Inclined viscous film with opposing surface shear
= Inclined viscous film with opposing surface shear
For a film of thickness $h$ flowing down a slope of angle $\alpha$, an upslope surface stress of magnitude $\tau$ gives
$$
u(z)=\frac{\rho g\sin\alpha}{\mu}
\left(hz-\frac{z^2}{2}\right)-\frac{\tau z}{\mu}.
$$
The surface velocity, total flux, and basal shear reverse at
$$
\tau=\frac12\rho gh\sin\alpha,
\qquad
\frac23\rho gh\sin\alpha,
\qquad
\rho gh\sin\alpha,
$$
respectively.