Objective of the Course:
The aim of the course is to give students an essence and some practical level of understanding and knowledge on how the different physical scales are to shape the problem's formulation in the two important areas very distinctive by nature, but unified by few deeply inherited features - semiconductor technology and biomechanical applications.

Tentative outline:
Starting from an atomic scale to different continuous measurable physical scales problems in solid state physics, biomechanics and medicine will be formulated and discussed for each one separate scale. Afterward, the course proceeds to more difficult tasks of description and selection of mathematics for interconnected physically multiscale problems.

It will be given an analysis of few two- and three scale engineering field's problems such as, for example, formulation of electrical conductivity and permittivity properties in superlattices, blood flow and sound waves propagation in tissue. Derivation, explanation in detail, and mathematical features of homogeneous Gauss-Ostrogradsky and Stocks theorems, as well as few of their counterpart theorems in heterogeneous science will be outlined in class.

The course will proceed to practical contemporary knowledge problems in nanoscale physics, semiconductor technology and biomechanics as for the scaled phenomena, which should be described at the every participating level (scale) of process (event). There will be examples and few problems (depending on the course length) analyzed in detail. This course may be taught as a sequential program due to large volume of contemporary science knowledge, which is not given and known in a core physics and mathematics curriculum at present time.

Required Textbooks: none

Recommended Textbooks:

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).

Gray, W.G., Leijnse, A., Kolar, R.L., and Blain, C.A., Mathematical Tools for Changing Spatial Scales in the Analysis of Physical Systems, CRC Press, Boca Raton, FL, (1993).

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).