Past exam of the mathematics course of the University of Cambridge / 2021 / ib / Paper 4 / 16A / a

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- Solution a

Solution

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}

. Since the flow is incompressible,

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.

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