Stability of the flow of a viscoelastic fluid past a deformable surface in the low Reynolds number limit

Abstract

The stability of the plane Couette flow of a viscoelastic fluid adjacent to a flexible surface is analyzed with the help of linear and weakly nonlinear stability theory in the limit of zero Reynolds number. The fluid is described by an Oldroyd-B model, which is parametrized by the viscosity η, the relaxation time γ, and the parameter β, which is the ratio of solvent-to-solution viscosity; β = 0 for a Maxwell fluid and β = 1 for a Newtonian fluid. The wall is modeled as an incompressible neo-Hookean solid of finite thickness and is grafted to a rigid plate at the bottom. The neo-Hookean constitutive model parametrized by the shear modulus G, augmented to include the viscous dissipation, is used for the solid medium. Previous studies for the Newtonian flow past a compliant wall predict an instability as the dimensionless shear rate Γ = (ηV/GR) is increased beyond the critical value Γ_c. The present analysis investigates the effect of fluid elasticity, in terms of the Weissenberg number W = γG/η, on the critical value of the imposed shear rate Γ_c for various parameters. The fluid elasticity is found to increase Γ_c, indicating the stabilizing influence of the polymer addition on the viscous instability. For dilute polymeric solutions with β ≥ 0.5, the flow is stable when the Weissenberg number is increased beyond a maximum value W_max, and W_max increases proportional to the ratio of solid-to-fluid thickness H. For concentrated polymer solutions and melts with β<0.5, the flow becomes unstable when the strain rate increases beyond a critical value for any large Weissenberg number. The weakly nonlinear analysis reveals that the bifurcation of the linear instability is subcritical when there is no dissipation in the solid. The nature of bifurcation, however, changes to supercritical when the viscous effects in the solid are taken into account and the relative solid viscosity η_r is large such that √η_r/H>1. The equilibrium amplitude and the threshold strain energy for the solid have been calculated, and the effect of parameters H, β, η_r, and interfacial tension on these quantities is analyzed.