Spatially periodic model

Saez, AE Perfetti, JC Rusinek, I, Prediction of Effective Diffusivities in Porous Media Using Spatially Periodic Models, Transport in Porous Media 6, 143, 1991. 

The foregoing results may be discussed in terms of spatially periodic suspensions, which represent the only exactly analyzable suspension models currently available for concentrated systems. Since spatially periodic models are discussed in the next section, the remainder of this section may be omitted at first reading. 

Second, the spatially periodic model suggests further interpretations and experiments. That no kink exists in the viscosity vs. concentration curve may be related to the fact that the average dissipation rate remains finite at the maximum kinematic concentration limit, ma>. Infinite strings of particles are formed at this limit. It may thus be said that although the geometry percolates, the resulting fields themselves do not, at least not within the context of the spatially periodic suspension model. 

Spatially periodic models of suspensions (Adler and Brenner, 1985a,b Adler et al., 1985 Zuzovsky et al, 1983 Adler, 1984 Nunan and Keller, 1984) constitute an attractive subject for theoretical treatment since their geometrical simplicity permits rigorous analysis, even in highly concentrated systems. In particular, when a unit cell of the spatially periodic arrangement contains but a single particle, the underlying kinematical problems can be... 

Finally, the self-reproducibility in time of the lattice configuration (for two-dimensional flows) must be addressed. In the elliptic streamline region (A < 0), the lattice necessarily replicates itself periodically in time owing to closure of the streamlines. For hyperbolic flows (A > 0), the lattice is not generally reproduced however, in connection with research on spatially periodic models of foams (Aubert et al., 1986 Kraynik, 1988), Kraynik and Hansen (1986, 1987) found a finite set of reproducible hexagonal lattices for the extensional flow case A = 1. It is not clear how this unique discovery can be extended, if at all. 

Spatially periodic porous media are made up of structural elements whose arrangement in space is completely described by a single unit cell (similar to the representative elementary volume concept of Bear, 1969), that is then repeated ad infinitum (Adler, 1992). The structural elements can be discrete voids in a continuous solid phase or vice versa. The simplest spatially periodic models are comprised... 

J. A. Ochoa-Tapia, P. Stroeve, and S. Whitaker, Diffusion Transport in TVo-Phase Media Spatially Periodic Models and Maxwell s Theory for Isotropic and Anisotropic Systems, Chem. Engng. Sci,... 

Ochoa-Tapia J.A., del Rio J.A. and Whitaker S. 1993. Bulk and surface diffusion in porous media An application of the surface averaging theorem, Chem. Eng. Sci., 48, 2061-2082. Ochoa-Tapia J.A., Stroeve P. and Whitaker S. 1994. Diffusive transport in two-phase media Spatially periodic models and Maxwell s theory for isotropic and anisotropic systems, Chem. Eng. Sci., 49, 709-726. 

In the spatially periodic model we incorporate the general features of these random walk analyses and assume that there is no ring resampling during the random walk of the excitation in PS/PVME... 

Figure 7. Concentration dependence of 1])/% for miscible PS/PVME blends cast from toluene (circles) and immiscible blends cast from tetrahydrofuran (squares). The solid line through the toluene cast film data is the best fit of the spatially periodic model, while the solid line through the THF results is the best fit of the two phase model. Taken from Figure I of reference 79.

Lattice models have been studied in mean field approximation, by transfer matrix methods and Monte Carlo simulations. Much interest has focused on the occurrence of a microemulsion. Its location in the phase diagram between the oil-rich and the water-rich phases, its structure and its wetting properties have been explored [76]. Lattice models reproduce the reduction of the surface tension upon adsorption of the amphiphiles and the progression of phase equilibria upon increasmg the amphiphile concentration. Spatially periodic (lamellar) phases are also describable by lattice models. Flowever, the structure of the lattice can interfere with the properties of the periodic structures. 

Two macromolecular computational problems are considered (i) the atomistic modeling of bulk condensed polymer phases and their inherent non-vectorizability, and (ii) the determination of the partition coefficient of polymer chains between bulk solution and cylindrical pores. In connection with the atomistic modeling problem, an algorithm is introduced and discussed (Modified Superbox Algorithm) for the efficient determination of significantly interacting atom pairs in systems with spatially periodic boundaries of the shape of a general parallelepiped (triclinic systems). 

When applied to spatially extended dynamical systems, the PoUicott-Ruelle resonances give the dispersion relations of the hydrodynamic and kinetic modes of relaxation toward the equilibrium state. This can be illustrated in models of deterministic diffusion such as the multibaker map, the hard-disk Lorentz gas, or the Yukawa-potential Lorentz gas [1, 23]. These systems are spatially periodic. Their time evolution Frobenius-Perron operator... 

One of the simplest models of deterministic diffusion is the multibaker map, which is a generalization of the well-known baker map into a spatially periodic system [1, 27, 28]. The map is two dimensional and mles the motion of a particle which can jump from square to square in a random walk. The equations of the map are given by... 

Theoretically we have employed a generalized Cahn-Hilliard model to describe the effects of stationary and traveling spatially periodic temperature-modulations... 

An underlying question remains in the quest for understanding the hydrophobic effect Can continuum models adequately describe this phenomenon Force measurements between mica surfaces suspended in KC1 solution clearly exhibit an oscillatory behavior with a spatial periodicity approximately the same as the diameter of a water molecule [17], suggesting a challenge to the use of continuum models. 

This section begins with an account of spatially periodic suspension models embodying a single particle (a solid sphere in most cases) per unit cell. Rigidity... 

To conclude this subsection, we expose an interesting paradox arising from the time dependence of the particle configuration. As discussed in Section III, Frankel and Acrivos (1967) developed a time-independent lubrication model for treating concentrated suspensions. Their result, given by Eq. (3.7), predicts singular behavior of the shear viscosity in the maximum concentration limit where the spheres touch. Within the spatially periodic framework, the instantaneous macroscopic stress tensor may be calculated for the lubrication limit, e - 0. The symmetric portion of its deviatoric component takes the form (Zuzovsky et al, 1983)... 

The previous analysis may be extended to spatially periodic suspensions whose basic unit cell contains not one, but many particles. Such models would parallel those employed in liquid-state theories, which are widely used in computer simulations of molecular behavior (Hansen and McDonald, 1976). This subsection briefly addresses this extension, showing how the trajectories of each of the particles (modulo the unit cell) can be calculated and time-average particle stresses derived subsequently therefrom. This provides a natural entree into recent dynamic simulations of suspensions, which are reviewed later in Section VIII. 

Accompanying the impeded particle rotation is the (kinematical) existence of an internal spin field 12 within the suspension, which is different from one-half the vorticity to = ( )V x v of the suspension. The disparity to — 2 between the latter two fields serves as a reference-frame invariant pseudovector in the constitutive relation T = ((to — 12), which defines the so-called vortex viscosity ( of the suspension. Expressions for (( ) as a function of the volume of suspended spheres are available (Brenner, 1984) over the entire particle concentration range and are derived from the prior calculations of Zuzovsky et ai (1983) for cubic, spatially-periodic suspension models. 

On the basis of this model one expects that if the spatial period of the grating pattern being recorded is increased, the achievable index modulation will drop off when the period becomes larger than the distance over which monomer can diffuse in the time before the fixing exposure. The experimental spatial-frequency response curve in Fig. 18 shows this expected low-spatial-frequency cutoff (37). Measured rates of monomer diffusion in polymer films are also consistent with the basic diffusion model (38). 

We take the motion below the boundary layer to be spatially periodic with up-welling and down-welling zones, and model the flowfield v just below the boundary layer as... 

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