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ID: past-exam-of-the-fluids-course-of-the-university-of-oxford/2021/4/b/i/solution

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Past exam of the fluids course of the University of Oxford / 2021 / 4 / b / i / Solution by

Codex  0 2026-10-25

Using

w = ϕ + i ψ

and the complex velocity convention

w^{'} = v_{x} - i v_{y}

w = β (x^{2} - y^{2}) + 2 i β x y, ψ = 2 β x y, v_{x} = 2 β x, v_{y} = - 2 β y .

(1)

The streamlines are

x y = ψ / (2 β)

, a family of rectangular hyperbolas whose asymptotes are the coordinate axes. The origin is a stagnation point. Fluid moves inward along the

y

axis and outward along the

x

axis, with the same directions approached by the neighbouring streamlines. This is a planar extensional incompressible flow, illustrated in the left panel below.

Figure 1.
Quadratic complex potentials give hyperbolic streamlines and area-preserving dye deformation
. Original streamlines for zero phase and phase $π /3$ , with arrows from the analytic velocity fields. The principal extension axis rotates by $- α /2$ . The right panel compares a unit dye circle with its image at $βt = 0.3$ for $α = π /3$ ; the ellipse has the same area. **Details**.

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