Flow of a class of fluids defined via implicit constitutive equation down an inclined plane: Analysis of the quasi-steady regime

We consider the motion of a rate-type fluid defined by an implicit constitutive equation down an inclined plane. We assume that the characteristic height of the layer is small in comparison to the characteristic length, so that lubrication approximation can be applied. After re-scaling the governing equations we focus on the leading order approximation and we consider the quasi-steady regime which occurs when the velocity of the advancing front and the velocity of the fluid at the inlet are substantially different. We write the differential equation for the evolution of the upper free surface and solve it numerically, plotting the profile of the layer together with the evolution of the advancing front. A comparison with the Newtonian model is also presented, with particular emphasis on the motion of the front.