Energy flux vectors

For most problems one needs to know how the components of the energy-flux vector are related to the space derivatives of the temperature. For conductive heat transfer the necessary relations are of the form... 

The viscous dissipation term is normally not important. Its significance has been considered in connection with lubrication theory (VI), flow through tubes (B20), extrusion of plastics melts (BIO), and viscometry in rotating-cylinder systems (W6). There is also an additional contribution to the energy flux vector describing energy transport by radiation. See discussion in connection with Eq. (29). 

Assuming spherical symmetry, the mass and energy flux vectors can be expressed in terms of scalar functions as... 

Equations [56] and [57] give the differential equations for the mass and energy flux vectors... 

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... 

In accordance with the First Law the total energy of a system can only be changed by energy transport across its boundaries. We therefore introduce a total energy flux vector 7i/ by the conservation law... 

The q term is the heat added to the system and almost always includes a conduction component of some form. We now define an energy flux vector, e, (J/m- s), to include both the conduction and convection of energy. 

The integral can be changed to that over energy with the introduction of a density of states D(t). The energy flux vector can then be written as... 

Theory [4] allows us to calculate the power conversion coefficients as ratios of the components of the energy flux vectors normal to the plane of the 2D system at zero order of diffraction as... 

In order to use the expressions for the mass flux vector, the stress tensor, and the energy flux vector in Table 1, it is necessary to know the singlet and doublet configuration-space distribution functions. For example, we need to solve Eq. (10.6) or Eq. (10.7) to get the distribution function for a single polymer molecule. This cannot, however, be done until something is inserted for the double-bracket... 

We now turn to the energy flux vector expression for gases. The dimensionless, kinetic contribution to the energy flux vector follows from... 

Up to an error proportional to the square of quantities assumed to be small, these expressions are identical to those for perfectly elastic spheres as given, for example, in Section 16.34 of Chapman and Cowling (1970). Consequently, so also are the expressions for the pressure tensor and the energy flux vector calculated by employing the velocity distribution function (4.7) and Enskog s extension of the assumption of molecular chaos. 

Derivations for the stress tensor and energy flux vector that appear in the momentum and energy balance equations (Equations 4.3 and 4.4) can be found in Chapman and Cowling (1970, Section 16.41). The stress tensor is... 

With the definitions of the stress tensor (Equation 4.9), energy flux vector (Equation 4.10), and dissipation y (Equation 4.13), this determines the granular temperature... 

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