Interdisciplinary Applied Mathematics

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Another approximation that allows computationally tractable models is linearization of the Boltzmann equation. The distribution function is then written as

f (x, v,t) = fo(no,To)[1 + h(x, v,t)],

where f0 is the absolute Maxwellian distribution corresponding to equilibrium state (no, To), and h(x, v, t) is the perturbation distribution function. The linearized Boltzmann equation is then

dh dh ~

— + v ■ w- — Qh = 0,



where the linearized collision term is

Qh = /    / fo(v*)(h’ + h* — h — h*) dn dv*.

Jr3 Js+

All macroscopic (continuum) parameters can then be written in terms of the perturbation function, i.e.,


n = no + (1, h);    n0u=(v,h); T = T0 + W (v2, h)—(1»,

бкв no    no

where (•, ■) denotes a weighted inner product with fo as the weight function.

A different linearized Boltzmann equation can be obtained by linearizing around the local Maxwellian distribution, i.e.,

f (x, v,t) = fioc(n,T)[1 + h(x, v,t)],

where the macroscopic local quantities n(x,t) and T(x,t) are involved. In this case the perturbation function satisfies an inhomogeneous equation:

dh    dh    ~    Г1 dn    ( mv2    3 1 dT

dt    dt    4    [ndx    2kBT    2) T dx

Further approximations are required for both the boundary conditions and the linearized collision operator Q to bring the Boltzmann equation to a computationally friendly form; see (Cercignani, 1988; Cercignani et al., 1994; Sharipov and Sleznev, 1998).

15-4-2 Sone’s Asymptotic Theory

When the Knudsen number is small, the contribution of the collision term in the Boltzmann equation becomes large, and the velocity distribution function approaches a local Maxwellian. Then, the behavior of the gas may be described as continuum. In this section, we present the work of Sone and collaborators (Sone, 2002) for the small Knudsen number limit, which is pertinent to microflows. In particular, we will consider small and large deviations from equilibrium. This is measured by the Mach number, and we recall that the Knudsen number, the Mach number, and the Reynolds number are related (see Chapter 2):

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