Instability of viscoelastic plane Couette flow past a deformable wall

Abstract

The stability of plane Couette flow of an upper-convected Maxwell (UCM) fluid of thickness R, viscosity η and relaxation time τ R past a deformable wall (modeled here as a linear viscoelastic solid fixed to a rigid plate) of thickness HR, shear modulus G and viscosity η w is determined using a temporal linear stability analysis in the creeping-flow regime where the inertia of the fluid and the wall is negligible. The effect of wall elasticity on the stable modes of Gorodtsov and Leonov [J. Appl. Math. Mech. 31 (1967) 310] for Couette flow of a UCM fluid past a rigid wall, and the effect of fluid elasticity on the unstable modes of Kumaran et al. [J. Phys. II (Fr.) 4 (1994) 893] for Couette flow of a Newtonian fluid past a deformable wall are analyzed. Results of our analysis show that there is only one unstable mode at finite values of the Weissenberg number, W= τ R V/ R (where V is the velocity of the top plate) and nondimensional wall elasticity, Γ= Vη/( GR). In the rigid wall limit, Γ≪1 and at finite W this mode becomes stable and reduces to the stable mode of Gorodtsov and Leonov. In the Newtonian fluid limit, W→0 and at finite Γ this mode reduces to the unstable mode of Kumaran et al. The variation of the critical velocity, Γ c, required for this instability as a function of W ̄ =τ R G/η (a modified Weissenberg number) shows that the instability exists in a finite region in the Γ c − W ̄ plane when Γ c> Γ c,Newt and W ̄ < W ̄ max , where Γ c,Newt is the value of the critical velocity for a Newtonian fluid. The variation of Γ c with W ̄ for various values of H are shown to collapse onto a single master curve when plotted as Γ c H versus W ̄ /H , for H≫1. The effect of wall viscosity is analyzed and is shown to have a stabilizing effect.