For <plane Poiseuille flow> between walls $y=\pm h$, let $G=-p_x>0$. Per unit out-of-plane depth,
$$
u(y)=\frac{G}{2\mu}(h^2-y^2),
\qquad
Q=\int_{-h}^h u\,dy
=\frac{2Gh^3}{3\mu}.
$$
For vessel length $\ell$,
$$
\mathcal P_{\rm pump}
=\Delta p\,Q
=\frac{3\mu\ell Q^2}{2h^3}.
$$
The maintained cross-sectional area per unit depth is $2h$, so
$$
\mathcal P_{\rm met}=2\alpha h\ell.
$$
Optimization gives
$$
-\frac{9\mu\ell Q^2}{2h^4}+2\alpha\ell=0,
\qquad
\boxed{Q=\frac23\sqrt{\frac{\alpha}{\mu}}\,h^2}.
$$
The two-dimensional Murray law is therefore
$$
\boxed{h_0^2=h_1^2+h_2^2}.
$$

Solution

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

 Solution

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