Flow-induced resonant shear-wave instability between a viscoelastic fluid and an elastic solid

Abstract

Linear stability analysis of plane Couette flow of a viscoelastic, upper-convected Maxwell (UCM) fluid past a deformable elastic solid is carried out in the low Reynolds number limit using both numerical and asymptotic techniques. The UCM fluid is characterized by its viscosity η, density ρ, and relaxation time τR, whereas the deformable solid is considered to be a linear elastic solid of shear modulus G. The asymptotic analysis is performed in the Re ≪ 1 limit, where Re = ρVR/μ is the Reynolds number, V is the top plate velocity, and R is the thickness of the fluid. Both asymptotic and numerical approaches are used to understand the effect of solid elasticity, represented by the dimensionless parameter Γ, and fluid elasticity, characterized by the Weissenberg number W, on the growth rate of a class of modes with high frequencies (compared to the imposed shear rate, termed high-frequency Gorodtsov-Leonov, or “HFGL” modes) in the Re ≪ 1 limit. Here, the dimensionless groups are defined as W = τRV/R and Γ = ηV/GR. The results obtained from the numerical analysis show that there is an interaction between the shear waves in the fluid and the elastic solid, which are coupled via the continuity conditions at the interface. The interaction is particularly pronounced when W = Γ, strongly reminiscent of resonance. The resonance-induced interaction leads to shear waves in the coupled system with a decay rate of ci = −1/[2k(W + Γ)]. In this case, it is not possible to differentiate the fluid and solid shear waves individually and the coupled fluid-solid system behaves as a single composite material. The leading order asymptotic analysis suggests that the growth rate of the HFGL modes is proportional to W2 for W ≪ 1. The asymptotic analysis, up to first correction, shows an oscillating behavior of ci with an increase in Γ, in agreement with the results from our numerical approach. In addition, we also carry out an asymptotic analysis in the no-flow, but nonzero inertia limits to illustrate the role played by the imposed flow in the instability of the shear waves. It is found that, at the leading order, the wave speed for the coupled fluid-solid problem is neutrally stable in the absence of flow. Thus, the unstable resonant modes in the coupled fluid-solid system are shown to be driven by the imposed flow.