Divergence, of flux

Note that the conservation equations can be distinguished from the transport equations since they do not contain any production or destruction terms. Nevertheless, the conservation equations may contain terms on the RHS expressing a divergence of fluxes related to transport phenomena. The way in which these flux terms are divided into divergence of transport fluxes or source terms is rather involved, but procedures exist based on a number of requirements on the two types of terms which determine this separation uniquely. 

As the considered domain may be arbitrarily chosen, this relationship between spatial charge density and divergence of flux density needs to be satisfied at every point. Thus, the differential form of Maxwell s equation can be obtained ... 

In traditional Fan-Beam CT the radiation emitted from the X-ray tube is collimated to a planar fan, and so most of the intensity is wasted in the collimator blades (Fig. 2a). Cone-Beam CT, where the X-rays not only diverge in the horizontal, but also in the vertical direction, allows to use nearly the whole emitted beam-profile and so makes best use of the available LINAC photon flux (Fig. 2b). So fast scanning of the samples three-dimensional structure is possible. For Cone-Beam 3D-reconstruction special algorithms, taking in consideration the vertical beam divergence of the rays, were developed. 

If there are no reactions, the conservation of the total quantity of each species dictates that the time dependence of is given by minus the divergence of the flux ps vs), where (vs) is the drift velocity of the species s. The latter is proportional to the average force acting locally on species s, which is the thermodynamic force, equal to minus the gradient of the thermodynamic potential. In the local coupling approximation the mobility appears as a proportionality constant M. For spontaneous processes near equilibrium it is important that a noise term T] t) is retained [146]. Thus dynamic equations of the form... 

In most cases, the term expressing the divergence of the molecular flux in Equation (40) (DV c,) can be neglected compared to the other two transport terms. Important excep-... 

The gradient of species concentration is in a direction opposite to the thermal gradient, see the green arrows. At places where the front is concave toward the unburnt gas, the species flux is locally divergent. The flux of reactive species into the reactive zone decreases, leading in turn to... 

To test the feasibility of obtaining submicron size patterns in the resist films, an exposure source was used which consisted of the X-ray continuous spectrum produced by synchrotron radiation from the 5 0 MeV storage ring of the University of Orsay (ACO) since synchrotron radiation had been shown previously (2.,8.) to be a suitable source for providing very high resolution due to the small divergence of the beam. The maximum output flux of ACO... 

Fig. 20.1. Control volume of an aquifer, showing the origin of divergence principle. Volume dimensions are dx x dy x dz. The rate at which a chemical component i accumulates within the volume depends on the divergence of the mass fluxes i.e., the rate at which the component s mass is transported into the volume along x and y, less the rate it is transported out.

Bear, 1972 1979 Freeze and Cherry, 1979), which is the negative divergence of the fluxes. This equation is a statement of the divergence principle, applied to solute... 

Mass conservation for component i implies that at a certain point in space, the time dependence of c, is related to the divergence of the flux of i. It is easily understood that a finite positive divergence in the flux will lead to depletion, i.e. to a lowering of the concentration. This can be expressed as ... 

For an incompressible liquid (i.e. a liquid with an invariant density which implies that the mass balance at any point leads to div v = 0) the time dependency of the concentration is given by the divergence of the flux, as defined by equation (13). Mathematically, the divergence of the gradient is the Laplacian operator V2, also frequently denoted as A. Thus, for a case of diffusion and flow, equation (10) becomes ... 

The divergence of the flux vector is therefore the net rate of accumulation of the quantity which is transported in and out of the volume element dK This can be integrated over an arbitrary volume Cl limited by the surface I to give the divergence theorem of Gauss... 

Thus, the divergence or flux of the electric field is directly related to the net charge at that point. The electric field is simply defined as the force acting on a unit charge ... 

Recognizing the terms in the parenthesis on the right-hand side as the divergence of the mass-flux vector and dV = rdrdOdz, it can be seen that the procedure has recovered the Gauss divergence theorem (Eq. 2.29). That is,... 

The right-hand side of the expression above can be recognized as the divergence of the heat-flux vector,... 

The momentum flux vector, which is the divergence of the stress tensor, appears in the Navier-Stokes equation (3.53) ... 

Accumulation of an extensive quantity arising from a divergence of its flux... 

This is the rate at which the flux causes the density of the quantity comprising the flux to decrease. The rate of accumulation of the extensive quantity s density is therefore minus the divergence of the flux of that quantity plus the rate of production. Alternatively, Eq. 1.14 could be derived directly from... 

Vapor transport differs from surface diffusional transport, where the flux is always in the surface plane. For both surface diffusion and vapor transport, the diffusion potential at the surface is proportional to the local value of 7sk if the surface free energy is isotropic. For surface diffusion, the interface normal velocity is related to a derivative (i.e., the divergence of the flux). Also, the total volume is conserved during surface diffusion. For vapor transport, the interface normal velocity is directly proportional to the vapor flux, and the total number of atoms is not necessarily conserved. 

The divergence of the flux is on the left-hand side, i.e., the difference between the amount of matter carried in and out by the flow per unit volume on the right-hand side is the change in the amount of the substance a per unit volume d(pa)/dt and the amount of a which forms per unit volume as a result of the chemical reaction. 

Let V be a region in space bounded by a closed surface S (of Lyapunov-type [24, 50]), and f (x) be a vector field acting on this region. A Lyapunov-type surface is one that is smooth. The divergence (Gauss) theorem establishes that the total flux of the vector field across the closed surface must be equal to the volume integral of the divergence of the vector (see Theorem 10.1.1). 

The charge balance and the material balance for species / in a volume element are the divergences of the fluxes of current and material respectively as... 

However, there are some important points that should be made about this existing source. First, the X-ray flux available for ultrafast experiments is severely limited due to the full divergence of the X-ray radiation [4]. This prevents the efficient development of applications. Second, the X-ray flux cannot be scaled-up due to the physics of the laser-matter interaction. Finally, the source is highly monochromatic and fairly tunable, which prevents any X-ray absorption applications. 

The aroma compound will diffuse in as well as out of the cube because of its perpendicular side surface areas. Due to the greater decrease in the aroma near the soap s external surface, the flux out of the side of the cube closer to the surface is greater than the flux into the side of the cube that lies deeper in the soap. The difference between the aroma diffusing in and out will be positive which means one can consider the cube as an aroma source. As a consequence of the flux out of the cube, the concentration in the cube decreases with time. The concentration is also a function of time, c = c(x, y, z, t) and its decrease with time, i.e. the partial derivative —dc/dl in the cubic volume AV = Ax Ay Az, represents the net flux out of the cube and designated div/ the divergence of the flux. 

---