Effective dispersion tensor

In general in complex media, the convection and diffusion of a solute is a difficult problem to analyze at the microscale and the moment method enables the analysis to be carried out at the macroscale leading to the replacement of the convective-diffusion problem by an effective global velocity and an effective dispersion tensor as in Eq. (4.6.30). Brenner s procedure analyzes the time evolution of the spatial moments (cf. Eq. 4.6.36) of the conditional probability density that a Brownian particle is located at a given position at a specific time knowing the position from which it was initially released into the fluid. 

In a circular bed, the effective dispersion tensor is anisotropic and is composed of the longitudinal and lateral dispersion coefficients D Jff and Djjf, respectively. The longitudinal dispersion coincides with the direction of the mean fluid flow with the lateral dispersion normal to this direction. At high Peclet numbers, the longitudinal dispersion is large in comparison with the lateral dispersion, since the component of the fluid velocity parallel to the mean flow direction has the largest gradients. The lateral dispersion D if is associated with the weaker lateral fluid motion, whence D fj. 

The form of the effective mobility tensor remains unchanged as in Eq. (125), which imphes that the fluid flow does not affect the mobility terms. This is reasonable for an uncharged medium, where there is no interaction between the electric field and the convective flow field. However, the hydrodynamic term, Eq. (128), is affected by the electric field, since electroconvective flux at the boundary between the two phases causes solute to transport from one phase to the other, which can change the mean effective velocity through the system. One can also note that even if no electric field is applied, the mean velocity is affected by the diffusive transport into the stationary phase. Paine et al. [285] developed expressions to show that reversible adsorption and heterogeneous reaction affected the effective dispersion terms for flow in a capillary tube the present problem shows how partitioning, driven both by electrophoresis and diffusion, into the second phase will affect the overall dispersion and mean velocity terms. 

Throughout this summary we have neglected the effect of dispersion on the overall transport of mass and heat. This is due to the fact that if dispersion is included, dispersion tensors must be determined before the equation can be solved. This can be done by solving the appropriate transport equation within a unit cell. Because a unit cell cannot be defined in most reinforcements used in polymer matrix composites, however, dispersion tensors cannot be accurately determined, so we have left dispersion effects out of our equations. In general, we anticipate dispersion to play a minor role in the IP, AP, and RTM processes. This assumption can be checked, however, by evaluating the dispersion terms using an approach similar to [16] where experiments and correlations are used to determine the importance of dispersion. 

In the two-medium treatment of the single-phase flow and heat transfer through porous media, no local thermal equilibrium is assumed between the fluid and solid phases, but it is assumed that each phase is continuous and represented with an appropriate effective total thermal conductivity. Then the thermal coupling between the phases is approached either by the examination of the microstructure (for simple geometries) or by empiricism. When empiricism is applied, simple two-equation (or two-medium) models that contain a modeling parameter hsf (called the interfacial convective heat transfer coefficient) are used. As is shown in the following sections, only those empirical treatments that contain not only As/but also the appropriate effective thermal conductivity tensors (for both phases) and the dispersion tensor (in the fluid-phase equation) are expected to give reasonably accurate predictions. 

After the formal derivations, the energy equation for each phase ((T)f and (T) ) can be written in a more compact form by defining the following coefficients. Note that both the hydrodynamic dispersion, that is, the influence of the presence of the matrix on the flow (noslip condition on the solid surface), as well as the interfacial heat transfer need to be included. The total thermal diffusivity tensors Dff, D , D/s, and Dv/ and the interfacial convective heat transfer coefficient hsf are introduced. The total thermal diffusivity tensors include both the effective thermal diffusivity tensor (stagnant) as well as the hydrodynamic dispersion tensor. A total convective velocity v is defined such that the two-medium energy equations become... 

The modelling of transport of a single component (HOC, DOC or HOC-DOC) is based on the following assumptions The porosity, the diffusion-dispersion-tensor and the advection are equal for all components and the water flow is not influenced by the transport of the components. These assumptions should be sensible, if one deals with very small concentrations of particles, with diameters that are very much smaller than the mean poresize. Further we assume, that transport of HOC has no influence on transport of DOC, i.e. the amount of HOC adsorbed on DOC has not any effect, neither on... 

Surface waves at an interface between two innniscible fluids involve effects due to gravity (g) and surface tension (a) forces. (In this section, o denotes surface tension and a denotes the stress tensor. The two should not be coiifiised with one another.) In a hydrodynamic approach, the interface is treated as a sharp boundary and the two bulk phases as incompressible. The Navier-Stokes equations for the two bulk phases (balance of macroscopic forces is the mgredient) along with the boundary condition at the interface (surface tension o enters here) are solved for possible hamionic oscillations of the interface of the fomi, exp [-(iu + s)t + i V-.r], where m is the frequency, is the damping coefficient, s tlie 2-d wavevector of the periodic oscillation and. ra 2-d vector parallel to the surface. For a liquid-vapour interface which we consider, away from the critical point, the vapour density is negligible compared to the liquid density and one obtains the hydrodynamic dispersion relation for surface waves + s>tf. The temi gq in the dispersion relation arises from... 

It can be seen that, in all cases, relaxation rates are directly proportional to (Aa). Because Aa reflects the anisotropy of the shielding tensor and because the chemical shift originates from the shielding effect, the terminology Chemical Shift Anisotropy is used for denoting this relaxation mechanism. Dispersion may be disconcerting because of the presence of Bq (proportional to cOq) in the numerator of and R2 (Eq. (49)). Imagine that molecular reorientation is sufficiently slow so that coo 1 for all considered values of coo from (49), it can be seen that R is constant whereas R2 increases when Bq increases, a somewhat unusual behavior. 

The static ZFS, which is present in low-symmetry complexes, affects mainly the energy level fine structure. It is described by axial and rhombic components, D and E. Its effects on nuclear relaxation depend on two angles, 9 and cj), defining the position of the nucleus with respect to the ZFS principal tensor axes. Figure 23 shows the dispersion profiles for different values of S, D, E and 9. Many such examples are reported in Chapter 2. 

Equation (I-l) is the general representation of the dispersion model. The dispersion coefficient is a function of both the fluid properties and the flow situation the former have a major effect at low flow rates, but almost none at high rates. In this general representation, the dispersion coefficient and the fluid velocity are all functions of position. The dispersion coefficient, D, is also in general nonisotropic. In other words, it has different values in different directions. Thus, the coefficient may be represented by a second-order tensor, and if the principal axes are taken to correspond with the coordinate system, the tensor will consist of only diagonal elements. 

Without loss of generality, we assume a bulk material where the major carriers are electrons with an effective mass inf. In general, the electron masses are anisotropic, and the effective mass is expressed as a symmetric second-rank tensor. The dispersion relation of the electrons is written as... 

In 1962 the first implicit prediction appeared of a cross-effect between natural and magnetic optical activity, which discriminates between the two enantiomers of chiral molecules [7]. This was followed independently by a prediction of magnetospatial dispersion in noncentrosymmetrical crystalline materials [8]. This cross-effect has been called magnetochiral anisotropy and has since been predicted independently several times more [9-12]. Its existence can be appreciated by expanding the dielectric tensor of a chiral medium subject to a magnetic field to first order in the wave vector k and magnetic field B [8] ... 

Kleinman symmetry (index permutation symmetry). Far from resonances of the medium where dispersion is negligible, the susceptibilities become to a good approximation invariant with respect to permutation of all Cartesian indices (without simultaneous permutation of the frequency arguments). This property is called Kleinman symmetry (Kleinman, 1962). It is important in the discussion of the exchange of power between electromagnetic waves in an NLO medium. In many cases approximate validity of Kleinman symmetry can be used effectively to reduce the number of independent tensor components of an NLO susceptibility. 

In the absence of electron dispersion and absorption, the tensor of second-order non-linear polarizability ft(— cog coi, tog) can be dealt with as totally symmetric, and the numbers of its non-zero and independent elements are to be found in Table 11. Static values of the nonJinear polarizability = i(0 0,0) have been calculated theoretically for some molecules. Tensor elements A" = fi(— Kerr effect or molecular light scattering in liquids. ... 

The traveling-wave excitation described by Eq. (21) affects the dielectric tensor, as described by Eq. (15). The effects can be detected by a variably delayed probe pulse that is phase matched for coherent scattering, that is, collinear (in practice, nearly collinear) with the excitation pulse and the vibrational wave vector. Since the probe pulse follows the excitation pulse through the sample at the same speed c/n (neglecting dispersion), it surfs along a crest or null of the vibrational wave. The probe pulse therefore encounters each region of the sample with identical coherent vibrational distortion. 

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