The Annals of Frontier and Exploratory Science
If the medium which overall properties are sought, known as being in dependence on the lower (smaller) scale physical phenomena then their physical and mathematical descriptions need to be considered and constructed in a way incorporating the interdependence of the higher (larger) scale descriptions and mathematical modeling into the lower and vice versa. A theoretical approach for treatment of Heterogeneous Phenomena problems in heterogeneous media considered as scaled subject was suggested first in 1967.
The Governing Equations for any kind of transport in heterogeneous media are described using the Hierarchical Non-local Volume Averaging Theory, Mathematics and Physics (HSP-VAT or HSP), which is based as in Homogeneous physics on the application of various the Volume - Surrounding Surface theorems. These theorems known in Homogeneous physics as Ostrogradsky-Gauss theorems (in 3D, while others are available for 2D) are different for Heterogeneous media. Most often the situation arising when the two levels of hierarchy and two - or three phases in media allowed connecting phenomena in the neighboring scales. The third phase is the interface (interphase, grain) surface, which is also being taken into account as a transport medium.
The hierarchical HSP-VAT governing mathematical equations for scaled heterogeneous medium resulting from the VAT based analysis in many fields are the parabolic partial differential equations (PPDE) and they have additional integral and integro-differential terms. This theory, among methods in use today, has special attributes that enable explanations of hierarchical phenomena to be obtained because it is able to combine scale, interfacial and morphology features simultaneously with mathematical rigor into governing equations and mathematical statements.
The texts in this website for variety of physical disciplines are not for the beginner or unfamiliar with the HSP-VAT researcher.
Experiences to remind me that the medium time for graduate student to have some good familiarity with the HSP-VAT's theorems, rules, procedures and practices is about
two (2) years. So, for the first time professional visitors (any rank, level) there are exist the texts, papers, books
by S.Whitaker, J.Slattery, W.Gray, M.Kaviany, K.Vafai with co-authors to understand the basics. I have mentioned only few, but outstanding researchers in the field, who spent more then 20 years each
studying these tasks. Those tasks are primarily the linear and half-linear HSP 1.5 scale
problems in Fluid Mechanics, Thermal Physics, Chemical Technology,
and Groundwater science modeling.
Seems only after that, a reader will be more or less qualified to understand this website pages, where among other new directions in HSP-VAT as - the presentation of the HSP-VAT models as the Two (2) Scale (at least) directly scaled physical problems, the True Scaled HSP-VAT models physical and mathematical Closure, the HSP-VAT Heterogeneous Experiments basics, and the HSP-VAT Heterogeneous Optimization basics have been advanced for the first time for many physical disciplines including the Non-linear, and Turbulent Hierarchical Scaled Theories in Continuum, Atomic and Sub-Atomic physics where models, and simulations have been described, advanced, and took substantial parts in the texts. And we are sorry, it cannot be helped with this including the texts for the first time education (see the "Education...." section in the right column of the main page for the scope of available now courses for undergraduate and graduate education). May be later on, that would blow-up the size of this website substantially. Still, everyone who is the undergraduate student in physics, mathematics, biology, medicine, ecology or good level (university) technical science major, can understand the general intentions at this website and for the variety of disciplines in it.
Then, one can go to the beginning in -
What are the basics of HSP-VAT Physics in Heterogeneous Media?