<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> gives
$$
\mathcal P_{\rm pump}
=\Delta p\,Q
=\frac{8\mu\ell Q^2}{\pi R^4}.
$$
If maintenance costs $\alpha$ per unit volume,
$$
\mathcal P_{\rm met}=\alpha\pi R^2\ell.
$$
Setting the derivative of $\mathcal P_{\rm pump}+\mathcal P_{\rm met}$ with respect to $R$ to zero gives
$$
Q=\frac{\pi}{4}\sqrt{\frac{\alpha}{\mu}}\,R^3.
$$
Thus $Q\propto R^3$. Conservation of volume flux at a bifurcation gives
$$
\boxed{R_0^3=R_1^3+R_2^3}.
$$

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/iii/paper-334/1/a/i/solution