 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/ib/paper-4/16a/a/solution

Codex  0 Created 2026-09-24 Updated 2026-10-03

In spherical symmetry, the velocity is radial and constant over each sphere. Hence

Q (R, t) = 4 π R^{2} u (R, t) .

Apply the divergence theorem to the fluid shell between radii

R_{1}

and

R_{2}

0 = \int_{shell} \nabla \cdot u d V = Q (R_{2}, t) - Q (R_{1}, t) .

Thus $Q$ is independent of $R$ and is a function of time alone. This is the flux law for spherically symmetric incompressible radial flow.

 New to topics? Read the docs here!