Instability of creeping Couette flow past a neo-Hookean solid

Fluid flow over a deformable solid can become unstable due to the fact that waves may propagate along the solid-fluid interface. In order to understand the role that nonlinear rheological properties of the solid play in these elastohydrodynamic instabilities, we apply linear stability analysis to investigate creeping Couette flow of a Newtonian fluid past an incompressible and impermeable neo-Hookean solid of finite thickness. As inertial effects are assumed to be negligible, the problem is governed by three dimensionless parameters: an imposed strain, a thickness ratio, and an interfacial tension. In the base state, there is a first normal stress difference in the neo-Hookean solid, and this leads to instability behavior that is significantly different from what is observed with a linear constitutive equation. In the absence of interfacial tension, the first normal stress difference gives rise to a shortwave instability. For sufficiently thin solids, a large range of high-wavenumber modes becomes unstable first as the strain that is imposed on the system increases, while for sufficiently thick solids, a small band of O(1) wavenumbers first becomes unstable. The presence of interfacial tension removes the shortwave instability and leads to larger critical imposed strains and smaller critical wavenumbers. In comparison to the linear elastic constitutive equation, the neo-Hookean model leads to smaller values of the critical imposed strain and larger values of the critical wavenumber, but the difference rapidly diminishes as the solid thickness increases. Analysis of the continuity of velocity boundary condition at the interface reveals that at the critical conditions, the mean flow tends to amplify horizontal interface perturbations, while horizontal velocity perturbations tend to suppress them. The results of this work highlight the importance of properly accounting for large displacement gradients when modeling elastohydrodynamic instabilities.