A.O. Maksimov V.I.Il’ichev Pacific Oceanological Institute, Far East Branch of the Russian Academy of Sciences, Vladivostok 690041, Russia
Abstract The paper reviews some aspects of application of symmetry groups for analysis of nonlinear dynamics of gas bubbles in liquid: scaling bubble pulsation, dynamics of tethered bubble, distribution of coagulating bubbles. Ó 2003 Elsevier B.V. All rights reserved. PACS: 43.25.Ts; 43.25.Yw; 02.20.Qs Keywords: Nonlinear acoustics; Symmetry; Lie groups; Bubble dynamics
1. Introduction An investigation on novel lines is made into the problem of nonlinear bubble dynamics. The use of method based on infinitesimal-transformation theory provides a systematic account of symmetries inherent to the problem. The complete symmetry group is found for the Rayleigh equation. The detail analysis of nonlinear bubble pulsation under scaling law is given [1]. Conformal symmetry of the Laplace equation provides an approach for finding exact solutions for the pulsation of gas bubble tethered to a rigid wall. An analog of the Rayleigh equation has been derived and the dependence of natural frequency on contact angle has been studied. The final example illustrating the approach shows that the symmetry of Ôscattering integralÕ in the kinetics of coagulating bubbles can be used to find an exact solution of this nonlinear equation [2].
2. Symmetries of the Rayleigh equation First observed 12 years ago ÔSingle-bubble sonoluminescenceÕ (SBSL)––the phenomenon that describes how a gas bubble in liquid, driven by a power acoustic wave, collapse and emit light 1007-5704/$ - see front matter Ó 2003 Elsevier B.V. All rights reserved. doi:10.1016/S1007-5704(03)00017-0
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attracts an additional interest after appearance of publications [3,4] on Ôbubble fusionÕ. Bubble pulsation are described by the Rayleigh equation € þ 3 R_ 2 ¼ ½Pi ðRÞ P ðtÞ ; RR 2 q0
Pi ðRÞ ¼ P0 ðR0 =RÞ3c ;
ð1Þ
where P is the pressure in liquid, q0 and P0 are the equilibrium density and pressure; R, R0 are the equilibrium and current bubble radii; c is the polytropic index. Traditionally [5] Eq. (1) has been analyzed for harmonic driving P ðtÞ ¼ P0 þ Pm sinðxtÞ, where Pm is the amplitude and x is the frequency of pumping wave. Three types of non-simple-harmonic waves, the rectangular, triangular and as well as the sinusoidal wave with a pulse, have been used to drive the SBSL in the study [6]. The triangular wave was the more effective, while the rectangular wave was the worst and the sinusoidal wave in the middle. Nevertheless it is not clear whether this driving is most effective (natural) for nonlinear bubble pulsation. Let us suppose that ÔnaturalÕ is a synonym of symmetric than we come to the problem of finding symmetry groups of Eq. (1). We begin our analysis of the symmetry of Eq. (1) with the construction of infinitesimal generators of the desired continuous groups. We represent Eq. (1) in terms of normalized variables u ¼ R=R0 , u_ ¼ du=ds, s ¼ tX0 , X20 ¼ ð3cP0 =q0 R20 Þ, and u1 ¼ P ðtÞ=P0 1 3 1 1 € u : ð2Þ u_ 2 þ u¼ 1 u 2 3c u3c The procedure of determining the tangent vector field of the group of symmetry V ¼ nðs; uÞos þ gðs; uÞou is given in monographs [7–9] and consists of the construction of the prolongations of the first and second kinds to the space s, u, u_ , €u prð1Þ V ¼ V þ f1 ðs; u; u_ Þou_ ; ð2Þ
f1 ¼ D1 ðgÞ u_ D1 ðnÞ;
pr V ¼ V þ f1 ðs; u; u_ Þou_ þ f2 ðs; u; u_ ; € uÞo€u ;
D1 ¼ os þ u_ ou
f2 ¼ D2 ðf1 Þ €uD2 ðnÞ;
and the solution of the group determining equation 1 3 2 1 1 ð2Þ pr V € u u_ þ u1 ¼0 u 2 3c u3c
D2 ¼ os þ u_ ou þ €uou_
ð3Þ
€ ¼ u1 ½ð3=2Þu_ 2 þ ð3cÞ1 ðu3c u1 Þ. Rewriting Eq. (3) in an explicit under the condition that u form and collecting the coefficients of equal powers of independent variables, we obtain 6c=ð2þ3cÞ 2 c0 ; ð4Þ u; u1 ðsÞ ¼ u1 ð0Þ n ¼ c0 þ c1 s; g ¼ c1 ð2 þ 3cÞ c0 þ c1 s where c0 and c1 are constants. In the case of a steady external pressure, we have ou1 =os ¼ 0, u1 ¼ const, and c1 ¼ 0, and the group generator is V1 ¼ os :
ð5Þ
This group is the group of time translation G1: ðu; u_ ; s þ Þ. Its presence results in that the Rayleigh equation has an integral of motion, i.e., a Hamiltonian [10,11]
A.O. Maksimov / Communications in Nonlinear Science and Numerical Simulation 9 (2004) 83–92
u3 u_ 2 1 1 u3 u1 þ H¼ ¼ const: 3c 3ðc 1Þu3ðc1Þ 2 3 The second nontrivial solution to Eq. (2) appears for 6c=ð2þ3cÞ s0 ; c0 =c1 ¼ s0 : u1 ðsÞ ¼ u1 ð0Þ s þ s0
85
ð6Þ
6c=ð2þ3cÞ
Changing the variable s~ ¼ ðs þ s0 Þ, we obtain u1 ð~ sÞ ¼ U s~6c=ð2þ3cÞ , where U ¼ u1 ð0Þs0 . An 6c=ð2þ3cÞ external perturbation of this form corresponds to the shock wave P ðtÞ ¼ Pm ½t0 =ðt þ t0 Þ with the pressure drop Pm at the leading front and with the characteristic decay time t0 ðs0 ¼ t0 X0 Þ. In this case, the group generator has the form V2 ¼ s~os~ þ
2 uou : 2 þ 3c
ð7Þ
sÞ, s~0 ¼ k~ s, which transforms This group is the group of scaling transformations G2: u0 ¼ k2=ð2þ3cÞ uð~ the solution of the Rayleigh equation to another solution; i.e., the function u0 is also a solution to s0 Þ ¼ k2=ð2þ3cÞ uðk1 s~0 Þ. Thus, only Eq. (2) if it is considered as a function of the new variable s~0 : u0 ð~ two point symmetry groups, namely, the time translation and the scaling transformations, can exist for the Rayleigh equation. Let us investigate the behavior of the solutions under the condition that the group of scaling s0 Þ ¼ k2=ð2þ3cÞ uðk1 s~0 Þ ¼ uð~ s0 Þ has the transformations is available. The invariant solution u0 ð~ form uð~ sÞ ¼ z~ s2=ð2þ3cÞ ;
ð8Þ
where z is defined by 18ðc 1Þcð2 þ 3cÞ2 z2 þ z3c U ¼ 0. To find the general solution, we will use the fact that the availability of certain continuous group of symmetry generally offers a possibility of reducing the order of the corresponding differential equation. Namely, it appears possible to introduce new coordinates ðw; zÞ in such a way that the vector field V2 and its prolongations prð1Þ V2 and prð2Þ V2 represent the shift: V2 ¼ o=ow [9]. Thus, to be invariant in the new coordinate system, differential equation (2) must be independent of w. The change of variables is constructed with the use of the group invariants; in our case, it is reduced to the substitution zðwÞ ¼ uð~ sÞ~ s2=ð2þ3cÞ , w ¼ ln s~. It is convenient to introduce the new 3c=ð2þ3cÞ variable z1 ¼ s~ u_ ð~ sÞ and rewrite the second-order differential equation (2) as a system of two first-order equations. It turns out that the variable w does not appear explicitly in the resulting system dz 2 ¼ z þ z1 ; dw 2 þ 3c dz1 3c 1 3 2 1 1 z1 þ ¼ U : z1 þ z 2 3c z3c dw 2 þ 3c From this system, we derive 1 dz1 3c 3 2 1 1 2 1 U z z1 : zz1 þ z1 þ z ¼ 2 þ 3c 2 3c z3c 2 þ 3c dz
ð9Þ
ð10Þ
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Indeed, Eq. (10) is an ordinary differential equation of the first-order. Let us analyze the phase portrait of dynamic system (9). Stationary states are determined from the condition that the right-hand sides of Eq. (9) vanish. These equations have no roots for 3c=ð2þ3cÞ ð3c2Þ=ð2þ3cÞ =½2ð2 þ 3cÞ , one root for U > Uc and c ¼ 1, and U < Uc , where Uc ¼ ½12ðc 1Þ two roots for U > Uc and c > 1. Explicit expressions for these stationary points can be found either if the polytropic exponent only slightly deviates from unity ðc 1Þ 1, or for c ¼ 4=3. The type of these singular points is determined from the stability analysis in linear approximation. Stationary states appear at U ¼ Uc . For U > Uc , two roots correspond to a nodal point and a saddle point. With further increase in U , the node is transformed to the focus at zn ¼ f½25 þ 18ðc 1Þ þ 9ðc 1Þ2 =½4ð2 þ 3cÞ2 g1=ð2þ3cÞ (this transformation occurs at U ¼ Uf 1 for c ¼ 4=3). In the case of c ! 1, the bifurcation point, at which the node and the saddle merge, tends to infinity. We will investigate the behavior of trajectories for infinite distances using the mapping of the phase plane onto the Poincare sphere (which is the unit radius sphere touching the ðz; z1 Þ plane at the origin). On the sphere, two points correspond to every point ðz; z1 Þ of the plane; they lie on the straight line passing through the center of the sphere and the point under consideration. The point of the plane located at the infinite distance is mapped onto the equator of the sphere. In this mapping, the integral curves on the plane are transformed to the corresponding curves on the sphere, and this transformation will not change the characteristic behavior of saddles, nodes, and focuses. However, new singular points will appear on the equator. The transformation z ¼ ð1=sÞ, z1 ¼ ðm=sÞ offers a possibility of studying singular points on the equator of the Poincare sphere. Two singular points are located on the equator. They are unstable (s ¼ 0, m ¼ 0) and stable (s ¼ 0, m ¼ 5=2) nodes. At long distances from the origin, all trajectories tend to this equilibrium state along certain directions, and these directions are the m-axis (along which only two trajectories arrive at the node) and the s-axis (along which an infinite set of semi-trajectories arrive at the node). On the initial ðz; z1 Þ phase plane, the angular coefficient of the direction along which the trajectories tend to the simple equilibrium state measures 2/5. In the case of time reversal, the trajectories will tend to the unstable node along the direction coinciding with the z-axis. Fig. 1 illustrates the above analysis. It shows the behaviors of trajectories calculated for characteristic values of the amplitude parameter U ¼ 10 and the polytropic exponent c ¼ 4=3. These values are chosen with the goal of characterizing the trajectory behaviors in the most complicated situation of large amplitude parameters. Note that, for U 1, the solution of the Rayleigh equation can be obtained in an explicit form. Indeed, since Pm P P0 ðu1 P 1Þ in the shock wave, the condition U 1 can be satisfied only in the case of short pulses t0 X0 1ðs0 1Þ and a not too intense wave, because only the inertial term compensates the varying external force. The radius will only slightly vary during the action of the pulse, and the velocity will also be small. The maximal speed u_ max , the minimal radius umin ¼ Rmin =R0 , and the collapse time s depend on the parameters of the acoustic pulse, as in the case of the exponential pulse P ðtÞ ¼ Pm expðt=t0 Þ that was studied in detail in [12]. This fact is not surprising, because only integral characteristics of the shock wave govern the effect in the case of short pulses. In the figure, the phase portrait for U ¼ 10 illustrates the bubble oscillations under the action of an intense shock wave ðU 1Þ. Note that, in this case, the singular points are spaced rather widely. Three calculated trajectories decorate the separatrixes of the saddle point and the
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Fig. 1. Phase portrait of dynamic system (9) for intense driving U ¼ 10.
attraction region of the focus. In this region of the parameter U , the bubble collapse ðz 1Þ is so intense that trajectories go beyond the frame of the figure and one has to follow the branches of a trajectory by using markers. This case also allows an analytical description. For s0 1, the external action on the bubble varies slowly, which results in the existence of an adiabatic invariant [11]. In the stationary regime two superimposed motions govern the variation of the bubble radius 1=4 1=3 P0 t þ t0 P0 RðtÞ ¼ R0 þ 0:01R0 ðt0 X0 Þ1=3 Pm t0 Pm ( ) 3=4 P0 sin ðt0 X0 Þ ln½X0 ðt þ t0 Þ þ a : ð11Þ Pm One of these motions is the increase in the radius according to the power law with the exponent noticeably different from the exponent in the Rayleigh law. The other motion is the oscillation with a constant amplitude and a logarithmically increasing period. The constant phase a cannot be calculated explicitly in the framework of the abiabatic invariant approach. It appears thus possible to analytically describe the nonlinear dynamics of a bubble driven by an external perturbation in the conditions ensuring the scale invariance of the Rayleigh equation.
3. Conformal symmetry and dynamics of tethered bubbles The numerical boundary integral method was a unique way to simulate the problem of interaction of a bubble with a boundary [13–15]. In this study we present an analytical approach by
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using the symmetry of the problem. The 10th-dimensional algebra of symmetry of the Laplace equation has the basis of the following form: Pj ¼ oj ¼ oxj ; j ¼ 1; 2; 3; J3 ¼ x2 o1 x1 o2 ;
J 1 ¼ x 3 o2 x 2 o3 ;
J2 ¼ x1 o3 x3 o1 ;
D ¼ ð1=2 þ x1 o1 þ x2 o2 þ x3 o3 Þ;
K1 ¼ x1 þ ðx21 x22 x23 Þo1 þ 2x1 x3 o3 þ 2x1 x2 o2 ; K2 ¼ x2 þ ðx22 x21 x23 Þo2 þ 2x2 x3 o3 þ 2x1 x2 o1 ; K3 ¼ x3 þ ðx23 x21 x22 Þo3 þ 2x1 x3 o1 þ 2x2 x3 o3 and leads to 17 coordinate systems admitting separation of variables [16]. The torus shaped coordinates is one of these systems (see Fig. 2). x ¼ R1 sinh n cos u;
y ¼ R1 sinh n sin u;
z ¼ R1 cos w;
R ¼ cosh n sin w:
The basis of the symmetry operators corresponding to the torus shaped coordinates has the form 2 S1 ¼ J32 , 4S2 ¼ ðP3 þ K3 Þ . Spherical segments are the coordinate surfaces of this system which are orthogonal to the torii surfaces. Thus dynamics of the tethered bubble in torus shaped coordinates can be analyzed by using an analytical approach and by analogy with dynamics of free spherical bubble. The volume pulsation of the tethered bubble are described by the modified Rayleigh equation of the form
Fig. 2. Torus shaped coordinates––natural coordinates for the tethered bubble.
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1 d2 DV cP0 Pm þ DV ¼ sinðxtÞ; 2 2R0 sin #c Cð#c Þ dt q0 V 0 q0 Z p p2 sin #c sin jdj Cð#c Þ ¼ pffiffiffi 2 2 2#c #c ð cos #c cos jÞ3=2 Z
89
ð12Þ
pn sinh 2# dn c ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1cos #c cos j pn 1cos # cos j c arccoshð cos #c cos j Þ cosh n cos #c cos j cosh 2#c 1
where DV is the variation in bubble volume, #c is the contact angle. It follows directly from this equation that the fundamental––the so-called ÔMinnaertÕ frequency for the tethered bubble is given by X20 ðR0 ; #c Þ ¼
2cR0 sin #c Cð#c ÞP0 XI2 0 ðR0 Þ4 sin #c Cð#c Þ ¼ : q0 V0 2p½4 ð1 cos #c Þ2 ð2 þ cos #c Þ
ð13Þ
The evident physical consequence of this formula is that the natural frequency of the tethered bubble depends on the contact angle and this dependence is not monotonic.
4. Distribution of coagulating bubbles A sound wave propagating in liquid containing gas bubbles causes the bubbles to coagulate due to the mutual attraction induced by the Bjerknes forces. The bubble-size distribution function gðr; t; RÞ is described by the kinetic equation og þ divðUgÞ ¼ IðgÞ; ot Z
IðgÞ ¼ I Iþ ;
1
dR0 rðR; R0 ÞUðRÞ UðR0 ÞjgðRÞgðR0 Þ; 0 Z R Iþ ¼ ð1=2Þ dR0 rðR0 ; R00 ÞjUðR0 Þ UðR00 ÞjðR2 =R02 ÞgðR0 ÞgðR00 Þ; I ¼
ð14Þ
0
where R is the radius of the bubble, U is the translational velocity U ¼ U0
ð1 s2 Þ½WrW þ W rW 6m½ð1 s2 Þ2 þ d2
id½WrW W rW 6m½ð1 s2 Þ2 þ d2
;
where U0 is the bubble velocity in the absence of sound field, s ¼ x0 =x, x0 , d are the natural frequency and logarithmic decrement of the bubble, x and W are the frequency and potential of the external field, m is the kinematic viscosity, and r is the collision cross-section of the bubbles. According to [17], rjU1 U2 j is equal to 4pk for k 0 and is equal to zero for k < 0, k ¼ ðR1 þ R2 Þð3mÞ1 ½ðs21 1Þðs22 1Þ þ d1 d2 ½ðs21 1Þ2 þ d21 jWj2 ½ðs22 1Þ2 þ d22 . In a standing-wave field bubbles with a radius smaller than resonance value ðR < R Þ–– the natural frequency of resonant bubble coincide with the frequency of the external field, come together in the vicinity of an antinode of the field, i.e., in the zone where the force of the
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radiation pressure of the standing wave equalizes the buoyancy. As a result of coagulation, the bubble cloud can acquire bubbles of large radius up to resonance value. For superresonance bubbles ðR > R Þ, the force of the radiation pressure change sign, and such bubbles escape from the cloud. The process is stationary in the generation of small bubbles. It is necessary to include the density of source Sðr; RÞ localized in the space of radii close to Rs ðRs R Þ in Eq. (14) in this case. Three intervals of characteristic radii are naturally distinguished in the description of this situation: source R Rs Sðr; RÞ IðgÞ ¼ divðUgÞ; sink R R IðgÞ ¼ divðUgÞ; the inertial interval Rs R R
IðgÞ ¼ 0:
The symmetry (homogeneity) of the collision operator can be used to find an exact solution of the nonlinear integral equation IðgÞ ¼ 0. The parameter s is large s 1 in the inertial interval, so that the kernel of the collision integral can be written in the simplified form 2
wðR1 ; R2 Þ rðR1 ; R2 ÞjU1 U2 j ’
4pðR1 þ R2 ÞjWj : 3ms21 s22
The symmetry allows one to consider the similar processes in I and Iþ terms of collision integral R0 þ R00 ! R, R þ Q0 ! Q00 , where Q0 ¼ l1 R00 , Q00 ¼ l1 R ¼ l21 R0 , l1 ¼ R=R0 . For them we have Z 1 dQ0 wðR; Q0 ÞgðRÞgðQ0 Þ I ðgÞ ¼ Z 1 Z0 1
0 ¼ dQ dQ00 wðR; Q0 ÞgðRÞgðQ0 Þ3Q002 d Q003 Q03 R3 Z0 1 Z 01
3 0 03 003 ¼ dR dR00 l31 wðR; R00 Þl1 gðl1 R0 Þgðl1 R00 Þ3R02 l41 l3 1 d R R R 0 Z0 R 0 dR wðR0 ; R00 Þgðl1 R0 Þgðl1 R00 Þl31 R2 =R002 ¼ 0
and analogous for Q0 ¼ l2 R0 , Q00 ¼ l2 R ¼ l22 R00 , l2 ¼ R=R00 Z R I ðgÞ ¼ dR0 wðR0 ; R00 Þgðl2 R0 Þgðl2 R00 Þl32 R2 =R002 : 0
As a result, the collision integral can be represented in the factorized form Z R IðgÞ ¼ ð1=2Þ dR0 wðR0 ; R00 ÞR2 =R002 ½gðR0 ÞgðR00 Þ l31 gðl1 R0 Þgðl1 R00 Þ l32 gðl2 R0 Þgðl2 R00 Þ: 0
ð15Þ a
For power distribution gðRÞ ¼ g0 ðR0 =RÞ the expression in brackets of Eq. (13) is proportional to ½1 ðR=R0 Þ32a ðR=R00 Þ32a . The condition of conservation of the volume of the gas in bubble coagulation determines the power exponent a ¼ 3. This solution corresponds to a constant flow over the spectrum of sizes.
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5. Conclusions In the presented work the study of Lie point symmetries in bubble dynamics and its applications to nonlinear acoustics has been reviewed. As the first step, explained in Section 2, the symmetry groups for the simplest, unrestricted form of single bubble oscillations are identified systematically by use of infinitesimal-transformation and prolongation theory. Then, as explained in Section 3, an appeal to symmetry as a tool for separation of variables provides an approach to study the dynamic of restricted (tethered) bubble. The symmetry of collision operator in the multibubble media has been used in Section 4 for finding exact form of distribution function of bubbles coagulating in a sound field. It appears thus possible to analytically describe the nonlinear dynamics of bubbles in a variety of physical situations. This fact offers the possibility of using the class of solutions obtained above as a convenient model for analyzing such phenomena as cavitation, sonoluminescence and shock wave propagation in liquids with phase inclusions.
Acknowledgements This work was supported by the Russian Foundation for Basic Research under project numbers 01-05-64915, 01-02-96901.
References [1] Maksimov AO. Symmetry of the Rayleigh equation and the analysis of nonlinear gas bubble oscillations in liquid. Acoust Phys 2002;48(6):713–9. [2] Maksimov AO. Distribution of bubbles coagulating in a sound field. Sov Phys Acoust 1985;31(4):329–40. [3] Teleyarkhan RP et al. Evidence for nuclear emission during acoustical cavitation. Science 2002;295:1868–73. [4] Shapira D, Saltmarsh M. Nuclear fusion in collapsing bubbles––is it there? An attempt to repeat the observation of nuclear emissions from sonoluminescence. Phys Rev Lett 2002;89(10):104392-1–4. [5] Brenner M, Hilgennfeldt S, Lohse D. Single-bubble sonoluminescence. Rev Mod Phys 2002;74(2):425–83. [6] Chen W et al. Single-bubble sonoluminescence driven by non-simple-harmonic ultrasounds. J Acoust Soc Am 2002;111(6):2632–7. [7] Ovsyannikov LV. Group analysis of differential equations. Moscow: Nauka; 1978 [Academic: New York; 1982]. [8] Ibragimov NH. Transformation groups applied to mathematical physics. Moscow: Nauka; 1983 [Reidel: Dordrecht; 1985]. [9] Olver P. Application of Lie groups to differential equations. New York: Springer-Verlag; 1986. [10] Sedov LI. A course in continuum mechanics. Moscow: Nauka; 1976 [Wolters-Noordhoff: Groningen; 1972]. [11] Smereka S, Binir B, Banerjee B. Regular and chaotic bubble oscillations in periodically driven pressure fields. Phys Fluids 1987;30(11):3342–50. [12] Kedrinskii VK. Hydrodynamics of explosions: experiment and models. Novosibirsk: Sibirian Branch Russian Akad Sciences; 2000 [in Russian]. [13] Blake JR, Taib BB, Doherty G. Transient cavities near boundaries. Part 1. Rigid boundary. J Fluid Mech 1986;170:479–97. [14] Blake JR, Taib BB, Doherty G. Transient cavities near boundaries. Part 2. Free surface. J Fluid Mech 1987;181:197–212. [15] Sato K, Tomita Y, Shima A. Numerical analysis of a gas bubble near a rigid boundary in an oscillatory pressure field. J Acoust Soc Am 1994;95(5):2416–24.
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[16] Miller Jr W. Symmetry and separation of variable. London: Addison-Wesley; 1977. [17] Kobelev Yu. Nonlinear dipole oscillations of the spherical particle in a sound field. Akust Zh 1983;29:783–9 [English translation in: Sov Phys Acoust 1983;29:463–7].
Symmetry in bubble dynamics A.O. Maksimov V.I.Il’ichev Pacific Oceanological Institute, Far East Branch of the Russian Academy of Sciences, Vladivostok 690041, Russia
Abstract The paper reviews some aspects of application of symmetry groups for analysis of nonlinear dynamics of gas bubbles in liquid: scaling bubble pulsation, dynamics of tethered bubble, distribution of coagulating bubbles. Ó 2003 Elsevier B.V. All rights reserved. PACS: 43.25.Ts; 43.25.Yw; 02.20.Qs Keywords: Nonlinear acoustics; Symmetry; Lie groups; Bubble dynamics
1. Introduction An investigation on novel lines is made into the problem of nonlinear bubble dynamics. The use of method based on infinitesimal-transformation theory provides a systematic account of symmetries inherent to the problem. The complete symmetry group is found for the Rayleigh equation. The detail analysis of nonlinear bubble pulsation under scaling law is given [1]. Conformal symmetry of the Laplace equation provides an approach for finding exact solutions for the pulsation of gas bubble tethered to a rigid wall. An analog of the Rayleigh equation has been derived and the dependence of natural frequency on contact angle has been studied. The final example illustrating the approach shows that the symmetry of Ôscattering integralÕ in the kinetics of coagulating bubbles can be used to find an exact solution of this nonlinear equation [2].
2. Symmetries of the Rayleigh equation First observed 12 years ago ÔSingle-bubble sonoluminescenceÕ (SBSL)––the phenomenon that describes how a gas bubble in liquid, driven by a power acoustic wave, collapse and emit light 1007-5704/$ - see front matter Ó 2003 Elsevier B.V. All rights reserved. doi:10.1016/S1007-5704(03)00017-0
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A.O. Maksimov / Communications in Nonlinear Science and Numerical Simulation 9 (2004) 83–92
attracts an additional interest after appearance of publications [3,4] on Ôbubble fusionÕ. Bubble pulsation are described by the Rayleigh equation € þ 3 R_ 2 ¼ ½Pi ðRÞ P ðtÞ ; RR 2 q0
Pi ðRÞ ¼ P0 ðR0 =RÞ3c ;
ð1Þ
where P is the pressure in liquid, q0 and P0 are the equilibrium density and pressure; R, R0 are the equilibrium and current bubble radii; c is the polytropic index. Traditionally [5] Eq. (1) has been analyzed for harmonic driving P ðtÞ ¼ P0 þ Pm sinðxtÞ, where Pm is the amplitude and x is the frequency of pumping wave. Three types of non-simple-harmonic waves, the rectangular, triangular and as well as the sinusoidal wave with a pulse, have been used to drive the SBSL in the study [6]. The triangular wave was the more effective, while the rectangular wave was the worst and the sinusoidal wave in the middle. Nevertheless it is not clear whether this driving is most effective (natural) for nonlinear bubble pulsation. Let us suppose that ÔnaturalÕ is a synonym of symmetric than we come to the problem of finding symmetry groups of Eq. (1). We begin our analysis of the symmetry of Eq. (1) with the construction of infinitesimal generators of the desired continuous groups. We represent Eq. (1) in terms of normalized variables u ¼ R=R0 , u_ ¼ du=ds, s ¼ tX0 , X20 ¼ ð3cP0 =q0 R20 Þ, and u1 ¼ P ðtÞ=P0 1 3 1 1 € u : ð2Þ u_ 2 þ u¼ 1 u 2 3c u3c The procedure of determining the tangent vector field of the group of symmetry V ¼ nðs; uÞos þ gðs; uÞou is given in monographs [7–9] and consists of the construction of the prolongations of the first and second kinds to the space s, u, u_ , €u prð1Þ V ¼ V þ f1 ðs; u; u_ Þou_ ; ð2Þ
f1 ¼ D1 ðgÞ u_ D1 ðnÞ;
pr V ¼ V þ f1 ðs; u; u_ Þou_ þ f2 ðs; u; u_ ; € uÞo€u ;
D1 ¼ os þ u_ ou
f2 ¼ D2 ðf1 Þ €uD2 ðnÞ;
and the solution of the group determining equation 1 3 2 1 1 ð2Þ pr V € u u_ þ u1 ¼0 u 2 3c u3c
D2 ¼ os þ u_ ou þ €uou_
ð3Þ
€ ¼ u1 ½ð3=2Þu_ 2 þ ð3cÞ1 ðu3c u1 Þ. Rewriting Eq. (3) in an explicit under the condition that u form and collecting the coefficients of equal powers of independent variables, we obtain 6c=ð2þ3cÞ 2 c0 ; ð4Þ u; u1 ðsÞ ¼ u1 ð0Þ n ¼ c0 þ c1 s; g ¼ c1 ð2 þ 3cÞ c0 þ c1 s where c0 and c1 are constants. In the case of a steady external pressure, we have ou1 =os ¼ 0, u1 ¼ const, and c1 ¼ 0, and the group generator is V1 ¼ os :
ð5Þ
This group is the group of time translation G1: ðu; u_ ; s þ Þ. Its presence results in that the Rayleigh equation has an integral of motion, i.e., a Hamiltonian [10,11]
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u3 u_ 2 1 1 u3 u1 þ H¼ ¼ const: 3c 3ðc 1Þu3ðc1Þ 2 3 The second nontrivial solution to Eq. (2) appears for 6c=ð2þ3cÞ s0 ; c0 =c1 ¼ s0 : u1 ðsÞ ¼ u1 ð0Þ s þ s0
85
ð6Þ
6c=ð2þ3cÞ
Changing the variable s~ ¼ ðs þ s0 Þ, we obtain u1 ð~ sÞ ¼ U s~6c=ð2þ3cÞ , where U ¼ u1 ð0Þs0 . An 6c=ð2þ3cÞ external perturbation of this form corresponds to the shock wave P ðtÞ ¼ Pm ½t0 =ðt þ t0 Þ with the pressure drop Pm at the leading front and with the characteristic decay time t0 ðs0 ¼ t0 X0 Þ. In this case, the group generator has the form V2 ¼ s~os~ þ
2 uou : 2 þ 3c
ð7Þ
sÞ, s~0 ¼ k~ s, which transforms This group is the group of scaling transformations G2: u0 ¼ k2=ð2þ3cÞ uð~ the solution of the Rayleigh equation to another solution; i.e., the function u0 is also a solution to s0 Þ ¼ k2=ð2þ3cÞ uðk1 s~0 Þ. Thus, only Eq. (2) if it is considered as a function of the new variable s~0 : u0 ð~ two point symmetry groups, namely, the time translation and the scaling transformations, can exist for the Rayleigh equation. Let us investigate the behavior of the solutions under the condition that the group of scaling s0 Þ ¼ k2=ð2þ3cÞ uðk1 s~0 Þ ¼ uð~ s0 Þ has the transformations is available. The invariant solution u0 ð~ form uð~ sÞ ¼ z~ s2=ð2þ3cÞ ;
ð8Þ
where z is defined by 18ðc 1Þcð2 þ 3cÞ2 z2 þ z3c U ¼ 0. To find the general solution, we will use the fact that the availability of certain continuous group of symmetry generally offers a possibility of reducing the order of the corresponding differential equation. Namely, it appears possible to introduce new coordinates ðw; zÞ in such a way that the vector field V2 and its prolongations prð1Þ V2 and prð2Þ V2 represent the shift: V2 ¼ o=ow [9]. Thus, to be invariant in the new coordinate system, differential equation (2) must be independent of w. The change of variables is constructed with the use of the group invariants; in our case, it is reduced to the substitution zðwÞ ¼ uð~ sÞ~ s2=ð2þ3cÞ , w ¼ ln s~. It is convenient to introduce the new 3c=ð2þ3cÞ variable z1 ¼ s~ u_ ð~ sÞ and rewrite the second-order differential equation (2) as a system of two first-order equations. It turns out that the variable w does not appear explicitly in the resulting system dz 2 ¼ z þ z1 ; dw 2 þ 3c dz1 3c 1 3 2 1 1 z1 þ ¼ U : z1 þ z 2 3c z3c dw 2 þ 3c From this system, we derive 1 dz1 3c 3 2 1 1 2 1 U z z1 : zz1 þ z1 þ z ¼ 2 þ 3c 2 3c z3c 2 þ 3c dz
ð9Þ
ð10Þ
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Indeed, Eq. (10) is an ordinary differential equation of the first-order. Let us analyze the phase portrait of dynamic system (9). Stationary states are determined from the condition that the right-hand sides of Eq. (9) vanish. These equations have no roots for 3c=ð2þ3cÞ ð3c2Þ=ð2þ3cÞ =½2ð2 þ 3cÞ , one root for U > Uc and c ¼ 1, and U < Uc , where Uc ¼ ½12ðc 1Þ two roots for U > Uc and c > 1. Explicit expressions for these stationary points can be found either if the polytropic exponent only slightly deviates from unity ðc 1Þ 1, or for c ¼ 4=3. The type of these singular points is determined from the stability analysis in linear approximation. Stationary states appear at U ¼ Uc . For U > Uc , two roots correspond to a nodal point and a saddle point. With further increase in U , the node is transformed to the focus at zn ¼ f½25 þ 18ðc 1Þ þ 9ðc 1Þ2 =½4ð2 þ 3cÞ2 g1=ð2þ3cÞ (this transformation occurs at U ¼ Uf 1 for c ¼ 4=3). In the case of c ! 1, the bifurcation point, at which the node and the saddle merge, tends to infinity. We will investigate the behavior of trajectories for infinite distances using the mapping of the phase plane onto the Poincare sphere (which is the unit radius sphere touching the ðz; z1 Þ plane at the origin). On the sphere, two points correspond to every point ðz; z1 Þ of the plane; they lie on the straight line passing through the center of the sphere and the point under consideration. The point of the plane located at the infinite distance is mapped onto the equator of the sphere. In this mapping, the integral curves on the plane are transformed to the corresponding curves on the sphere, and this transformation will not change the characteristic behavior of saddles, nodes, and focuses. However, new singular points will appear on the equator. The transformation z ¼ ð1=sÞ, z1 ¼ ðm=sÞ offers a possibility of studying singular points on the equator of the Poincare sphere. Two singular points are located on the equator. They are unstable (s ¼ 0, m ¼ 0) and stable (s ¼ 0, m ¼ 5=2) nodes. At long distances from the origin, all trajectories tend to this equilibrium state along certain directions, and these directions are the m-axis (along which only two trajectories arrive at the node) and the s-axis (along which an infinite set of semi-trajectories arrive at the node). On the initial ðz; z1 Þ phase plane, the angular coefficient of the direction along which the trajectories tend to the simple equilibrium state measures 2/5. In the case of time reversal, the trajectories will tend to the unstable node along the direction coinciding with the z-axis. Fig. 1 illustrates the above analysis. It shows the behaviors of trajectories calculated for characteristic values of the amplitude parameter U ¼ 10 and the polytropic exponent c ¼ 4=3. These values are chosen with the goal of characterizing the trajectory behaviors in the most complicated situation of large amplitude parameters. Note that, for U 1, the solution of the Rayleigh equation can be obtained in an explicit form. Indeed, since Pm P P0 ðu1 P 1Þ in the shock wave, the condition U 1 can be satisfied only in the case of short pulses t0 X0 1ðs0 1Þ and a not too intense wave, because only the inertial term compensates the varying external force. The radius will only slightly vary during the action of the pulse, and the velocity will also be small. The maximal speed u_ max , the minimal radius umin ¼ Rmin =R0 , and the collapse time s depend on the parameters of the acoustic pulse, as in the case of the exponential pulse P ðtÞ ¼ Pm expðt=t0 Þ that was studied in detail in [12]. This fact is not surprising, because only integral characteristics of the shock wave govern the effect in the case of short pulses. In the figure, the phase portrait for U ¼ 10 illustrates the bubble oscillations under the action of an intense shock wave ðU 1Þ. Note that, in this case, the singular points are spaced rather widely. Three calculated trajectories decorate the separatrixes of the saddle point and the
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Fig. 1. Phase portrait of dynamic system (9) for intense driving U ¼ 10.
attraction region of the focus. In this region of the parameter U , the bubble collapse ðz 1Þ is so intense that trajectories go beyond the frame of the figure and one has to follow the branches of a trajectory by using markers. This case also allows an analytical description. For s0 1, the external action on the bubble varies slowly, which results in the existence of an adiabatic invariant [11]. In the stationary regime two superimposed motions govern the variation of the bubble radius 1=4 1=3 P0 t þ t0 P0 RðtÞ ¼ R0 þ 0:01R0 ðt0 X0 Þ1=3 Pm t0 Pm ( ) 3=4 P0 sin ðt0 X0 Þ ln½X0 ðt þ t0 Þ þ a : ð11Þ Pm One of these motions is the increase in the radius according to the power law with the exponent noticeably different from the exponent in the Rayleigh law. The other motion is the oscillation with a constant amplitude and a logarithmically increasing period. The constant phase a cannot be calculated explicitly in the framework of the abiabatic invariant approach. It appears thus possible to analytically describe the nonlinear dynamics of a bubble driven by an external perturbation in the conditions ensuring the scale invariance of the Rayleigh equation.
3. Conformal symmetry and dynamics of tethered bubbles The numerical boundary integral method was a unique way to simulate the problem of interaction of a bubble with a boundary [13–15]. In this study we present an analytical approach by
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using the symmetry of the problem. The 10th-dimensional algebra of symmetry of the Laplace equation has the basis of the following form: Pj ¼ oj ¼ oxj ; j ¼ 1; 2; 3; J3 ¼ x2 o1 x1 o2 ;
J 1 ¼ x 3 o2 x 2 o3 ;
J2 ¼ x1 o3 x3 o1 ;
D ¼ ð1=2 þ x1 o1 þ x2 o2 þ x3 o3 Þ;
K1 ¼ x1 þ ðx21 x22 x23 Þo1 þ 2x1 x3 o3 þ 2x1 x2 o2 ; K2 ¼ x2 þ ðx22 x21 x23 Þo2 þ 2x2 x3 o3 þ 2x1 x2 o1 ; K3 ¼ x3 þ ðx23 x21 x22 Þo3 þ 2x1 x3 o1 þ 2x2 x3 o3 and leads to 17 coordinate systems admitting separation of variables [16]. The torus shaped coordinates is one of these systems (see Fig. 2). x ¼ R1 sinh n cos u;
y ¼ R1 sinh n sin u;
z ¼ R1 cos w;
R ¼ cosh n sin w:
The basis of the symmetry operators corresponding to the torus shaped coordinates has the form 2 S1 ¼ J32 , 4S2 ¼ ðP3 þ K3 Þ . Spherical segments are the coordinate surfaces of this system which are orthogonal to the torii surfaces. Thus dynamics of the tethered bubble in torus shaped coordinates can be analyzed by using an analytical approach and by analogy with dynamics of free spherical bubble. The volume pulsation of the tethered bubble are described by the modified Rayleigh equation of the form
Fig. 2. Torus shaped coordinates––natural coordinates for the tethered bubble.
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1 d2 DV cP0 Pm þ DV ¼ sinðxtÞ; 2 2R0 sin #c Cð#c Þ dt q0 V 0 q0 Z p p2 sin #c sin jdj Cð#c Þ ¼ pffiffiffi 2 2 2#c #c ð cos #c cos jÞ3=2 Z
89
ð12Þ
pn sinh 2# dn c ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1cos #c cos j pn 1cos # cos j c arccoshð cos #c cos j Þ cosh n cos #c cos j cosh 2#c 1
where DV is the variation in bubble volume, #c is the contact angle. It follows directly from this equation that the fundamental––the so-called ÔMinnaertÕ frequency for the tethered bubble is given by X20 ðR0 ; #c Þ ¼
2cR0 sin #c Cð#c ÞP0 XI2 0 ðR0 Þ4 sin #c Cð#c Þ ¼ : q0 V0 2p½4 ð1 cos #c Þ2 ð2 þ cos #c Þ
ð13Þ
The evident physical consequence of this formula is that the natural frequency of the tethered bubble depends on the contact angle and this dependence is not monotonic.
4. Distribution of coagulating bubbles A sound wave propagating in liquid containing gas bubbles causes the bubbles to coagulate due to the mutual attraction induced by the Bjerknes forces. The bubble-size distribution function gðr; t; RÞ is described by the kinetic equation og þ divðUgÞ ¼ IðgÞ; ot Z
IðgÞ ¼ I Iþ ;
1
dR0 rðR; R0 ÞUðRÞ UðR0 ÞjgðRÞgðR0 Þ; 0 Z R Iþ ¼ ð1=2Þ dR0 rðR0 ; R00 ÞjUðR0 Þ UðR00 ÞjðR2 =R02 ÞgðR0 ÞgðR00 Þ; I ¼
ð14Þ
0
where R is the radius of the bubble, U is the translational velocity U ¼ U0
ð1 s2 Þ½WrW þ W rW 6m½ð1 s2 Þ2 þ d2
id½WrW W rW 6m½ð1 s2 Þ2 þ d2
;
where U0 is the bubble velocity in the absence of sound field, s ¼ x0 =x, x0 , d are the natural frequency and logarithmic decrement of the bubble, x and W are the frequency and potential of the external field, m is the kinematic viscosity, and r is the collision cross-section of the bubbles. According to [17], rjU1 U2 j is equal to 4pk for k 0 and is equal to zero for k < 0, k ¼ ðR1 þ R2 Þð3mÞ1 ½ðs21 1Þðs22 1Þ þ d1 d2 ½ðs21 1Þ2 þ d21 jWj2 ½ðs22 1Þ2 þ d22 . In a standing-wave field bubbles with a radius smaller than resonance value ðR < R Þ–– the natural frequency of resonant bubble coincide with the frequency of the external field, come together in the vicinity of an antinode of the field, i.e., in the zone where the force of the
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radiation pressure of the standing wave equalizes the buoyancy. As a result of coagulation, the bubble cloud can acquire bubbles of large radius up to resonance value. For superresonance bubbles ðR > R Þ, the force of the radiation pressure change sign, and such bubbles escape from the cloud. The process is stationary in the generation of small bubbles. It is necessary to include the density of source Sðr; RÞ localized in the space of radii close to Rs ðRs R Þ in Eq. (14) in this case. Three intervals of characteristic radii are naturally distinguished in the description of this situation: source R Rs Sðr; RÞ IðgÞ ¼ divðUgÞ; sink R R IðgÞ ¼ divðUgÞ; the inertial interval Rs R R
IðgÞ ¼ 0:
The symmetry (homogeneity) of the collision operator can be used to find an exact solution of the nonlinear integral equation IðgÞ ¼ 0. The parameter s is large s 1 in the inertial interval, so that the kernel of the collision integral can be written in the simplified form 2
wðR1 ; R2 Þ rðR1 ; R2 ÞjU1 U2 j ’
4pðR1 þ R2 ÞjWj : 3ms21 s22
The symmetry allows one to consider the similar processes in I and Iþ terms of collision integral R0 þ R00 ! R, R þ Q0 ! Q00 , where Q0 ¼ l1 R00 , Q00 ¼ l1 R ¼ l21 R0 , l1 ¼ R=R0 . For them we have Z 1 dQ0 wðR; Q0 ÞgðRÞgðQ0 Þ I ðgÞ ¼ Z 1 Z0 1
0 ¼ dQ dQ00 wðR; Q0 ÞgðRÞgðQ0 Þ3Q002 d Q003 Q03 R3 Z0 1 Z 01
3 0 03 003 ¼ dR dR00 l31 wðR; R00 Þl1 gðl1 R0 Þgðl1 R00 Þ3R02 l41 l3 1 d R R R 0 Z0 R 0 dR wðR0 ; R00 Þgðl1 R0 Þgðl1 R00 Þl31 R2 =R002 ¼ 0
and analogous for Q0 ¼ l2 R0 , Q00 ¼ l2 R ¼ l22 R00 , l2 ¼ R=R00 Z R I ðgÞ ¼ dR0 wðR0 ; R00 Þgðl2 R0 Þgðl2 R00 Þl32 R2 =R002 : 0
As a result, the collision integral can be represented in the factorized form Z R IðgÞ ¼ ð1=2Þ dR0 wðR0 ; R00 ÞR2 =R002 ½gðR0 ÞgðR00 Þ l31 gðl1 R0 Þgðl1 R00 Þ l32 gðl2 R0 Þgðl2 R00 Þ: 0
ð15Þ a
For power distribution gðRÞ ¼ g0 ðR0 =RÞ the expression in brackets of Eq. (13) is proportional to ½1 ðR=R0 Þ32a ðR=R00 Þ32a . The condition of conservation of the volume of the gas in bubble coagulation determines the power exponent a ¼ 3. This solution corresponds to a constant flow over the spectrum of sizes.
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5. Conclusions In the presented work the study of Lie point symmetries in bubble dynamics and its applications to nonlinear acoustics has been reviewed. As the first step, explained in Section 2, the symmetry groups for the simplest, unrestricted form of single bubble oscillations are identified systematically by use of infinitesimal-transformation and prolongation theory. Then, as explained in Section 3, an appeal to symmetry as a tool for separation of variables provides an approach to study the dynamic of restricted (tethered) bubble. The symmetry of collision operator in the multibubble media has been used in Section 4 for finding exact form of distribution function of bubbles coagulating in a sound field. It appears thus possible to analytically describe the nonlinear dynamics of bubbles in a variety of physical situations. This fact offers the possibility of using the class of solutions obtained above as a convenient model for analyzing such phenomena as cavitation, sonoluminescence and shock wave propagation in liquids with phase inclusions.
Acknowledgements This work was supported by the Russian Foundation for Basic Research under project numbers 01-05-64915, 01-02-96901.
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[16] Miller Jr W. Symmetry and separation of variable. London: Addison-Wesley; 1977. [17] Kobelev Yu. Nonlinear dipole oscillations of the spherical particle in a sound field. Akust Zh 1983;29:783–9 [English translation in: Sov Phys Acoust 1983;29:463–7].