Nonlinear evolution of the interface between two immiscible fluids in a micro channel subjected to an electric field

Abstract

In this work, the nonlinear evolution of the interface between two immiscible, Newtonian, leaky dielectric uids in a micro channel subjected to a pressure-driven base ow in the presence of an electric eld applied normal to the interface is investigated. The long wave analysis of the interface is performed and the evolution equations of the interface position and the surface charge density are derived. The evolution equations are solved using a numerical procedure, in which the Fourier transform and the 4th order Runge-Kutta formulation are used to discretize the space and the time derivatives, respectively. A major result indicates that the presence of a pressure-driven base ow prevents the interface from reaching the walls of the channel and also results in loss of symmetry of the interface. Stronger base ow reduces the vertical distance between the maximum and the minimum of the interface, i.e. the amplitude of the interface. As linear theory suggests, the early times of the evolution of the interface do not depend on the presence or the strength of the base ow. In addition to the base ow analysis, a parametric study, which involves the depth, the viscosity and the conductivity ratios, and the dimensionless groups, namely the electric number and the ratio of the uid to electric time scales, is conducted. It is revealed that increasing the electric number leads to a more symmetrical interface position pro le with a larger amplitude, which means that it has the exact opposite e ect on the interface compared to the strength of the base ow. The parametric study also shows that when the viscosity ratio is constant, the largest interface amplitude occurs for the depth ratio that gives a base state velocity pro le, for which the maximum is at the at interface between the uids.