Wall-induced motion of a planar potential dipole ($\dot{\bar{a}}=w_{\mathrm{reg}}^{\prime }(a)$)

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 $w^{\prime }=v_x-i{v}_y$. Reflected image parameters are held fixed when taking the spatial derivative, then evaluated at the instantaneous 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.