Interdisciplinary Applied Mathematics

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15.4 The Boltzmann Equation

The Boltzmann equation can be derived rigorously from Newton’s laws at    least    for    the low-density limit,    but    it is    also    used    for    the    dense    limit,

although there is no totally rigorous theoretical basis for the latter (Cer-cignani et    al.,    1994). Research    efforts    in    the    1990s    focused    on    obtaining

most of the fluid-dynamics limits of the Boltzmann equation including the incompressible limit;    see    (Bardos    et    al.,    1991;    Bardos    et    al.,    1993),    and

(Desjardins et al., 1999; DiPerna and Lions, 1991), and references therein.

Here we assume that microscale fluidic and thermal gas transport in the entire Knudsen regime (0 < Kn < ж) is governed by the Boltzmann equation (BE). The Boltzmann equation describes the evolution of a velocity distribution function by molecular transport and binary intermolecular collisions. An analysis of microflows based on the Boltzmann equation has been presented in series of papers that study silicon accelerometer motion and squeezed film damping as a function of the Knudsen number and the time-periodic motion    of    the    surfaces    (Veijola    et    al.,    1995a;    Veijola    et    al.,

1998; Fukui and Kaneko, 1988; Fukui and Kaneko, 1990); see Section 6.1. More applications using solutions of the Boltzmann equation are presented in (Aoki, 2001).

To illustrate some of the complexities in performing simulations based on the Boltzmann equation, we provide a brief review next. Let us consider monoatomic gases, the state of which can be described by a velocity distribution function f (t, x, v), where x denotes the position and v denotes the velocity of the molecules. The distribution function represents the number of particles in the six-dimensional phase space dx dv at time t. This distribution function obeys the Boltzmann equation (Sone, 2002; Cercignani, 1988; Bird, 1994)

df df df


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