Mathematical prerequisites
• Linear algebra and the notion of the echelon form of a matrix.
• The concept of expansion in series in terms of a given (usually small) parameter or in terms of gauge functions.
• The concept of a singular perturbation problem.
• Matching and patching between the inner and outer solutions.
Some of these mathematical topics will be explained in the chapter. For additional reading on asymptotic and perturbation methods the book by Bender and Orszag (1978) is a good starting reference.
We have presented by now a detailed introduction to the analysis and modeling of transport phenomena. Basic differential equations were developed and examples were shown for all three modes of transport and some coupled problems in transport. We also studied how models at higher levels can be developed by either cross-sectional averaging or volume averaging. The goal of this chapter is to look into the details of some mathematical underpinning of the transport equations as well as some novel methods for solving transport equations. The chapter examines a number of important techniques for the analysis of transport models. Three tools are examined here.
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