Past exam of the options I course of the University of Oxford / 2011 / K (Fluid Dynamics and Waves)

A streamline is a curve whose tangent is parallel to the instantaneous fluid velocity field. In steady flow one may parametrize it by $d\mathbf{r}/ds=\mathbf{u}(\mathbf{r})$. Dotting the steady Euler equations with $\mathbf{u}$ eliminates the cross-product term, since $\mathbf{u}\cdot ((\nabla \times \mathbf{u})\times \mathbf{u})=0$. Hencewhich is Bernoulli's principle along each streamline.

For the supplied field, $xu+yv=0$, so along a streamlineWriting $r^2=x^2+y^2$, the tangential speed is $v_{\theta }=-br+c/r$ and the radial speed is zero. Both $r$ and $|\mathbf{u}{|}^2=(-br+c/r{)}^2$ are constant around each nonstagnant circular streamline. The Bernoulli quantity is constant there, so the pressure is constant on every such circle.

If a circle is wholly stagnant, the same conclusion follows directly from the steady Euler equations: on it $\mathbf{u}=0$ and $\nabla |\mathbf{u}{|}^2=0$, hence $\nabla p=0$. Equivalently the radial momentum equation gives $\nabla p=\rho (-br+c/r{)}^2{\mathbf{e}}_r/r$ and no angular pressure variation, including at zero tangential speed. Thus $p$ is constant on $x^2+y^2=a^2$ for every nonzero $a$, with no exception for a stagnation circle.

Use the convention that positive vortex strength is counterclockwise and complex velocity is $u-iv$. A complex potential of a two-dimensional potential flow for the superposed uniform flow and two point vortices isLocal choices of logarithm branch change no velocity. Differentiation givesFor nonzero vortex strength and $U>0$, its two zeros areThe number under the square root is nonzero, so these are distinct stagnation points, neither at a vortex.

A free point vortex moves with the velocity induced by the other vortex and the background, omitting its own singular field. At either vortex, the other induces vertical velocity $-\Gamma /(4\pi a)$, while the background contributes $(U,0)$. Their velocities are therefore equal. Their separation stays $(2a,0)$, making this induced velocity constant for all time. Both travel along parallel straight lines with