Instability of high-frequency modes in viscoelastic plane Couette flow past a deformable wall at low and finite Reynolds number

The linear stability of plane Couette flow of an upper convected Maxwell (UCM) fluid of viscosity η, density ρ, relaxation time τR and thickness R flowing past a linear viscoelastic solid of shear modulus G and thickness HR is analyzed using a combination of asymptotic analysis (at low Reynolds number) and numerical solution (at finite Reynolds number). The asymptotic analysis is used to analyze the effect of wall deformability on a class of high frequency modes in the UCM fluid with c∼O(Re−1/2) at Re≪1 first studied by Gorodtsov and Leonov [J. Appl. Math. Mech. 31 (1967) 310-319; abbreviated GL here], who showed that these modes are stable in a rigid channel. Here, c is the wavespeed of perturbations (nondimensionalised by V), Re=ρVR/η is the Reynolds number, V is the velocity of the moving plate. Our asymptotic results show that all the high frequency-low Re modes of GL are rendered unstable by solid wall deformability. The variation of the growth rate with the nondimensional solid elasticity parameter Γ=Vη/(GR) shows an oscillatory behavior alternating between stable and unstable regions, and the variation for upstream traveling waves is completely antiphase with that for downstream waves. The parameter Γ is shown to be proportional to Re1/4 at Re≪1 for neutrally stable downstream waves. Numerical continuation shows that the instability at low Re continues to finite Re, and typically the instability ceases to exist above a critical Re. This critical Re increases with an increase in the Weissenberg number W=τRV/R. In some cases, however, the instability continues to very high Re, and Γ∝Re−1 in that limit for neutrally stable modes. Neutral stability curves in the Γ-W plane show that the predicted instability of the high frequency GL modes exists only at finite and large W, and is absent in the Newtonian fluid (W→0) limit. The asymptotic analysis for the Oldroyd-B model shows that the ratio of solvent viscosity to total viscosity of the solution should be O(Re1/2) in order for the instability to exist at low Re, meaning the instability is absent for realistic values of solvent viscosity at low Re. However, numerical results show that the instability of the high-frequency GL modes is present at finite Re in an Oldroyd-B fluid, and increasing solvent viscosity ratio has a stabilizing effect on the instability. Similarly, the solid to fluid viscosity ratio ηr also has a stabilizing effect at finite Re, but in the limit of low Re, ηr∼O(Re1/2) in order for the instability to exist. Our study thus demonstrates the presence of an instability in viscoelastic plane Couette flow past a deformable wall which is primarily due to the viscoelastic nature of the fluid, and is absent in Newtonian fluids.