Stability of a plane Couette flow over inhomogeneously stressed solids

Abstract

In this paper, we investigate the stability of a plane Couette flow over non-linear solids undergoing homogeneous or inhomogeneous initial stresses. Almost all biological and engineering structures are subjected to complex initial residual stresses. We probe here the stability characteristics of a plane Couette flow over these stressed solid surfaces. Since initial stress is commonly inhomogeneous in Nature, our main focus is to analyze the stability of flow over inhomogeneously stressed solids at all Reynolds number regimes, which was not investigated earlier. We apply an initially stressed neo-Hookean solid model which precludes the necessity of using the stress-free reference. A linear perturbation analysis based on an Eulerian–Lagrangian formulation determines the stability associated with short and finite wave modes for zero, low, and high Reynolds number flows. In presence of a uniform initial stress field, the short wave modes are stabilized by tensile stresses and destabilized by compressive stresses in the limit of creeping flow. A similar influence of pre-stress is observed in buckling of columns. On the other hand, the case of spatially varying initial stress is far more intricate due to the inhomogeneous shearing of the solid in the unperturbed state. A linear perturbation introduced to this complicated unperturbed state presents several interesting observations. We find multiple unstable modes for low Reynolds number (Re≪1) creeping flows which were not reported earlier to the best of our knowledge. While literature demonstrates only downstream unstable modes for a plane Couette flow, both upstream and downstream unstable modes are observed for inhomogeneously stressed solids in various Reynolds number regimes. In the creeping flow limit (Re=0), the stability depends mainly on the tensile or compressive initial stress at the solid–fluid interface. When this initial stress is compressive at the interface, only the finite wave modes determine the stability for all thicknesses instead of the short-wave modes. At higher Reynolds number, we report many more interesting results for spatially varying stress-fields which includes an unusual scaling of wall-mode validated through eigenmode analysis.