The momentum flux is related to the gas mixture viscosity and the velocities by

(5–38)

where is the velocity vector, is the dyadic product, is the transpose of the dyadic product, and is the unit tensor.[33] The Ansys Chemkin Transport package provides average values for the mixture viscosity, , but no information on the bulk viscosity, .

The energy flux is given in terms of the thermal conductivity by

(5–39)

where,

(5–40)

The multicomponent species flux is given by

(5–41)

where are the mass fractions and the diffusion velocities are given by

(5–42)

The species molar masses are denoted by and the mean molar mass by . are the ordinary multicomponent diffusion coefficients, and are the thermal diffusion coefficients.

By definition in the mixture-average formulations, the diffusion velocity is related to the species gradients by a Fickian formula as,

(5–43)

The mixture diffusion coefficient for species is computed as [42]

(5–44)

A potential problem with this expression is that it is not mathematically well defined in the limit of the mixture becoming a pure species. Even though diffusion itself has no real meaning in the case of a pure species, the numerical implementation must ensure that the diffusion coefficients behave reasonably and that the program does not "blow up" when the pure species condition is reached. We circumvent these problems by evaluating the diffusion coefficients in the following equivalent way.

(5–45)

In this form the roundoff is accumulated in roughly the same way in both the numerator and denominator, and thus the quotient is well behaved as the pure species limit is approached. However, if the mixture is exactly a pure species, the formula is still undefined.

To overcome this difficulty we always retain a small quantity of each species. In other words, for the purposes of computing mixture diffusion coefficients, we simply do not allow a pure species situation to occur; we always maintain a residual amount of each species. Specifically, we assume in the above formulas that

(5–46)

where is the actual mole fraction and is a small number that is numerically insignificant compared to any mole fraction of interest, yet which is large enough that there is no trouble representing it on any computer. A value of for works well.

In some cases (for example, Warnatz [43] and Coltrin, et al [. ]) it can be useful to treat multicomponent diffusion in terms of an equivalent Fickian diffusion process. This is sometimes a programming convenience in that the computer data structure for the multicomponent process can be made to look like a Fickian process. To do so supposes that a mixture diffusion coefficient can be defined in such a way that the diffusion velocity is written as Equation 5–43 rather than Equation 5–42 . This equivalent Fickian diffusion coefficient is then derived by equating Equation 5–42 and Equation 5–43 and solving for as

(5–47)

Unfortunately, this equation is undefined as the mixture approaches a pure species condition. To help deal with this difficulty a small number () may be added to both the numerator and denominator to obtain

(5–48)

Furthermore, for the purposes of evaluating the "multicomponent" , it may be advantageous to compute the in the denominator using the fact that . In this way the summations in the numerator and the denominator accumulate any rounding errors in roughly the same way, and thus the quotient is more likely to be well behaved as the pure species limit is approached. Since there is no diffusion due to species gradients in a pure species situation, the exact value of the diffusion coefficient is not as important as the need for it to be well defined, and thus not cause computational difficulties.

In practice we have found mixed results using the equivalent Fickian diffusion to represent multicomponent processes. In some marching or parabolic problems, such as boundary-layer flow in channels,[44] we find that the equivalent Fickian formulation is preferable. However, in some steady state boundary value problems, we have found that the equivalent Fickian formulation fails to converge, whereas the regular multicomponent formulation works quite well. Thus, we cannot confidently recommend which formulation should be preferred for any given application.