Stability of plane Couette flow past an initially stressed solid

Abstract

In this paper, we carry out linear stability analysis of a coupled solid–fluid system in a plane Couette flow past an initially stressed neo-Hookean solid at creeping flow limit (Re=0) and arbitrary Reynolds number (Re), as well as for different solid thicknesses. In biological and engineering systems, we often encounter flows past initially stressed deformable solids. For these cases, initial strain or initial stress appears as an additional symmetry representing tensor in the constitutive relation. Consequently, initial stress induces anisotropy (transverse isotropy in the present case) and affects material properties. We employ a hyperelastic model for the initially stressed deformable solid to study the effects of both uni-axial and equi-biaxial initial stresses on flow instability. This study also presents a case where anisotropy in the solid interferes with the stability of the solid–fluid coupled system. In contrast with the previous formulations with two or three state Eulerian–Lagrangian formulations, the present formulation requires four configurations for the initially stressed solid, viz. stress-free, initially stressed, deformed, and the perturbed state. This formulation applies to any fluid–solid coupled problems where the solid contains initial/residual stress. Our results show that initial tensile stress stabilizes the flow, while compressive initial stress destabilizes the same. Fortuitously, this observation of coupled fluid–solid stability is in agreement with the stability of solid column-like structures, where compressive axial stress leads to buckling. We report similar results for uni-axial and equi-biaxial initial stress, for different wave modes at creeping flow limit and their extensions at higher Re, and various solid thicknesses. Two modes of instability, viz., short-wave and finite-wave modes present in the creeping-flow limit are significantly affected by tensile/compressive initial stress for both uni-axial and equi-biaxial cases. For uni-axial tensile initial stress, the finite-wave mode does not extend to high Re. Instead of them, we observe multiple upstream modes to become unstable for higher Re, which show the wall mode scaling as O(Re−1/3). These upstream modes are not observed for stress-free solid. This behavior is triggered by the additional coupling terms between initial stress and perturbed variable in the governing equations of initially stressed solids. However, for the equi-biaxial case, the finite-wave mode does extend to higher Re to show wall mode scaling. This mode further stabilizes/destabilizes depending on the tensile/compressive nature of the initial stress present in the solid.