 Solution

ID: past-exam-of-the-fluids-and-waves-course-of-the-university-of-oxford/2023/1/b/ii/solution

Put

A = R^{2} \dot{R}

ϕ_{t} = \dot{G} - \frac{A ˙}{r}, ∣\nabla ϕ ∣^{2} = \frac{A ^{2}}{r ^{4}} .

At infinity the velocity vanishes and

p \to p_{\infty}

F (t) = \dot{G} + p_{\infty} / ρ

. Substitution cancels the arbitrary

G

\frac{p ( r , t )}{ρ} = \frac{1}{r} \frac{d}{d t} (R^{2} \dot{R}) - \frac{R ^{4} R ˙ ^{2}}{2 r ^{4}} + \frac{p _{\infty}}{ρ} .

The distinction between

\partial_{t} ϕ

at fixed

r

and the material derivative along the moving boundary is essential here.

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