  Consider the flow of a thin layer of non-Newtonian fluid over a solid
surface. I model the case of a viscosity that depends nonlinearly on the
shear-rate; power law fluids are an important example, but the analysis here is
for general nonlinear dependence. The modelling allows for large changes in
film thickness provided the changes occur over a large enough lateral length
scale. Modifying the surface boundary condition for tangential stress forms an
accessible base for the analysis where flow with constant shear is a neutral
critical mode, in addition to a mode representing conservation of fluid.
Perturbatively removing the modification then constructs a model for the
coupled dynamics of the fluid depth and the lateral momentum. For example, the
results model the dynamics of gravity currents of non-Newtonian fluids even
when the flow is not very slow.
