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

Table of contents
- Solution i

Solution

Murray's law minimizes the sum of the power needed to pump a Newtonian fluid and the metabolic power needed to maintain blood volume. For a cylindrical vessel of radius

R

ℓ

, and prescribed volume flux

Q

, Hagen-Poiseuille flow gives

P_{pump} = Δ p Q = \frac{8 μ ℓ Q ^{2}}{π R ^{4}} .

If maintenance costs

α

per unit volume,

P_{met} = α π R^{2} ℓ .

Setting the derivative of

P_{pump} + P_{met}

with respect to

R

to zero gives

Q = \frac{π}{4} \frac{α}{μ} R^{3} .

Thus

Q \propto R^{3}

. Conservation of volume flux at a bifurcation gives

R_{0}^{3} = R_{1}^{3} + R_{2}^{3} .

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