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

Abstract

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 η_fV/R) are comparable with the elastic stresses in the solid (of the order G). Here, η_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 (η_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 Γ=(η_fV/GR) is increased beyond the critical value Γ_c. The role of viscous dissipation in the solid medium on the stability behavior is examined. The effect of solid-to-fluid viscosity ratio η_r on the critical shear rate Γ_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<√η_r, the neo-Hookean model predicts an instability for all values of η_r and H. The value of Γ_c increases upon increasing η_r from zero up to √η_r/H≈1, at which point the value of Γ_c attains a peak and any further increase in η_r results in a decrease in Γ_c. The weakly nonlinear analysis indicated that the bifurcation is subcritical for most values of H when η_r=0. However, upon increasing η_r, there is a crossover from subcritical to supercritical bifurcation for √η_r/H≈1. Another crossover is observed as the bifurcation again becomes subcritical at large values of η_r. A plot in H versus √η_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 srolid, have been calculated, and the effect of parameters H, η_r, and interfacial tension on these quantities are analyzed.