Shear viscosity collisional contribution

These expressions for the shear viscosity are compared with simulation results in Fig. 5 for various values of the angle a and the dimensionless mean free path X. The figure plots the dimensionless quantity (v/X)(x/a2) and for fixed y and a we see that (vkin/A,)(x/a2) const A, and (vcol/A)(r/u2) const/A. Thus we see in Fig. 5b that the kinetic contribution dominates for large A since particles free stream distances greater than a cell length in the time x however, for small A the collisional contribution dominates since grid shifting is important and is responsible for this contribution to the viscosity. 

Solids shear viscosity also comprises a kinetic contribution and collisional contributions. Commonly used expressions for viscosity are ... 

Just as for the pressure, there are both kinetic and collisional contributions to the transport coefficients. We present here a heuristic discussion of these contributions to the shear viscosity, since it illustrates rather clearly the essential physics and provides background for subsequent technical discussions. 

Collisional contribution to the shear viscosity At small mean free paths, X/a< I, particles stream only a short distance between colUsions, and the multi-particle collisions are the primary mechanism for momentum transport. These collisions redistribute momenta within cells of linear size a. This means that momentum hops an average distance a in one time step, leading to a momentum diffusion coefficient a /At. The general form of the colUsional contribution to the shear viscosity is therefore... 

The other approach uses kinetic theory to calculate the transport coefficients in a stationary non-equilibrium situation such as shear flow. The first application of this approach to SRD was presented in [21], where the collisional contribution to the shear viscosity for large M, where particle number fluctuations can be ignored, was calculated. This scheme was later extended by Kikuchi et al. [26] to include fluctuations in the number of particles per cell, and then used to obtain expressions for the kinetic contributions to shear viscosity and thermal conductivity [35]. This non-equilibrium approach is described in Sect. 5. 

Table 1 Theoretical expressions for the kinematic shear viscosity v, the thermal diffusivity, Dr, and the self-diffusion coeflScient, D, in both two d = 2) and three (d = 3) dimensions. M is the average number of particles per ceU, a is the coUision angle, 1 b is Boltzmann s constant, T is the temperature, At is the time step, m is the particle mass, and a is the cell size. Except for selfdiffusion constant, for which there is no coUisional contribution, both the kinetic and collisional contributions are listed. The expressions for shear viscosity and self-diffusion coefficient include the effect of fluctuations in the number of particles per cell however, for brevity, the relations for thermal diffusivity are correct only up to 0(1/M) and 0(1/M ) for the kinetic and collisional contributions, respectively. For the complete expressions, see [28,53,54]...

The collisional contribution to the shear viscosity is proportional to a /At as discussed in Sect. 3.2, it results from the momentum transfer between particles in a cell of size a during the collision step. Consider again a collision cell of linear dimension... 

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