Objective of the Course:
This course will suggest students the knowledge of how the abstract mathematical constructions from different mathematical fields can be applied toward practical problems in everyday life and in engineering. The second objective is to give students some mathematical flavor of scaled consideration of physical, biological and environmental systems. There is the very limited number of universities in the country where some basics of this science being taught.
The course starts with the general provisions for mathematics used for development of the governing equations for most of the contemporary physical, chemical, and engineering disciplines. The number of examples from science and engineering will be discussed. Problems raging from contemporary atomic physics, materials science to groundwater pollution and urban area terrorist attack with toxic gases will be formulated using mathematics as the major tool.
We will talk in the class on details of connection between differential and integral formulation of technical problems and their possible solutions. The definitions of continuous, discontinuous, piecewise, and generalized functions will be treated in class with detail. Within the core of the course we will be dealing with the problems involving formulation of the simplest forms of governing equations in different sciences and disciplines as - fluid mechanics, solid state physics, electrodynamics, acoustics, earth science, materials science, etc.
Will be given a brief observation of homogeneous theorems of Gauss-Ostrogradsky and Stocks. There will be an intermediate part on numerical analysis and methods of numerical approximations for classical 1-3D problems. Few software packages will be discussed in terms of how the mathematics in governing equations transformed into the codes and software packages. Then, there will be made emphasis on some features of the development of governing equations for the two scale systems and processes. We will discuss the new heterogeneous theorems of Gauss-Ostrogradsky and Stocks type. Consequently, the new types of governing equations for applications will be developed and analyzed. We compare few scaling theories and discuss practical implications of scaled science mathematics.
Required Textbooks: none
Travkin, V.S. and Catton, I., Chap. 1, ''TRANSPORT PHENOMENA IN HETEROGENEOUS MEDIA BASED ON VOLUME AVERAGING THEORY'', in Advances in Heat Transfer, Vol. 34, pp.1-144, (2001).
Kaviany, M., Principles of Heat Transfer in Porous Media, 2nd. edition, Springer, (1995).
Slattery, J.C., Momentum, Energy and Mass Transfer in Continua, Krieger, Malabar, (1980).
Whitaker, S., ''Volume Averaging of Transport Equations'', Chap. 1, in Fluid Transport in Porous Media, Computational Mechanics Publications, Southampton, UK, (1997).