Why is it Different from Homogeneous and other Theories and Methods of Heterogeneous Media Mechanics/(other Sciences) Description?
The main differences are due to different governing equations for the UPPER scale physics description. These equations are mainly different because the Heterogeneous Whitaker-Slattery-Anderson-Marle (WSAM) kind of theorems applied to problem's formulation bring out equations with few (many) additional, often nonlinear terms.
| Cross-Section of Phase #1 on the Bounding Surface | |
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| Homogeneous Gauss-Ostrogradsky Theorems | Heterogeneous WSAM Whitaker-Slattery-Anderson-Marle "Gauss-Ostrogradsky" Theorems |
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Later efforts in 70s-90s (see among others including ours, the great publications by Gray, et al., 1993; Gray and Hassanizadeh, 1989) brought in the substantial number of various averaging HSP-VAT theorems for 1D - 3D cases.
The Detailed Micro-Modeling - Direct Numerical Modeling (DMM-DNM) method used predominantly now as the most "full and correct" method for Upper scale models, is mostly incorrect for this purpose. Especially, when the effective properties and the upper scale characteristics sought as the calculated ones based on these Lower Scale model's solution fields or when compared with experiment or sought to be used for experimental basics.
The DMM-DNM is good for the Lower scale, but averaging and other operations for and over the Upper scale functions, operators, etc. done with the Homogeneous GO theorem are mostly wrong.
On the other hand, we would like to state again and again that the Homogeneous subject physics is just the genuine constituent part of the broader business of the Heterogeneous description of matter. And there is no contradiction in this statement.
What physics was and is using for the Homogeneous matter - on this or that particular scale, is the internal part of the next Upper or the outer part of the preceding Lower scale of the substance described. This is nobody seems object to! But when the connection, interaction between the scales are in need to be assessed precisely and definitely, then the conventional contemporary "Homogeneous" physics has failed so far to address the issue.
The formulation of heterogeneous medium transport equations has evolved a great deal since the 1950-s. Even so, the proper form of the governing equations for a heterogeneous (including porous) media is still a source of frequent discord.
Determination of the flow-variables and the magnitude of the scalar fields transport for problems involving heterogeneous porous media is difficult, even when subject to simplifications allowing specification of the medium periodicity or regularity. Linear or linearized models fail to intrinsically account for transport phenomena, requiring dynamic coefficient models to correct for short-comings in the governing models. Additionally, when attempting to describe processes in a heterogeneous media the correct form of the governing equations remains an area of debate among various researchers (see, for example, Whitaker, [7, 8, 9, 10]; Koch and Brady, [11]; Travkin and Catton, [4,5]). Allowing inhomogeneities to be of random or stochastic character further compounds the already daunting task of properly identifying pertinent transport mechanisms and predicting transport phenomena.
Mathematical simulation of physical processes in a highly non-homogeneous media, in general, calls for obtaining averaged characteristics of the medium and, consequently, the averaged equations. Those equations would be of the Upper spatial scale. The averaging of processes in a randomly organized media can be performed with different levels of rigorousness. If a physical model has several interdependent structurally organized levels of processes underway, it is expedient to employ one of the hierarchical methods of simulation ( for example, see Kheifets and Neimark [12], and Cushman [13] among others). The hierarchical principle of simulation consists of successively studying the processes at a number of structural levels.
One first deals with the smallest scale element, for example a small smooth capillary or globular media. Next, various types of capillary wall morphology are incorporated. This is followed by studies of a range of diameters, first smooth then rough, and then networking. Regular variations of the parameters are treated first, followed by random. This is done at each level. This approach is used for capillary morphologies as well as granular or other morphologies. The process leads one to find ways to deal with the large number of closure expressions that result from the Volume Averaging Theory (VAT) used to obtain the original governing set of equations for Upper scale. Although of a common form, the resulting usable form depends on the media morphology and the local boundary conditions at the each level of the hierarchy. A particular closure expression will be different for energy, mass or momentum transfer between the fluid and the solid matrix, in part because of their different boundary conditions.
There are many disagreements about the applicability of models based on conventional diffusivity type models of transport phenomena in Heterogeneous (and porous) media to media with the following features: 1) multi-scaled media; 2) media with non-linear physical characteristics; 3) polydisperse morphologies; 4) materials with phase anisotropy; 5) media with non-constant or field dependent phase properties; 6) transient problems; 7) presence of imperfect interface surfaces; 8) presence of internal (mostly at the interface) physico-chemical phenomena, etc.
The most common way to treat such problems has been to seek a solution by doing numerical experiments over more or less the exact morphology of interest. This leads to heavy use of large computers to solve large algebraic statements. The treatment and analysis of the results of such a Direct Numerical Modeling (DNM) is difficult and needs to follow some guidance "known" for averaging, whether ensemble or volumetric.
What is vital, that these Detailed Micro-Modeling - Direct Numerical Modeling (DMM-DNM) numerical approaches, methods are using as a common value tool the Homogeneous Gauss-Ostrogradsky Theorems when summarizing their Lower scale calculations or even for analytical solution fields for the Upper scale, and that is erroneous.
We have shown in many places of this website, by means of the Two scale solutions, especially with the exact Two scale solutions of those few common textbooks known problems, see in -
that this new kind of Mathematical Physics old problems can be successfully tackled and solved either.
While also obtained after 2002 the analytical solutions of the following classical problems that have not been solved for many decades by other methods (given in textbooks the Lower Homogeneous scale "solutions" are wrongly attributed to the Upper Heterogeneous scale averaged fields)
These solutions leave no chances for calculations or comparison with experiment of the Upper scale characteristics using the basis of Homogeneous GO. This has no sense, invalid for Heterogeneous problems.
In various parts of this website we also have been analyzing the used now, proclaimed as the multiscaled, physically sounded etc. methods, approaches dealing with Heterogeneous, scaled media. We need to do this as long as the large population in physics scientific and engineering communities studying these issues have no desire to learn something more mathematically challenging than what they had been taught in their educational years. And on the other hand, most of mathematicians have no qualification, knowledge and understanding of genuine physical problems, but considering mathematical tasks in established fields with mathematical statements known for decades.
Meanwhile, the VAT presents an incredibly powerful tool for dealing with complex heterogeneous media problems having features like those enumerated above. The equations resulting from the use of VAT have strange additional integro-differential terms that are not usually seen in Lower scale equations. One needs to ask whether or not these new terms are small enough to ignore. In the above mentioned studies and exact solutions, as well as throughout the whole this website information we have shown that they are not. In fact, they are of the same order of magnitude as the terms that are normally kept.
An important aspect of heterogeneous medium transport theory is the development of appropriate boundary condition equations for both (at least) scales in addition to the equations governing transport in the internal region. Most existing treatments conveniently rely upon the first (I), second (II) or third (III) kind boundary conditions for heat, potential, mass, momentum or other field transport. However, the I, II and III kind boundary statements are insufficient in portions of the near-boundary regions and as a basis for upper scale boundary conditions direct formulation. There are numerous ways to account for the additional including jump terms between phases - porous medium - homogeneous fluid, two heterogeneous (porous) media interface, other heterogeneous media interactions, which need to be addressed.
Few of well understood scaled VAT disciplines with some advancement are discussed in our publications
and in
These publications embrace only the part of known today HSP-VAT knowledge base. The main features described only in these publications are - nonlinearities in governing equations, advancements in experimental applications of HSP-VAT, electrodynamics and engineering applications of two-scale description of scaled processes, few aspects of atomic scale physics outlined using employment of the HSP-VAT, etc.
References:
4. Travkin, V. S. and Catton, I.,
"Porous Media Transport Descriptions -
Non-Local, Linear and Non-linear Against Effective Thermal/Fluid
Properties," Advances in Colloid and Interface Science, Vol. 76-77, pp. 389-443, (1998).
5. Travkin, V.S. and Catton, I., Transport phenomena in heterogeneous media
based on volume averaging theory// Advances in Heat Transfer, New York,
Academic Press, Vol. 34., pp.1-144, (2001).
7. Whitaker, S., "Diffusion and Dispersion in Porous Media," AIChE Journal, Vol.13, No. 3, pp. 420-427, (1967).
8. Whitaker, S.,
"Simultaneous Heat, Mass and Momentum Transfer in Porous Media: a Theory of Drying," Advances in Heat Transfer, Vol. 13, pp. 119-203, (1977).
9. Whitaker, S.,
"Flow in Porous Media I: A Theoretical Derivation of Darcy's Law," Transport in Porous Media, Vol. 1, No. 1, pp. 3-25, (1986a).
10. Whitaker, S.,
"Flow in Porous Media II: The Governing Equations for Immiscible, Two-Phase Flow," Transport in Porous Media, Vol. 1, No. 2, pp. 105-125, (1986b).
11. Koch, D. L. and Brady J. F.,
"Dispersion in Fixed Beds," Journal of Fluid Mechanics, Vol. 154, pp. 399-427, (1985).
12. Kheifets, L. I. and Neimark, A. V.,
Multiphase Processes in Porous Media, Nadra, Moscow, 288 pages, (in Russian), (1982).
13. Cushman, J.H.,
"Hierarchial Problems: Some Conceptual Difficulties in the Development of Transport Equations,"
Presented at the International Seminar of the International Centre for Heat and Mass Transfer,
Dubrovnik, Yugoslavia, May 20-24, 13 pgs., (1991).
Gray, W.G. and Hassanizadeh, S.M.,
"Averaging Theorems and Averaging Equations for Transport of Interface Properties in Multiphase Systems," Int. J. Muliphase Flow, Vol. 15, No. 1, pp. 81-95, (1989).
Gray, W.G., Leijnse, A., Kolar, R.L., Blain,C.A.,
Mathematical Tools for Changing Spatial Scales in the Analysis of Physical Systems, CRC Press, Boca Raton, (1993).
In the Bibliography Section below are given more references to publications on Fundamentals of the HSP-VAT. Meanwhile, we need here to take note of that the Bibliographies and References in this website are not maintained up to the latest fresh publications. There is no necessity in this so far. The noticeable publications on VAT and HSP-VAT are rare events. For that reason we do comments, analysis of some works, often just related to HSP-VAT topics and intentions, selectively.
Copyright © 2001...2010 V.S.Travkin, Hierarchical Scaled Physics and Technologies™