Spherically symmetric adiabatic flow
= Spherically symmetric adiabatic flow
A smooth radial <compressible flow> with a fixed <adiabatic exponent> conserves $r^2\rho u$ and $p/\rho^\gamma$. In a prescribed <Newtonian gravitational potential>, its <Bernoulli function> is $u^2/2+c^2/(\gamma-1)+\Phi$, with $c$ the <adiabatic sound speed>. Eliminating the density derivative yields $(u-c^2/u)u'=2c^2/r-\Phi'$.