= Point vortex in stagnation flow beside a wall
{title2=$\dot x=\kappa x+\Gamma/(4\pi y),\quad\dot y=-\kappa y$}
In the half-plane $y>0$ with an impermeable wall, place a <point vortex> of circulation $\Gamma$ in the background <velocity> $(\kappa x,-\kappa y)$, where $\kappa>0$. The <image vortex at a plane wall> has circulation $-\Gamma$ at the reflected location and contributes horizontal velocity $\Gamma/(4\pi y)$. Excluding the vortex's own singular velocity gives the displayed motion equations.
Put $c=\Gamma/(4\pi)$. Their first integral is $\kappa xy+c\log y$, since its time derivative is $\kappa[(\kappa x+c/y)y-\kappa xy]+c(-\kappa)=0$. The explicit trajectory is $y=y_0e^{-\kappa t}$ and $x=e^{\kappa t}(x_0+ct/y_0)$. These formulas show how image-induced drift changes the hyperbolic trajectories of the background flow while preserving a modified stream-function-type invariant.
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