Fields flux vector

In unsteady states the situation is less satisfactory, since stoichiometric constraints need no longer be satisfied by the flux vectors. Consequently differential equations representing material balances can be constructed only for binary mixtures, where the flux relations can be solved explicitly for the flux vectors. This severely limits the scope of work on the dynamical equations and their principal field of applicacion--Che theory of stability of steady states. The formulation of unsteady material and enthalpy balances is discussed in Chapter 12, which also includes a brief digression on stability problems. 

As argued by Reed [4], the Beltrami vector field originated in hydrodynamics and is force-free. It is one of the three basic types of field solenoidal, complex lamellar, and Beltrami. These vector fields originated in hydrodynamics and describe the properties of the velocity field, flux or streamline, v, and the vorticity V x v. The Beltrami field is also a Magnus force free fluid flow and is expressed in hydrodynamics as... 

It should be emphasized that the flux vectors for which expressions have been given in Eqs. (28) through (36) are all defined here as fluxes with respect to the mass average velocity. Not all authors use this convention, and considerable confusion has resulted in the definition of the energy flux and the mass flux. Mass fluxes with respect to molar average velocity, stationary coordinates, and the velocity of one component (such as the solvent, for example) are all to be found in the literature on diffusional processes. Research workers in the field of diffusion should be meticulous in specifying the frame of reference for fluxes used in writing up their research work. In the next section this important matter is considered in detail for two-component systems. 

Up to now, the mass-continuity equations (e.g., Eq. 3.124) have been written in terms of the mass-flux vector j, which is a function of the species composition field. As noted in Section 3.5.2, different levels of theory can used to specify the functional relationship between flux and composition gradients, and mass flux can also depend on temperature or... 

The generic balance relations and the derived relations presented in the preceding section contain various diffusion flux tensors. Although the equation of continuity as presented does not contain a diffusion flux vector, were it to have been written for a multicomponent mixture, there would have been such a diffusion flux vector. Before any of these equations can be solved for the various field quantities, the diffusion fluxes must be related to gradients in the field potentials . 

This step requires some care under an external gradient positive and negative charges move in opposite directions, but the particle flux vectors are multiplied by e and by —e, respectively. Hence, in both cases the currents point along the direction of the electric field vector, but the latter points in a direction opposite to that of the increasing electrochemical potential gradient for holes. 

Since cations and anions move in opposite directions under the influence of an electrical field, the vector for a cation is in the opposite direction to that for an anion /x- This means that the flux /, multiplied by the ionic valence z, has the same sign for all ions as a result each ion contributes to the solution conductivity in such a way that it becomes larger. 

We calculated the flux vectors Jjfe/) = 0,(z.OVi(z,/), where v is the mean flow velocity at time 1 and depth z, and the index i denote macropore or micropore, respectively. Figures 4-6 and 4 7 show the computed flux fields for one of the numerical experiments. During rainlall, infiltration is dominated by the macropore. I ateral infiltration into the matrix occurs as water advances within the macropore (see the direction ol vectors in Fig. 4 6). As soon as the nearby walls are saturated. 

Centrifugal Force. Robertson et al40 designed an apparatus in which the centrifugal force vector was perpendicular to the membrane surface but opposite (and parallel) to the flux vector. Solutions of casein and dextran (60,400 dal-tons) were passed over the membrane in laminar flow. When the apparatus was spun, centrifugal field strengths from 100 to 600g resulted in flux improvement factors of 3 to 16. 

Species mass concentration Tensor in heat-flux vector expression Species contribution to extra stress tensor Potential energy for all molecules in liquid Tensor used in heat-flux expression Potential energy for single molecule Potential energy for single molecule in external field... 

Implicit in the derivation of Eq. (15.12) is the assumption of a homogeneous flow field - that is, one in which the velocity gradient (and therefore also the stress tensor and the mass-flux vector) is constant throughout space. If we regard this assumption as valid only over a length scale of the order of the polymer molecules, but allow the flow field to be nonhomogeneous on the scale of the fluid flow pattern, we can proceed to examine Eq. (15.12), on the basis that the stress tensor and velocity gradients may be spatially dependent (see, however, the caveat at the end of Sect. 14.1). If we take that point of view then Eq. (15.12) can be written as ... 

In crossed molecular beam scattering experiments, the direction of approach of the reactants is fixed by the configuration of the colliding beams [41]. Since by convention the incoming flux vector is directed along the space-fixed z-axis (and parallel to the field vectors E and B), we have = 0, and the general expression for the scattering amplitude (4.14) reduces to... 

Finally, Fig. 8.36 shows the evolution of the rotor flux vector in the (x, y)- and (d, <7)-planes from time t = 0.4 s to t = Is. The vectors noted as ri and r2 represent the rotor flux at steady state before and after the supply frequency reversal, respectively. The figure on the left clearly shows the rotating field phenomenon typical in AC-machines. Particularly, the effect of the supply frequency reversal on the rotor flux vector can be appreciated the counter-clockwise rotation gives place to a transient that finally results in a clockwise rotation, including amplitude variation during the transient and at the new steady state as well. 

As in previous chapters we work in the continuum limit employing quantities averaged over macroscopically infinitesimal volume elements and disregarding microscopic local variations associated with the molecular structure (see Brown 1956). These considerations will be limited to processes sufficiently slow to restrict the treatment to time independent or quasistatic fields. The validity of Maxwell s equations of electrostatics is presupposed. The basic electric state variables are the electric field strength vector E, the electric flux density (or electric displacement) vector D, and the electric polarization vector P, related by... 

To the left is the heat flux vector Jq (heat flow rate per unit area A) and to the right is the spatial change of the temperature field. The thermal conductivity X combines both of these vectors, which makes it a tensor. In general, X depends on the direction inside a solid (depending on its crystal structure) moreover, it is a function of temperature. Equation (4.1) can be solved in a closed form in only a few special cases because the initial and boundary conditions, that is, the temperature field at the initial time and the temperature of the boundary, as well as the geometry of the arrangement are included in the solution. Somewhat less complex are the relationships in the one-dimensional treatment under such circumstances, Eqs. (4.1) and (4.2) become... 

Fig. 8.18 Bond order flux (arrows) and bond order density (contour lines) plotted on xz plane induced by a pulse laser circularly polarized on this plane. The central frequency tc and field strength Eg shined are 0.057 and 0.03, respectively. The peak time and the characteristic time of pulse decay tn, are 4.84 fs and 2.42, respectively. The solid and dotted contour lines denote the positive and negative bond order densities, respectively. The increment of the contour lines is 0.01. Snapshot times are (a) 3.25 fs (b) 4.00 fs (c) 4.50 fs (d) 5.25 fs. The positions of all atoms are projected onto the the xz plane. The overlapped H atoms outside the B-B bonds are not shown. The flux vectors are multiplied by 20 for presentation. (Reprinted with permission from T. Yonehara et al, Chem. Phys. 366, 115 (2009)).

EXAMPLE 17.5 Computing the flux. Suppose you have a field of vectors that... 

Chapter 17 describes the flux of a flowing fluid. You can also define the flux for electric field vectors. Why is it useful to define a flux for electric fields The electric field flux has the important general property that it is independent of... 

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