39A (Fluid Dynamics II)

H / L ≪ 1

, small wall slope, and negligible inertia, for example

ρ U H / μ ≪ L / H

. To leading order,

p_{z} = 0, μ u_{zz} = \nabla p, \nabla \cdot u + w_{z} = 0.

With no slip at

z = 0, h

u = U (1 - \frac{z}{h}) + \frac{z ( z - h )}{2 μ} \nabla p .

Integration gives

q = \frac{U h}{2} - \frac{h ^{3}}{12 μ} \nabla p .

Integrated incompressibility gives

\nabla \cdot q = 0

, so locally

q = \nabla \land (0, 0, ψ)

. Solving the flux law for

\nabla p

and taking its curl yields

\nabla \cdot (h^{- 3} \nabla ψ) = - U h^{- 3} h_{y} .

b

q = - \frac{h _{0}^{3}}{12 μ} \nabla p = \frac{h _{0}^{3} G}{12 μ} e_{x},

ψ \sim A r sin θ, A = \frac{h _{0}^{3} G}{12 μ} .

Continuity of

q_{r} = r^{- 1} ψ_{θ}

gives

[ψ_{θ}]_{-}^{+} = 0

. Since

\nabla p = - 12 μ h^{- 3} q

, continuity of tangential pressure derivative gives

[h^{- 3} ψ_{r}]_{-}^{+} = 0

The harmonic solutions regular at zero and with the prescribed far field are

ψ_{>} = (A r + \frac{B}{r}) sin θ, ψ_{<} = C r sin θ .

The interface conditions give

B = A a^{2} \frac{h _{1}^{3} - h _{0}^{3}}{h _{0}^{3} + h _{1}^{3}}, C = \frac{2 A h _{1}^{3}}{h _{0}^{3} + h _{1}^{3}} .

Integrating the outward flux across the appropriate semicircle gives

2 a C = \frac{4 A a h _{1}^{3}}{h _{0}^{3} + h _{1}^{3}} .

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