Interdisciplinary Applied Mathematics

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One such approach is the method of Bhatnagar, Gross, and Krook (Bhat-nagar et al., 1954), the so-called BGK model. In this method, the collision integral is approximated as

QBOK(f, f* ) = v*(floc — f),

where v* is the collision frequency, which is assumed to be independent of the molecular velocity v, but it is a function of spatial coordinates and time. We can obtain a relation of the collision frequency from the mean thermal velocity v = ■sJWk^TjJwm) and the mean free path Л = /2/(2mid2), i.e.,

v    4 p(x,t)

А 7Г p{T)

where p(x,t) is the pressure obtained from the distribution function

p(x,t)

m

3~

c2f (x, v, t)dv.

However, this expression for the collision frequency leads to expressions for the dynamic viscosity (also for the thermal conductivity) that are inconsistent with those derived using the Chapman-Enskog method to obtain solutions of the BGK model equation. Specifically, the Chapman-Enskog solution leads to

P(x,t)

KT) ’

v*

which is    the    same    equation    as the    one    corresponding to    solutions    of    the

Boltzmann equations with the full collision term. Other models have been proposed for the collision frequency, including corrections that allow dependence on the molecular velocity, since full simulations suggest this to be the case; the interested reader can find appropriate references in (Sharipov and Sleznev, 1998).

In general, numerical evidence from full solutions of the Boltzmann equation suggests that the BGK model is an accurate method for isothermal flows. However, for nonisothermal flows, corrections for the Prandtl number (and collision frequency) need to be introduced.

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