이중 에멜젼 액적 주변에서 전기장에 의해 형성된 흐름에 관한 연구

하종욱, 양승만

한국과학기술원 화학공학과

Flow Patterns generated by electric field around a double emulsion droplet

Jong-Wook Ha, Seung-Man Yang

Department of Chemical Engineering

Korea Advanced Institute of Science and Technology

Introduction

Fundamental studies on the dynamics of encapsulated particles or drops have appeared recently in the fluid mechanics literature. The interest in the effect of an electric field on the fluid dynamics of the drops goes far back as the early experimental investigations of O'Konski and Thacker [1]. The difference in the dielectric constants and the electrical resistivities across the drop interface causes electrostatic charges to build up. These charges in the presence of an electric field lead to surface stresses which create surface mobility and hence fluid motions [2].

The introduction of direct-contact heat and mass transfer processes has revived the interest in the subject owing to the fact that the heat and mass transfer rate is enhanced by the internal circulation created by an electric field. In many direct contact systems, however, three immiscible phases can be present and they form so-called compound multiple drops. Obviously, the solution given by Taylor [3] for the single drop is not applicable to the complex geometry of such drops [4].

Our concern in the present study is the effect of electric field on the flow patterns formed inside and outside drop. In particular, the case of two non-intersecting concentric fluid spheres is considered as shown in Figure 1. The concentric geometry is simpler than eccentrically configured globules and more straightforward than the case of multiple subdrops, but it is not just a mathematical diversion. In practical, direct-contact heat exchange is achieved by passing drops of one liquid through anothar immiscible liquid. A complete analysis of multiple drops would not only have to include hydrodynamic factors, but would also have to take account of thermodynamic consideration (including breakdown mechanism) and the role of surfactants [5]. In fact, by considering solely on the hydrodynamics, we severely limit the applicability of our results to real system.

Governing equations

Consider the double emulsion droplet illustrated in Figure 1. We restrict our attention to the case where the center of mass of the two droplets remain coincident at the origin of the spherical polar coordinate. The electric field part of the problem is independent of the flow field since the electrical charge convection by the motion of the fluid can be neglected. In this case the governing equation for the electric field part problem is the Laplace's equation for the three distinct phases. The fluids involved in this problem are neither perfect dielectrics nor perfect conductors. The boundary conditions for the two interfaces are:

at 12-interface

[eqn: {bold t}_2 CDOT {bold E}_1 = {bold t}_2 CDOT {bold E}_2]

[eqn: R_12 {bold n}_2 CDOT {bold E}_1 = {bold n}_2 CDOT {bold E}_2]

at 23-interface

[eqn: {bold t}_3 CDOT {bold E}_2 = {bold t}_3 CDOT {bold E}_3]

[eqn: R_23 {bold n}_3 CDOT {bold E}_2 = {bold n}_3 CDOT {bold E}_3]

where [eqn: R_ij] is a ratio of electrical resistivity of phase i to that of phase j. In these equations, [eqn: {bold n}_i] and [eqn: {bold t}_i] is the unit normal directed outward from phase i and the unit tangential respectively. [eqn: {bold E}_i] denotes the electric fields developed in fluid phases and can be related to the electrostatic potential as [eqn: {bold E}_i = - DEL phi _i]. The electric traction can be expressed in terms of Maxwell's electric stress.

[eqn: {bold T}_{E,i}=(DEL phi_i DEL phi_i - 1 over 2 {bold I} LEFT | DEL phi_i RIGHT |^2 )]

In the steady-state, fluid motions induced by the surface charge should be such that it has exactly the same viscous stresses as the electrostatic florces at the drop interfaces. It is assumed that the fluid motion is rather slow and the flow field is in the Stokes regime. Thus, the fluid motions in each phase are governed by the quasi-steady Stokes equations. The dimensionless governing equations for phase i are

[eqn: DEL^2 {bold u}_i = DEL p_i]

[eqn: DELCDOT u_i = 0]

Here [eqn: ({bold u}_i, {p_i})] represent the dimensionless velocity and pressure fields in phase i. The boundary conditions are:

at the 12-interface

[eqn: {bold t}_2 CDOT {bold u}_1={bold t}_2 CDOT {bold u}_2]

[eqn: {bold n}_2 CDOT {bold u}_1 = {bold n}_2 CDOT {bold u}_2 =0]

[eqn: {bold T}_{E,1} : {bold n}_2 {bold t}_2 -S_12 {bold T}_{E,2} : {bold n}_2{bold t}_2 + {bold T}_{H,1}: {bold n}_2 {bold t}_2 - lambda _12 {bold T}_{H,2}: {bold n}_2 {bold t}_2 =0]

at the 23-interface

[eqn: {bold t}_3 CDOT {bold u}_2={bold t}_3 CDOT {bold u}_3]

[eqn: {bold n}_3 CDOT {bold u}_2 = {bold n}_3 CDOT {bold u}_3 =0]

[eqn: {bold T}_{E,2} : {bold n}_3 {bold t}_3 -S_23 {bold T}_{E,3} : {bold n}_3{bold t}_3 + {bold T}_{H,2}: {bold n}_3 {bold t}_3 - lambda _23 {bold T}_{H,3}: {bold n}_3 {bold t}_3 =0]

in which [eqn: S_ij] and [eqn: lambda_ij] is a dielectric constant ratio and viscosity ratio respectively. The viscous stress tensor [eqn: {bold T}_{H,i}] has the usual definition

[eqn: {bold T}_{H,i}=-p_i{bold I}+ DEL {bold u}_i+{(DEL {{bold u}_i})}^T]

Results and Discussion

The problem formulated in the previous section is nonlinear, in spite of the linearity of governing equations. This nonlinearity comes solely from the fact that the boundary conditions. In order to make analytical approach, the drops consisting of the double emulsion droplet are treated as concentric and their shapes are spherical. Consequently, the analysis is limited to the small deformation case in which the restoring interfacial force is much larger than the electric force.

In Figure 2, the circulatory flow generated by electric field inside and outside the double emulsion drop is represented. With regard to the electrical properties of fluids, biaxial or uniaxial straining flow can be induced. In Figure 3, uxexpectedly, secondary flow can be produced in the intermediate phase for a certain combination of electrical properties of fluids. Obviously, when the double emusion drop is subjected to an external electric field, more complex flow patterns are formed comparing to that of translating double emulsion drop without electric field. The patterns of electric field induced-flow is completely symmetric.

Conclusion

In this study, the flow patterns generated by external electric field is considered. The purely analytical approach is carried out with assumption of concentric double emulsion drop. The results show that the internal circulation is built up inside double emulsion drop. Occasionally, secondary flow can be produced. This complex fluid motions are expected to enhance heat or mass transfer rate in the direct-contact processes.

References

[1] C.T. O'Konski and H.C. Thacher, J. Phys. Chem. 57, 955 (1953)

[2] J.R. Melcher and G.I. Taylor, Annu. Rev. Fluid Mech. 1, 111 (1969)

[3] G.I. Taylor, Proc. R. Soc. Lond. A291, 159 (1966)

[4] S.S. Sadhal, P.S. Ayyaswamy and J.N. Chung, Transport Phenomena with Drops and Bubbles, Springer-Verlag, New York (1997)

[5] S. Torza and S.G. Mason, J. Colloid Interface Sci. 33, 67 (1970)

Figure 1. A schematic representation of a concentric double emulsion droplet in an external electric field

Figure 2. Flow patterns generated by electric field around a double emulsion drop

Figure 3. Secondary flow generated by electric field