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 Incoming links: Half-contraction free streamline at a horizontal canal lip

Past exam of the perturbation methods and applied complex variables course of the University of Oxford / 2008 / B / 6 / iv / Solution 2026-10-31

Integrate real and imaginary parts from the lip value

z (- π) = i

:

x (θ) = - \frac{a}{π} \int_{- π}^{θ} cos s cot (s /2) d s, y (θ) = 1 - \frac{a}{π} \int_{- π}^{θ} sin s cot (s /2) d s .

The identities

sin s cot (s /2) = 1 + cos s

and

\int cos s cot (s /2) d s = lo g (1 - cos s) + cos s

give the explicit parametrization

x (θ) = - \frac{a}{π} [lo g (1 - cos θ) + cos θ - lo g 2 + 1],

y (θ) = 1 - \frac{a}{π} (θ + π + sin θ), - π < θ < 0.

θ ↑ 0

,

x \to + \infty

and

y \to 1 - a

. This must be the downstream height

a

, so

a = \frac{1}{2} .

The surface initially moves slightly left of the lip before turning right. Its leftmost point occurs at

θ = - π /2

, with

x_{m i n} = - (1 - lo g 2) / (2 π)

. This is the half-contraction free streamline at a horizontal canal lip.

Figure 1.
The fluid free boundary contracts to half the lip height and traces the upper hodograph plane
. Details.

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