A non-equilibrium kinetic description of shock wave structure

A formulation for the shock wave structure is devised by viewing the transition as a phenomenon in which non-equilibrium effects play an important role. The essence of the method is the approximation of Boltzmann's equation by a simpler kinetic model. Initially, the distribution function in Boltzmann's collision integral is expressed in terms of a function of deviation from local equilibrium. Then, by suitably transforming the complete collision term, the molecular velocities after collision are eliminated. At this stage the formulation of the method is specialized to hard sphere molecules and the problem of deriving a model equation thus reduces to one of assigning an expression for the deviation function. In the first instance, this function is chosen to be zero and an exploratory model is obtained which, when its variable collision frequency is replaced by its mean value, reduces identically to the Bhatnagar-Gross-Krook model. However, it is found that the exploratory model provides a somewhat crude representation of Boltzmann's equation and is shown to imply a Prandtl number very nearly equal to unity. A more accurate model is then derived by choosing for the deviation function the first order term of Chapman-Enskog’s sequence, leading to the Navier-Stokes equations. Here, the specific form of Boltzmann's collision term is represented more accurately than hitherto and the model is found to possess all the known features of the Boltzmann equation. It is shown that this model contains a description of a gas in non-equilibrium state.