Nonlinear waves in viscous multilayer shear flows in the presence of interfacial slip

The stability of immiscible two-fluid Couette flows is considered when slip is present at the liquid–liquid interface. The phenomena are modelled by incorporating a Navier slip condition at the interface to replace that of no slip. A nonlinear asymptotic theory is developed when the flow geometry consists of a thin layer slipping over a thick fluid layer that scales with the channel height. A nonlocal, nonlinear evolution equation is derived that is valid at finite Reynolds numbers, slip lengths, viscosity and density ratios. The nonlocal term arises from the coupling between the phases and its Fourier symbol is calculated in closed form in terms of Airy functions, thus generalising past results by the inclusion of slip. The linear spectrum is calculated and it is shown that in geometries containing a thin layer, the role of slip is to introduce dispersion and reduce instability or enhance stability