 Solution

ID: past-exam-of-the-advanced-fluid-dynamics-course-of-the-university-of-oxford/2023/2/a/solution

Expand the given translating-sphere Stokes flow into its Stokeslet and potential dipole parts:

u = \frac{3 a}{4} (\frac{U}{r} + \frac{( U \cdot x ) x}{r ^{3}}) + \frac{a ^{3}}{4} (\frac{U}{r ^{3}} - \frac{3 ( U \cdot x ) x}{r ^{5}}) .

The second parenthesis is

\nabla [(U \cdot x) / r^{3}]

and has zero curl outside the sphere. For the first,

\nabla \times \frac{U}{r} = \frac{U \times x}{r ^{3}}, \nabla \times \frac{( U \cdot x ) x}{r ^{3}} = \frac{U \times x}{r ^{3}} .

ω = \nabla \times u = \frac{3 a}{2} \frac{U \times x}{r ^{3}} .

The result is independent of the irrotational finite-size dipole correction. It holds in the exterior

r > a

; the singular solid harmonics are used only outside their source.

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