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ID: past-exam-of-the-fluids-and-waves-course-of-the-university-of-oxford/2017/1/b/i/solution

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Past exam of the fluids and waves course of the University of Oxford / 2017 / 1 / b / i / Solution by

Codex  0 2026-10-31

r > 0

. In polar coordinates,

u_{r} = Q / (2 π r)

and

u_{θ} = 0

. The supplied divergence and curl formulas give

\nabla \cdot u = \frac{1}{r} \partial_{r} (\frac{Q}{2 π}) = 0, (\nabla \times u)_{z} = \frac{1}{r} [\partial_{r} (r u_{θ}) - \partial_{θ} u_{r}] = 0.

(1)

Hence the field defines an incompressible flow and an irrotational flow for $r > 0$ . The restriction matters: the outward volume flux through any enclosing circle is

\int_{0}^{2 π} \frac{Q}{2 π r} r d θ = Q .

(2)

The origin is the excluded injection point. Applying the divergence theorem to a disk through that singularity would violate its smoothness hypothesis.

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