Stability of gas flow past viscoelastic compliant solid

Stability of a high-speed gas flow past a compliant solid is impacted by two distinct features: high solid-to-fluid density ratio ($\rho_r$), and flow compressibility when flow speeds are comparable to acoustic speed. This study investigates the linear stability of a shear-driven compressible gas flow past a compliant substrate modelled as a continuum Neo-Hookean solid. Numerical solutions of the eigenvalue problem reveal that at high density ratios, the dominant instabilities are the elastic shear-waves of the solid. Our study shows that flow compressibility exerts a non-monotonic effect on the growth rate of the elastic modes; the growth rate increases with increase in Mach number up to $\mbox{Ma} \approx 2$ before subsequently decreasing. Furthermore, for compressible flows, strong thermal-coupling renders the base state highly sensitive to both the solid-to-fluid thermal conductivity ratio and the substrate's bottom-surface temperature. Numerical results demonstrate that increasing the conductivity ratio destabilizes the system, whereas increasing the bottom wall temperature is stabilizing. The stability equations, analyzed in the asymptotic limit of $\rho_r \mbox{Re} \gg 1$, reveals that the fluid-solid system de-couples at the leading order with the elastic modes emerging as a solution of the linear elasticity equations under free-shear condition at the interface. We derive a closed-form expression for the leading-order growth rate of the instability, which shows an excellent agreement with the numerical solution. This expression explicitly quantifies the influence of fluid stresses at the interface, which can in-turn be expressed as integrals of the flow solution isolating the distinct physical mechanisms driving the instability.