ABSTRACT
This paper is devoted to a discussion of the profiles of shock waves using the full nonlinear Burnett equations of hydrodynamics as they appear from the Chapman-Enskog solution to the Boltzmann equation. The system considered is a dilute gas composed of rigid spheres. The numerical analysis is carried out by transforming the hydrodynamic equations into a set of four first-order equations in four dimensions. We compare the numerical solutions of the Burnett equations, obtained using Adam's method, with the well known direct simulation Monte Carlo method for different Mach numbers. An exhaustive mathematical analysis of the results offered here has been done mainly in connection with the existence of heteroclinic trajectories between the two stationary points located upflow and downflow. The main result of this study is that such a trajectory exists for the Burnett equations for Mach numbers greater than 1. Our numerical calculations suggest that heteroclinic trajectories exist up to a critical Mach number ( approximately 2.69) where local mathematical analysis and numerical computations reveal a saddle-node-Hopf bifurcation. This upper limit for the existence of heteroclinic trajectories deserves further clarification.
ABSTRACT
In 1982 Bobylev [A.V. Bobylev, Sov. Phys. Dokl. 27, 29 (1982)] made a linear stability analysis of the Burnett equations and showed that beyond a certain critical reduced wave number there exist normal modes that grow exponentially, concluding that the Burnett equations are linearly unstable. We have partially extended his analysis, originally made for Maxwellian molecules, for any interaction potential and argue that his results can be reinterpreted as to give a bound for the Knudsen number above which the Burnett equations are not valid.