Role of fluid and wall elasticities on the instabilities in plane Couette flow of a White-Metzner fluid past a neo-Hookean solid

The stability of plane Couette flow, past a deformable wall, of a shear-thinning White-Metzner fluid augmented with Carreau-Yasuda model for shear-rate dependence of viscosity, is analyzed with the aim of exploring the presence of qualitatively new unstable modes that are absent in the flow of the same fluid past a rigid surface. The deformable solid is assumed to be described by a purely elastic neo-Hookean model without any dissipation. In the limit of a rigid solid layer, it is recently predeicted that the combined effects of shear-thinning and elasticity destabilize plane Couette flow. While wall elasticity does not affect this shear-thinning elastic instability at finite wavenumbers in inertia-less regime, it has a significant destabilizing effect at high wavenumbers, thus rendering the flow past a deformable surface to be more unstable compared to its rigid counterpart. In contrast, fluid elasticity is shown to have a stabilizing effect on the instability present in Newtonian Couette flow past a deformable surface. These two qualitatively different unstable modes, however, remain unaffected by each other, and are even shown to coexist in some windows of the parameter regimes explored in this work.