No Go Theorem for Kinematic Self-Similarity with A Polytropic Equation   of State
  We have investigated spherically symmetric spacetimes which contain a perfect fluid obeying the polytropic equation of state and admit a kinematic self-similar vector of the second kind which is neither parallel nor orthogonal to the fluid flow. We have assumed two kinds of polytropic equations of state and shown in general relativity that such spacetimes must be vacuum. 
Hideki Maeda, Tomohiro Harada, Hideo Iguchi and Naoya Okuyama