Relativistic gasdynamics in two dimensions

Steady two-dimensional flow of an ideal compressible fluid is studied in the context of special-relativistic gasdynamics. The Newtonian equations for potential flow, including the equation of characteristics, Chaplygin’s equation, and the Euler–Tricomi equation, are generalized. It is found that these equations can have the same form in the Newtonian and the relativistic regimes if their parameters are defined in the local rest-frame of the fluid. The Mach number thus defined, ℳ≡β/(1−β2)1/2×βs/(1−β2s)1/ 2−1, where β and βs are, respectively, the speed of the fluid and the speed of sound relative to the fluid (in units of the speed of light), is shown to have the same properties as M=β/βs in Newtonian theory. The Newtonian expressions for oblique plane shock waves in a perfect gas can similarly be generalized in certain cases (which include, in particular, the extreme-relativistic limit), and it is shown that the adiabatic index Γ in these expressions is generalized by Γ̂≡ Γ/(1−β2s). Some applications of these results are discussed. In addition, it is shown that the analogy between Newtonian sound speeds and the corresponding relativistic proper speeds extends to the expressions for hydromagnetic wave speeds.