Fluid Dynamics | Part IB, 2002

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

DωDt=0\frac{D \omega}{D t}=0

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

Consider now the flow generated in unbounded fluid by two point vortices of strengths κ1\kappa_{1} and κ2\kappa_{2} at x1(t)=(x1,y1)\mathbf{x}_{1}(t)=\left(x_{1}, y_{1}\right) and x2(t)=(x2,y2)\mathbf{x}_{2}(t)=\left(x_{2}, y_{2}\right), respectively. Show that in the subsequent motion the quantity

q=κ1x1+κ2x2\mathbf{q}=\kappa_{1} \mathbf{x}_{1}+\kappa_{2} \mathbf{x}_{2}

remains constant. Show also that the separation of the vortices, x2x1\left|\mathbf{x}_{2}-\mathbf{x}_{1}\right|, remains constant.

Suppose finally that κ1=κ2\kappa_{1}=\kappa_{2} and that the vortices are placed at time t=0t=0 at positions (a,0)(a, 0) and (a,0)(-a, 0). What are the positions of the vortices at time tt ?

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