Weakly nonlinear analysis of viscous instability in flow past a neo-Hookean surface

We analyze the stability of the plane Couette flow of a Newtonian fluid past an incompressible deformable solid in the creeping flow limit where the viscous stresses in the fluid (of the order $\eta_fV/R)$ are comparable with the elastic stresses in the solid (of the order G). Here, $\eta_f$ is the fluid viscosity, V is the top-plate velocity, R is the channel width, and G is the shear modulus of the elastic solid. For $(\eta_fV/GR)$ =O(1), the flexible solid undergoes finite deformations and is, therefore, appropriately modeled as a neo-Hookean solid of finite thickness which is grafted to a rigid plate at the bottom. Both linear as well as weakly nonlinear stability analyses are carried out to investigate the viscous instability and the effect of nonlinear rheology of solid on the instability. Previous linear stability studies have predicted an instability as the dimensionless shear rate $\Gamma=(\eta_fV/GR)$ is increased beyond the critical value $\Gamma_c$. The role of viscous dissipation in the solid medium on the stability behavior is examined. The effect of solid-to-fluid viscosity ratio $\eta_r$ on the critical shear rate $\Gamma_c$ for the neo-Hookean model is very different from that for the linear viscoelastic model. Whereas the linear elastic model predicts that there is no instability for $H<\sqrt\eta_r$, the neo-Hookean model predicts an instability for all values of $\neta_r$ and H. The value of $\Gamma_c$ increases upon increasing $\eta_r$ from zero up to $\sqrt\eta_r/H \approx 1$, at which point the value of $\Gamma_c$ attains a peak and any further increase in $\eta_r$ results in a decrease in $\Gamma_c$. The weakly nonlinear analysis indicated that the bifurcation is subcritical for most values of $H$ when $\eta_r$=0. However, upon increasing $\eta_r$, there is a crossover from subcritical to supercritical bifurcation for $\sqrt\eta_r/H \approx 1$. Another crossover is observed as the bifurcation again becomes subcritical at large values of $\eta_r$. A plot in $H$ versus $\sqrt\eta_r/H$ space is constructed to mark the regions where the bifurcation is subcritical and supercritical. The equilibrium amplitude and some physical quantities of interest, such as the total strain energy of the disturbance in the solid, have been calculated, and the effect of parameters $H$, $\eta_r$, and interfacial tension on these quantities are analyzed