Electrokinetic instability: The sharp interface limit

Abstract

An instability between two miscible liquid regions of identical mechanical properties but different electrical conductivities stressed by an external electric field parallel to the interface is studied. The problem is of interest due to its applications to mixing in microchannels. It is shown that the problem can be modeled by considering a sharp interface and an appropriate jump condition for the electrical conductivity. The transport of the electrical conductivity is governed by a diffusive equation. An infinite domain case and a shallow channel case are considered. It is shown that any velocity perturbation at the interface leads to a varying electrical conductivity in its vicinity due to the electromechanical coupling in the jump condition for the electrical conductivity. This in turns leads to a bulk charge density that gives a body force in the fluid equations. The body force generates a cellular motion that results in the instability. The results compare favorably with the experimental data and the numerical analysis for the diffuse interface case by Chen et al. [J. Fluid Mech. 524, 263 (2005)]. The critical condition for the instability is given in terms of a nondimensional parameter PΣ, which is a product of the Péclet number and another nondimensional parameter that depends on the conductivity ratio of the two liquids.