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Natural, engineered, and biological systems often contain multiple phases with complex boundaries that change in space and time. Mathematical models used to understand fundamental transport phenomena, make predictions, guide designs, and inform policies are used to represent such systems in a simplified manner. A common, and often necessary, modeling approach for many multiphase systems does not resolve the boundaries between phases or the properties of such boundaries, which greatly reduces the computational complexity. Such macroscale models are often posited phenomenologically and not rigorously derived from established microscale continuum mechanical and thermodynamic principles. The thermodynamically constrained averaging theory (TCAT) has been advanced as a method to ensure a rigorous connection between the microscale and larger scales, ensure thermodynamic consistency, and more accurately represent the operative physics than traditional approaches. An overview of the TCAT approach is provided, available modeling hierarchies are summarized, and some specific examples are considered. One example considered in detail is non-Newtonian flow of a single fluid through a porous medium, which is modeled as a generalized Newtonian fluid (GNF) with a viscosity that depends upon the rate of strain tensor. A TCAT approach is used to derive a macroscale model for GNFs, and these flows are predicted accurately with only rheological characterization of the fluid, and readily accessible properties of the porous medium. The case of two-fluid flow through porous media is considered as a second example, a closed model advanced, and evaluations of the model shown. Other recent applications include sediment transport in surface waters, open channel flow, tumor growth, and separation processes.
Professor Miller is educated as an environmental engineer earning his PhD from the University of Michigan in 1984 and beginning his academic career soon after that as an assistant professor in the department of environmental sciences and engineering at the University of North Carolina, where he has remained on the faculty ever since. He is currently the Okun Distinguished Professor of Environmental Engineering with involvement in multiple departments at UNC (Environmental Sciences and Engineering, Mathematics, and Applied Physical Sciences). He has served as the primary advisor of about 100 post docs and graduate students and produced a total of more than 725 published works, including a book on the thermodynamically constrained averaging theory, which he coauthored with Bill Gray. Professor Miller has served in a wide range of scientific leadership positions including more than two decades as an editor of leading journals in the field---Environmental Science & Technology, and Advances in Water Resources. Professor Miller was an organizer of the bi-annual conference on Computational Methods in Water Resources for several years. A distinguishing feature of the research of Professor Miller is his joint, synergistic use of theory, computational methods, and experimental approaches to advance understanding of transport phenomena in complex systems.
A Pedagogical Approach to the Thermodynamically Constrained Averaging Theory