Fundamentals of Hierarchical Scaled Physics (HSP-VAT) Description of Transport and Phenomena in Heterogeneous and Scaled Media

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 (HSP-VAT), which is based as in homogeneous physics on the application of 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.

Here at this site we would like to state, argue and pursue the fundamental thesis that at present time there exists no other theory, or approach than the HSP-VAT which suggests better, more accurate and mathematically correct means (ways) to consider, model and simulate the scaled transport physics and processes of any kind or nature in heterogeneous media.

Many distinguished professionals and groups have been surprised and first wanted to raise their voices to defend what they are doing or valued during their involvement in their respected fields and defend their achievements. We invite the arguments and will post out the most appealing and substantial points.

So far we have not heard loud enough on that and we are waiting since 1980s, because many issues surfaced here on various fields of physics are quite important or crucial.

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 problems in Fluid Mechanics, Thermal Physics, Chemical Technology, and Groundwater science modeling.

And 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 Scale (at least) directly scaled physical problems, the True Scaled HSP-VAT models physical Closure, the HSP-VAT Heterogeneous Experiments basics, and the HSP-VAT Heterogeneous Optimization basics have been advanced first time for many physical disciplines, the Non-linear, and Turbulent HSP-VAT physics, models, and simulations have been described, advanced, and took substantial parts in the texts. And I am sorry, I can not help with this including the texts for the first time education (see the "Education...." section in the right column of the home page for the scope of available now courses for undergraduate and graduate education). May be later on, but it would blow-up the size of this website substantially. Still, everyone who is the undergraduate student in mathematics, physics, biology, medicine, ecology or good level (university) technical science major, can understand the general intentions at this website and this discipline.

2011. Regarding the some history and priorities, we need to write, since it is only a short time when I learned that my former student on HSP-VAT and collaborator I.Catton started to name himself as the "starter"/"founder" of the HSP-VAT, VAT? Which is the self-PR and a wrong statement. I am still alive and well versed. Well, this also should be announced and corrected.

Some of the prehistory of HSP-VAT in the USSR and in the USA should be spelled out, and it is now in a few places here in this and other websites as, for start, in -

Announcements.

Then, one can go to basics in -

What are the basics of HSP-VAT Physics in Heterogeneous Media?