Shock waves in Stokes flows down an inclined plate

We consider a viscous flow on an inclined plate, such that the liquid's depth far upstream is larger than that far downstream, resulting in a "smoothed-shock wave" steadily propagating downstream. Our numerical simulations show that in a large section of the problem's parameter space all initial conditions overturn (i.e., the liquid's surface becomes vertical at some point) and thus no steady solution exists. The overturning can only be stopped by a sufficiently strong surface tension.