선형 전단 흐름에 놓여 있는 캡슐의 변형과 배향에 대한 균일 전기장의 영향

하종욱, 양승만

한국과학기술원 화학공학과

Effect of uniform electric field on the Deformation and Orientation of a capsule freely suspended in a linear shear flow

Jong-Wook Ha, Seung-Man Yang

Department of Chemical Engineering,

Korea Advanced Institute of Science and Technology

INTRODUCTION

If one takes into account the fact that the membrane does have a finite viscosity, a membrane element is subjected to a cyclic time-dependent load, and thus viscosity plays an important role because of the conjugated effects of the tank-treading motion and of the steady global shape. Barthes-Biesel and Sgaier [1] solved the fluid-mechanical problem based on the Kelvin-Voigt type viscoelastic cell membrane. With a purely viscous membrane (infinite relaxation time) the capsule deforms into an ellipsoid and has a continuous flipping motion. When the membrane relaxation time is of the same order as the shear time, the particle reaches a steady ellipsoidal sphae which is oriented with respect to streamlines at an angle that varies between 45°and 0°and decreases with increasing shear rates.

In the first part of this study, we considered that the electric-field-induced deformation of a capsule freely suspended in a quiescent fluid. Interior and exterior of the capsule was assumed as leaky dielectric materials and the membrane was modelled as the Kelvin-Voigt type. The electric-field-induced flow inside and outside the capsule could be calculated by utilizing Taylor's leaky dielectric theory and the time-dependent deformation of the capsule was discussed. Since the electric-field-induced flow is vorticity-free and purely staining, at leading order, the axis of symmetry of the capsule is alway parallel to the direction of externally applied electric field. Hence, the capsule is oriented parallel to the streamlines. However, when the capsule is suspended in a simple shear flow, one should expect the above results to become different and to depend on the ratio of viscous to electric forces acting in the membrane in addition to the viscoelastic properties of membrane. It is the objective of the present part of study the influence of electric field on the deformation and orientation of an initially spherical capsule suspended in a linear shear flow.

DEFORMATION OF THE CAPSULE

In the case of a viscoelastic membrane the time evolution of the deformation is

[eqn: beta {DOT {BOLD J}} = 5(1+beta) {BOLD E}^h +xi (2lambda+3)(a_0 +a_1 )(1+beta) {BOLD E} + {BOLD L} + {5 OVER 8} {BOLD M}]              (1)

[eqn: beta{DOT {BOLD K}} = {5 Over 2} (1+beta) {BOLD E}^h + {xi OVER 2} (2lambda+3) (a_0 + a_1 )(1+beta) {BOLD E} + {1 OVER 2}{BOLD L} + {1 OVER 4}{BOLD M}]      (2)

In which, dot means the Jaumann derivative defined by

[eqn: DOT {BOLD J} = {{PARTIAL {BOLD J}} OVER {PARTIAL t}}- omega CDOT {BOLD J} + {BOLD J} CDOT omega]

for any second order tensor. Because of the presence of vorticity in flow field, it follows that J and K are not parallel to E or [eqn: {BOLD E}^h]. As a consequence, it may be expected that the capsule will orient itself at an angle with respect to the principal direction of flow and electric field.

Once the type of elastic behaviour is specified, the above equations describe the time evolution of the capsule shape for any given flow field [eqn: {BOLD E}^h] and for electric-field-induced flow field E. Since the electric-field-induced flow is purely axisymmetric, the flow has a zero vorticity. However, the situation of interest clearly occurs when the flow has a non-zero vorticity due to the rotation of the membrane creates a time-dependent load on each material point.

Since it is assumed that the electric field is applied in the [eqn: x_2]-direction as illustrated in Fig. 1, consequently, the solution to (1) and (2) will be sought in the case of a steady linear flow given by

[eqn: E_{11}=-{1 Over 2}E_{22} = E_{33} =1]

[eqn: {E^h}_{12}={E^h}_{21}= OMEGA _{12} =-OMEGA_{21} = {1 OVER 2}]

all other components being zero. The components of interest for J and for K are the 11, 22, and 33 ones, since the others vanish.

Since the resulting profile of the capsule is non-symmetric, its deviation from sphericity cannot be measured by a single quantity. In the classical way, the degree of the deformation is

[eqn: D = varpi ({J_{11}}^2 +{J_{12}}^2 ) ^ {1/2} + O(varpi^2 )]

and the orientation angle is

[eqn: tan theta =J_{12}/J_{11}]

The types of motion are now studied for membranes with different constitutive behaviors.

GENERAL SOLUTIONS FOR THE DISCRETE MODEL OF THE CAPSULE MEMBRANE

1. Three-dimensional isotropic incompressible membrane

[eqn: J_{11} = {{{15 Over 2}beta(1+beta) - {1 Over 3} xi (2lambda+3)(a_0 +a_1 ) (1+beta)} Over {4({1 Over 9}+beta^2 )}}]

[eqn: +{{15beta(1+beta)-2 xi (2lambda+3)(a_0 +a_1 )(1+beta)} OVER {24(1+beta^2 )}}]

[eqn: J_{12} = {{{5 Over 2}beta(1+beta) +beta xi (2lambda+3)(a_0 +a_1 ) (1+beta)} Over {4({1 Over 9}+beta^2 )}}]

[eqn: +{{15beta(1+beta)+ 2beta xi (2lambda+3)(a_0 +a_1 )(1+beta)} OVER {24(1+beta^2 )}}]

2. The RBC-type elasticity

[eqn: J_{11} = {{{15 Over 2}beta(1+beta)-{{ xi Over 2}(2lambda+3)(a_0 + a_1 )(1+beta)}} OVER {(1+4beta^2 )}}]

[eqn: J_{12} = {{{15 Over 4}(1+beta)+{beta xi (2lambda+3)(a_0 + a_1 )(1+beta)}} OVER {(1+4beta^2 )}}]

DISCUSSION

In fig. 2, the effect fo uniform electric field on the orientation of the capsule is shown. In the case of purely elastic membrane ([eqn: beta~ ->~ 0]), at least first order, the capsule orients to the direction of principal axis in the asence of electric field (when R = 1). When the resistivity ratio R is smaller than 1, the electric-field-induced flow is uniaxially straining so that the orientation of the capsule tends to parallel to the direction of electric field as the strength of electric field increases. On the other hand, when R > 1, the biaxially straining flow is induced by electric field, and thus the capsule deforms into an oblate spheroid. The resulting orientation of the capsule is perpendicular to the direction of electric field when the electric field exceeds shear flow.

When the membrane is a viscoelastic material, the capsule is oriented between 0°and 45°even in the absence of electric field depending upon [eqn: beta]. In Figs. 3 and 4, the orientation of the capsule in the presence of the electric field is represented for isotropic incompressible membrane and the RBC-type membrane, respectively. Again, when R = 1, there is no contribution of electric field on the orientation or the deformation of the capsule. As expected, when the electric-field-induced flow is uniaxial, the orientation of capsule is parallel to the direction of electric field and in the case of biaxially strainging flow, the orientation of the capsule is decreased. Of course, when the external flow field is dominant than the electric field, the results are agreed with those of Barthes-Biesel and Sgaier [1] for the capsule in a linear shear flow.

CONCLUSION

In this study, we consider the combined effect of a uniform electric field and a simple shear flow in the orientation of the viscoelastic capsule. As special model cases, the isotropic incompressible membrane and the RBC-type membrane are considered. When the uniaxial strainging flow is induced by electric field, the orientation of the capsule approaches to the direction of the electric field as the electric field strength increases and reverse trend is observed when the biaxially straining flow is induced.

REFERENCE

[1] Barthes-Biesel, D. and Sgaier, H., J. Fluid Mech. 160, 119 (1985)

[eqn: {BOLD E} = V {BOLD x}_2]

Fig. 2 Orientation of elastic membrane.

(S = 1, □ = 1)

[eqn: x_2]

[eqn: x_1]

Fig.1 A shematic representation of

capsule in an electric field

Fig. 3 Orientation of isotropic

incompressible membrane.

(S = 1, □ = 1, □ = 10)

Fig. 4 Orientation of RBC-type membrane

(S = 1, □ = 1, □ = 10)