Elliptic crack pressure dipole ($\delta p\sim \frac{\alpha \mu U{a}^2}{2(1+\alpha )k}\frac{\cos \theta }{r}$)

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

For dipole pressure $\delta p\sim Cx/(x^2+y^2)$, choose a rectangular control volume whose transverse size tends to infinity faster than its streamwise size. Each end face contributes $\pi C$ to $\int \delta p{n}_{x}ds$, so fixed-throughflow boundary work gives $\Delta \mathcal{D}=-2\pi UC$, the displayed result. This control-surface choice removes the integrated far-field velocity correction. A circular cutoff of the pressure-only functional gives $-\pi UC$ instead and does not implement the same throughflow normalization.