Boundary-layer flow of the power-law fluid over a moving wedge: a linear stability analysis

We study the stability of the two-dimensional boundary-layer flow of a power-law (Ostwald de Waele) non-Newtonian fluid over a moving wedge. The mainstream velocity is assumed to have a power of distance from the leading boundary layer, such that the system admits to the self-similar solutions. We discuss the problem in question for both shear-thickening and shear-thinning fluids which lead to a non-uniqueness (double solutions) in the base flow solutions. We then address an issue of the stability of the non-unique solutions. A linear eigenvalue analysis of the double solution reveals that the basic flow represented by the first solution is always stable, and this flow is practically encountered. The system becomes unstable to the second solutions which have the mode-two perturbations with larger boundary-layer thickness. The first and second solutions form a tongue-like structure in the solution space. Furthermore, the modification of the viscosity for the power-law fluids reveals that the system predicts an infinite viscosity in the confinement of the boundary-layer region. Extensive comparisons of the solutions with the existing models with Newtonian fluid are made, and a physical explanation behind these solutions is proposed.

     

Numerical solution of shear-thinning and shear-thickening boundary-layer flow for Carreau fluid over a moving wedge

This paper investigates the linear stability of the flow in the two-dimensional boundary-layer flow of the Carreau fluid over a wedge. The corresponding rheology is analysed using the non-Newtonian Carreau fluid. Both mainstream and wedge velocities are approximated in terms of the power of distance from the leading edge of the boundary layer. These forms exhibit a class of similarity flows for the Carreau fluid. The governing equations are derived from the theory of a non-Newtonian fluid which are converted into an ordinary differential equation. We use the Chebyshev collocation and shooting techniques for the solution of governing equations. Numerical results show that the viscosity modification due to Carreau fluid makes the boundary layer thickness thinner. Numerical results predict an additional solution for the same set of parameters. Thus, a further aim was to assess the stability of dual solutions as to which of the solutions can be realized. This leads to an eigenvalue problem in which the positive eigenvalues are important and intriguing. The results from eigenvalues form tongue-like structures which are rather new. The presence of the tongue means that flow becomes unstable beyond the critical value when the velocity ratio is increased from the first solution.