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_{f}VR ) 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_{f}VGR)=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_{f}VGR) 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 eta_{r} and H . The value of Gamma_{c} increases upon increasing eta_{r} from zero up to sqrt[eta_{r}]H approximately 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 approximately 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.