= Image-induced motion of a planar doublet
{title2=$\dot b=Ma^2/(|b|^2-a^2)^2$}
For a <two-dimensional potential doublet> of fixed complex strength $M$ outside a stationary circle of radius $a$, the <Milne-Thomson circle theorem> gives the image contribution $\bar M z/(a^2-\bar b z)$. If the singularity is stipulated to follow the regular fluid <velocity> at its position, differentiating this term and using the <complex velocity> convention gives the displayed motion law. Thus $\operatorname{Im}(b\bar M)$ is constant: the trajectory is a straight line parallel to $M$. With $M=me^{i\alpha}$ and initial location $b_0>a$ real, its <polar coordinates> satisfy $r\sin(\theta-\alpha)=-b_0\sin\alpha$. This statement assumes fixed orientation; the translational rule alone supplies no equation for orientation evolution.
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