Murray's law

Remarks

By Poiseuille's law the pressure drop is $\Delta p = 8 \mu \ell Q / (\pi r^4)$, so the pumping power is $Q \Delta p$, the first term of $P$. Then $P'(r) = -32 \mu \ell Q^2/(\pi r^5) + 2 b \pi r \ell$, which vanishes exactly when $Q^2 = \pi^2 b r^6/(16 \mu)$; $P'$ is negative before and positive after this point, so it is the unique minimum. Part 2 follows from $Q_0 = Q_1 + \cdots + Q_n$ and $Q_k \propto r_k^3$ with the same constant. Example: a vessel splitting into two equal daughters has $r_1 = r_2 = 2^{-1/3} r_0 \approx 0.79 \, r_0$. The hypotheses matter: for turbulent flow or for vessels with very different costs, the same optimisation gives a different exponent. Cecil D. Murray derived the law in 1926 for blood vessels.

Going deeper

For a student

It is a simple optimisation built on the Hagen–Poiseuille law, and it is a basic example of optimal transport networks in biology. Since $Q \propto r^3$, the mean speed $Q / (\pi r^2)$ is proportional to $r$, and the wall shear stress $4 \mu Q / (\pi r^3)$ is the same in all vessels of an optimal network.