Fluxes Functions

B.4.2. Condition on Magnetic Flux According to Eq. (B.3), the magnetic flux function becomes... 

The surface integral in Eq. (5.70) may be expressed as a volume integral if we include the delta function S(S(q)) in the integrand such that we only get contributions when the coordinates satisfy Eq. (5.72). We also need the flux of system points across the surface. Both of these may be introduced by considering the flux function F(p, q) given in terms of the Heaviside step function as... 

Equations [88] and [89] provide the differential equations for the mass and energy flux vectors in Region II. Invoking the continuity of mass and energy fluxes at the interface of Regions I and II, the scalar mass and energy flux functions at the sphere of influence are given by... 

For the free-transport problem in Eq. (8.98), the realizable numerical flux function is... 

In the following sections we describe how the reconstructed NDF at the cell faces can be used to define a numerical flux function G(ni, Hr) for each of the spatial-transport terms in Eqs. (B.2)-(B.5) that guarantees that the updated NDF n"jC found from Eq. (B.ll) (and thus the volume-average transported moment set is realizable. The extension of... 

To conclude this section, we discuss three technical points that arise when applying the realizable high-order scheme for free transport. The first point is how to choose the value of N given that iV = is used for multivariate EQMOM. To answer this question, we first note that the highest-order transported moment in any one direction is of order 2n (which corresponds to 2n -t-1 pure moments in any one direction). The extra (even-order) moment is used in the EQMOM to determine the spread of the kernel density functions (see Section 3.3.2 for details). When free transport is applied to the moment of order 2n, the flux function involves the moment of order 2n -i- 1. Therefore, in order to exactly predict the flux function for free transport using the half-moment sets, we must have fV > (n-i-1). The obvious choice for N is thus fV = (n -i- 1). ... 

In order to solve the moment-transport equation, a closure must be provided for Using the QMOM, the first IN moments can be used to find N weights and N abscissas a- The flux function can then be closed using... 

General Expression for Flux Distribution of the Form (1 - u2)1. Yovanovich [130] chose the general flux function f(u) = (1 - u2y with parameter p, which gives (1) the isoflux contact when p = 0, (2) the equivalent isothermal strip when p = -Vi, and (3) the parabolic flux distribution when x = Vi to develop another general solution ... 

As a second example, it is instructive to derive the Kramers stationary flux function which serves as a basis for practical application in the Rayleigh quotient variational method (34,35). In principle there are an infinity of stationary flux functions, as any function in phase space which is constant along a classical trajectory will be stationary. Kramers imposed in addition the boundary condition that the flux is associated with particles that were initiated in the infinite past in the reactant region. Following Pechukas (69), one defines (68) the characteristic function of phase points in phase space Xr, which is unity on all phase space points of a trajectory which was initiated in the infinite past at reactants and is zero otherwise. By definition x, is stationary. The distribution function associated with the characteristic function Xr projected onto the physical phase space is then... 

Figure 7. The brightness function (H2(y)) and the flux function (G (y)) for a bending magnet source. Both functions fall slowly for energies less than the critical energy, and fall rapidly for energies above the critical energy.

This cross section can be readily evaluated for the case of 7 = 0, that is, a rotationally cold sample. When the sum in Eq. (9.59) is replaced by an integration and the approximation that 7 = 0 so that j ( is made, the flux function [Eq. (9.54)] reduces to... 

The development and application of generalized perturbation theory (GPT) has made considerable progress since its introduction by Usachev (i(S). Usachev developed GPT for a ratio of linear flux functionals in critical systems. Gandini 39) extended GPT to the ratio of linear adjoint functionals and of bilinear functionals in critical systems. Recently, Stacey (40) further extended GPT to ratios of linear flux functionals, linear adjoint functionals, and bilinear functional in source-driven systems. A comprehensive review of GPT for the three types of ratios in systems described by the homogeneous and the inhomogeneous Boltzmann equations is given in the book by Stacey (41). In the present review we formulate GPT for composite functionals. These functionals include the three types of ratios mentioned above as special cases. The result is a unified GPT formulation for each type of system. 

As was just mentioned, quantum theory limits the accuracy with which the classical variables of position and velocity can be specified. On the other hand, it introduces a new characteristic for particles their spin. Properly speaking, this should also be one of the variables of the flux O, or, more concretely, two flux functions are actually needed to specify the neutron distribution completely. One of these, 4>r, would describe the flux due to neutrons of right helicity (spin parallel to velocity), the other, Oz, would describe the flux due to neutrons of left helicity (spin antiparallel to velocity). There are transitions in which the helicity of neutrons changes so... 

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