Dyadics diffusion

Equations (8.10)—(8.12), tensorial ranks and boundary conditions (8.14)-(8.15) notwithstanding, embody a structure similar in format and symbolism to their counterparts for the transport of passive scalars, e.g., the material transport of the scalar probability density P (Brenner, 1980b Brenner and Adler, 1982), at least in the absence of convective transport. As such, by analogy to the case of nonconvective material transport, the effective kinematic viscosity viJkl of the suspension may be obtained by matching the total spatial moments of the probability density Pu to those of an equivalent coarse-grained dyadic probability density P j, valid on the suspension scale, using a scheme (Brenner and Adler, 1982) identical in conception to that used to determine the effective diffusivity for material transport at the Darcy scale from the analogous scalar material probability density P. In particular, the second-order total moment M(2) (sM, ) of the probability density P, defined as... 

Ru -moiety described by a magnetic field independent relaxation time ( s) was also important for their MFEs. They carried out numerical simulation of the observed AR E) values with Eq. (12-39), where fe, and bet were only treated as empirical parameters. We can see from Fig. 12-13 that the simulated curves can well reproduce the observed AR B) values. The parameters determined with this method for the four complexes are listed in Table 12-6. It is noteworthy from this table that the reaction processes described by Reactions (12-34a) - (12-34e) occur in ps-time region. Although such ps-processes of the RIPs in fluid solutions can not be measured directly by ps-laser photolysis techniques due to diffusion-controlled formation of the RIPs, it is a great experimental challenge to observe directly the MFEs of the reaction processes of dyadic RIPs in ps time-resolved experiments. Such investigations have recently been carried out [19]... 

As implied by the subscript notation in Eqs. (327)-(328), the translational and coupling diffusivity dyadics vary with choice of origin. This dependence can be quantitatively established. By invoking appropriate kinematic argu-... 

The rotational diffusivity dyadic is origin-independent. These origin-displacement theorems are the counterparts of the comparable hydrodynamic relations set forth in Eqs. (51) and (52). 

As in the hydrodynamic case, there exists a unique origin, say P, at which the coupling diffusivity dyadic is symmetric that is,... 

If the value of the coupling diffusivity dyadic is known at some point 0, and if the value of the rotational diffusivity dyadic is also known, the location of P (relative to O) may be determined from the relation... 

Expressed in terms of the individual resistance and diffusion dyadics, Eq. (350) may be written as... 

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