Instability of the interface between thin fluid films subjected to electric fields

Abstract

The effect of an externally applied electric field on the stability of the interface between two thin leaky dielectric fluid films of thickness ratio β and viscosity ratio μ r is analyzed using a linear stability analysis in the long-wave limit. A systematic asymptotic expansion is employed in this limit to derive the coupled nonlinear differential equations describing the evolution of the position of the interface between the fluids and the interfacial free charge distribution. The linearized stability of these equations is determined and the effect of the ratio of the conductivities, dielectric constants, thicknesses, and viscosities on the wavenumber of the fastest growing mode, k max, and the growth rate of the most unstable mode, s max, is examined in detail. Specific configurations considered in previous studies, such as a perfect dielectric–air interface, leaky dielectric–air interface, etc., emerge as limiting cases from the general formulation developed in this paper. Our results show that the viscosity ratio, μ r , does not have any significant effect on k max for the interface between perfect and leaky dielectric fluids. In marked contrast, however, μ r is shown to have a significant effect on the interface between two leaky dielectrics. Increasing μ r from 0.1 to 10 could decrease k max up to a factor of 5. In general, our results show that the presence of nonzero conductivity in either one or both of the fluids has a profound influence on the length-scale characteristic of the linear instability: a reduction even by a factor of 1/50 in the length scale can be effected when compared to the interface between two perfect dielectrics. These predictions could have important implications in pattern formation applications in thin fluid films that employ electric fields. The variation of k max and s max on the thickness ratio, β, indicates in general that k max∝ β − α , and s max∝ β − θ , where the exponents α and θ (both >0) are found to depend only on the ratio of conductivities, and are largely independent of other system parameters.