Past exam of the fluids and waves course of the University of Oxford / 2017 / 2 / b / iv

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- Solution iv

Solution

For the same counterclockwise line vortex strength

κ

, its local velocity potential and streamfunction are

ϕ = κ θ / (2 π)

and

ψ = - κ lo g r / (2 π)

. Consequently

w (z) = - \frac{iκ}{2 π} lo g z, w^{'} (z) = - \frac{iκ}{2 π z}, b_{1} = - \frac{iκ}{2 π} .

Substitution gives

2 πi b_{1} = κ = Γ + i M, M = 0,

in agreement with the direct circulation and volume flux calculation. The sign of the imaginary coefficient is essential for a positive counterclockwise circulation.

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