Stability of plane Couette flow of a power-law fluid past a neo-Hookean solid at arbitrary Reynolds number

Abstract

The linear stability of plane Couette flow of a power-law fluid past a deformable solid is analyzed at arbitrary Reynolds number (Re). For flow of a Newtonian fluid past a deformable solid, at high Re, there are two different modes of instability: (i) “wall modes” (Γ∝Re−1∕3) and (ii) “inviscid modes” (Γ∝Re−1) where Γ=VμfGR is the non-dimensional shear-rate in the fluid (V, μf, G, and R denote the top-plate velocity, fluid viscosity, shear modulus of the solid, and fluid thickness, respectively). In this work, we consider the power-law model for the fluid to elucidate the effect of shear-thickening/shear-thinning behaviour on the modes of instability present in the flow, especially at moderate and high Re. At high Re, our numerical results show that wall modes exhibit different scalings in Γ (VηfGR, where ηf is Newtonian-like constant viscosity) vs Re for different values of the power-law index (n), and the scaling exponents are different from that for a Newtonian fluid. This drastic modification in the scaling of wall modes is not observed in viscoelastic (modelled as upper-convected Maxwell or Oldroyd-B fluids) plane Couette flow past a deformable solid. We show that the difference in scaling exponents can be explained by postulating that the wall modes in a power-law fluid are determined by the actual viscosity corresponding to the shear rate of the laminar flow denoted by ηapp. A non-dimensional shear rate based on this viscosity Γapp=VηappGR can be defined, and we show that the postulate Γapp∼Re−1∕3 (motivated by the wall-mode scaling in a Newtonian fluid) captures all the numerically observed scalings for Γ vs Re for different values of n>0.3, which is found to be Γ∼Re−12n+1. Further, we numerically evaluated the wall layer thickness and this agreed with the theoretical scaling of δ∼Re−n2n+1. Interestingly, the theoretical and numerical prediction of wall modes is found to be valid for power-law index, n≥ 0.3. For n≤ 0.3, there is a marked departure from the wall-mode scalings, and our results show a scaling of Γ ∼ Re−1 corresponding to inviscid modes. The variation of the power-law index (n) can stabilise/destabilise the inviscid mode when compared with Newtonian fluid, and this result is observed only in the power-law model and is not seen in the flow of viscoelastic fluid past deformable surfaces. Thus, the present study shows that the shear-rate dependence of viscosity has a significant impact on both the qualitative and quantitative aspects of stability of non-Newtonian fluid flow past deformable surfaces.