Linear stability analysis of viscous multi-layer shear flows with interfacial slip

The stability of two superposed viscous, incompressible, immiscible fluid layers sheared in a plane Couette flow configuration is considered when slip is present at the deforming liquid-liquid interface. Guided by experiments and molecular dynamics simulations in the literature, slippage is modelled by employing a Navier-slip boundary condition at the liquid-liquid interface, and the arising novel instabilities are studied in detail. The linear stability of the system is addressed asymptotically for long- and short-waves as well as with a combination of analytical and numerical calculations for arbitrary wavenumbers and other parameter values. Slip is found to be capable of destabilising perturbations of all wavelengths due to the presence of a velocity jump at the interface, a phenomenon that appears to be a viscous analogue of classical Kelvin-Helmholtz instabilities. In regimes where the flow is stable to perturbations of all wavelengths when slip is absent, the presence of slip and a favorable combination of the physical parameters, induces a Turing-type instability by destabilisation of a small band of finite wavenumber perturbations. In the case where the underlying layer is asymptotically thin, the results are found to agree with the linear properties of a weakly non-linear asymptotic model.