Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 314 / 2 / b

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- Solution b

Solution

The radial equation can be written

\frac{d}{d r} (\frac{u ^{2}}{2} + c_{s}^{2} ln ρ - \frac{GM}{r}) = 0.

Thus the Bernoulli function is constant on each streamline:

B = \frac{u ^{2}}{2} + c_{s}^{2} ln ρ + Φ, Φ = - \frac{GM}{r} .

The boundary conditions

u \to 0

and

ρ \to ρ_{0}

r \to \infty

fix

B = c_{s}^{2} ln ρ_{0}

.

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