Solution

Let

x

point downslope and

z

point normally away from the plane, with

0 \leq z \leq h

. Write the parallel velocity as

u (z) e_{x}

and take

τ > 0

to be the magnitude of the air's upslope stress. The steady equations are

0 = - p_{x} + ρ g sin α + μ u^{''}, 0 = - p_{z} - ρ g cos α,

u (0) = 0, p (h) = p_{0}, μ u^{'} (h) = - τ .

The free-surface condition makes

p_{x} = 0

, and integration gives

p (x, z) = p_{0} + ρ g cos α (h - z),

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

The surface velocity, downslope shear exerted by the fluid on the plane, and volume flux per unit width are

u_{h} = \frac{h}{μ} (\frac{ρ g h sin α}{2} - τ),

τ_{0} = μ u^{'} (0) = ρ g h sin α - τ,

q = \int_{0}^{h} u (z) d z = \frac{ρ g h ^{3} sin α}{3 μ} - \frac{τ h ^{2}}{2 μ} .

quantity directed upslope u_{h} q τ_{0} condition τ > \frac{1}{2} ρ g h sin α τ > \frac{2}{3} ρ g h sin α τ > ρ g h sin α .

The order of increasing required air stress is therefore

surface velocity, volume flux, stress on the plane .

Solved by gpt-5.6-sol high.

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