 Adiabatic superbubble similarity solution

For constant injected power $\dot{E}$ and ambient mass density ${\rho }_1$, dimensional analysis gives $R=\alpha (\dot{E}/{\rho }_1{)}^{1/5}{t}^{3/5}$. In a similarity solution, set $\eta =r/R$, $\rho ={\rho }_{1}f(\eta )$, $p={\rho }_1{\dot{R}}^{2}g(\eta )$ and $u=\dot{R}h(\eta )$. The strong-shock values on the inner side of $\eta =1$ are $f=(\gamma +1)/(\gamma -1)$ and $g=h=2/(\gamma +1)$.