General formulation conditional moments

The scalar mean conditioned on the mixture-fraction vector can be denoted by 

By definition, the unconditional scalar means can be found from Q(C x, t) and the mixture-fraction-vector PDF  

The higher-order conditional scalar moments are defined in terms of the conditional fluctuations. For example, the conditional scalar covariances are defined by 

Extending (5.286) to conditional scalar moments of arbitrary order, the conditional chemical source term can be approximated by 

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Summary. The first three balance equations are formulated in this section. The balances are necessary conditions to be fulfilled not only in thermodynamics but generally (in continuum mechanics). The balance of mass was formulated locally in several alternatives—(3.62), (3.63), or (3.65). The most important consequence of the balance of momentum is the Cauchy theorem (3.72), which introduces the stress tensor. The local form of this balance is then expressed by (3.76) or (3.77). The most relevant outcome of the balance of moment of momentum is the symmetry of the stress tensor (3.93). Note that in this section also an important class of quantities— the specific quantities—was introduced by (3.66) note particularly their derivative properties (3.67) and (3.68). 

Aiming to find a particular solution for differential equations system (1), i.e. particular expression of each indicator as on time dependent function, we need to formulate initial conditions for differential equations system. Constants C), / = 1,..., fV of general solution may be found by solving system of differential equations (1) in time moment... 

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