Dynamical rules notation

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 ... 

The rest of this chapter is organized as follows. In the next section, we provide the relevant literature review. We then outline the notation and basic assumptions of our model and develop a dynamic program with the aim of minimizing the total expected delivery cost. Next, we propose three heuristic decision rules, which we evaluate by numerical simulation. We wrap up the chapter with a summary of our main results and some ideas for future research. 

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