Non-Newtonian Shear Viscosity in a Dense System of Hard Disks

Kinetic theory can be viewed as an intermediate step between a detailed microscopic analysis of a many-body system and the corresponding phenomenological macroscopic description. In kinetic theory the main objective is to derive and solve the kinetic equation for the one-particle distribution function, thus obtaining information about the system properties. In the context of dilute gases, the Boltzmann equation (BE) [1] provides the adequate framework for studying states arbitrarily far from equilibrium. Exact solutions to this equation are rare, but a great deal of information can be obtained from simplified kinetic models [2] or from simulation Monte Carlo methods [3]. On the other hand, the assumptions implicit in the BE are only physically justified in the low-density limit. As the density increases, structural effects become important, potential contributions to the fluxes dominate, and the BE is no longer adequate. There is no general kinetic equation valid for finite densities. A singular exception, however, is the idealized system of hard spheres of diameter σ, for which Enskog proposed a semi-phenomenological equation [1] by introducing two crucial changes in the Boltzmann collision integral: (a) the centers of two colliding particles are separated by a distance equal to σ; (b) the collision frequency is increased by a factor that accounts for the spatial correlation between the two colliding molecules. Although the Enskog equation (EE) also ignores the correlations in the velocities before collision (stosszahlansatz), it leads to transport coefficients that are in good agreement with experimental and simulation values for a wide range of densities. In addition, the revised Enskog theory (RET) [4] is asymptotically exact at short times and therefore has no limitations on density or space scale in that limit. Moreover, it admits both fluid and crystal equilibrium states as stationary solutions.