= Wall-induced motion of a planar potential dipole
{title2=$\dot{\bar a}=w'_{\mathrm{reg}}(a)$}
A transported <planar potential dipole> follows the surrounding regular <velocity>, so its position satisfies the displayed equation after its own singular term is removed. The <conjugation> comes from the convention $w'=v_x-iv_y$. Reflected image parameters are held fixed when taking the spatial <derivative>, then evaluated at the instantaneous <planar potential dipole>[dipole] position.
For a unit circular wall, the <Milne-Thomson circle theorem> gives image potential $\bar m z/(1-\bar a z)$ and therefore $\dot{\bar a}=\bar m/(1-|a|^2)^2$. For a horizontal straight wall the image is $\bar m/(z-\bar a)$, giving $\dot{\bar a}=-\bar m/(a-\bar a)^2$. For the line $y=-x$, reflected <conjugation> gives image $-i\bar m/(z+i\bar a)$ and hence $\dot{\bar a}=i\bar m/(a+i\bar a)^2$. In each case the image lies outside the fluid region and makes the wall a <streamline>; differentiation of the regular image proves the motion formula. Several compatible straight-wall reflections can be summed before the same operation is applied.
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