Other Methods Of Solution

In the previous chapters we reviewed some methods of solution for the Boltzmann equation based on perturbation expansions, i.e., the Hilbert and Chapman—Enskog expansions and the linearization of the Boltzmann equation. The latter procedure has usually been coupled with the use of kinetic models. These models, however, have been shown to be capable of arbitrarily approximating not only the linearized Boltzmann equation, but also its solutions (Chapter VI); hence the procedures presented in Chapter VII can be considered to be exact, as long as the use of the linearized Boltzmann equation is justified. The complexity of the results obtained in Chapter VII even for relatively simple problems suggests that for more complicated problems, of linear or nonlinear nature, one should look for less sophisticated procedures yielding approximate but essentially correct results. Such procedures can be easily constructed for linearized problems or in the limit of either large or small Knudsen numbers; the intermediate range of Knudsen numbers (transition region) in nonlinear situations is at present a matter of interpolation procedures of more or less sophisticated nature. In addition, it is to be noted that a good procedure does not necessarily mean a good method of solution, since in many cases the procedure consists in deducing a system of nonlinear partial differential equations. The latter have to be solved in correspondence with particular problems, and in general they are tougher than the Navier—Stokes equations for the same problem. As a consequence one has to resort to numerical procedures to solve them. The approximation procedures can be grouped under two general headings: moment methods and integral equation methods. In the former case one constructs certain partial differential equations as mentioned above, and in the latter one tries to obtain either expansions valid for large Knudsen numbers or numerical solutions. In connection with both methods one can simplify the calculations by the use of models (sometimes in an essential manner), but one has to remember that the accuracy of kinetic models in nonlinear problems is less obvious than in the linearized ones. Finally, in connection with both methods one can apply variational procedures; again, the latter are more significant, and probably much more accurate for linearized problems.