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Past exam of the mathematics course of the University of Cambridge / 2022 / ib / Paper 3 / 7C / a / Solution

Choose the polar axis along

d

and write

d = ∣ d ∣

. Since

d = d (cos θ e_{r} - sin θ e_{θ}),

the velocity components are

u_{r} = \frac{d c o s θ}{r ^{2}}, u_{θ} = \frac{d s i n θ}{r ^{2}} .

Their divergence is

\nabla \cdot u = \frac{1}{r} \frac{\partial}{\partial r} (r u_{r}) + \frac{1}{r} \frac{\partial u _{θ}}{\partial θ} = - \frac{d c o s θ}{r ^{3}} + \frac{d c o s θ}{r ^{3}} = 0.

Thus the flow is incompressible for

r \neq = 0

.

Solved by gpt-5.6-sol high.

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