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

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

Codex  0 2026-10-25

Taking the imaginary part of the complex potential gives the streamfunction

ψ = β [(x^{2} - y^{2}) sin α + 2 x y cos α] = β r^{2} sin (2 θ + α) .

The corresponding velocity field is

(v_{x} v_{y}) = 2 β (cos α - sin α - sin α - cos α) (x y) .

Put

ζ = z e^{i α /2} = ξ + i η

. Then

w = β ζ^{2}

and

ψ = 2 β ξ η

. Thus the streamlines are the previous rectangular hyperbolas rotated clockwise by

α /2

. Their separatrices have angles

θ = - α /2 + nπ /2

; the extension axis is

θ = - α /2

and the compression axis is perpendicular to it. The middle panel of the figure shows this rotation.

For pure advection, a marked fluid element obeys

\dot{ξ} = 2 β ξ, \overset{η}{˙} = - 2 β η, ξ (t) = e^{2 βt} ξ (0), η (t) = e^{- 2 βt} η (0) .

Consequently the dye circle becomes an ellipse with semiaxes

R e^{2 βt}

and

R e^{- 2 βt}

. Its area is

A (t) = π (R e^{2 βt}) (R e^{- 2 βt}) = π R^{2} .

Stretching in one principal direction exactly compensates compression in the other. Equivalently the flow-map Jacobian determinant is one, as required by incompressibility. Negligible diffusion is needed to regard the sharp dye boundary as a material curve.

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