Asymptotic matching constraints for a boundary-layer flow of a power-law fluid

Abstract

We reconsider the three-dimensional boundary-layer flow of a power-law (Ostwald–de Waele) rheology fluid, driven by the rotation of an infinite rotating plane in an otherwise stationary system. Here we address the problem for both shear-thinning and shear-thickening fluids and show that there are some fundamental issues regarding the application of power-law models in a boundary-layer context that have not been mentioned in previous discussions. For shear-thickening fluids, the leading-order boundary-layer equations are shown to have no suitable decaying behaviour in the far field, and the only solutions that exist are necessarily non-differentiable at a critical location and of ‘finite thickness’. Higher-order effects are shown to regularize the singularity at the critical location. In the shear-thinning case, the boundary-layer solutions are shown to possess algebraic decay to a free-stream flow. This case is known from the existing literature; however here we shall emphasize the complexity of applying such solutions to a global flow, describing why they are in general inappropriate in a traditional boundary-layer context. Furthermore, previously noted difficulties for fluids that are highly shear thinning are also shown to be associated with the imposition of incorrect assumptions regarding the nature of the far-field flow. Based on Newtonian results, we anticipate the presence of non-uniqueness and through accurate numerical solution of the leading-order boundary-layer equations we locate several such solutions.