Electrohydrodynamic instability of the interface between two fluids confined in a channel

The stability of the interface between two dielectric fluids confined between parallel plates subjected to a normal electric field in the zero Reynolds number limit is studied analytically using linear and weakly nonlinear analyses, and numerically using a thin-layer approximation for long waves and the boundary element technique for waves with wavelength comparable to the fluid thickness. Both the perfect dielectric and leaky dielectric models are studied. The perfect dielectric model is applicable for nonconducting fluids, whereas the leaky dielectric fluid model is applicable to fluids where the time scale for charge relaxation, $\epsilon \epsilon_o/\sigma$, is small compared to the fluid time scale $(\mu R/\Gamma)$, where $\epsilon_o$ is the dielectric permittivity of the free space,\epsilon and \sigma are the dielectric constant and the conductivity of the fluid,\mu and \Gamma are the fluid viscosity and surface tension, and R is the characteristic length scale. The linear stability analysis shows that the interface becomes unstable when the applied potential exceeds a critical value, and the critical potential depends on the ratio of dielectric constants, electrical conductivities, thicknesses of the two fluids, and surface tension. The critical potential is found to be lower for leaky dielectrics than for perfect dielectrics. The weakly nonlinear analysis shows that the bifurcation is supercritical in a small range of ratio of dielectric constants when the wavelength is comparable to the film thickness, and subcritical for all other values of dielectric constant ratio in the long-wave limit. The thin-film and boundary integral calculations are in agreement with the weakly nonlinear analysis, and the boundary integral calculation indicates the presence of a secondary subcritical bifurcation at a potential slightly larger than the critical potential when the instability is supercritical. When a mean shear flow is applied to the fluids, the critical potential for the instability increases, and the flow tends to alter the nature of the bifurcation from subcritical to supercritical