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

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

For plane Poiseuille flow between walls $y=\pm h$, let $G=-p_x>0$. Per unit out-of-plane depth,For vessel length $\ell$,The maintained cross-sectional area per unit depth is $2h$, soOptimization givesThe two-dimensional Murray law is therefore

The wall shear stress in the planar vessel isThe optimized relation $Q\propto h^2$ therefore makesthroughout an ideal network. Murray's optimization can equivalently be interpreted as selecting a uniform wall shear stress. The familiar three-dimensional law $Q\propto R^3$ has the same interpretation because cylindrical Poiseuille flow has ${\tau }_w=4\mu Q/(\pi R^3)$.