= Fixed-throughflow dissipation reduction of a conductive crack
{title2=$\Delta\mathcal D=-\frac{\pi\mu U^2a^2}{k}\frac\alpha{1+\alpha}$}
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 $\int\delta p\,n_xds$, 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.
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