Constitutive Relations for Stress and Heat Flux in a Shock Wave

A form of Maxwell's moment method is used to obtain constitutive relations for stress and heat flux for a gas passing through a shock wave. The cx2 moment of the Boltzmann equation together with several physical approximations are used to obtain a nonlinear expression relating stress and deformation rate, with Mach number as a parameter. It is shown that the constitutive relations for both stress and heat flux reduce to the Navier-Stokes relations for the case M = 1 and |τ/p| ≪ 1, and differ from these relations by almost a factor of 2 for M = 1 and |τ/p| = 1. Theoretical density profiles for shock-wave Mach numbers between 2.8 and 8 are compared with experimental data for shock waves in argon, and the agreement is shown to be excellent, thus supporting the use of the proposed constitutive relations. The physical significance of the approximations used and areas for further improvement in the theory are discussed.