Microscopic Equation of Change for Total Energy

Once again, the equation of continuity is used to replace 9p/9t in (25-22), and the mass transfer equation for species i replaces dcoifdt. The substitutions are 

The final form of the microscopic equation of change for internal energy is 

The fourth step of the general strategy in Section 25-1 is to add the microscopic equations of change given by (25-19) and (25-24). The kinetic energy and internal energy equations are repeated here, for comparison  

Notice that the fourth term on the right side of (25-25) cancels with the third term on the right side of (25-26). Application of the following vector identity for scalar a and vector b allows one to combine a few terms in (25-25) and (25-26)  

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This is a statement of the product rule for the divergence of the vector dot product of a tensor with a vector, which is valid when the tensor is symmetric. In other words, r = r, where is the transpose of the viscous stress tensor. Synunetry of the viscous stress tensor is a controversial topic in fluid dynamics, bnt one that is invariably assumed. is short-hand notation for the scalar double-dot product of two tensors. If the viscous stress tensor is not symmetric, then r must be replaced by in the second term on the right side of the (25-29). The left side of (25-29), with a negative sign, corresponds to the rate of work done on the fluid by viscous forces. The microscopic equation of change for total energy is written in the following form ... 

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