Conservation of Energy and the Entropy Inequality

In view of the differential form of the equation of motion, Eq. (2-31), and the fact that (7 ) is arbitrary, we see that the angular acceleration principle, (2-33), requires that 

Note that the condition of stress symmetry is not valid if there is a significant body couple per unit mass c in the field. In this case, we can easily show, following the same steps that we used in going from (2-35) to (2-40), that 

This gives a relationship between c and the off-diagonal components of T, but the stress is clearly not symmetric. We hereafter restrict our attention to the case in which c = 0. 

We see that application of the angular acceleration principle does reduce, somewhat, the imbalance between the number of unknowns and equations that derive from the basic principles of mass and momentum conservation. In particular, we have shown that the stress tensor must be symmetric. Complete specification of a symmetric tensor requires only six independent components rather than the full nine that would be required in general for a second-order tensor. Nevertheless, for an incompressible fluid we still have nine apparently independent unknowns and only four independent relationships between them. It is clear that the equations derived up to now - namely, the equation of continuity and Cauchy s equation of motion do not provide enough information to uniquely describe a flow system. Additional relations need to be derived or otherwise obtained. These are the so-called constitutive equations. We shall return to the problem of specifying constitutive equations shortly. First, however, we wish to consider the last available conservation principle, namely, conservation of energy. 

The rate at which the total energy changes with time is determined by the principle of energy conservation for the material volume element, according to which 

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