Absolute and convective instabilities in combined Couette-Poiseuille flow past a neo-Hookean solid

Abstract

Temporal and spatio-temporal stability analyses are carried out to characterize the occurrence of convective and absolute instabilities in combined Couette-Poiseuille flow of a Newtonian fluid past a deformable, neo-Hookean solid layer in the creeping-flow limit. Plane Couette flow of a Newtonian fluid past a neo-Hookean solid becomes temporally unstable in the inertia-less limit when the parameter Γ = V η/(GR) exceeds a critical value. Here, V is the velocity of the top plate, η is the fluid viscosity, G is the shear modulus of the solid layer, and R is the fluid layer thickness. The Kupfer-Bers method is employed to demarcate regions of absolute and convective instabilities in the Γ-H parameter space, where H is the ratio of solid to fluid thickness in the system. For certain ranges of the thickness ratio H, we find that the flow could be absolutely unstable, and the critical Γ required for absolute instability is very close to that for temporal instability, thus making the flow absolutely unstable at the onset of temporal instability. In some cases, there is a gap in the parameter Γ between the temporal and absolute instability boundaries. The present study thus shows that absolute instabilities are possible, even at very low Reynolds numbers in flow past deformable solid surfaces. The presence of absolute instabilities could potentially be exploited in the enhancement of mixing at low Reynolds numbers in flow through channels with deformable solid walls.