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Inclined viscous film with opposing surface shear

For a film of thickness

h

flowing down a slope of angle

α

, an upslope surface stress of magnitude

τ

gives

u (z) = \frac{ρ g s i n α}{μ} (h z - \frac{z ^{2}}{2}) - \frac{τ z}{μ} .

The surface velocity, total flux, and basal shear reverse at

τ = \frac{1}{2} ρ g h sin α, \frac{2}{3} ρ g h sin α, ρ g h sin α,

respectively.

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Solution