17D (Fluid Dynamics)

Take

x

vertically downward along the wall and

y

normally outward, with the wall at

y = 0

y = h

. The expected steady velocity is

u = u (y) e_{x}

: it vanishes at the wall and is opposed by an upward air shear

k u (h)

at the surface. Gravity supplies the downward body force

ρ g e_{x}

, while the surface normal stress is the atmospheric pressure

p_{0}

b

The steady parallel-flow Navier-Stokes equations reduce to

p_{y} = 0, μ u^{''} + ρ g = 0.

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

The last condition balances the liquid's tangential traction with the opposing air stress. Hence

p = p_{0}

and

u (y) = - \frac{ρ g}{2 μ} y^{2} + \frac{ρ g h}{μ + kh} (1 + \frac{kh}{2 μ}) y .

Substitution of

y = h

gives

u (h) = \frac{ρ g h ^{2}}{2 μ} (1 + \frac{kh}{μ})^{- 1} .

Q = \int_{0}^{h} u (y) d y = \frac{ρ g h ^{3}}{12 μ} \frac{4 + kh / μ}{1 + kh / μ} .

Therefore

Q ⟶ \frac{ρ g h ^{3}}{3 μ} (k \to 0), Q ⟶ \frac{ρ g h ^{3}}{12 μ} (k \to \infty) .

The first is the stress-free film; the second has an effectively immobilized free surface.

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