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Fluids course of the University of Oxford

Past exam of the fluids course of the University of Oxford

2021

{c}

Using $w=\phi+i\psi$ and the <complex velocity> convention $w'=v_x-iv_y$,
$$
w=\beta(x^2-y^2)+2i\beta xy,
\qquad
\boxed{\psi=2\beta xy,\quad v_x=2\beta x,\quad v_y=-2\beta y.}
$$
The <streamlines> are $xy=\psi/(2\beta)$, 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.

\Image[/past-exam-of-the-fluids-course-of-the-university-of-oxford/2021-straining-flow-streamlines.png]
{title=Quadratic complex potentials give hyperbolic streamlines and area-preserving dye deformation}
{description=Original streamlines for zero phase and phase $\pi/3$, with arrows from the analytic velocity fields. The principal extension axis rotates by $-\alpha/2$. The right panel compares a unit dye circle with its image at $\beta t=0.3$ for $\alpha=\pi/3$; the ellipse has the same area.}

Solution

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

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

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