Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 334 / 1 / a / 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 $R$, length $\ell$, and prescribed volume flux $Q$, Hagen-Poiseuille flow givesIf maintenance costs $\alpha$ per unit volume,Setting the derivative of ${\mathcal{P}}_{\mathrm{pump}}+{\mathcal{P}}_{\mathrm{met}}$ with respect to $R$ to zero givesThus $Q\propto R^3$. Conservation of volume flux at a bifurcation gives