Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 334 / 1 / a / ii

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- Solution ii

Solution

For plane Poiseuille flow between walls

y = \pm h

, let

G = - p_{x} > 0

. Per unit out-of-plane depth,

u (y) = \frac{G}{2 μ} (h^{2} - y^{2}), Q = \int_{- h}^{h} u d y = \frac{2 G h ^{3}}{3 μ} .

For vessel length

ℓ

,

P_{pump} = Δ p Q = \frac{3 μ ℓ Q ^{2}}{2 h ^{3}} .

The maintained cross-sectional area per unit depth is

2 h

, so

P_{met} = 2 α h ℓ .

Optimization gives

- \frac{9 μ ℓ Q ^{2}}{2 h ^{4}} + 2 α ℓ = 0, Q = \frac{2}{3} \frac{α}{μ} h^{2} .

The two-dimensional Murray law is therefore

h_{0}^{2} = h_{1}^{2} + h_{2}^{2} .

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