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 ... 

In the limit of maximum packing, granular flow becomes incompressible. Under such conditions, the kinetic contribution to viscosity is replaced by a friction contribution. Theories of soil mechanics may be used to estimate such friction contributions (Schaeffer, 1987). 

The first term on the RHS in the viscosity closure denotes the kinetic contribution and dominates in the dilute regime. The second term on the RHS denotes the collisional contribution and dominates in the dense flow regime. 

The Coefficient of (3c0/9r)+° has earlier been defined as —2 77 where the viscosity coefficient 17 is now the sum of the following collisional and kinetic contributions... 

As noted by Pooley and Yeomans [35] and confirmed in [28], the macroscopic stress tensor of SRD is not symmetric in davp. The reason for this is that the multiparticle collisions do not, in general, conserve angular momentum. The problem is particularly pronounced for small mean free paths, where asymmetric collisional contributions to the stress tensor dominate the viscosity (see Sect. 4.1.1). In contrast, for mean free paths larger than the cell size, where kinetic contributions dominate, the effect is negligible. 

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 . 

Fig. 1 a Normalized kinetic contribution to the viscosity, Atk-gT), in three dimensions as a function of the collision angle a. Data were obtained by time averaging the GK relation over 75,000 iterations using XJa = 2.309 for Af = 5 (filled squares) and M = 20 (filled circles). The lines are the theoretical prediction, (32). Parameters L/a = 32, At = 1. From [53]. b Normalized colUsional contribution to the viscosity, At jcfi-, in three dimensions as a function of the... 

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. 

For MPC-AT, the viscosities have been calculated in [32] using the methods described in Sects. 5.1 and 5.2. The total viscosity of MPC-AT is given by the sum of two terms, the collisional and kinetic contributions. For MPC-AT—a, it was found for both two and three dimensions that [32]... 

MPC-AT-a and MPC-AT- -a both have the same kinetic contribution to the viscosity in two dimensions however, imposing angular-momentum conservation makes the collisional contribution to the stress tensor synunetric, so that the asynunetric conttibution, V2, discussed in Sect. 4.1.1 vanishes. The resulting collisional contti-bution to the viscosity is then reduced by a factor close to 2. 

The transport coefficients can be determined using the same GK formalism as was used for the original SRD algorithm [21,51]. Alternatively, the non-equilibrium approach described in Sect. 5 can be used. Assuming molecular chaos and ignoring fluctuations in the number of particles per cell, the kinetic contribution to the viscosity is found to be... 

The transport coefficients can be calculated in the same way as for the one-component non-ideal system. The resulting kinetic contribution to the viscosity is... 

The global rate of the process is r = rj + r2. Of all the authors who studied the whole reaction only Fang et al.15 took into account the changes in dielectric constant and in viscosity and the contribution of hydrolysis. Flory s results fit very well with the relation obtained by integration of the rate equation. However, this relation contains parameters of which apparently only 3 are determined experimentally independent of the kinetic study. The other parameters are adjusted in order to obtain a straight line. Such a method obviously makes the linearization easier. 

The friction coefficient of a large B particle with radius ct in a fluid with viscosity r is well known and is given by the Stokes law, Q, = 67tT CT for stick boundary conditions or ( = 4jit ct for slip boundary conditions. For smaller particles, kinetic and mode coupling theories, as well as considerations based on microscopic boundary layers, show that the friction coefficient can be written approximately in terms of microscopic and hydrodynamic contributions as ( 1 = (,(H 1 + (,/( 1. The physical basis of this form can be understood as follows for a B particle with radius ct a hydrodynamic description of the solvent should... 

Photosensitization of diaryliodonium salts by anthracene occurs by a photoredox reaction in which an electron is transferred from an excited singlet or triplet state of the anthracene to the diaryliodonium initiator.13"15,17 The lifetimes of the anthracene singlet and triplet states are on the order of nanoseconds and microseconds respectively, and the bimolecular electron transfer reactions between the anthracene and the initiator are limited by the rate of diffusion of reactants, which in turn depends upon the system viscosity. In this contribution, we have studied the effects of viscosity on the rate of the photosensitization reaction of diaryliodonium salts by anthracene. Using steady-state fluorescence spectroscopy, we have characterized the photosensitization rate in propanol/glycerol solutions of varying viscosities. The results were analyzed using numerical solutions of the photophysical kinetic equations in conjunction with the mathematical relationships provided by the Smoluchowski16 theory for the rate constants of the diffusion-controlled bimolecular reactions. 

For hydrophilic water-soluble polymers, hydration is the first step of dissolution in aqueous solutions, followed by dissolution of the hydrated phase. The latter step involves disentanglement of polymer molecules. In general, the dissolution kinetics follow Eq. (5.2), suggesting that the solubility of polymers and the viscosity of the hydrated phase are the major variables affecting the dissolution rate. Diffusion of dissolved drug molecules through the hydrated polymer layer also may contribute to the overall release kinetics. 

The generalized relation for the pressure drop for flows through a packed bed was formulated by Ergun (1952). The pressure loss was considered to be caused by simultaneous kinetic and viscous energy losses. In Ergun s formulation, four factors contribute to the pressure drop. They are (1) fluid flow rate, (2) properties of the fluid (such as viscosity and density), (3) closeness (such as porosity) and orientation of packing, and (4) size, shape, and surface of the solid particles. 

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