Past exam of the mathematics course of the University of Cambridge / 2024 / ib / Paper 2 / 5D (Fluid Dynamics)

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- Solution 5D

Solution

Writing the velocity as

(u, v) = (y, a x)

gives

\partial_{x} u + \partial_{y} v = 0,

so the flow is incompressible. With the convention

u = ψ_{y}, v = - ψ_{x},

a stream function is

ψ (x, y) = \frac{1}{2} (y^{2} - a x^{2}) .

The streamlines are its level sets. For

a > 0

, they are the hyperbolas

y^{2} - a x^{2} = constant,

with separatrices

y = \pm a x

and a saddle at the origin. For

a < 0

, they are concentric ellipses

y^{2} + ∣ a ∣ x^{2} = constant,

traversed clockwise. This is the streamline classification of a planar linear saddle or centre.

The scalar vorticity is

\partial_{x} v - \partial_{y} u = a - 1.

Hence the flow is irrotational exactly when

a = 1 .

Then

(u, v) = (y, x) = \nabla ϕ

, and a velocity potential is

ϕ (x, y) = x y + constant .

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