Constitutive relations for

The polarization P is given in tenns of E by the constitutive relation of the material. For the present discussion, we assume that the polarization P r) depends only on the field E evaluated at the same position r. This is the so-called dipole approximation. In later discussions, however, we will consider, in some specific cases, the contribution of a polarization that has a non-local spatial dependence on the optical field. Once we have augmented the system of equation B 1.5.16. equation B 1.5.17. equation B 1.5.18. equation B 1.5.19 and equation B 1.5.20 with the constitutive relation for the dependence of Pon E, we may solve for the radiation fields. This relation is generally characterized tlirough the use of linear and nonlinear susceptibility tensors, the subject to which we now turn. 

Pseudoplastic fluids are the most commonly encountered non-Newtonian fluids. Examples are polymeric solutions, some polymer melts, and suspensions of paper pulps. In simple shear flow, the constitutive relation for such fluids is... 

Ishii, M., andT. C. Chawla, 1978, Multichannel Drift-Flux Model and Constitutive Relation for Transverse Drift-Velocity, Rep. ANL/RAS/LWR 78-2, Argonne Natl. Lab., Argonne, IL. (3)... 

Nonlinear second order optical properties such as second harmonic generation and the linear electrooptic effect arise from the first non-linear term in the constitutive relation for the polarization P(t) of a medium in an applied electric field E(t) = E cos ot. 

Of course, we have to express f, g, and h in terms of a common variable, u for example, by means of a constitutive relation for the material under study often u = h. 

As with any theory of material behavior, we have to make constitutive assumptions in order to define the peculiar mixture of a poroelastic material and a compressible bi-component fluid. Among other quantities, we must state constitutive relations for the mass supply a and the momentum supply m, which give rise to adsorption/desorption and to diffusion, respectively. 

As a first approximation, the stresses for the solid, ice and gel water can be formulated with the help of a linearized Hookean type law, where the depression of the gel water below the macroscopic freezing point of water must be considered. This can be done by including the micro-ice-lens model of Set-zer [1] in the constitutive relations for the aforementioned stress tensor. The gas phase can be described as an ideal gas. Concerning the constitutive assumptions for the liquid stresses, the heat flux and the interactions, the reader is referred to de Boer et al. [4], There a ternary model for the numerical simulation of freezing and thawing processes is discussed. 

In the above mentioned field equations the number of unknown quantities does not correspond to the number of equations, thus we have to conclude the problem with the constitutive equations for the partial stress tensors T , the interaction forces p", the partial internal energies ea and the partial heat flows q . From the evaluation of the entropy inequality of the saturated porous body, see de Boer [4], we obtain for the solid phase and the mobile phases with Index j3 = L, G the constitutive relations for T and p ... 

To yield a constitutive relation for the mass flux of particles, it is convenient to begin with the equation of motion of a single particle, which is expressed by... 

However, in the earlier times, the constitutive relation for a viscoelastic liquid were formulated when the equations for relaxation processes could not be written down in an explicit form. In these cases the constitutive relation was formulated as relation between the stress tensor and the kinetic characteristics of the deformation of the medium (Astarita and Marrucci 1974). 

There is another subtle but fundamental issue in coupling of hybrid models that has to do with differences in constitutive relations in various subdomains. In particular, models at various scales correspond (upon passing to the continuum limit) to different constitutive relations. For example, in the continuum model on a terrace, Eq. (2), there are no interactions between molecules. Consequently, Fick s first law... 

The resistor function is mostly nonlinear and approaches a constant value only in the vicinity of equilibrium. Combining Eqs. (14.5) and (14.38), the constitutive relation for the capacitative element C, is... 

Write the constitutive relations for the medium to relate stress to strain, assuming elastic linearity (Hooke s Law) ... 

Closure of such differential equations requires the definitions of both constitutive relations for hydrodynamical functions and also kinetic relations for the chemistry. These functions are specified by recourse both to theoretical considerations and to rheological measurements of fluidization. We introduce the ideal gas approximation to specify the gas phase pressure and a caloric equation-of-state to relate the gas phase internal energy to both the temperature and the gas phase composition. It is assumed that the gas and solid phases are in local thermodynamic equilibrium so that they have the same local temperature. 

Jenkins, J.T, and Mancini, F. (1987), Balance laws and constitutive relations for plane flows of a dense binary mixture of smooth, nearly elastic circular disks, J. Appl. Meek, 54, 27. 

Johnson, PC. and Jackson, R. (1987), Frictional collisional constitutive relations for granular materials with application to plain shearing, J. Fluid Meek, 176, 67. 

The setting up of the constitutive relation for a binary system is a relatively easy task because, as pointed out earlier, there is only one independent diffusion flux, only one independent composition gradient (driving force) and, therefore, only one independent constant of proportionality (diffusion coefficient). The situation gets quite a bit more complicated when we turn our attention to systems containing more than two components. The simplest multicomponent mixture is one containing three components, a ternary mixture. In a three component mixture the molecules of species 1 collide, not only with the molecules of species 2, but also with the molecules of species 3. The result is that species 1 transfers momentum to species 2 in 1-2 collisions and to species 3 in 1-3 collisions as well. We already know how much momentum is transferred in the 1-2 collisions and all we have to do to complete the force-momentum balance is to add on a term for the transfer of momentum in the 1-3 collisions. Thus,... 

A more rigorous kinetic theory than that in Chapter 2 not only supplies us with the proper form of the constitutive relations for multicomponent diffusion, it also provides an explicit relation for the binary diffusion coefficient. A slightly simplified version of the kinetic theory result is... 

If the driving forces for mass transfer are taken to be the difference in compositions between the interface (x j) and bulk fluid (x ), then the constitutive relations for (/) may be written as... 

As we have already seen, the analysis of a diffusion problem proceeds by solving the conservation equations together with appropriate constitutive relations for the diffusion fluxes. Use of Eq. 6.1.1 for the diffusion flux with Eq. 1.3.9 for the conservation of mass leads to... 

BALANCE AND CONSTITUTIVE RELATIONS FOR TURBULENT MASS TRANSPORT... 

The constitutive relation for j y, taking account of the molecular diffusion and turbulent eddy contributions, is... 

The analysis of turbulent eddy transport in binary systems given above is generalized here for multicomponent systems. The constitutive relation for j y in multicomponent mixtures taking account of the molecular diffusion and turbulent eddy contributions, is given by the matrix generalization of Eq. 10.3.1... 

CONSTITUTIVE RELATIONS FOR SIMULTANEOUS HEAT AND MASS TRANSFER... 

The most appropriate starting point for setting up the constitutive relation for q is the... 

For diffusion in isothermal multicomponent systems the generalized driving force was written as a linear function of the relative velocities (m/ — My). In the general case, we must allow for coupling between the processes of heat and mass transfer and write constitutive relations for and q in terms of the (m — My) and V(l/r). With this allowance, the complete expression for the conductive heat flux is... 

Equations 11.2.5 and 11.2.6 are the complete forms of the constitutive relations for simultaneous mass and energy transport. The reader is referred to the treatise by Hirschfelder et al. (1964) and to the papers by Merk (1960) and Standart et al. (1979) for further background to these derivations. 

Chapter 1 serves to remind readers of the basic continuity relations for mass, momentum, and energy. Mass transfer fluxes and reference velocity frames are discussed here. Chapter 2 introduces the Maxwell-Stefan relations and, in many ways, is the cornerstone of the theoretical developments in this book. Chapter 2 includes (in Section 2.4) an introductory treatment of diffusion in electrolyte systems. The reader is referred to a dedicated text (e.g., Newman, 1991) for further reading. Chapter 3 introduces the familiar Fick s law for binary mixtures and generalizes it for multicomponent systems. The short section on transformations between fluxes in Section 1.2.1 is needed only to accompany the material in Section 3.2.2. Chapter 2 (The Maxwell-Stefan relations) and Chapter 3 (Fick s laws) can be presented in reverse order if this suits the tastes of the instructor. The material on irreversible thermodynamics in Section 2.3 could be omitted from a short introductory course or postponed until it is required for the treatment of diffusion in electrolyte systems (Section 2.4) and for the development of constitutive relations for simultaneous heat and mass transfer (Section 11.2). The section on irreversible thermodynamics in Chapter 3 should be studied in conjunction with the application of multicomponent diffusion theory in Section 5.6. 

Jenkins JT, Mancini F (1987) Balance Laws and Constitutive Relations for Plane Flows of a Dense, Binary Mixture of Smooth, Nearly Elastic, Circular Disks. Journal of Applied Mechanics 54 27-34... 

Johnson PC, Jackson R (1987) Erictional-coUisional constitutive relations for granular materials, with application to plane shearing. J Fluid Mech 176 67-93 Johnson PC, Nott P, Jackson R (1990) Frictional-colhsional equations of motion for particulate flows and their application to chutes. J Fluid Mech 210 501-535 Jung J, Gidaspow D, Gamwo IK (2006) Bubble Computation, Granular Temperatures, and Reynolds Stresses. Chem Eng Comm 193 946-975... 

Reeks MW (1993) On the constitutive relations for dispersed particles in nonuniform flows. I. Dispersion in simple shear flow. Phys Fluids A 5 750-761 Reyes Jr JN (1989) Statistically Derived Conservation Equations for Fluid Particle Flows. Nuclear Thermal Hydraulics 5th Winter meeting, Proc ANS Winter Meeting. 

Due to experimental diffculties, turbulence closures exist only for the turbulent momentum transport phenomena. In particular, approximate semi-empirical models for the turbulent viscosity parameter have been derived. To achieve constitutive relations for the corresponding heat and mass transport... 

---