 Solution

ID: past-exam-of-the-fluids-and-waves-course-of-the-university-of-oxford/2017/2/b/ii/solution

Multiply the complex velocity by the complex displacement:

\frac{d w}{d z} d z = (u - i v) (d x + i d y) = (u d x + v d y) + i (u d y - v d x) .

The real term is the circulation integrand. With a counterclockwise curve,

n d s = (d y, - d x)

is outward, so the imaginary term is the outward volume flux integrand. Integrating gives

\oint_{C (t)} \frac{d w}{d z} d z = Γ (t) + i M (t) .

This also fixes the sign of

M

: a positive line source has

w^{'} = Q / (2 π z)

and the integral is

i Q

, so its outward volume flux is

Q

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