Since 1990s years we have been witnessing the phenomenon when professionals from different branches of science - different artisan guilds we can say, started to "create," make up their own different versions of "averaging" and "multiscaling." Often it is just "multiscaling." Actually, it is the old and known method to proclaim something of what is wanted to be created, and then name this thing as if it is right now has been invented the new method or "theory." Nothing bad with this when the novelty is real and worthwhile.
We have to note here that what people call usually with a variety of names just to designate that the phenomena of known technical set-up (physical model) at a certain scale can (could be, should be) be connected to the sought description at the other physical scale mostly have to bare the features of somehow averaged events, phenomena, mathematical models, etc. That is - the phenomena should be somewhat arranged in a multiple fashion to claim the generality of a tool.
In most cases nowadays (and always) this is done because workers wanted to be the "Inventors" of New type of "averaging" method? Instead or sometimes because of studying the published sources. We found few varieties of these "methods" with regard of averaging definition made in physics (while some are in biology). In mathematics more or less stricter situation, when people can not declare what is not working. Because this activity is of no innocent content, we will pay attention to these publications and make some comments on those from time to time.
We distinguish:
1) A Pseudo-averaging: when averaging is being declared but not existing.
2) A Quasi-averaging: when averaging has been done incorrectly, intentionally or because of ignorance. Most often this is when the Homogeneous Gauss-Ostrogradsky theorem being applied for situations, media, volumes with the Heterogeneous features.
3) An Ad-hoc averaging: when researchers plainly saying that they want to do this or that and no matter what mathematics would say, they continue to name this or that the "averaging" (Scaling, Hierarchy) !?
4) Other types of "averaging": when a "new kind of averaging" (Scaling, Hierarchy) has been made up to fill the gap and because of the current trend, fashion for a multiscaling and an averaging.
To the class of pseudo-averaging we have attributed the type of "averaging" done in the Atomic physics and Electrodynamics, for example, when the same kind of equations as for continuum Homogeneous matter are being applied to the completely different atomic (nanoscale often, and other scales) matter. This happens because in the first half-of the last century there were no mathematical tools to do the multibody treatment and the correct spatial averaging with detailing of spatial structure and specifics. There was probably no belief that this is possible. And no belief in that now as well.
The many-body theories used right now are of the same homogeneous one-scale GO theorem origin and can not be representative for scaled and hierarchical physics. We speak out on that in few places of this website.
The great significance continue to play the notion by Lorentz that the Maxwell's equations are suitable for any kind of matter as well as the huge number of models and theoretical constructions made by the Lorentz's theory of dielectric materials.
Concerning the Continuum Mechanics, Thermal Physics and Fluid Mechanics one might note that Bejan's "Constructal theory" claim (Bejan, 2000) is the factual attempt to develop the meaningful, sound and mathematically correct theory for scalable, and hierarchical description of Hierarchical matters in physics, so far mostly in Fluid Mechanics and Heat Transfer. Meanwhile, some declarations have been made regarding applicability of this "universal" theory also for Biology and even to the Physics as a whole (Bejan, A., Morega, A., West, G.B., and Brown, J.H., 2005)!
That is done at the conferences and in publications, while being fiercely promoted (advertised) in numerous periodicals throughout the last like 18 or more years - http://www.constructal.org/.
We studied these claims in 90s and made remarks and published on insufficient background, actually no mathematical background can be found - serious mathematical basics. Again, as in most such claims there are good intentions and programs to fulfill, just not enough foundation, followed with the Qualitative program. More - there is the logistics to support the intention - like in any marketing program - there should be the written publicized activity to present this intention.
We won't be interested addressing this topic unless so intense industrial type marketing campaigning on the claim of a "Universal" value and force for physics and biology in terms of Hierarchy and Polyscaling (Multiscaling), that is not supported by a good, unabridged mathematics as a whole.
Well, this is raising an attention, when one has to account for the value of the claim. We considered many publications of 90s and some recent publications of 2000s. Looking into the implementation (not onto the claim itself) one might notice that:
1) There is the wrong understanding and modeling mathematical statements as long as the GE's (governing equations) are used from the Lower scale phenomena while the heat transfer goals have taken for the Upper (averaged should be) scale phenomena - as the heat transfer rate, etc.
2) The optimization statement and criteria are formulated incorrectly - as for a one scale task.
3) The mathematics of 3D and 2D Distributed parameters (sources) 2 scale Optimization problems is incorrect, etc., etc.
The Bejan's "Constructal theory" also is the one that is based and using the basis and language of semi-logistic, semi-mathematical (Homogeneous) algorithms for explanation of the different scales phenomena obviously connected by scales and by physics.
See more mathematics and physics insights in our -
where given in depth analysis and critics of some issues of "Constructal theory" thermal transport for medium of heat exchangers; and in
where we talk on some procedures for the "Constructal theory" optimization two-scale thermal transport in porous medium of heat exchangers.
We would allocate this theory to the kind of a Ad-hoc averaging, Ad-hoc science supposition. In spite that Bejan's techniques do not involve the averaging over the any sub-phase, nevertheless, the intention of theory is, by no means, toward the modeling and calculation of the Upper scale (or interscale) momentum and heat transfer characteristics.
Whether author does not want to recognize this openly, he still tries to develop some modus operandi to approach the goals those achieving with the averaging and scaling techniques in the HSP-VAT.
Among many published manuscripts those claiming the "special" and "successful" kind of averaging we might look into the book by Nakorchevskiy and Basok (2001), neither too old, nor yet "multiscaling." This is the example of a book with superficial claim for averaging of processes - the Ad-hoc averaging.
The case is based on the statement of authors that while taking the size of the averaging volume (REV) as small as we can do staying within the continuum mechanics, no matter of what size of particle or an element of dispersive phase one might has - one would still has the ability to perform the "dot" averaging. What authors mean is that the temporal consideration of any phase being or having within itself the REV and waiting for this phase temporarily going through the "dot" (REV) would equal to the averaging of the multiphase medium in any way - either as of phase averaging or as of bulk volume averaging. And that corresponds to the averaging process of this unique "dot" averaging.
In this way authors claiming to prove the theorem that is opposite to the WSAM theorem - that the derivative of averaged function is equal to the averaged derivative?? Which theorem is correct? We "guess" that the WSAM theorem is correct, going through many efforts of verifications and even having the exact proof through the exact solution on two scales of the classical problems, see our texts on that in the website -
Of course, using this false theorem authors in this way of doing "averaging" obtained the Homogeneous type of governing equations for multiphase media. The same kind of equations as all other authors using the GO theorem and homogeneous description.
In reality the claimed "dot" averaging brings us to the same large volume of the REV while we keeping for the "dot" going through the medium. This means - that the WSAM theorem should be applied and etc., etc.
The kind of "in between" of the pseudo-averaged and ad-hoc type averaging is presented in the very known in Russian language literature (translated into English as well) books by Nigmatulin (1987, 1991, for example). We provide the thorough breakdown while looking into equation after equation in these books to return readers to reality regarding these mathematical formulations that effect the whole body of research in Russian on Continuum Mechanics (CM), and many sub-disciplines connected to CM.
For example, when author comes to formulation of applications we can see that he writes in (1.3.3), (1.3.38), (1.3.45), (1.3.52), (1.3.63), 1.3.66) the just pseudo-averaged, ad-hoc averaged equations.
These and further equations in paragraphs 4 and 5, 6,7,8,9 (the governing equations for porous media are just unacceptable), and § 10 of chapter one are the mostly well known and used homogeneous type equations for heterogeneous media.
Thus, unfortunately, we observe the huge gap between the mostly correct concepts and their mathematical formulation in §1,2 and their miss-implementations in §3 -10 in chapter one. The same non-implementation of the Upper scale averaging theory reader can find in chapters 4,5 in book 1 and in other chapters of book 2 (Nigmatulin, 1987b; 1991b - English edition).
More can be seen in -
This Continuum Mechanics subsection's content has been written for this website first of all to depict the ways used in contemporary CM to avoid consideration of Heterogeneous, Scaled media and problems in that media. Nevertheless, the result can be seen also and as the short review of models of "quasi-" and "pseudo-" averaging methods widely accepted in many physical disciplines.
Related to this and covering more recent paper publications on pseudo-multiscaling in area of Continuum Mechanics is the next subsection -
Closely related to this field is the now famous Homogenization theory with claims on multiscaling and, of course, averaging of the problem's mathematical statement, which is not correct, see many arguments in the close subsection -
There exists a large layer of "quasi - averaging" manuscripts in chemistry and chemical technologies, in Russian and in English language scientific literature.
A lot of examples of "quasi - averaging" in Fluid Mechanics and Meteorology we analysed and uploaded few years ago in our part -
Particularly sweet areas considered by workers trying to overcome, while they almost celebrating the next in turn a "triumph", the difficulties of turbulent phenomena modeling in porous media -
It was a gloomy picture watching the world gathering of mostly meteorologists, when researchers are so confined within their artisan guild crafts that do not notice what's going on in some neighboring disciplines (fields), as that happened at NATO Advanced Study Institute 980064 -
and in
As long as the publications in these modes continue to be published as the "scientific literature" in a variety of disciplines - this phenomenon should be placed in public eyes like an information on the roadblocks for drivers - in our case, for students and for some advanced industries customers as for the generations of users of "free" scientific production needs to exist some explanatory guide to heterogeneous media approaches and on their value.
The complex picture on averaging and "multiscaling" exists in biology with its stalled through the century the Qualitative and experimental mostly progress. The matter of fact is that biological issues are more multifaceted and could not be described with the contemporary traditional tools and theories supplied by Homogeneous orthodox physics and chemistry. Because the biology phenomena and samples are more polyscaled than any other in physics and visual observation gives the first support to this simple statement.
Our continued from 80s effort in this field found some form of web present analysis of biology related methods of also "in between" of the pseudo-averaged and ad-hoc type matter description and actual averaging is presented in -
All of this is happening because in the other way around - many authors actually know that the strict averaging (Scaling, Hierarchy) methods, studies exist and written in papers, monographs. Unfortunately, accepting this would mean for most of the surrogate Upper scale, "averaging" methods the end of their authors carriers or, at least, a huge setback, because they can not consume (utilize) such a large volume of new knowledge and techniques! And what they wrote?! Students have to "utilize" that. This is only a part of the miserable story side.
Another side of the depiction is connected to traditional education which in physics, chemistry and following them biology is based on perception of the one scale-one separate description for everything. Then for another phenomenon the separate one scale-one disconnected description for that, etc. This way of doing science and engineering brought
much of harm not only to physics, chemistry, other sciences, but mainly to biology fields.
Because Life is not functioning at one scale per case. The many methods used in biological sciences
are actually physical and chemical methods and approaches.
Biology disciplines so far are not functioning as like based on
Quantitative Theories by themselves.
In biology used the material of one scale homogeneous theories that physics and chemistry suggest
to its professionals, or they are familiar with from their graduate student years.
That is one more obstacle in progress that most of funding agencies have no desire to consider.
Bejan, A., Morega, A., West, G.B., and Brown, J.H., "Constructing a Theory for Scaling and More," Physics Today, Vol. 58, Iss. 7, pp. 20-21, (2005).
Bejan, A., Shape and Structure: From Engineering to Nature, Cambridge U. Press, Cambridge, (2000), and references therein.
Bejan, A., "Theory of Heat Transfer From a Surface Covered with Hair," Trans. ASME, J. Heat Transfer, Vol. 112, pp. 662- 667, (1990).
Bejan, A., "How Nature Takes Shape," Mech. Enginering, No. 10, pp. 90-92, (1997).
Ellis, George F. R., "Physics and the Real World," Physics Today, Vol. 58, Iss. 7, pp. 49-54, (2005).
Nakorchevskiy, A.I. and Basok, B.I., Hydrodynamics and Heat- and Mass Transport in Heterogeneous Systems and Pulsating Flows, Kiev, Naukova Dumka, (2001), (in Russian).
Nigmatulin, R.I., Dynamics of Multiphase Media, Part I, Moscow, Nauka, (1987), 464 p., (in Russian).
Nigmatulin, R.I., Dynamics of Multiphase Media, Vol. I, Revised and Augmented Edition, English edit. J.C.Friedly, New York, Hemisphere Publish. Corp., (1991). 507 p.
West, Geoffrey B. and Brown, James H., "Life's Universal Scaling Laws," Physics Today, Vol. 57, Issue 9, pp. 36-42, (2004).
We will be adding to this subsection more our comments on publications and trends.