Stability of plane Couette–Poiseuille flow of shear-thinning fluid

A linear stability analysis of the combined plane Couette and Poiseuille flow of shear-thinning fluid is investigated. The rheological behavior of the fluid is described using the Carreau model. The linearized stability equations and their boundary conditions result in an eigenvalue problem that is solved numerically using a Chebyshev collocation method. A parametric study is performed in order to assess the roles of viscosity stratification and the Couette component. First of all, it is shown that for shear-thinning fluid, the critical Reynolds number for a two-dimensional perturbation is less than for a three dimensional. Therefore, it is sufficient to deal only with a modified Orr–Sommerfeld equation for the normal velocity component. The influence of the velocity of the moving wall on the critical conditions is qualitatively similar to that for a Newtonian fluid. Concerning the effect of the shear thinning, the computational results indicate that this behavior leads to a decrease in the phase velocity of the traveling waves and an increase in stability, when an appropriate viscosity is used in the definition of the Reynolds number. Using a long-wave version of the Orr–Sommerfeld equation, the cutoff velocity is derived. The mechanisms responsible for the changes in the flow stability are discussed in terms of the location of the critical layers, Reynolds stress distribution, and the exchange of energy between the base flow and the disturbance.