Transport coefficients collisional contribution

Here ji(qa) is the spherical Bessel function of order l,g(a) is the radial distribution function at contact, and f = /fSmn/Anpo2g a) is the Enskog mean free time between collisions. The transport coefficients in the above expressions are given only by their Enskog values that is, only collisional contributions are retained. Since it is only in dense fluids that the Enskog values represents the important contributions to transport coefficient, the above expressions are reasonable only for dense hard-sphere fluids. Earlier Alley, Alder, and Yip [32] have done molecular dynamics simulations to determine the wavenumber-dependent transport coefficients that should be used in hard-sphere generalized hydrodynamic equations. They have shown that for intermediate values of q, the wavenumber-dependent transport coefficients are well-approximated by their collisional contributions. This implies that Eqs. (20)-(23) are even more realistic as q and z are increased. 

In this representation, particular emphasis has been placed on a uniform basis for the electron kinetics under different plasma conditions. The main points in this context concern the consistent treatment of the isotropic and anisotropic contributions to the velocity distribution, of the relations between these contributions and the various macroscopic properties of the electrons (such as transport properties, collisional energy- and momentum-transfer rates and rate coefficients), and of the macroscopic particle, power, and momentum balances. Fmthermore, speeial attention has been paid to presenting the basic equations for the kinetie treatment, briefly explaining their mathematical structure, giving some hints as to appropriate boundary and/or initial conditions, and indicating main aspects of a suitable solution approach. 

For large particles, is given very accurately by > e with = 6 even for small particles, however, ) e given a good estimate of D. In fact, Ie with = 4, is amazingly accurate for of a dense pure fluid. It follows that the mode-mode contribution to must always be important in dense fluids, even for small particles. This state of affairs holds for other types of diffusion, but not for nondiffusive processes. Diffusion is the slowest of transport processes, since it may only proceed by actual physical transport of particles, while other transport utilizes this mechanism plus collisional transport of the quantity of interest from particle to particle. However, the mode-mode part of all transport coefficients is comparable to D e, i e., to D, Thus, mode-mode coupling becomes important in nondiffusive transport only when some phenomenon, such as a critical point, slows down the processes that usually dominate the transport. 

In addition to restoring Galilean invariance, this grid-shift procedure accelerates momentum transfer between cells and leads to a collisional contribution to the transport coefficients. If the mean free path A is larger than a/2, the violation of Galilean invariance without grid shift is negligible, and it is not necessary to use this procedure. 

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 kinetic contribution dominates for A 2> a, while the collisional contribution dominates in the opposite limit. Two other transport coefficients of interest are the thermal diffusivity, Dt, and the single particle diffusion coefficient, D. Both have the dimension square meter per second. As dimensional analysis would suggest, the kinetic and collisional contributions to Dj exhibit the same characteristic depen-... 

Two complementary approaches have been used to derive the transport coefficients of the SRD fluid. The first is an equilibrium approach which utilizes a discrete projection operator formalism to obtain GK relations which express the transport coefficients as sums over the autocorrelation functions of reduced fluxes. This approach was first utilized by Malevanets and Kapral [19], and later extended by Ihle, Kroll and Tiizel [20,27,28] to include collisional contributions and arbitrary rotation angles. This approach is described in Sect. 4.1. 

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. 

Collisional contributions Explicit expressions for the collisional contributions to the viscous transport coefficients can be obtained by considering various choices for k and a and in (25), (27), and (29). Taking k in the y-direction and a = 15 = I yields... 

Overlaps reduce the frequency of collisions while chattering replaces strong binary collisions with numerous weak ones. In a range of volume fractions between 0.49 and 0.60 and coefficients of restitution greater than 0.60, the influence of the correlations on the fluxes of momentum and energy is compensated for by nonlocal transport associated with the correlated motion. Consequently, overlaps and chattering first influence the collisional rate of dissipation. For denser and/or more inelastic flows, anisotropic and rate-independent contributions to the pressure and shear stress, associated with chains or clusters that eventually span the flow, are anticipated to develop (Goldman and Swinney 2006, Hatano et al. 2007, Schroter et al. 2007, Chialvo et al. 2012). 

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