Stability of plane Couette flow of Carreau fluids past a deformable solid at arbitrary Reynolds numbers

Abstract

The linear stability of the plane Couette flow of both power-law and Carreau fluids past a deformable, neo-Hookean solid is analyzed at arbitrary Reynolds numbers. An algebraic error in the mathematical formulation of the earlier studies (for the power-law fluid) is corrected and is shown to result in quantitative differences in the predictions for the stability of the flow. Due to the lack of a proper (zero-shear) viscosity scale and a time scale for the onset of shear thinning in the power-law model, we show that the stability analysis of the flow yields vastly different scalings for the unstable mode depending on the way the problem is scaled to render it dimensionless. When the deformable solid properties are used to non-dimensionalize, we show that for the unstable modes (the so-called “wall modes” at high Re) Γc∝Re−1(2n+1), while when flow properties are used to non-dimensionalize, Γc∝Re−13 much akin to a Newtonian fluid, where Γ=Vm*η*/G*R* is the dimensionless shear rate in the flow, and Γc denotes the minimum value required for instability. Here, Vm* is the velocity of the top plate, G* is the shear modulus of the solid, R* is the fluid thickness, and η* is the (arbitrary) viscosity scale in the power-law model. Within the framework of the power-law model, it is not possible to discriminate between the two predicted scalings. To resolve this in an unambiguous manner, we used the Carreau model to account for shear thinning and to study its role on the stability of flow past deformable solid surfaces. The Carreau model has a well-defined zero-shear viscosity η0* as well as a time scale λ* that characterizes the onset of shear thinning. For fixed λ*η0*/(ρ*R*2), we show that the unstable wall modes scale as Γc∼Re(1−2n)3 at high Re, thus providing a resolution to the ambiguity in the results obtained using the power-law model. The present work thus shows that, at moderate to high Re, shear thinning has a strongly stabilizing effect on the wall mode instability in flow past deformable solid surfaces.