Past exam of the options I course of the University of Oxford / 2012 / K / K2 / c / Solution

For a point vortex in stagnation flow beside a wall, a freely moving point vortex is transported by the regular velocity due to the background and its image, excluding its own singular velocity. Differentiate the potential and evaluate that regular part at $z_0=x+iy$:Thus the equations of motion areWriting $c=\Gamma /(4\pi )$, differentiate the proposed path function:HenceFor initial position $(x_0,y_0)$, an explicit check is $y(t)=y_0{e}^{-t}$ and $x(t)=e^t(x_0+ct/y_0)$. Their substitution makes the linear terms in $t$ cancel in the path function and also shows that the vortex stays above the wall at every finite time.