Electrohydrodynamic instability of a capacitive elastic membrane

The electrohydrodynamic instability of a leaky (weakly conducting) capacitive elastic membrane driven by a direct current electric field, both perpendicular and parallel to the membrane in a micro-fluidic channel, is investigated theoretically. In the leaky dielectric framework, electric charges can accumulate on either side of the membrane, and the effect of the accumulated surface charge depends on the ratio of charge relaxation time in the bulk to the membrane charging time. Under a parallel electric field, a non-conducting membrane can become unstable while under a perpendicular electric field a non-conducting capacitive membrane is always stable and membrane conductance is essential for the membrane instability. The effects of membrane conductance, bending modulus, and charge relaxation time on the membrane instability are elucidated for several combinations of conductivity ratio and permittivity ratio in the bulk fluids. Regions of instability are computed for both the parallel and perpendicular electric fields. The tangential electric field acts similarly to the membrane tension in terms of its damping effects at small length scales (high wave number), while either bending or membrane tension is needed to damp out the small-scale perturbations under a perpendicular electric field.