= Central energy input in a similarity solution
For point injection of constant power, the outward <energy flux> through a sphere tending to the source must tend to $\dot E$. In the <adiabatic superbubble similarity solution>, this requires
$$
\lim_{\eta\downarrow0}\eta^2h\left(\frac12fh^2+\frac{\gamma g}{\gamma-1}\right)=\frac{125}{108\pi\alpha^5}.
$$
Imposing finite regular central pressure and density with $h=O(\eta)$ would instead give zero flux and fail to supply the prescribed power. A resolved source region or an inner contact-boundary model can replace the point-source condition.
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