Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 3 / 38C / b

Solution

0 = - p_{x} + μ u_{yy},

with

p = p (x)

u (x, 0) = U, u (x, h (x)) = 0.

Twice integrating in

y

u (x, y) = U (1 - \frac{y}{h}) + \frac{p _{x}}{2 μ} y (y - h) .

Its volume flux per unit span is

Q (x) = \int_{0}^{h} u d y = \frac{U h}{2} - \frac{h ^{3} p _{x}}{12 μ} .

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