Self-similar collapse of collisional gas in an expanding Universe
  Similarity solutions are found for the adiabatic collapse of density perturbations $\delta M/M \propto r^{-s}$ $(s>0)$ in a flat universe containing collisional gas only. The solutions are obtained for planar, cylindrical, and spherical perturbations with zero initial pressure. For adiabatic index $\gamma\ge 4/3$, a shock develops at a fixed fraction of the current turnaround distance. Near the center of a spherical perturbations with $\gamma>4/3$ and $s>1/2$, the gas is in quasi-hydrostatic equilibrium (pressure supported) and has an asymptotic power law density profile, $\rho\sim r^{-3s/(s+1)}$, independent of $\gamma$. For $s\le 1/2$, the profile depends on $\gamma$, the pressure is finite, the temperature decreases inward, and gravity dominates pressure causing a continuous inward flow. Although for $1/2<s<2$ the temperature decreases at the center, the gas is pressure supported. The pressure is finite in cylindrical perturbations for $s\le 2(\gamma-1)/(3\gamma-4)$, and in planar perturbations for any $s>0$. We also derive the asymptotic behaviour of the gas variables near the center in a universe dominated by collisionless matter. In such a universe, the gas in a spherical perturbation with $s<2$ cannot be pressure supported and the temperature approaches a constant near the center. The solutions and the asymptotic behaviour are relevant for modelling the gas distribution in galaxy clusters and pancake-like superclusters, and determining the structure of haloes of self-interacting dark matter with large interaction cross section. 
Leonid Chuzhoy and Adi Nusser