| dc.description |
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. |
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