Nonlinear electrokinetic flow about a polarized conducting drop

In the thin-double-layer limit κ 1, electrokinetic flows about free surfaces are driven by a combination of an electro-osmotic slip and effective shear-stress jump. An intriguing case is that of a highly conducting liquid drop of radius a, where the inability to balance the viscous shear by Maxwell stresses results in an O(κa) velocity amplification relative to the familiar electro-osmotic scale. To illuminate the inherent nonlinearity we consider uncharged drops, where the induced surface-charge distribution results in a fore-aft symmetric electrokinetic flow profile with no attendant drop translation. This problem is analyzed using a macroscale model, where the double layer is represented by effective boundary conditions. Because of the intense flow, ionic convection within the O(1/κ)-wide diffuse-charge layer is manifested by a moderate-zeta-potential surface-conduction effect. The drop deforms to a prolate shape in response to the combination of hydrodynamic forces and the effective electrocapillary reduction of the surface-tension coefficient, both mechanisms being asymptotically comparable. The flow field and the concomitant drop deformation are calculated using both a weak-field approximation and numerical simulations of the nonlinear macroscale model.