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 Incoming links: Pressure field of a pulsating sphere

Past exam of the fluids and waves course of the University of Oxford / 2023 / 1 / b / ii / Solution 2026-10-25

Put

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

. The partial derivative in the Unsteady Bernoulli equation is taken at fixed spatial position, so

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

and yields the pressure field of a pulsating sphere:

\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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Past exam of the fluids and waves course of the University of Oxford / 2023 / 1 / c / i / Solution 2026-10-25

With zero internal pressure and negligible surface tension, pressure continuity sets the liquid-side boundary value to

p (R, t) = 0

. Evaluating the pressure field of a pulsating sphere there gives

0 = \frac{1}{R} (R^{2} \ddot{R} + 2 R \dot{R}^{2}) - \frac{1}{2} \dot{R}^{2} + \frac{p _{\infty}}{ρ} .

Thus the Rayleigh equation for an inviscid spherical bubble, together with its initial conditions, is

R \ddot{R} + \frac{3}{2} \dot{R}^{2} = - \frac{p _{\infty}}{ρ}, R (0) = a, \dot{R} (0) = 0 .

For

p_{\infty} > 0

, initially

\ddot{R} (0) = - p_{\infty} / (ρ a) < 0

, so the physically relevant branch describes Rayleigh collapse of a spherical cavity.

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