= Elliptic crack pressure dipole
{title2=$\delta p\sim\frac{\alpha\mu Ua^2}{2(1+\alpha)k}\frac{\cos\theta}{r}$}
Subtracting the uniform-flow pressure $-\mu Ux/k$ leaves $(\mu Ua/k)\alpha e^{-\xi}\cos\eta/(1+\alpha)$. The coordinate map has $r\sim ae^\xi/2$, so $e^{-\xi}\sim a/(2r)$, giving the displayed dipole coefficient. The one-half is essential. In the perfect-conductor limit the independent expression $p=-(\mu U/k)\operatorname{Re}\sqrt{z^2-a^2}$ has the same far-field expansion.
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