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Past exam of the mathematics course of the University of Cambridge / 2021 / ib / Paper 4 / 16A / b / Solution

The radial velocity and velocity potential are

u (r, t) = \frac{Q ( t )}{4 π r ^{2}}, ϕ (r, t) = - \frac{Q ( t )}{4 π r} .

Choose the gauge in the Unsteady Bernoulli equation by evaluating it at infinity, where

u = 0

,

ϕ = 0

, and

p = p_{0}

:

ϕ_{t} + \frac{1}{2} u^{2} + \frac{p}{ρ} = \frac{p _{0}}{ρ} .

At the cavity surface the vacuum pressure is zero, and the kinematic boundary condition gives

u (a, t) = \overset{a}{˙}

. Since

ϕ_{t} (a, t) = - \frac{1}{4 πa} \frac{d Q}{d t},

Bernoulli's equation becomes

- \frac{1}{4 πa} \frac{d Q}{d t} + \frac{a ˙ ^{2}}{2} = \frac{p _{0}}{ρ} .

Therefore

\frac{1}{4 πa} \frac{d Q}{d t} - \frac{a ˙ ^{2}}{2} = - \frac{p _{0}}{ρ} .

Solved by gpt-5.6-sol high.

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