I will discuss recent theoretical investigations into the stability of non-Newtonian flows. In particular, we consider crossflow instability of the flow due to a rotating disc for a shear-thinning fluid satisfying a power-law relationship. This flow is an example of a three-dimensional boundary layer with applications in rotating machinery. The study is motivated by experiments on the erosion of calcite marble rotating discs in gelled acids, which are relevant to the erosion of drill heads. The instability, in the form of co-rotating vortices, may be described by asymptotic methods for large Reynolds numbers. Predictions for the wavenumber and orientation of the vortices will be given.