Nonlinear interfacial instability in two-fluid viscoelastic Couette flow

Weakly-nonlinear interfacial instabilities in two-fluid planar Couette flow are investigated for the case where one layer is thin. Taking this thin-layer thickness as a small parameter, asymptotic analysis is used to derive a nonlinear evolution equation for the interface height valid for wavelengths that scale with the channel height. Consequently, the influence of the thick layer is felt through a non-local coupling term which is obtained by solving a system of linear equations which are a simplified viscoelastic analogue to the Orr-Sommerfeld equation. The evolution equation allows for the clear identification of the influence of normal stresses at the interface on both the initial instability and the subsequent nonlinear dynamics. Results from numerical simulations illustrate: (1) an array of non-stationary states including traveling waves and chaos, (2) competition between elastic instability and instability due to viscosity stratification, and (3) the accuracy of a simplified 'localized' evolution equation (derived using a long-wave approximation to the coupling term) when either the elasticity of the thick-layer fluid is sufficiently weak or the elasticities of the two fluids are sufficiently well-matched.