Effect of surface charge convection and shape deformation on the dielectrophoretic motion of a liquid drop

The dielectrophoretic motion and shape deformation of a Newtonian liquid drop in an otherwise quiescent Newtonian liquid medium in the presence of an axisymmetric nonuniform dc electric field consisting of uniform and quadrupole components is investigated. The theory put forward by Feng [J. Q. Feng, Phys. Rev. E 54, 4438 (1996)] is generalized by incorporating the following two nonlinear effects—surface charge convection and shape deformation—towards determining the drop velocity. This two-way coupled moving boundary problem is solved analytically by considering small values of electric Reynolds number (ratio of charge relaxation time scale to the convection time scale) and electric capillary number (ratio of electrical stress to the surface tension) under the framework of the leaky dielectric model. We focus on investigating the effects of charge convection and shape deformation for different drop-medium combinations. A perfectly conducting drop suspended in a leaky (or perfectly) dielectric medium always deforms to a prolate shape and this kind of shape deformation always augments the dielectrophoretic drop velocity. For a perfectly dielectric drop suspended in a perfectly dielectric medium, the shape deformation leads to either increase (for prolate shape) or decrease (for oblate shape) in the dielectrophoretic drop velocity. Both surface charge convection and shape deformation affect the drop motion for leaky dielectric drops. The combined effect of these can significantly increase or decrease the dielectrophoretic drop velocity depending on the electrohydrodynamic properties of both the liquids and the relative strength of the electric Reynolds number and electric capillary number. Finally, comparison with the existing experiments reveals better agreement with the present theory.

Figure 1

Schematic representation of the physical system under consideration: a neutrally buoyant liquid drop of undeformed radius $a$ is suspended in an otherwise quiescent, unbounded liquid medium and encounters an axisymmetric nonuniform electric field consisting of uniform and quadrupole components. A spherical coordinate system $\left(r,\theta ,\varphi \right)$ is attached to the drop centroid.

Figure 2

(a) Variation of drop velocity $\left(U_d\right)$ as a function of viscosity ratio $\left(\lambda \right)$. (b) Drop shape for $\lambda =0.0013$ and $\mathrm{C}{\mathrm{a}}_E=0.05$, and (c) variation of the drop deformation parameter $\left(\mathcal{D}\right)$ with the capillary number $\left(\mathrm{C}{\mathrm{a}}_E\right)$ for $\lambda =0.0013$. Other properties have the following values: $R\to \infty$ and $E_u=E_q=1$.

Figure 3

(a) Variation of drop velocity $\left(U_d^{\left(0\right)},U_d^{\left(\mathrm{C}{\mathrm{a}}_E\right)}\right)$ as a function of permittivity ratio $\left(S\right)$. Other properties have the following values: $E_u=E_q=\lambda =1$.

Figure 4

(a) Variation of drop velocity $\left(U_d\right)$ as a function of viscosity ratio $\left(\lambda \right)$ for $E_u=E_q=1$. (b) Drop shape for $\mathrm{C}{\mathrm{a}}_E=0.1$ and $\lambda =1$. (c) Variation of the drop deformation parameter $\left(\mathcal{D}\right)$ up to $O\left({\mathrm{Ca}}_E^2\right)$ with the capillary number $\left(\mathrm{C}{\mathrm{a}}_E\right)$ for $\lambda =1$. Other properties have the following values: $R=S=5$.

Figure 5

(a) Variation of drop velocity $\left(U_d\right)$ as a function of viscosity ratio $\left(\lambda \right)$ for $E_u=1, E_q=1.5$. (b) Drop shape for $\mathrm{C}{\mathrm{a}}_E=0.2$ and $\lambda =1$. (c) Variation of the drop deformation parameter $\left(\mathcal{D}\right)$ up to $O\left({\mathrm{Ca}}_E^2\right)$ with the capillary number $\left(\mathrm{C}{\mathrm{a}}_E\right)$ for $\lambda =1$. Other properties have the following values: $R=S=0.1$.

Figure 6

Variation of drop velocity $\left(U_d\right)$ as a function of viscosity ratio $\left(\lambda \right)$ for (a) $\left(R,S\right)=\left(2.5,0.1\right)$ and (b) $\left(R,S\right)=\left(0.02,0.5638\right)$. Here we have neglected the $O\left(\mathrm{C}{\mathrm{a}}_E\right)$ correction in drop velocity. Different parameters considered are $E_u=1$ and $E_q=1$.

Figure 7

(a) Variation of the leading order tangential electric stress jump for system B. (b) Variation of the leading order tangential surface velocity for system B. (c) Leading order streamline pattern for system B. (d) Variation of the leading order tangential surface velocity for system A. Different parameters have the following values: for system A: $R=2.5, S=0.1$, and $\lambda =0.01$; for system B: $R=0.02, S=0.5638$, and $\lambda =0.4359$. Other parameters are taken as $E_u=1$ and $E_q=1$.

Figure 8

(a) Surface charge distribution on the drop surface for system A. (b) Surface charge distribution on the drop surface for system B. Different parameters have the following values: for system A: $R=2.5, S=0.1$, and $\lambda =0.01$, for system B: $R=0.02, S=0.5638$, and $\lambda =0.4359$. Other parameters are taken as $E_u=1$ and $E_q=1$.

Figure 9

Variation of drop velocity $\left(U_d\right)$ as a function of viscosity ratio $\left(\lambda \right)$ for (a) $\left(R,S\right)=\left(2.5,0.1\right)$ and (b) $\left(R,S\right)=\left(0.02,0.5638\right)$. The inset in Fig. 9 depicts the drop shape for $\mathrm{C}{\mathrm{a}}_E=0.2$ and $\lambda =1$. The inset in Fig. 9 depicts the drop shape for $\mathrm{C}{\mathrm{a}}_E=0.2$ and $\lambda =0.4359$. Here we have neglected the $O\left({\mathrm{Re}}_E\right)$ correction in drop velocity. Different parameters considered are $E_u=1$ and $E_q=1$.

Figure 10

Variation of drop velocity $\left(U_d\right)$ as a function of viscosity ratio $\left(\lambda \right)$ for (a) $\left(R,S\right)=\left(2.5,0.1\right)$, and (b) $\left(R,S\right)=\left(0.02,0.5638\right)$. Here we have considered the combined effect of charge convection and shape deformation. Different parameters considered are $E_u=1$ and $E_q=1$.

Figure 11

(a) Leading order drop velocity $U_d^{\left(0\right)}$ on the $R-S$ plane. (b) Drop velocity with due consideration of charge convection and shape deformation. Other parameters have the following values: ${\mathrm{Re}}_E=0.1, \mathrm{C}{\mathrm{a}}_E=0.1, \lambda =1$, and $E_u=E_q=1$. The contour levels and color bar represent the magnitude of the drop velocity.

Figure 12

Variation of interface deformation $\left(\mathcal{D}\right)$ as a function of capillary number $\left(\mathrm{C}{\mathrm{a}}_E\right)$ for (a) $\left(R,S,\lambda \right)=\left(2.5,0.1,0.1\right)$, and (b) $\left(R,S,\lambda \right)=\left(0.02,0.5638,0.4359\right)$. Other parameters have the following values: $E_u=E_q=1$ and ${\mathrm{Re}}_E=0.2$ for $O\left(\mathrm{C}{\mathrm{a}}_E{\mathrm{Re}}_E\right)$ deformation.

Figure 13

Variation of drop deformation parameter with the vertical position for (a) $\overline{\phi }=800\mathrm{V}$, (b) $\overline{\phi }=600\mathrm{V}$, and (c) $\overline{\phi }=400\mathrm{V}$. The drop radius $a=250\mu \mathrm{m}$ and the distance between the electrodes is $2.6\mathrm{mm}$. Electrohydrodynamic properties are given in Table 1.

Figure 14

Variation of drop deformation parameter with the vertical position for (a) $a=425\mu \mathrm{m}$, (b) $a=275\mu \mathrm{m}$, and (c) $a=200\mu \mathrm{m}$. The applied potential is $\overline{\phi }=600\mathrm{V}$ and the distance between the electrodes is $2.6\mathrm{mm}$. Electrohydrodynamic properties are given in Table 1.