Past exam of the mathematics course of the University of Cambridge / 2020 / ib / Paper 1 / 17C / c

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- Solution c

Solution

Using

γ = λ

and the Pythagorean trigonometric identity,

∣ u ∣^{2} = u_{r}^{2} + u_{θ}^{2} = A^{2} γ^{2} r^{2 γ - 2},

so the speed is independent of

θ

. Since

γ = π / α > 1

,

∣ u (r) ∣ = u_{0} (\frac{r}{r _{0}})^{γ - 1}

and the speed tends to zero at the vertex. The steady irrotational Bernoulli equation gives

p + \frac{1}{2} ρ ∣ u ∣^{2} = p_{0} + \frac{1}{2} ρ u_{0}^{2} .

Thus the fluid pressure is also independent of

θ

, with

p (r) = p_{0} + \frac{1}{2} ρ u_{0}^{2} [1 - (\frac{r}{r _{0}})^{2 γ - 2}] .

In particular, the pressure difference between the vertex and

r_{0}

is

p (0) - p_{0} = \frac{1}{2} ρ u_{0}^{2} .

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