선형 전기장 하의 3차원 기포 진동의 비선형 해석

오정민, 강인석

포항공과 대학교 화학공학과, 기계산업공학부

Nolinear Analysis of 3-dimensional Bubble Oscillation

under Linear Electric Fields

Jung Min Oh and In Seok Kang

Department of Chemical Engineering, Division of Mechanical Engineering,

Pohang University of Science and Technology

Introduction

The Rayleigh-Plesset (RP) equation and the Prosperetti-Seminara (PS) equations have been developed continuously (Plesset and Prosperetti 1977, Feng and Leal 1997, Prosperetti 1977).  Applying electromagnetic force is an important part of bubble/drop dynamics in recent.  A bubble in an electric field may be deformed from spherical shape due to the nonuniform normal stress imbalance over the spherical bubble surface.  These electrohydrodynamic phenomena have attracted much attention of researchers since the seminal work of Taylor.  The electric force due to the permittivity difference between inside and outside a bubble makes a bubble oscillation dynamic and complicated.  Lee and Kang (1999) investigated the linear electric field effects in natural frequencies of a bubble in a dielectric liquid.  Oh et al. derived the extended RP equation and PS equations including electric field effects.  They considered two kinds of linear electric fields : uniform field and axisymmetric straining field.  When the viscosity effect is considerable, both the volume and shape mode oscillations can be chaotic at the same time.  In the present work, Oh et al.'s results are extended to bubble oscillations in general linear electric fields.  Oscillation equations are derived by using the domain perturbation method with Taylor and Melcher's leaky dielectric model and weak viscosity assumption.  The shape of a bubble can be expressed in terms of volume averaged radius [eqn: R`(t)``] and surface spherical harmonics [eqn: Y_n^m (theta, phi)``].  As noted by Lee and Kang (1999), to the first order, the shape of a bubble in arbitrary linear electric field can be represented by a linear combination of finite number of [eqn: Y_n^m ``], where [eqn: 0 <= l <= 4``] and [eqn: |m| <= l``].

It is important in bubble dynamics to understand the exchange of gases across the interface between free state within the bubble and the dissolved state outside the bubble.  It is well known that in the volume oscillation of a bubble the usual tendency to dissolve in undersaturated liquids may be reversed through a phenomenon known as rectified diffusion.  Most researches in rectified diffusion are restricted to volume oscillations and the effect of bubble shape deformation in rectified diffusion has been less investigated though a lot of progress in understanding mass transport in bubble oscillations have been made.  If the shape of oscillating bubble is deformed, the problems become much complex.  One of the difficulties lies in solving the diffusion equation on the domain deformed from spherical shape, in which spherical coordinate cannot be applied.  Instead of solving the transport equation directly, we concentrate on the dynamical behavior of the surface area of oscillating bubble.  The surface area of deformed bubble can be calculated analytically without loss of degree of accuracy by using the orthogonality of surface harmonics.  The detail of calculation is explained in the present work.  Two kinds of measures relating to dynamical behaviors of surface area of oscillating bubble are also defined. In this study, the surface area dynamics is investigated by observing the defined measures for three kinds of electric fields : axisymmetric straining electric field, 2-dimensional hyperbolic field, and combination of uniform and hyperbolic fields.  And the effects of deformation of oscillating bubble in time-periodic electric fields are considered.

Theory

Consider a bubble, which consists of vapor and gas, in perfect dielectric liquid under linear electric field.  No free charge in medium is assumed and the leaky dielectric model is used.  General linear electric field can be expressed as

[eqn: {rm bold E}_infty ~=~{rm bold E}_0 + {rm bold G}cdot {rm bold x}``]where

(1)

[eqn: {rm bold E}_0 ~=~E_0 (gamma, delta, zeta)``] and [eqn: {rm bold G} = G left[ matrix{beta &0 &0#0 &beta-1 &0#0&0&1}right]``]

with [eqn: gamma^2 + delta^2 + zeta^2 = 1``] and [eqn: 0 <= beta <= 1 over2 ``].  [eqn: E_0 ``] and [eqn: G``] are functions of time.  General solution of electric field around a spherical bubble can be obtained in analytic form (Lee and Kang, 1999).

[eqn: {rm bold E} ~=~A_1 {rm bold E}_0 - A_2 ({rm bold E}_0 cdot {rm bold n}){rm bold n}+B_1{rm bold G cdot bold n}-B_2 ({rm bold n cdot bold G cdot bold n}){rm bold n} ``]  [eqn: {rm bold E}_{i n} ~=~A_1 {rm bold E}_0 +B_1{rm bold G cdot bold n} ``]

(2)

(3)

where [eqn: A_1 = {3}over{2+q},~~A_2 ={3(1-q)}over{2+q},~~ B_1 ={5R}over{3+2q},{rm and}~ B_2 ={5R`(1-q)}over{3+2q}].  Here, [eqn: q``] is the ratio between permittivities of inside and outside a bubble and[eqn: rm bold n ``] is the outward normal vector.  The shape of a bubble is determined from the normal stress balance.  The Maxwell stress tensor is defined as

[eqn: {rm bold T}^e ~=~epsilon left( {rm bold E`` bold E}- 1over2 E^2 {rm bold I}right).``]

(4)

Here, [eqn: epsilon``] is the permittivity of medium.  And the jump of normal components of Maxwell stress on the bubble surface can be represented by

[eqn: [[ {rm bold n}cdot({rm bold n}cdot{rm bold T}^e )]] =epsilon {1-q}over2 left[ (1-q) left{ A_1 ({rm bold E}_0 cdot {rm bold n})+ B_1 ({rm bold n}cdot{rm bold G}cdot{rm bold n}) right}^2 ##- |A_1 {rm bold E}_0 +B_1 {rm bold G}cdot {rm bold n}|^2 right].``]

(5)

Equation (5) can be expressed in terms of spherical harmonics and is appleid to the normal stress balance to determine the shape of a bubble.

[eqn: [[{rm bold n}cdot({rm bold n}cdot {rm bold T}^h )]] + [[{rm bold n}cdot({rm bold n}cdot {rm bold T}^e )]] = gamma`` ({bold Del}cdot {rm bold n}) ``]

(6)

Here, [eqn: gamma``] is the surface tension and is different from [eqn: gamma``] in linear electric field description.  In derivation, no translation of a bubble is considered so that the component which induces translational motion must be removed. Derived equations are verified by comparing with the special cases derived in the previous work (Oh et al. 2000).  From the derived oscillation equations, it can be seen that the translational motion of a bubble is originated from the interaction between uniform field and straining field.  Under the same parameters, a bubble oscillation is more vigorous in combined field, which includes uniform electric field.  But uniform field effect becomes weak as a bubble grows.  Straining field effect is dominant when a bubble size is large.  The subharmonic peaks rather than superharmonic peaks are conspicuous when the frequency of electric field varies.

When the shape of a bubble can be expressed in terms of surface harmonics, the surface area of a bubble can also be expressed in analytical way without loss of degree of accuracy compared with the oscillation equations.  The surface area element of a bubble with arbitrary shape can be given in spherical coordinates as

[eqn: dA ~=~ {r^2 sin theta d theta d phi``}over{({rm bold n cdot bold e}_r )}``]

(7)

When the shape function is given as [eqn: F = r - f``(theta, phi``)``], the surface area element is expressed as

[eqn: dA ~&=&~| {bold nabla} F``|f``^2 sin theta d theta d phi## &=&~sqrt{1 + 1 over {f``^2} left( {partial f}over{partial theta}right)^2 + 1 over{f``^2 sin^2 theta``}left( {partial f}over{partial phi}right)^2}``f``^2 sin theta d theta d phi``.]

(8)

Then the surface area of a slightly deformed bubble can be expressed by using the orthogonality of surface harmonics as

[eqn: S ~=~ 4 pi R^2 + 1 over 2 in^2 sum_{nm} left{ n(n+1)R^2 + 2 right} alpha_{nm}^2 + O(in^3 `)``]

(9)

The derivation is given in detail in the present work.  It is interesting to note that there is no [eqn: O( in)``] term.  Since the volume and shape modes are highly nonlinear, the surface area also shows complicated nonlinear behaviors in simulations.  The time derivative of the surface area is given as follows.

[eqn: dot{S} ~=~8 pi R dot{R} + in^2 sum_{nm} left[ n(n+1)R dot{R} alpha_{nm}^2 + left{ n(n+1)R^2 +2 right} alpha_{nm} dot{alpha}`_{nm} right]``]

(10)

[eqn: (S, dot{S}``)] makes a phase space and is observed in the study. Dynamic behaviors of surface area are observed and compared for three cases : axisymmetric straining field, 2-dimensional hyperbolic field and the combination of uniform and hyperbolic fields.  In the present work, two kinds of surface measures are defined to observe how much largely and how fast the surface area of a bubble changes.

[eqn: M_1 ~=~ |S_max - S_min |``][eqn: M_2 ~=~ max |dot{S}``|``]

(11)

(12)

[eqn: M_1 ``] , which is the maximum surface area change, represents how much largely a bubble changes and [eqn: M_2 ``], which is the maximum surface area change rate, how fast a surface area changes.  In the phase space of surface area, [eqn: M_1 ``] is the width of oscillation orbit and [eqn: M_2 ``] is the maximum height of the orbit as shown in Figure 1.  By observing the measures, the characteristics of a bubble oscillation can be understood in another view point.

Conclusion

The generalized Rayleigh-Plesset equation and Properetti-Seminara equations, which include general linear electric field effects, are derived by using the domain perturbation method.  The shape of a bubble is represented in finite number of spherical harmonics.  And the surface area of a bubble is calculated in analytical form without loss of degree of accuracy and the dynamical behaviors of surface area are investigated by observing the surface measures, which are defined in the present work.  We hope that the present work gives the understanding of effects of shape deformation in surface area change and the insight of transport mechanism in nonlinear oscillations of a bubble in linear electric fields to enhance the heat/mass transfer.  The derived oscillation equations in the present work can be used to simulate the translation of a bubble under linear electric fields.

References

1. M. S. Plesset and A. Prosperetti, Bubble Dynamics and Cavitation, Annu. Rev. Fluid. Mech., 9, 145-85 (1977)

2. Z. C. Feng and L. G. Leal, Nonlinear Bubble Dynamics, Annu. Rev. Fluid. Mech. 29, 201-43 (1997)

3. A. Prosperetti, Viscous Effects on Perturbed Spherical Flows, Q. Appl. Math, 35, 339-52 (1977)

4. S. M. Lee and I. S. Kang, Three-dimensional Analysis of the Steady-state Shape and Small-amplitude Oscillation of a Bubble in Uniform and Non-uniform Electric Fields, J. Fluid Mech., 384, 59-91 (1999)

5. J. M. Oh, P. J. Kim and I. S. Kang, Chaotic Oscillation of a Bubble in a Weakly Viscous Dielectric Fluid under Electric Fields, Phys. Fluids, Submitted

Acknowledgement

This work has been supported by the grant from the Ministry of Education of Korea through Brain Korea 21 (BK21) program at the Pohang University of Science and Technology (POSTECH).  The authors wish to acknowledge the financial support.

Figure  Phase space of a bubble surface