The flow of two stratified viscous immiscible perfect dielectric fluids in a channel with
topographically structured walls is investigated. The flow is driven by a streamwise
pressure gradient and an electric field across the channel gap. This problem is
explored in detail by deriving and studying a nonlinear evolution equation for the
interface valid for large-amplitude long waves in the Stokes flow regime. For flat
walls, the electrified flow is long-wave unstable with a critical cutoff wavenumber
that increases linearly with the magnitude of the applied voltage. In the nonlinear
regime, it is found that the presence of pressure-driven flow prevents electrostatically
induced interface touchdown that has been observed previously – time-modulated
nonlinear travelling waves emerge instead. When topography is present, linearly stable
uniform flows become non-uniform spatially periodic steady states; a small-amplitude
asymptotic theory is carried out and compared with computations. In the linearly
unstable regime, intricate nonlinear structures emerge that depend, among other
things, on the magnitude of the wall corrugations. For a low-amplitude sinusoidal
boundary, time-modulated travelling waves are observed that are similar to those
found for flat walls but are influenced by the geometry of the wall and slide over it
without touching. The flow over a high-amplitude sinusoidal pattern is also examined
in detail and it is found that for sufficiently large voltages the interface evolves to
large-amplitude waves that span the channel and are subharmonic relative to the wall.
A type of ‘walking’ motion emerges that causes the lower fluid to wash through the
troughs and create strong vortices over the peaks of the lower boundary. Non-uniform
steady states induced by the topography are calculated numerically for moderate and
large values of the flow rate, and their stability is analysed using Floquet theory.
The effect of large flow rates is also considered asymptotically to find solutions that
compare very well with numerical computations.