Effect of nonuniform electric field on the electrohydrodynamic motion of a drop in Poiseuille flow

Abstract

The effect of a nonuniform electric field on the electrohydrodynamic motion of a leaky dielectric suspended drop in the presence of background Poiseuille flow is investigated analytically. Considering the nonuniform electric field to be a linear combination of uniform and quadrupole fields, the velocity of a force-free drop positioned at the flow centerline is obtained. The drop velocity is strongly influenced by the surface charge distribution and drop shape. In the Stokes flow limit, we employ an asymptotic method considering weak surface charge convection and small shape deformation. The present study shows the importance of type of nonuniform electric field (converging or diverging in the direction of the Poiseuille flow), strength of the electric field relative to the Poiseuille flow, and material property ratios on the magnitude and direction of drop motion in the presence of flow curvature. In the presence of a nonuniform electric field, the flow curvature can increase or decrease the drop velocity as compared with a uniform flow case. The converging electric field always drives a perfectly conducting drop in the direction of the Poiseuille flow with increased velocity, while the diverging electric field can drive the drop in either direction, depending on the relative strength of the applied electric field. Shape deformation increases the velocity of a perfectly conducting drop in the converging electric field, while shape deformation increases/decreases the velocity magnitude of a perfectly conducting drop in the diverging electric field. The converging electric field always drives a perfectly dielectric drop in the direction of the Poiseuille flow with increased (or decreased) velocity when the drop phase permittivity is greater (or less) than the medium phase permittivity. The diverging electric field can move a perfectly dielectric drop in either direction, depending on the strength of the electric field relative to the Poiseuille flow and drop-to-medium permittivity ratio. Shape deformation increases the velocity magnitude of a perfectly dielectric drop for larger permittivity ratios. For leaky dielectric drops, both surface convection and shape deformation can increase or decrease the drop velocity in nonuniform electric field, depending on the electrohydrodynamic properties of the drop and the suspending medium.