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For an inclined viscous film with opposing surface shear, the steady parallel velocity automatically satisfies the continuity equation and the advective acceleration vanishes. The two components of the Navier-Stokes equations reduce toThe uniform thickness and constant atmospheric pressure make $p_x=0$. The boundary conditions areThese are respectively no-slip boundary condition, imposed tangential traction on the surface with outward normal $+{\mathbf{e}}_z$, and the normal stress condition on the flat free surface. The normal viscous stress is zero because the normal velocity vanishes. Integrating givesIn particular $\mu u^{\prime }(0)=\rho gh\sin \alpha -S$ is the downslope tangential traction exerted by the fluid on the wall. The wall's tangential traction on the fluid has the opposite sign, because the outward normal of the fluid at its bottom is $-{\mathbf{e}}_z$. This fixes the potentially confusing wall sign convention.

Let $x$ point downslope and $z$ point normally away from the plane, with $0\le z\le h$. Write the parallel velocity as $u(z)e_x$ and take $\tau >0$ to be the magnitude of the air's upslope stress. The steady equations arewith boundary conditionsThe free-surface condition makes $p_x=0$, and integration gives

The surface velocity, downslope shear exerted by the fluid on the plane, and volume flux per unit width areThus the inclined viscous film with opposing surface shear reverses in the three senses whenThe order of increasing required air stress is therefore