The self-similar laminar boundary layer of power law non-Newtonian fluids

Many fluids of practical interest are non-Newtonian, which influences many aspects of their behaviour, particularly the structure and properties of the laminar boundary layer, that differs qualitatively and quantitatively from the classical solution of Prandtl. We study this problem for a power law rheology, that describes with fair accuracy the behaviour of many fluids in an important range of shear. Since this type of rheology depends on a single-dimensional parameter, the solution of the problem is self-similar and can be obtained by a generalization of the usual procedure for the case of Newtonian fluids. A phase-plane formalism is introduced and the solutions are obtained as a function of the rheological index. It is found that the boundary layer of a dilatant fluid is spatially localized, i.e. it ends abruptly at a finite distance from the wall; this is a consequence of the non-linear character of the vorticity diffusion of a non-Newtonian fluid. A, very good analytical approximation is obtained for dilatant fluids.