Electroosmotic flow of a two-layer fluid in a slit channel with gradually varying wall shape and zeta potential

•Electroosmotic flow of two-layer fluid in non-uniform channel is investigated.•Conducting lubricating and non-conducting working fluids are immiscible.•Wall shape and zeta potential vary slowly and periodically in a sinusoidal manner.•Induced pressure gradient and deformed shape of interface are part of the solution.•Effects of various factors on flow structure are investigated.