Shear viscosity kinetic 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 ... 

The pressure tensor can be written as the sum of kinetic and potential terms. The potential part arises from the direct intermolecular forces and is therefore a function of the local fluid microstructure. In a dense liquid the shear viscosity is overwhelmingly dominated by the potential contribution. ... 

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. 

A standard result from kinetic theory is that the kinetic contribution to the shear viscosity in simple gases is [46]... 

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. 

Kinetic contributions Kinetic contributions to the transport coefficients dominate when the mean free path is larger than the cell size, i.e., A > a. As can be seen from (24) and (26), an analytic calculation of these contributions requires the evaluation of time correlation functions of products of the particle velocities. This is straightforward if one makes the basic assumption of molecular chaos that successive collisions between particles are not correlated. In this case, the resulting time-series in (24) is geometrical, and can be summed analytically. The resulting expression for the shear viscosity in two dimensions is... 

The kinetic contribution to the stress tensor is symmetric, so that = 0 and the kinetic contribution to the shear viscosity is = vf . 

The kinetic contributions to the transport coefficients presented in Table 1 have all been derived under the assumption of molecular chaos, i.e., that particle velocities are not correlated. Simulation results for the shear viscosity and thermal diffusivity have generally been found to be in good agreement with these results. However, it is known that there are correlation effects for A/a smaller than unity [15,55]. They arise from correlated collisions between particles that are in the same collision cell for more than one time step. 

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]...

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