 Solution

The wall shear stress vanishes when $\mu u^{\prime }(0)=\rho gh\sin \alpha -S=0$. ThereforeFor a genuinely inclined plane the entire moving layer now flows upslope. The profile touches zero at the wall with zero derivative and becomes increasingly negative towards the surface. In the plot with velocity horizontal and height vertical, the zero derivative $du/dz=0$ at the wall is represented by a vertical tangent. The resulting volume flux is $-\rho g{h}^3\sin \alpha /(6\mu )$, so zero wall shear stress does not mean zero flux.