Past exam of the options I course of the University of Oxford / 2011 / K / 2

Solution

Use the convention that positive vortex strength is counterclockwise and complex velocity is

u - i v

W (z) = U z + \frac{Γ}{2 πi} lo g (z - a) - \frac{Γ}{2 πi} lo g (z + a) .

W^{'} (z) = U - \frac{i Γ a}{π ( z ^{2} - a ^{2} )} .

For nonzero vortex strength and

U > 0

, its two zeros are

z_{\pm} = \pm a^{2} + \frac{i Γ a}{π U} .

The number under the square root is nonzero, so these are distinct stagnation points, neither at a vortex.

A free point vortex moves with the velocity induced by the other vortex and the background, omitting its own singular field. At either vortex, the other induces vertical velocity

- Γ/ (4 πa)

, while the background contributes

(U, 0)

. Their velocities are therefore equal. Their separation stays

(2 a, 0)

, making this induced velocity constant for all time. Both travel along parallel straight lines with

V = (U, - \frac{Γ}{4 πa}), ∣ V ∣ = U^{2} + \frac{Γ ^{2}}{16 π ^{2} a ^{2}} .

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