Exact solutions can be found for steady fluid flow under constant shear, even if the stress–strain rate relation is nonlinear and shear heating connects the material properties to thermal behaviour. We present such solutions for a Newtonian material in which viscosity decreases exponentially with temperature, and for two empirical equations valid for high temperature creep. The onset of melting limits the range in which these solutions are applicable. If we assume that the region of the low velocity zone for shear waves is close to melting and that drag on this region by plates appears to a first approximation as a constant stress, we can predict surprisingly reasonable values for the plate velocity with respect to the mantle. The low viscosity of the zone becomes a consequence of melt induced by shear heating. Such melt would also explain a low Q and a reduction in shear velocity. A final solution is then given for an inhomogeneous material whose viscosity increases with depth. This can be interpreted as a material whose melting point increases with depth at a faster rate than the temperature of the material.