Scalar flux definition

Vector analysis. The author learnt vector analysis in a 1950 postgraduate course, based on the German book of 1932, Classical Electricity and Magnetism, by M. Abraham and R. Becker, published by Blackie and Sons, Glasgow. The book was written so that its definitions of scalar flux (flow in all directions) and of vector current (flow in one direction) fitted both hydrodynamics and electromagnetism. The same definitions were carried over unmodified into the scalar neutron fluxes and vector neutron currents of the nuclear power reactor. In electrochemistry, however, the term exchange current attaches to what is more properly described (see above) as a local, somewhat anisotropic, scalar flux. 

By definition, any plane 0 — constant is a plane of symmetry. In other words, there are always two elementary masses, which are equal to each other, and located at opposite sides of this plane but at the same distance. As is seen from Fig. 1.5d, the sum of 0-components, caused by both masses is equal to zero. Representing the total mass as a sum of such pairs we conclude that the 0-component, gg, due to the spherical mass is absent at every point outside and inside the body. In the same manner we can prove that — 0. Of course, volume integration, Equation (1.6), can prove this fact, but this procedure is much more complicated. Thus, the attraction field has only a radial component, g, and the field is directed toward the origin, 0. In order to determine this component we will proceed from Equation (1.26) and consider a spherical surface with radius R, as is shown in Fig. 1.5c. Inasmuch as dS — dSiR and the scalar component g is constant at points of the spherical surface, we have for the flux ... 

The second term on the right-hand side of (A.41) represents the flux of scalar dissipation into the scalar dissipation range, and can be rewritten in terms of known quantities. From (A.39), it can be seen that Taa(icD, t) = TD(t). Likewise, using the definition of kd, it follows that 2raK = CD(e/v)1/2. The scalar-dissipation flux term can thus be expressed as... 

The diffusion equation in an anisotropic medium is complicated. Based on the definition of the diffusivity tensor, the diffusive flux along a given direction (except along a principal axis) depends not only on the concentration gradient along this direction, but also along other directions. The flux equation is written as F = —D VC (similar to Fick s law F= -DVC but the scalar D is replaced by the tensor D), i.e.. 

Positive definite means that the matrix when left- and right-multiplied by an arbitrary vector will yield a nonnegative scalar. If the matrix multiplied by a vector composed of forces is proportional to a flux, it implies that the flux always has a positive projection on the force vector. Technically, one should say that Lap is nonnegative definite but the meaning is clear. 

Mathematical optimization deals with determining values for a set of unknown variables x, X2, , x , which best satisfy (optimize) some mathematical objective quantified by a scalar function of the unknown variables, F(xi, X2, , xn). The function F is termed the objective function bounds on the variables, along with mathematical dependencies between them, are termed constraints. Constraint-based analysis of metabolic systems requires definition of the constraints acting on biochemical variables (fluxes, concentrations, enzyme activities) and determining appropriate objective functions useful in determining the behavior of metabolic systems. 

In these equations fi is the coluirm mass of dry air, V is the velocity (u, v, w), and (jf) is a scalar mixing ratio. These equations are discretized in a finite volume formulation, and as a result the model exactly (to machine roundoff) conserves mass and scalar mass. The discrete model transport is also consistent (the discrete scalar conservation equation collapses to the mass conservation equation when = 1) and preserves tracer correlations (c.f. Lin and Rood (1996)). The ARW model uses a spatially 5th order evaluation of the horizontal flux divergence (advection) in the scalar conservation equation and a 3rd order evaluation of the vertical flux divergence coupled with the 3rd order Runge-Kutta time integration scheme. The time integration scheme and the advection scheme is described in Wicker and Skamarock (2002). Skamarock et al. (2005) also modified the advection to allow for positive definite transport. 

However, without showing all the lengthy details of the method by which the two scalar functions are determined, we briefly sketch the problem definition in which the partial solution (2.247) is used to determine expressions for the viscous-stress tensor o and the heat flux vector q. 

This precise definition refers to the current intensity which is a scalar corresponding to the overall charge flux or flow rate (1 A = 1 Cs ). Naturally, it must be defined with reference to a surface. However, in certain conditions including those fulfilled in this document , the absolute value of the current intensity in a conductor can be defined by considering any section of the conductor. For these reasons, which is moreover usual practice in numerous documents, we will use current to stand for current intensity across a section of area S. ... 

Finally, solutions to the integral flux equations like (2.189) and (2.197) are then obtained by expressing the scalar functions A C, n, T) and B(C, n, T) in terms of certain polynomials (i.e., Sonine polynomials). However, without showing all the lengthy details of the method by which the two scalar functions are determined, we briefly sketch the problem definition in which the partial solution (2.267) is used to determine expressions for the viscous-stress tensor a and the heat flux vector q. 

Transport processes in electrochemical systems should be analyzed with vector analysis, a part of calculus. The main definitions of vector analysis are terms like scalar, vector, gradient, divergence, and curl values. The reader is encouraged to refresh his or her memory for definitions of these values. There are two key equations that are fundamental to transport processes in electrochemical systans. The first describes the flux vector, of the ith species [1] ... 

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