4.II.16C

Use Euler's equation to show that in a planar flow of an inviscid fluid the vorticity $\boldsymbol{\omega}$ satisfies

$\frac{D \omega}{D t}=0$

Write down the velocity field associated with a point vortex of strength $\kappa$ in unbounded fluid.

Consider now the flow generated in unbounded fluid by two point vortices of strengths $\kappa_{1}$ and $\kappa_{2}$ at $\mathbf{x}_{1}(t)=\left(x_{1}, y_{1}\right)$ and $\mathbf{x}_{2}(t)=\left(x_{2}, y_{2}\right)$, respectively. Show that in the subsequent motion the quantity

$\mathbf{q}=\kappa_{1} \mathbf{x}_{1}+\kappa_{2} \mathbf{x}_{2}$

remains constant. Show also that the separation of the vortices, $\left|\mathbf{x}_{2}-\mathbf{x}_{1}\right|$, remains constant.

Suppose finally that $\kappa_{1}=\kappa_{2}$ and that the vortices are placed at time $t=0$ at positions $(a, 0)$ and $(-a, 0)$. What are the positions of the vortices at time $t$ ?

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