Effects of a Uniform Magnetic Field on a Growing Or Collapsing Bubble in a Weakly Viscous Conducting Fluid K

Effects of a uniform magnetic field on a growing or collapsing bubble in a weakly viscous conducting fluid K. H. Kang, I. S. Kang, and C. M. Lee

Citation: Phys. Fluids 14, 29 (2002); doi: 10.1063/1.1425410 View online: http://dx.doi.org/10.1063/1.1425410 View Table of Contents: http://pof.aip.org/resource/1/PHFLE6/v14/i1 Published by the American Institute of Physics.

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Downloaded 07 May 2012 to 132.236.27.111. Redistribution subject to AIP license or copyright; see http://pof.aip.org/about/rights_and_permissions PHYSICS OF FLUIDS VOLUME 14, NUMBER 1 JANUARY 2002

Effects of a uniform magnetic field on a growing or collapsing bubble in a weakly viscous conducting fluid K. H. Kang Department of Mechanical Engineering, Pohang University of Science and Technology, San 31, Hyoja-dong, Pohang 790-784, Korea I. S. Kang Department of Chemical Engineering, Pohang University of Science and Technology, San 31, Hyoja-dong, Pohang 790-784, Korea C. M. Lee Department of Mechanical Engineering, Pohang University of Science and Technology, San 31, Hyoja-dong, Pohang 790-784, Korea ͑Received 13 May 1999; accepted 14 September 2001͒ The effects of a uniform magnetic field on a growing or collapsing bubble are investigated. The governing equations for the volume and shape modes of oscillation are derived. To obtain the pressure correction due to the electromagnetic field, the perturbed flow by the electromagnetic force is analyzed with the aid of the domain perturbation method. The viscous effect is assumed to be confined to a thin layer adjacent to the bubble surface. It is shown that the electromagnetic field exerts a damping force on the volume mode so that both the growing and collapsing speeds of a bubble are reduced. The magnetic field also affects the shape mode by contributing to a forcing term. Due to the forcing term, the shape of a growing–collapsing bubble becomes unstable even in the case of no initial disturbance. © 2002 American Institute of Physics. ͓DOI: 10.1063/1.1425410͔

I. INTRODUCTION be significantly reduced by a magnetic field. A similar experiment was conducted by Kamiyama et al.9 in tap water. The violent growth or collapse of bubbles, in the form of Although the result is contrary to that of Perel’man and Go- either cavitation or vapor explosion, has the potential to vorskii, it is clearly shown that the magnetic field can give damage the surrounding objects.1 The most common ex- visible changes to the dynamics of cavitation cloud and ero- ample is the erosion of hydraulic machinery due to hydrody- sion area even in tap water. Kamiyama and Yamasaki10 also namic cavitations. In particular, the cavitation in flowing liq- investigated the effect of a magnetic field on hydraulic cavi- uid metal is an important subject in the design and safety tation of flowing mercury by using two types of venturi tubes analysis of the liquid–metal cooled fast breeder reactor. In which have a step expansion and a gradual expansion, re- addition, a catastrophic accident could occur due to the direct spectively. They found that the inception of cavitation in the fuel–coolant interaction followed by the loss of coolant flow, case of step expansion is significantly influenced by a mag- or a large leak of chemically reactive liquid metal sub- netic field. stances, such as sodium.2–4 In a certain liquid metal Hammitt et al.11 also investigated the effect of a mag- magnetohydrodynamic ͑MHD͒ energy conversion system, netic field on cavitation inception by using a vibratory appa- the circulation of conducting fluids is driven by the volume ratus in tap water. However, they could not find any differ- expansion of gases or water. The direct contact boiling of ence in threshold pressure-amplitude for cavitation inception. liquid can be explosive enough to cause damage to the sur- Another piece of evidence on the effect of a magnetic field rounding objects.5 on bubble dynamics was given by the experiment of Wong It has been anticipated that the magnetic field can be et al.,12 in which the effect of a magnetic field on the growth used to control the growth or collapse of bubbles in conduct- of gas bubbles supplied through an aperture was investi- ing fluids. Such an idea is based on past experimental obser- gated. They showed that if the magnetic field is applied per- vations. Shalobasov and Shal’nev6 showed, in a cavitation pendicular to the direction of gravity, the bubble departure tunnel filled with tap water, that a magnetic field of less than frequency from a plate in a liquid metal can be significantly 1 tesla can significantly change the degree of cavitation ero- reduced. sion. Later, Shal’nev et al.7 showed that the growth rate of a There are other situations in which bubble dynamics is spark-generated cavitation bubble in stationary tap water is influenced by a magnetic field. An electromagnetic force is significantly increased by an applied magnetic field. applied during the solidification of alloys for grain refine- Perel’man and Govorskii8 showed, by using a vibratory ment and degassing. Vive`s13 showed that, when an alloy is apparatus in a liquid metal ͑probably sodium͒, that the solidified in the presence of well-developed cavitation in- weight loss of a metal specimen due to cavitation erosion can duced by a time-varying electromagnetic force, a very fine

Downloaded 07 May 2012 to 132.236.27.111. Redistribution subject to AIP license or copyright; see http://pof.aip.org/about/rights_and_permissions 30 Phys. Fluids, Vol. 14, No. 1, January 2002 Kang, Kang, and Lee

compressibility of the surrounding fluid, condensation, evaporation, and heat transfer, which in some cases become important, are neglected. Furthermore, the effect of a solid wall is not considered. As is well known, the existence of a wall can significantly change the dynamics of a bubble, especially when the distance from the center of the bubble to the wall is in the order of the maximum bubble radius.15 However, the presence of the wall can only complicate the FIG. 1. The coordinate system adopted in this study. analysis of the problem without aiding to the main objective of the present investigation. The spherical coordinate system (r,␪,␸) is adopted and and homogeneous microstructure can be obtained. Another the unit vectors in r, ␪, and ␸ directions are denoted by e , example is found in the continuous casting process of steel. r e␪ , and e␸ , respectively. The uniform magnetic field is ap- Inert argon gas is injected in the molten steel to capture plied in the direction parallel to the positive x axis as impurities, while a magnetic field is applied to stabilize the ϭ ϭ ␪Ϫ ␪͒ ͑ ͒ flow of molten steel in the mold. B0 B0ex B0͑er cos e␪ sin , 1 As described above, numerous problems are associated where e denotes the unit vector parallel to the x axis. The with the dynamics of bubbles in a magnetic field. However, x flow exterior to the bubble satisfies the following momentum very few theoretical results are known for the influence of equation:16 the magnetic field on the dynamics of bubbles in conducting ץ fluids, especially on a cavitation and boiling bubble. Wong u p 1 FL ϩ⍀ϫuϭϪٌͩ ϩ u2 ͪ ϩ Ϫ␯ٌϫ⍀, ͑2͒ 12 t ␳ 2 ␳ץ -et al. studied the effect of a magnetic field on the deforma tion of a bubble attached to a wall by using spheroidal ap- ϭ proximation. However, authors did not provide a complete where u (ur ,u␪,0) denotes the fluid velocity, p the dynamic -analysis of the magnetic-field effects on the bubble dynamics pressure and ⍀(ϭٌϫu) the vorticity, respectively. The Lor in conducting fluids. The purpose of the present work, there- entz force FL is given by fore, is to investigate the effects of a uniform magnetic field F ϭJϫB, ͑3͒ on a growing or collapsing bubble in a weakly viscous con- L ducting fluid. where J is the electric current density. From the Maxwell equations and Ohm’s law, the current density becomes II. PROBLEM FORMULATION 1 Jϭ ٌϫBϭ␴͑EϩuϫB͒, ͑4͒ In the present work, we are concerned with the axisym- ␮ metric dynamics of a bubble in a weakly viscous, incom- pressible, and electrically conducting fluid subject to a uni- where E represents the electric field. ͑ ͒ form magnetic field, as shown in Fig. 1. The external fluid In the momentum equation, 2 , the force due to the has the density ␳, the kinematic viscosity ␯, the conductivity gravitational acceleration is neglected. It can be shown with- ␴, and the magnetic permeability ␮. All these material prop- out difficulty that the ratio of the gravity force with respect to ␳ ␴ ˙ 2 erties are assumed to be uniform in the external fluid phase. the electromagnetic force becomes g/( RB0) in which g The interior of the bubble is assumed to be filled with a represents the gravitational acceleration, and R˙ the radial ve- vapor and a permanent gas of low density so that the effect locity of the bubble wall. The ratio is much less than unity of the fluid motion inside the bubble on the external flow for typical liquid metals such as mercury, sodium, molten field can be neglected. The surface of the bubble is assumed steel, and liquid aluminum, under a normal condition of, say, ϭ ͉ Ϫ ͉ϭ to be characterized by a uniform surface tension ␥. B0 1 tesla and pvap pϱ 1 bar. Here, pvap is the vapor ϭ␴␮ The magnetic Reynolds number Rm lcuc , where lc pressure of the surrounding fluid, and pϱ the constant pres- and uc are the characteristic length and the velocity scales, is sure at a distance large enough to be independent of bubble assumed to be sufficiently small. If interior gas phase and motion. exterior liquid phase are assumed to be nonmagnetizable, The shape of an axisymmetric bubble can be represented both phases have the same magnetic permeability. In this by the function ⌿(r,␪,t) defined as case, the magnetic field is not perturbed by the presence of a ⌿͑r,␪,t͒ϭrϪ␨͑␪,t͒ϭ0. ͑5͒ bubble, as long as the magnetic field associated with induced current due to bubble motion is neglected. At the bubble surface, the following kinematic condition and Assumption of the fluids being nonmagnetizable does the normal stress conditions should be satisfied: not impose any severe limitation on the range of application ␨ץ ␨ u␪ץ of the present analysis. This is because many practically im- ϭ ϩ ͑ ͒ ur , 6 ␪ץ • t rץ ,portant fluids, such as water, air, sodium, and potassium belong to this category.14 n͑n Tem͔͔͒ϩ͓͓n ͑n Th͔͔͒ϭ␥ٌ n, ͑7͓͓͒ The main purpose of the present analysis is to firstly • • • • ͓͓͑ ͔͔͒ elucidate the effect of a magnetic field on the dynamics of a where the symbol • denotes the external quantity minus bubble. Thus, the effects of other parameters such as the the internal quantity, and n the outward unit normal vector

on the bubble surface. In ͑7͒, Tem and Th represent the elec- The momentum equation can be rewritten in the nondimen- tromagnetic and the hydrodynamic stress tensors, and they sional form as u 1ץ are given by ͒ ͑ ϩ⍀ϫ ϭϪٌ⌽ϩ ͑ ϫ ͒ϫ Ϫ ٌϫ⍀ u N u ex ex , 14 ץ ͒ ͑ ͔ emϭ ͓ Ϫ 1 2 ͔ϩ␮͓ Ϫ 1 2 T e EE 2E I HH 2H I , 8 t Re

h T ⌽ϭ ϩ ͑ ͒ T ϭϪpIϩ␳␯ٌ͓uϩٌu ͔, ͑9͒ where p u•u/2. If the curl and divergence of 14 are taken, respectively, the following vorticity and pressure ϭ ␮ where H B/ and e is the electric permittivity of the sur- equations are obtained: rounding fluid. ⍀ץ As we can see in ͑7͒, the applied magnetic field can ϩٌϫ͑⍀ϫu͒ tץ affect the bubble dynamics in two ways. One is the direct contribution denoted by Tem and the other is the indirect 1 ͒ ͑ contribution through the change in Th. It can be shown with- ϭ ٌϫ ϫ ͒ϫ Ϫ ٌϫٌϫ⍀ N ͓͑u ex ex͔ , 15 out difficulty that the electric field E in ͑4͒ vanishes to the Re first-order effect of the magnetic field, under the assumption ٌ2⌽ϭNٌ ͓͑uϫe ͒ϫe ͔Ϫٌ ͑⍀ϫu͒. ͑16͒ of no free electric charge inside the conducting fluid medium • x x • ϭ ͒ ϭ The velocity vector can be decomposed into two parts ٌ ͑ i.e., •J 0 . Since E 0 and the magnetic permeabilities are the same both inside and outside the bubble, we can see which describe the potential flow field and the rotational flow from ͑8͒ that ͓͓Tem͔͔ vanishes. field, respectively as ͑ ͒ The normal stress condition 7 at the bubble surface is uϭٌ␾ϩ˜uϭٌ␾ϩٌϫA. ͑17͒ reduced to the following form: Here, ␾ is the velocity potential, and the vector potential A ͒ ͑ ٌ␥Ϫ ϩ ␶͉ ϭ pin pout n•n• out •n. 10 represents the velocity field due to the rotational component Here ␶ denotes the viscous stress tensor. The pressure inside of the fluid motion. This vector potential has only ␸ bubble p can be represented as -directional component, which is denoted by A␸ , owing to in the axisymmetry of the flow assumed in this study. This ␬ R 3 function is related to the stream function ␺ and the r, ␪ ϭ ϩ ͩ o ͪ ͑ ͒ p p p , 11 17 in vap go R components of the velocity vector as ␺ in which pgo is the initial gas pressure, Ro the initial radius of A␸ϭ , ͑18a͒ the bubble, R the radius of the bubble, and ␬ the specific heat r sin ␪ ratio. ͒ ͑ץ ץ From the above argument, we have also seen that the 1 1 rA␸ ˜u ϭ ͑sin ␪ A␸͒, ˜u␪ϭϪ . ͑18b͒ rץ ␪ rץ ϭ␴ ϫ r r sin ␪ Lorentz force is given approximately by FL (u B0) ϫ B0 . This Lorentz force modifies the flow field outside the The vorticity also has only ␸-directional component ␻ ͑i.e., bubble, and this flow-field change results in the change of ⍀ϭ(0,0,␻)͒ which is related to A␸ as hydrodynamic stress at the bubble surface. In the following A␸ ͒ ͑ ͪ section, the effect of the magnetic field on the hydrodynamic ␻ϭٌ͑ϫ ͒ ϭϪͩ ٌ2 Ϫ ˜u ␸ A␸ , 19 stress at the bubble surface will be discussed. Before pro- r2 sin2 ␪ ceeding further, we first want to derive an equation for modi- where the subscript ␸ denotes the ␸-directional component. fication of the fluid velocity due to the magnetic field. For convenience, we nondimensionalize the equations III. HYDRODYNAMIC STRESS CHANGE by introducing the following nondimensional variables: DUE TO A MAGNETIC FIELD ͉ Ϫ ͒ ␳͉1/2 r ͑pvap pϱ / t r*ϭ , t*ϭ , It is assumed that the axisymmetric bubble dynamics can Ro Ro be described by the shape function u rϭ␨͑␪,t͒ϭR͑t͒ϩ␧g͑␪,t͒ ϭ ͑ ͒ u* ͉ Ϫ ͒ ␳͉1/2 , 12 ͑pvap pϱ / ϱ ϭ ͒ϩ␧ ͒ ␪͒ ͑ ͒ R͑t ͚ an͑t Pn͑cos , 20 where the asterisk denotes the dimensionless quantities. nϭ2 Hereafter, the asterisk is dropped for convenience’s sake. For ␧ later usage, the magnetic interaction parameter N, the Rey- where is a small perturbation parameter representing the nolds number Re, and the Weber number We are introduced distortion of the bubble from the spherical shape, and ␪ and defined, respectively, by Pn(cos ) denote the Legendre polynomials. The case of n ϭ1 represents the translation of the bubble. Since the mag- ␴B2R ϭ o o netic field has no influence on this mode of motion, as ex- N ͉␳ Ϫ ͉͒1/2 , ͑ ͒ ϭ ͑pvap pϱ plained below 49 , n 1 is excluded from the series given in ͑20͒. ͉͑ Ϫ ͒ ␳͉1/2 ͉ Ϫ ͉ Ro pvap pϱ / pvap pϱ Ro If it is assumed that Reӷ1, the effect of viscosity is Reϭ , Weϭ . ͑13͒ ␯ ␥ confined to a thin boundary layer adjacent to the bubble sur-

Downloaded 07 May 2012 to 132.236.27.111. Redistribution subject to AIP license or copyright; see http://pof.aip.org/about/rights_and_permissions 32 Phys. Fluids, Vol. 14, No. 1, January 2002 Kang, Kang, and Lee face. Thus, the flow domain can be divided into two regions. ͑ ͒ ٌ ϭ O N : •um 0, One is the inviscid region where the viscous effect can be ץ neglected, and the other is the thin boundary layer near the um , ϩ⍀ ϫu ϭϪٌ⌽ ϩ͑u ϫe ͒ϫe t m 0 m 0 x xץ .bubble wall where the viscous effect should be considered The effect of viscosity can be included by adding the viscous ϭ ϭ ͑ ͒ pressure correction and the viscous stress term in the normal umr 0atr R, 28 stress balance, while the external flow field is computed us- ⌽ ϭ ϩ ␧ where m pm u0•um . The O(1) and O( ) problems ing the inviscid approximation. yield the solutions18 For the inviscid region, the governing equations for the flow field are given by R2R˙ ͒ ͑ ϭٌ␾ ␾ ϭϪ u0 0 , 0 , 29 u rץ uϭ0, ϩ⍀ϫuϭϪٌ⌽ϩN͑uϫe ͒ϫe . ͑21͒ ٌ t x x ϭٌ␾ץ • u1 1 , Here we perform a perturbation analysis based on the as- ϱ r R nϩ2 2a R˙ sumptions of 0Ͻ␧Ӷ1 and 0ϽNӶ1 implying small distur- ␾ ϭϪ ͩ ͪ ͩ ϩ n ͪ ͑ ␪͒ ͑ ͒ 1 ͚ a˙ n Pn cos . 30 bances by the fluid motion and electromagnetic field, respec- nϭ1 nϩ1 r R tively. The perturbation expansion of u and p are in the form The O(N) problem can be solved as follows. The vor- of ticity and the pressure equations for this order are uϭu ϩ␧u ϩNu ϩ , pϭp ϩ␧p ϩNp ϩ . ⍀ץ ¯ m ¯ 0 1 m 1 0 ͑ ͒ m ϩٌϫ͑⍀ ϫu ͒ϭٌϫ͓͑u ϫe ͒ϫe ͔, ͑31͒ 22 t m 0 0 x xץ The first terms in the expressions (u0 ,p0), describe the po- ͒ ͑ ͒ tential flow field due to the radial motion of the bubble wall. ٌ2⌽ ϭٌ ͓͑ ϫ ͒ϫ ͔Ϫٌ ͑⍀ ϫ m • u0 ex ex • m u0 . 32 The second terms (␧u ,␧p ) represent the potential flow 1 1 ⍀ ϭ ␻ field due to the distortion of the bubble from the spherical When the vorticity vector is given as m (0,0, m), the vorticity equation for the perturbed flow due to the electro- shape. The third terms, (Num ,Npm) represent the rotational flow due to the electromagnetic force. Thus the vorticity in magnetic force is obtained as ͑ ͒ ␻ u ץ ␻ץ can be represented as 21 m ϩ ͩ m ͪ ϭ 0r 1͑ ␪͒ ͑ ͒ ⍀ϭٌϫ ϭ ٌϫ ϵ ⍀ ͑ ͒ ru0r P2 cos , 33 r r rץ tץ u N um N m , 23 and ⌽ can be expanded as ϭ 2 ˙ 2 1 ␪ ϭ ␪ ␪ where u0r R R/r and P2(cos ) 3 cos sin . Under the ⌽ϭ͑ ϩ 1 ͒ϩ␧͑ ϩ ͒ϩ ͑ ϩ ͒ assumption that the vorticity is initially zero everywhere, i.e., p0 2u0 u0 p1 u0 u1 N pm u0 um • • • ␻(r,␪,0)ϭ0, the solution of ͑33͒ can be obtained as ϩ . ͑24͒ ¯ r On the other hand, the kinematic condition ͑6͒, which should ␻ ͑r,␪,t͒ϭͫ Ϫ1ͬP1͑cos ␪͒. ͑34͒ m ͑r3ϪR3ϩ1͒1/3 2 be satisfied at rϭ␨ϭRϩ␧g, can be transformed to an equivalent condition at rϭR as It is noteworthy that the vorticity distribution is depen- dent only on the bubble radius R and independent of the rate ץ ץ u0r g u ϩ␧u ϩNu ϩ␧g ϩ ϭR˙ ϩ␧ of radius change R˙ . This result may seem to be a little tץ ¯ rץ 0r 1r mr strange at first glance. However, it is possible, because the viscous effect is neglected. Furthermore, we can see from ץ u0␪ g ϩ␧ ϩ . ͑33͒ that ¯ ␪ץ R ͑ ͒ ˙ ␻ ␻ 2 ץ ␻ 25 D m m m R R .ϭ ͩ ͪ ϩu ٌͩ ͪ ϭ P1͑cos ␪͒ ͪ ͩ t r 0• r r4 2ץ Now, we have the following continuity equation, mo- Dt r mentum equation, and kinematic boundary condition, for the ␻ problems of O(1), O(␧), and, O(N), respectively, for the The above equation indicates that D( m /r)/Dt changes its region rуR: sign when R˙ changes sign. Now the perturbed velocity field due to the magnetic ץ u0 1 ͑ ͒ ͑ O͑1͒: ٌ u ϭ0, ϩ ٌ͑u u ͒ϭϪٌp , field, um , can be obtained by computing A␸ in 19 as note ͒ t 2 0• 0 0 ϭ ␻ϭ ␻ץ 0 • ˜u Num and N m to the first order of N

u ϭR˙ at rϭR, ͑26͒ A␸ r ͒ ͑ 0r ٌ2 Ϫ m ϭͫ Ϫ ͬ 1͑ ␪͒ A␸ 1 P cos , 35 m r2 sin2 ␪ Z͑r,R͒1/3 2 uץ ␧͒ ٌ ϭ 1 ϩٌ͑ ͒ϭϪٌ͑ O : u1 0, u1 u0 p1 , ϭ ϭ 3Ϫ 3ϩ -t • where A␸ NA␸m and Z(r,R) r R 1. From the kineץ • matic boundary condition that ˜u ϭNu ϭ0atrϭR, and the r mr ץ ץ ץ g u0r u0␪ g condition at inﬁnity, the function A␸ should satisfy the fol- u ϭ Ϫg ϩ at rϭR, ͑27͒ m :␪ lowing boundary conditionsץ r Rץ tץ 1r

The stream function for the magnetically induced flow can be obtained from ͑18a͒ and ͑37͒. In Fig. 2, the stream- ␺ ␻ lines ( m) and vorticity contours ( m) are shown for R ϭ ϭ ͑ ␺ϭ ␺ ͒ 1.01 and R 0.99 note N m . It is found that the flow direction changes as the value of R decreases from greater than unity to less than unity. For a growing bubble (RϾ1), the perturbed flow by the magnetic field is incoming near the ␪ϭ 1␲ ␪ϭ ␲ equator ( 2 ) and outgoing near the poles ( 0, ). On the other hand, in the case of a collapsing bubble (RϽ1), the flow direction is reversed. Therefore, the magnitude of the overall velocity in the inviscid region to the first order of ␧ Х ϩ␧ ϩ and N, i.e., (u u0 u1 Num), becomes smaller near the equator than near the poles, regardless of the direction of bubble wall motion. Next, the pressure correction due to the magnetic field is obtained to determine the bubble shape by using the normal ⌽ ϭ ϩ stress balance. Since m pm u0•um for which we know ⌽ ͑ ͒ u0 and um , pm can be obtained if m is known. From 32 , we have

͒ ͑ ͒ 2⌽ ϭٌ ͓ ͑ ϫ ͒ϫ ͔Ϫٌ ͑␻ٌ m • u0r er ex ex • mu0re␪ . 39 ␻ ͑ ͒ We substitute m of 34 into the above equation to have the reduced pressure equation

4u 3 r FIG. 2. The streamlines and vorticity contours of the perturbed flow due to 2 0r ϭ ͑ ͒ ٌ ⌽ ϭ ͫ1Ϫ ͬP ͑cos ␪͒. ͑40͒ ͒ ͑ a magnetic field: a growing bubble (R 1.01); b collapsing bubble (R m r 2 Z͑r,t͒1/3 2 ϭ0.99). The solution of the above equation can be expressed as follows: ϭ ϭ A␸m 0atr R, ϱ dn A␸ →0asr→ϱ. ͑36͒ ⌽ ͑ ␪ ͒ϭ ͑ ͒ ͑ ␪͒ϩ ͑ ␪͒ m m r, ,t h r,t P2 cos ͚ nϩ1 Pn cos . nϭ0 r The solution of Eq. ͑35͒ has the following form: ϱ Ϫ 3 1/3 b The power-series expansion of 1 2•r/Z(r,t) is intro- ͑ ␪ ͒ϭ ͑ ͒ 1͑ ␪͒ϩ n 1͑ ␪͒ A␸m r, ,t f r,t P2 cos ͚ nϩ1 Pn cos . duced to find h(r,t). The unknown coefficients dn can be ϭ r n 1 determined by using the r-directional momentum equation of 1/3 ͑ ͒ ⌽ The binomial expansion of 1Ϫr/Z(r,R) is introduced to 14 to the O(N). Then m becomes find f (r,t), and then the boundary conditions of ͑36͒ are ץ used to determine the unknown coefficients b , so that 2 R2R˙ h͑r,t͒ 2 n ⌽ ͑ ␪ ͒ϭ ϩ ˙ Ϫ ͓ 3 ͑ ͔͒ m r, ,t RRͭ R f R ͮ Rץ R 3 3 r ˙ r3R ͑ ␪ ͒ϭͫ ͑ ͒Ϫ ͑ ͒ͩ ͪ ͬ 1͑ ␪͒ ͑ ͒ RR A␸ r, ,t f r,t f R P cos , 37 m r 2 ϫ ␪͒ ͑ ͒ P2͑cos , 41 in which, for RϽ1 ϱ in which, for RϽ1 r2 1 1 S 5/3 5 8 8 r3 ͑ ͒ϭ ͩ 3 ͪ ͑Ϫ ͒nͭ ͩ ͪ ͩ Ϫ ͪ f r,t ͚ 1 2F1 , n; ; ϱ 5 nϭ1 n 5 Z 3 3 3 Z R2R˙ R2R˙ 1 h͑r,t͒ϭ ϩ ͚ ͩ 3 ͪ 1 S n S 3r r nϭ1 n ϩ ͩ ͪ ͩ ϩ ͪͮ ͑ ͒ 2F1 n,1;n 1; , 38a 3n Z Z 3 S 2/3 2 5 5 r3 ϫ͑Ϫ1͒nͭ ͩ ͪ F ͩ , Ϫn; ; ͪ and for RϾ1 5 Z 2 1 3 3 3 Z 5 3 2/3 Ϫ2r ϩ͑2r Ϫ3S͒Z 2 S n S f ͑r,t͒ϭ ϩ ͩ ͪ ͩ ϩ ͪ ͑ ͒ 3 F n,1;n 2; ͮ , 42a 50r 5͑nϩ1͒ Z 2 1 Z 1 S 4 S Ͼ Ϫ F ͩ 1,1, ;2,2;Ϫ ͪ , ͑38b͒ and for R 1 45 r 3 2 3 r3 2 5/3 5 where F and F represent hypergeometric functions,19 2 R R˙ 3 Z Ϫr 2 1 3 2 h͑r,t͒ϭϪ ϩ . ͑42b͒ and Sϵ1ϪR3. When Rϭ1, f (r,t) becomes null. 3 r 5 r3S

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ϭ⌽ Ϫ Since pm m u0rumr , the electromagnetic correction of the pressure, to the ﬁrst order of N, becomes

2 R2R˙ h͑r,t͒ ͑ ␪ ͒ϭ ϩ ˙ Npm r, ,t N NRRͭ 3 r RR˙

ץ 2 Ϫ ͓ 3 ͑ ͔͒ ͑ ␪͒ R f R ͮ P2 cos Rץ r3R

3 6N R Ϫ ͭ f ͑r,t͒Ϫ f ͑R͒ͩ ͪ ͮ P ͑cos ␪͒. 2 r r ͑43͒ Thus, the electromagnetic correction of pressure at the bubble wall is ͑ ␪͒ϭ 2 ˙ ϩ ˙ ͑ ͒ ͑ ␪͒ ͑ ͒ Npm R, 3NRR NRRQ R P2 cos , 44 in which, for RϽ1 ϱ 1 1 3S2/3 2 5 5 ͑ ͒ϭ ϩ ͩ 3 ͪ ͑Ϫ ͒nͭ ͩ Ϫ 3 ͪ Q R ͚ 1 2F1 , n; ;R 3 nϭ1 n 5 3 3 3 2Sn ϩ F ͑n,1;nϩ2;S͒ 5͑nϩ1͒ 2 1 2R3S2/3 5 8 8 ϩ F ͩ , Ϫn; ;R3 ͪ 5 2 1 3 3 3 2Sn Ϫ F ͑n,1;nϩ1;S͒ͮ , ͑45a͒ 3n 2 1 and for RϾ1 R2ϩRϪ2 2 S 4 S Q͑R͒ϭ ϩ F ͩ 1,1, ;2,2;Ϫ ͪ . 3͑R2ϩRϩ1͒ 9 R3 3 2 3 R3 FIG. 3. The function Q(R) defined in ͑45͒: ͑a͒ growing bubble (R˙ Ͼ0); ͑b͒ ͑45b͒ collapsing bubble (R˙ Ͻ0). The function Q(R) is shown in Fig. 3 for the cases of a ͓ ͑ ͔͒ ͓ growing bubble Fig. 3 a and a collapsing bubble Fig. induced flow, is most noticeable near the equator ͑see Fig. 2͒. ͑ ͔͒ 3 b . The function Q is negative for all R in the case of a This will result in the increase of the pressure near the equa- growing bubble (R˙ Ͼ0, RϾ1). On the contrary, Q is positive tor, regardless of the bubble wall motion. Thus, even in the in the case of a collapsing bubble (R˙ Ͻ0, RϽ1). Conse- case of a collapsing bubble, the pressure near the equator ␪ ͑ ͒ ␪ϭ 1␲ ␪ϭ ␲ quently, the coefficient of P2(cos )in 44 is always nega- ( 2 ) has larger values than near the poles ( 0, ). tive for both cases. This means that the magnetically induced pressure has greater value near the equator of the bubble than ϭϪ ␪ϭ␲ ϭ ␪ IV. BUBBLE SHAPE near the poles, since P2 1/2 at /2 and P2 1at ϭ0, ␲. The normal stress condition given by ͑10͒ at the bubble ͓ ϭ ϩ␧⌺ϱ ␪ ͔ This result can be interpreted as follows. In the momen- surface r R(t) nϭ2an(t)Pn(cos ) is given in dimen- tum equation, ͑14͒, the Lorentz force acting on fluid ele- sionless form as ments around a bubble can be expressed to the first order of ␾ 1ץ 1 N as Ϫͩ ϩ ϩ ͪ ϩͩ ϩ ٌ␾ ٌ␾ͪ p pϱ p␯ Np • t 2ץ in Re m ϭϪ 2 ␪ ϩ ␪ ␪ ͒ FL Nu0r͑sin er sin cos e␪ . 2␾ 1ץ 2 It is evident that the magnitude of the r-directional force is ϩ ϭ ٌ n, ͑46͒ • r2 Weץ Re greatest near the equator, and the radial direction of the force ␾ϭ␾ ϩ␧␾ is always opposite to the direction of bubble wall motion. So, where 0 1 ,(1/Re)p␯ the viscous pressure correc- without considering the effect of magnetically induced flow, tion, and Npm the pressure correction due to the magnetically the pressure near the equator is increased for a growing induced flow. The viscous pressure correction can be ob- bubble, and decreased for a collapsing bubble. However, the tained, by analyzing the flow under the boundary layer as- deceleration of the flow velocity, due to the magnetically sumption, as follows:20

FIG. 4. The effect of a magnetic ﬁeld on the volume mode of a growing bubble in the case of Reϭ20, Weϭ2.

ϱ 1 2␧ n p␯ϭ ͚ Re Re nϭ2 nϩ1

ϩ ͒ Ϫ ͒ ˙ ͑n 2 a˙ n ͑n 1 Ran ϫͫ Ϫ ͬ P ͑cos ␪͒. ͑47͒ R R2 n Substituting ͑29͒, ͑30͒, ͑44͒, and ͑47͒ into ͑46͒,wecan obtain the equations of motion of the bubble wall. The pro- cedure is straightforward and here we present only the results. For the spherically symmetric motion, the following equation is obtained: 3 4 R˙ 2 2 1 1 3␬ RR¨ ϩ R˙ 2ϩ ϩ NRR˙ ϩ Ϫp ͩ ͪ 2 Re R 3 We R go R

p ϪPϱ ϭ vap ͑ ͒ ͉ Ϫ ͉ . 48 pvap pϱ FIG. 5. The effect of a magnetic field on the P2-mode shape change of a ϭ ϭ ͑ ͒ •••••• ϭ The fourth term in the left-hand side of the above equation growing bubble in the case of Re 20 and We 2: a N 0, a20 ϭ ϭ ϭ ͑ ͒ ϭ ϭ represents the contribution from the magnetic field. When N 1; –––––- N 0.1, a20 1; b N 0.1, a20 0; –•–•–•–• N ϭ0.1, a ϭϪ1. is null, the above equation becomes identical to the 20 Rayleigh–Plesset equation. From ͑48͒, we can also see that the magnetic field exerts a damping force to the bubble. This result is consistent with the well-known fact that the static magnetic field retards the fluid motion when the magnetic- magnetic field. It should be noted that the term is always field direction is not parallel to the flow direction. positive as explained below ͑45b͒. This means that the mag- For nу2 ͑shape modes͒, the following equation is ob- netic field results in a positive forcing term in the P2-mode tained: of bubble dynamics. The magnetic field has a destabilizing effect on bubble shape for both growing and collapsing 3R˙ 1 2͑2nϩ1͒͑nϩ2͒ a¨ ϩͫ ϩ ͬa˙ ϩ͑nϪ1͒ bubbles, irrespective of the initial condition of bubble shape, n R Re R2 n which will be demonstrated in detail in this section. As explained below ͑48͒, however, the magnetic field has a damp- 1 ͑nϩ1͒͑nϩ2͒ R¨ 2͑nϩ2͒ R˙ ϫͫ Ϫ ϩ ͬa ing effect on the volume mode of motion. A distinction We R3 R Re R3 n should be made for the influence of the magnetic field on the two modes of bubble motion. N ϭϪ ͑ ϩ ͒ ͑ ͒ ˙ ␦ ͑ ͒ In Figs. 4 and 5, the solutions of R and a for a growing ␧ n 1 Q R R n2 . 49 2 bubble are shown for the case of Reϭ20, Weϭ2, R˙ (0) ␦ ϭ ϭ ϭ ͑ ͒ The symbol n2 is the Kronecker delta. To obtain a mean- 0.01, and a˙ 2(0) 0. In the case of N 0 in Fig. 5 a ,an ␧ ϵ ϭ ingful result, the small parameter for shape deformation initial disturbance of a2(0) a20 1 is given. For compari- should be O(N). Thus in this study, we set ␧ϭN. The term son with the results of Prosperetti and Seminara,21 the pa- in the right-hand side represents the contribution from the rameters are intentionally chosen to be identical to those of

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FIG. 7. The effect of a magnetic ﬁeld on a collapsing bubble for the volume mode in the case of Reϭ1000 and Weϭ2000.

1 1 1 1 ͑R˙ ͒2ϭ␭2ͩ 1Ϫ ͪ Ϫ ͩ 1Ϫ ͪ R3 We R R2

3͑␬Ϫ1͒ pgo 1 1 2 ϩ␭2 ͫͩ ͪ Ϫ1ͬ, ␭2ϭ . ͑1Ϫ␬͒ R3 R 3 For a growing bubble, the velocity of the bubble wall asymp- totically approaches the constant ␭ as R becomes larger. Then, for a large value of R, ͑49͒ can be approximated to ϭϪ ␭ a¨ 2 3 Q(R). For a large value of R, therefore, a2 increases monotonically even without the initial disturbance ͓note Q(R)Ͻ0 for a growing bubble and see Fig. 5͑b͔͒. Thus, the shape of the bubble is significantly deformed from the spherical shape as shown in Fig. 6͑b͒ in which ϭ a20 0. It is evident that the P2-mode of a growing bubble FIG. 6. The temporal evolution of the surface of a growing bubble in the becomes unstable, due to the magnetic field, regardless of ϭ ϭ case of Re 20 and We 2. initial conditions. As mentioned earlier, the magnetically in- ␪ϭ 1␲ duced pressure near the equator ( 2 ) has greater value than near the poles (␪ϭ0,␲) of the bubble. The uniform magnetic field, therefore, has a tendency to change the grow- Fig. 5 of Prosperetti and Seminara.21 It is confirmed that ing bubble into a prolate shape in the direction of the mag- identical results to those of Prosperetti and Seminara are ob- netic field and eventually make the bubble unstable, as illus- ͑ ͒ tained in the case of no magnetic field. trated in Fig. 6 b . If the volumes of two bubbles in Figs. ͑ ͒ ͑ ͒ The growth rate of the bubble for volume mode is re- 6 a and 6 b are compared, the bubble which is under the duced due to the damping effect of the magnetic field, as influence of a uniform magnetic field has smaller volume due to the damping effect of the magnetic field for volume mode shown in Fig. 4. The shape mode is in general stable for a of motion. growing bubble in the absence of external disturbances, and For illustration purposes, the time history of bubble therefore the initial disturbance soon diminishes, as shown in shape in Fig. 6͑b͒ is intentionally extended to the final stage Fig. 5͑a͒. That is, the shape of the bubble becomes spherical of large distortion. The present linear analysis for a large ͑ ͒ ⌬ as shown in Fig. 6 a , in which t denotes the time interval distortion may not be valid. However, based on the previous between each solid line. study of collapsing bubbles obtained from the linear To see the magnetic-field effect on the shape mode, the analysis,22 it can be assumed that the linear theory may be solution of a2(t) is obtained and shown in Fig. 5 for N extended even for large deformation. The accurate limit of ϭ ϭϪ 0.1 with three different initial conditions of a20 1, 0, the linear analysis can only be confirmed by properly con- and 1. Before discussing the behavior of the P2-shape mode ducted experiment. in the magnetic field, the asymptotic form of ͑49͒ is dis- In Fig. 7, the temporal evolution of radius of a collapsing cussed for better understanding of the results. If the initial and bouncing bubble is shown for three different interaction growing velocity of the bubble is neglected, the growing parameters. The other parameters are chosen to be identical speed of the bubble wall can be approximated as to those of Fig. 2 of Prosperetti and Seminara,21 in which

FIG. 8. The effect of a magnetic ﬁeld on the P2-mode shape change of a collapsing bubble in the case of Reϭ1000 and Weϭ2000: •••••• Nϭ0, ϭ ϭ ϭ ϭ ϭ a20 1; –––––- N 0.1, a20 1; N 0.1, a20 0; –•–•–•–• N ϭ ϭϪ 0.1, a20 1.

Reϭ1000, Weϭ2000, and R˙ (0)ϭ0. As shown in Fig. 7, the magnetic force damps the volume mode of bubble motion. The shape mode of a collapsing bubble exhibits more complicated behaviors than that of a growing bubble. To en- hance understanding, an asymptotic analysis for a collapsing bubble is carried out. If the initial collapsing velocity of the bubble is neglected, the collapsing speed of the bubble wall can be approximated from ͑48͒ as 1 1 1 1 2 ͑R˙ ͒2ϭ␭2ͩ Ϫ1ͪ ϩ ͩ Ϫ1ͪ , ␭2ϭ . R3 We R R2 3 Applying this to ͑49͒, the asymptotic form for sufficiently small R obtained as FIG. 9. The temporal evolution of the surface of a collapsing bubble in the case of Reϭ1000 and Weϭ2000. 3␭ 1 a¨ Ϫ ˙a ϩ a ϭϪ3R˙ Q͑R͒. ͑50͒ 2 R5/2 2 R5 2 The above equation represents an equation of negatively forcing of the magnetic field ͓note R˙ Ͻ0 and Q(R)Ͼ0͔. damped forced oscillator. From ͑50͒, we can see also that the However, as the bubble size decreases further, it changes 5 ͑ ͒ term a2 /R becomes important when R is very small. sign as shown in Fig. 8. As mentioned below 50 , the term ͑ ͒ 5 In Fig. 8, a2(t) obtained from 49 is plotted against a2 /R becomes important for small R and the bubble exhib- R(t) for a collapsing bubble ͑Rϭ1 corresponds to tϭ0 and its oscillatory behavior. The absolute magnitude of the ͒ R decreases as t increases . The Reynolds and Weber num- P2-shape mode oscillation of this case is relatively smaller, ϭϮ bers are identical to those in Fig. 7. As the case of Prosperetti compared to the cases of a20 1. This is because Q is ϭ and Seminara, it is assumed that a˙ 2(0) 0. In the case of relatively small at the initial stage for a collapsing bubble ϭ ϭ ͓ ͓ ͑ ͔͒ ͉ ͉ N 0, an initial disturbance of a20 1 is given note that see Fig. 3 b . However, even in this case, a2 /R becomes there is no forcing term in ͑50͒ in the case of no magnetic very large, especially at the final stage of the collapse, and field͔. As predicted by the asymptotic form, ͑50͒, a collaps- the shape of the bubble is significantly deformed, as shown ing bubble is inherently unstable to external disturbances in in Fig. 9͑b͒. It is noteworthy that the bubble evolves from a the shape mode. The corresponding shape evolution is shown prolate shape to an oblate shape at the final stage of collapse. ͑ ͒ in Fig. 9 a . This corresponds to the sign change of a2 shown in Fig. 8. To see the effect of a magnetic field on the shape mode There follows a sample problem to demonstrate the ef- of a collapsing bubble, we considered the case of Nϭ0.1 fect of the magnetic field on the bubble motion more clearly. ϭϪ ͑ with three different initial shapes of a20 1, 0, and 1 see Firstly, the growth of an initially spherical bubble in liquid Fig. 8͒. When an initial disturbance is given, the shape of the sodium is considered when it enters a low pressure region, bubble becomes unstable regardless of the sign of the distur- such as a venturi tube. For the sake of simplicity, it is as- ϭ ϭ bance. When a20 0 and N 0.1, a2 has initially positive sumed that pϱ is abruptly changed and kept constant at ͑ Ͻ Ͻ Ͻ р ͒ Ϫ ϭ value 0 a2 1.16 for 0.041 R 1 due to the positive pvap pϱ 1 bar. The properties of liquid sodium at 100 °C

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shape. It is shown that the bubble is significantly deformed within a short time. If the bubble grows further, the bubble Ϫ will become unstable. When pvap pϱ becomes smaller, the interaction parameter is increased, and the bubble may be distorted more significantly. Next, the case of collapse of an initially spherical cavitation bubble in mercury is considered. It is assumed in this case too that pϱ is abruptly increased to some positive values such as the case in which a cavitation bubble leaves the venturi tube. The pressure is assumed to be kept constant so Ϫ ϭϪ that pvap pϱ 1 bar. The properties of mercury at room temperature are ␴ϭ106 ⍀Ϫ1 mϪ1, ␳ϭ13 550 kg/m3, and ␯ ϭ ϫ Ϫ7 2 ϭ ϭ ϫ Ϫ3 1.15 10 m /s. Let B0 1 tesla and R0 5 10 m, and then Nϭ0.135 and Reϭ1.19ϫ105. The temporal evolution of bubble shape near the final stage of the collapse process is shown in Fig. 10͑b͒. It is confirmed that the effect of Weber number is, in this case, negligible within the range of 102 –104. The overall feature of the shape deformation is similar to that shown in Fig. 9͑b͒, due to the similar interaction parameter and little dependency of cavitation-bubble dynamics on viscosity and surface tension during most of the stage of the collapse process.

V. COMPARISON WITH THE EXISTING EXPERIMENTAL RESULTS

It will be meaningful to examine the existing experimental results associated with bubble dynamics in a magnetic field with the results of the present investigation. The existing experimental results to investigate the effect of a magnetic field on cavitation bubble and cavitation erosion are summarized in Table I. As shown in the table, this subject has been a very controversial one.11 It should be noted that the interaction parameter, which FIG. 10. The temporal evolution of bubble shape: ͑a͒ growing bubble in sodium; ͑b͒ collapsing bubble in mercury. represents the relative magnitude of the electromagnetic force with respect to the inertia force, is very different in each investigation. In particular, for the results obtained by are ␴ϭ1.03ϫ107 ⍀Ϫ1 mϪ1, ␳ϭ928 kg/m3, vϭ7.39 using tap water, the interaction parameter is too small to ϫ Ϫ7 2 ␥ϭ ϭ 10 m /s, and 0.484 N/m. Let B0 1 tesla, and R0 influence the behavior of a cavitation bubble within the con- ϭ1.0ϫ10Ϫ4 m. Then, Nϭ0.106, Reϭ1,414, and We text of the present investigation. Nevertheless, the experi- ϭ20.9. Figure 10͑a͒ shows the temporal evolution of bubble mental result of Shal’nev et al.7 and Kamiyama et al.9

TABLE I. Summary of previous experimental results concerning the effect of a magnetic ﬁeld on cavitation bubbles and cavitation erosion.

Reference Working ﬂuid N Result

Shalobasov Tap water Ͻ10Ϫ7 Transverse magnetic field w.r.t. flow and Shal’nev ͑Ref. 6͒ direction significantly increases the weight loss due to cavitation erosion. Shal’nev et al. ͑Ref. 7͒ Tap water Ͻ10Ϫ7 Growth rate of a spark-generated bubble is increased due to magnetic field. Kamiyama et al. Tap water Ͻ10Ϫ7 Weight loss due to cavitation erosion ͑Ref. 9͒ is increased by magnetic field. Hammitt et al. ͑Ref. 11͒ Tap water Ͻ10Ϫ7 Little effect on cavitation inception. Perel’man ͑Probably͒ Ͻ0.1 Weight loss due to cavitation erosion and Govorskii ͑Ref. 8͒ Sodium11 is significantly reduced by magnetic field. Kamiyama Mercury Ͻ0.1 Cavitation inception can be much and Yamasaki ͑Ref. 10͒ delayed by transverse magnetic field w.r.t. flow direction.

Downloaded 07 May 2012 to 132.236.27.111. Redistribution subject to AIP license or copyright; see http://pof.aip.org/about/rights_and_permissions Phys. Fluids, Vol. 14, No. 1, January 2002 Effects of a uniform magnetic field on a bubble 39 showed that the behavior of the bubble is significantly influ- bubble. Additionally, the magnetic field destabilizes the col- enced by the magnetic field of NϽ10Ϫ7. Moreover, the re- lapsing bubble, as shown in Fig. 9͑b͒. A bubble can eventu- sults of Shal’nev et al.7 are evidently opposed to the well- ally be disintegrated during the collapse phase. This may known fact that the magnetic field usually tends to suppress reduce the peak amplitude of impulse force on a solid surface the motion of conducting fluids, which is consistent with the generated by the rupture of a bubble attached to the surface. result of the present investigation. This will certainly reduce the erosion of the metal surface This suggests that there may exist an unknown mecha- and is consistent with the experimental results of Perel’man nism controlling the bubble behavior in an effectively dielec- and Govorskii. tric liquid such as tap water in a magnetic field. For example, At the final stage of bubble collapse near a solid bound- the Coulomb force exerted on each ion on the bubble surface ary, a jet flow is usually formed towards the boundary. The may change the behavior of a bubble to a certain degree, as erosion of a metal surface by cavitation bubbles is mainly discussed by Shalobasov and Shal’nev.6 It needs further in- due to the impact force of the jet flow towards the metal vestigation, considering the influence of charges existing on surface. Magnetic fields in general exert a damping force the bubble surface, to obtain a reliable explanation for such against the flow in a conducting fluid. The magnitude of the anomalous behavior of a bubble in nonconducting liquids. damping force is proportional to the cross product of the On the other hand, the results of Hammitt et al.1,11 are magnetic flux vector and velocity vector. Then, the strength seemingly contradictory to the result of Shal’nev et al.7 and of the impulsive force due to the jet flow will certainly be Kamiyama et al.9 However, the influence of a magnetic field, reduced.23 This fact also supports the results of Perel’man through the results of either the present investigation or Sha- and Govorskii of reduced weight loss in a magnetic field. lobasov and Shal’nev,6 is basically driven by the motion of This could become an additional benefit of using a magnetic the bubble. As shown in ͑48͒ and ͑49͒, the influence of a field to reduce the damage of metal surfaces caused by cavi- magnetic field on bubble motion is proportional to the radial tation bubbles. velocity of the bubble wall. Initially, the radial velocity of a growing bubble is very small, so that the magnitude of the VI. CONCLUSIONS electromagnetic force becomes insufficiently small to sup- The governing equations for the volume and shape press the bubble growth. Thus, the magnetic field may have modes of oscillation are derived for a bubble under a uni- little effect during the initial growing stage of the bubble, form magnetic field. It is shown that the magnetic-field ex- unless the magnetic field is overwhelmingly strong. In the erts a damping force on the volume mode. Thus, both the case of nuclei induced cavitation, this may contribute to the 1,11 growing and collapsing speeds in volume mode are reduced reason why Hammitt et al. could not find any visible ef- by the magnetic field. The magnetic field also affects the fect of a magnetic field on cavitation inception. 10 P2-mode motion of the bubble wall by contributing to a The result of Kamiyama and Yamasaki of delayed forcing term. Within the validity of the present linearized cavitation inception by a magnetic field in a mercury flow is solution for the shape mode, the P2- mode of an either grow- interesting. As discussed above, a weak magnetic field may ing or collapsing bubble becomes unstable by the magnetic have little influence on cavitation inception. The delay of field, regardless of the initial conditions. inception in this case is conjectured to be due to the increase in static pressure at the venturi tube. The static pressure can ACKNOWLEDGMENTS be increased by the retarding flow velocity by the Lorentz force directed against the oncoming flow, owing to the mag- The authors wish to acknowledge the funding support of netic field applied transversely to the flow. the Korea Science and Engineering Foundation and the Ad- To check the result of Perel’man and Govorskii,8 the vanced Fluids Engineering Research Center of the Pohang volume mode of motion of a bubble in the case of ultrasonic University of Science and Technology. cavitation of liquid sodium is calculated by using ͑48͒.Inthe calculation, the interaction parameter was chosen to be 0.1, 1F. G. Hammitt, Cavitation and Multiphase Flow Phenomena ͑McGraw- which is a little greater than the value used by Perel’man and Hill, New York, 1980͒. 2D. L. Frost, ‘‘Dynamics of explosive boiling of a droplet,’’ Phys. Fluids Govorskii in their experiment. According to the calculated 31, 2554 ͑1988͒. results, the volume of the bubble is hardly influenced by the 3P. Selvaraj, K. N. Seetharamu, and G. Vaidyanathan, ‘‘Large leak sodium– magnetic field except near the final collapsing stage. This water reaction analysis of an LMFBR steam generator using a variable should not be construed as there being no effect of a mag- temperature spherical bubble model,’’ Nucl. Eng. Des. 123,87͑1990͒. 4R. Jonas, W. Schu¨tz, and A. Reetz, ‘‘Model calculations and experimental netic field on a growing bubble. One possible reason for such verification of the bubble behavior in a Bethe-Tait accident,’’ Nucl. Eng. a result is that much greater exciting pressure generated by Des. 118,71͑1990͒. ultrasonic vibrator than the pressure induced by the Lorentz 5S. Lesin, A. Baron, I. Zilberman, H. Branover, and J. C. 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