TABLE OF CONTENTS The Boltzmann equation can be used to describe the micro gas flow behavior at the microscopic level. The equation describes the rate of change of the number of particles due to convection in the physical and velocity space, and due to molecular collisions. In the equation, nf, the total number of particles of a given velocity class per unit volume, is the only dependent variable. The independent variables are the time, the phase space variables, including the three physical coordinates and the three velocity components. For a one-dimensional problem in the physical space, there are three independent variables. The number of the independent variables becomes five and seven in the two-dimensional and three-dimensional problems, respectively. The mathematical difficulty associated with the dimensions of the problem is further compounded by the integral form of the nonlinear collision term. As a result, it is not yet possible to find analytical solutions to the Boltzmann equation for realistic, complex flow problems such as those in microdevices. Direct numerical solutions would require discretization in the seven independent variables. The mathematical and the numerical efforts will be quite involved.
On the other hand, it is possible to develop equations that describe the macroscopic quantities of microflows by using the microscopic Boltzmann equation. For a certain limited class of microflows at small Knudsen number, it will be shown that the approximate forms of these moment equations bear much resemblance to the equations used in the continuum fluid mechanics, i.e., the Euler and the...
