Kinetic Model for Uniform Shear Flow

Abstract

The results obtained in Chapter 2 hold for the Boltzmann equation in the case of Maxwell molecules. While the description generally applies to arbitrary values of the shear rate a, it has two limitations. On the one hand, no explicit form of the velocity distribution function f (V) is known for finite a. On the other hand, the analysis cannot be extended to interactions different from the Maxwell one, unless additional approximations are introduced (Loose and Hess, 1988; Loose, 1988). These two limitations can be overcome by employing numerical simulations and/or kinetic models. This chapter is mainly concerned with the latter route. Specifically, we will focus on the well-known Bhatnagar-Gross—Krook (BGK) model kinetic equation (Bhatnagar et al., 1954; Welander, 1954; Cercignani, 1990) introduced in Section 1.7.