Seeking to address the widely spread among theorists (mostly mathematicians) the belief and the hope onto the Homogenization theory as the TOOL suited for scaled problems, we consider commenting here on the capability of this theory with regard to model and to simulate the heterogeneous scaled problems in physics.

Governing Equations and "Averaging" in Homogenization Theory

In Bakhvalov and Panasenko (1984,1989) book as the characteristic problem for Homogenization Theory covered in their book let's select the problem on page 56 (Russian edition) or page 34 in English edition. The equation of 1D vibrations in the layered structure MATH

within the 2N layers with periods j=1,2....,N. For these equations the problem formulates with transformed coefficients as MATH which solution is sought via the expansion

MATH

The solution is sought using the equation called "averaged" in homogenization theory

MATH

where averaged over the entire unit cell coefficients are

MATH

MATH

Averaged equation for heat conductivity has the same view (p. 64)

MATH

averaged equation for electric field $E^{j}$ in layered media is (p.66)

MATH

where $\varepsilon _{d}$ here is the dielectric permittivity, MATH is the magnetic permeability.

Averaged equation for the layered heat conductivity two-phase problem is depicted as

MATH

But these equations are not the averaged equations - for derivation used the Homogeneous GO theorem, in spite that the averaging domain is the heterogeneous unit cell. We have no further comparisons need to be used.

In the paper by Auriault and Boutin (1994) authors derived the double porosity acoustics equations based on their "scaling" homogenization considerations.

We are interested here mainly in illustration of their results and further comparison with the VAT governing equations for porous medium.

Referring at the beginning to the

Quasi-static development by Biot (1941-62),

where single porosity governing equations were obtained in the form

MATH

MATH

where $\QTR{bf}{c,}$ $\alpha ,$ $\beta $ are elastic tensors; $\QTR{bf}{U}_{s}$ is the displacement in solid phase; $\QTR{bf}{K}_{f}$ is the permeability; and MATH is the deformation tensor, authors (Auriault and Boutin, 1994) got almost the same macroscopic description using their scaling normalization technique.

Using the full homogeneous Navier-Stokes and Navier equations for movements in porous media.

When authors used the full statement homogenous governing equations - in fluid phase

MATH MATH MATH

and in solid phase the governing equations

MATH MATH MATH MATH where $\QTR{bf}{c}^{E}$ is the elastic stiffness at constant electric field; $\QTR{bf}{\ \tau }$ is the elastic stress tensor; MATH is the elastic strain tensor; $c_{f}$ is the sound speed in fluid; $\QTR{bf}{U}_{s}$ is the displacement vector in solid; $\QTR{bf}{V}$ is the velocity in fluid.

They did not use the second equation used in linear acoustics as Burridge and Keller (1981) did, for example,

MATH

After scaling they got the macroscopic description of linear acoustics momentum in porous medium

MATH

and

MATH

which can be combined to form the equation

MATH

and the second governing equation presented as the momentum balance for the bulk medium

MATH

which are close in form (equivalent as authors stated) to the governing equations by Biot (1956)

MATH

MATH

Here we see the strange mix of volume averaged and homogeneous (one scale) variables, for example, as - MATH and $\QTR{bf}{U}_{s}$. With many features unknown to be included into these governing equations.

The volume averaged velocity determined by authors as

MATH

The paper by Auriault (1991) is the paper to be used as an example of the misunderstanding - or misrepresenting of the efforts to obtain the scaled description of the process in heterogeneous medium. The method author used is the Homogenization Method.

Still, in the general pot of this kind of methods, theories the author thrown also the VAT which is presented by one of the Whitaker's papers. To do this means do not understand or misrepresent the basics.

On page 785 one can read: ''An old, classic idea is to replace the heterogeneous medium by a continuous, equivalent one which gives the average behavior of the medium submitted to the excitation, at the macroscopic scale, i.e. at the scale of the volume containing a large number of heterogeneities. As a matter of fact, the existence of such equivalent media is the basic assumption when investigating them directly at the macroscopic scale, using experimental or phenomenological approaches. Another route to obtain macroscopic descriptions is to pass from the description at the heterogeneity scale (we will subsequently call it the microscopic scale or the local scale) to the macroscopic one. These processes are named homogenization processes.''

Not all of them!

''They are extremely numerous and fruitful, depending on the different mathematical techniques. Let us note among others the homogenization for fine periodic structures [1,2], the statistical modeling [3], the self-consistent method (for example [4]), and generally speaking all methods using the average theorem [5-8].'' Completely wrong statement - the [7] is the paper by Howes and Whitaker !!

To illustrate the homogenization method used in the paper, we take some preliminary definitions available, as they given in the paper. We use few sentences.

On page 787 and 788 few notations describe the scale coordinates - fast $\QTR{bf}{y}$ and slow (macroscopic) $\QTR{bf}{x}$ one

MATHand the main supposition to have the asymptotic expansion for the sought function

MATH

''Equations to be solved are of the form

MATH

''This is a (local) balance for MATH where divMATH appears as a source term. Consequently, since the MATH are locally periodic or stationary, the source must be of zero average:

MATH

This means the author doesn't know the difference between the GO theorem and the WSAM theorem.

The Darcy formulation treatment

''It has been established experimentally that the macroscopic equivalent description is the Darcy law: MATH while the mass conservation equation on the macroscopic scale shown as MATH

Meanwhile the both equations at the upper scale should be for impermeable interface written as MATH

MATH

MATH

As we can see there are no scaling effects in the macroscopic equations presented by Auriault (1991) as in the proper VAT upper scale formulation.

The homogenization procedures performed for the Darcy formulation of the creeping flow in porous medium above got the following equations set to solve MATH

MATHwith the set of equations

MATH

MATH

MATH

MATH

MATH

MATH

As we can see these procedures have nothing common with the VAT upper scale equations, methods of representation and semantics. The homogenization method as I pointed out in the ''Acoustics'' section of this website in -

  • - "Linear Acousto-Elasticity in Porous Medium"

    has nothing to describe the upper scale physical features while keeping the lower scale in communication and interdependency of physical properties.

    In the book edited by Hornung (1997) collected the studies on homogenization theory used for solutions of periodic mathematical physics problems. There are 10 Chapters and no simulation results, but in one chapter #10, where the Darcy problem simulated for a periodic medium.

    It is symptomatic for the book and we again bring here the kind of essential for the theory equation as it is given on the page 6 for the layered media

    MATH

    which solution is being sought as via the asymptotic expansion

    MATH

    Obviously it is the one scale method and equations.

    In Chapter 5 (by Alain Bourgeat) there are the amazing sentences on page 95 - "For single-phase flow through a porous medium, by using homogenization theory, we are now able to rigorously derive filtration laws according to the porosity range (see, for instance, Chapter 3) or according to the Reynolds number as for instance in [BMP95] or in [Mik95]."

  • WHAT ABOUT THE SOLUTIONS OF CLASSICAL PROBLEMS ON THE UPPER SCALE - for capillary, or for globular, or for layered media? Not heard about this kind.

  • BUT THE VAT has provided these solutions and even in exact mode.

  • CAN IT BE DEMONSTRATED THE KIND OF EXACT EXPRESSIONS FOR THE DRAG RESISTANCE OR PERMEABILITY COEFFICIENTS AS IT IS DONE WITH THE VAT ? I doubt this, it was not done through the 60-something years the homogenization theory exists.

  • WHAT ABOUT THE AVERAGED CHARACTERISTICS OF FIELDS ON THE "UPPER" AVERAGED SCALE? No way, only intrinsic static each phase related volumetric characteristics.

  • Well, in the VAT they are easily formulating any kind of Upper scale field's characteristics and compute them.

    Page 95 - "Unlike one-phase flow, deriving the filtration law from the behavior of the two-phase mixture at the pore level is still not clearly and totally rigorously understood, although there have been several attempts (see, for instance, [ASP86], [EP87], [Mar82], [Whi86]) leading to partial results."

    NO COMMENTS, BUT only to say - that only Whitaker and Gray with their co-authors published the most profound and correct models for two-phase creeping and laminar regime transport in porous media.

    And on page 96 - "But due to the complexity of the phenomena involved in describing a multiphase multicomponent flow at the pore level, until now there has been no mathematical proof of the validity for volume averaging, and some heuristics have to be used to obtain a macroscopic model from microscopic behavior."

    This author definitely lives in an ivory tower, he doesn't need to read other then his own studies.

    On page 98 this author writes "the macroscopic mass balance equation obtained" in this "way" - by "spatial" averaging !!


    MATH

    this equation is the parody on the spatial averaging obtained equations. Besides, in this and in any found in the like equations the non-linear terms (as MATH and/or MATH) are just proclaimed as "averaged". In reality they are not averaged.

    We can make to repeating here the summary written in the subsection -

  • - "Linear Acousto-Elasticity in Porous Medium"

  • 1) The homogenization approach (Homogenization Theory -HT) is the useful theory for those situations for which it was invented - as for the two scale inhomogeneous media with fluctuations. Nevertheless, the method and like "averaged" equations do not display the "Upper" scale correct equations, in the method used the second scale some properties to solve the lower scale problem.

    Meanwhile, most of Heterogeneous media are more complicated and having the phenomena within the interface surfaces, and through the interface, though when phases have the discontinuities or sharp transitions at the , then the HT is not enough. It can not treat the sharp phase change, interface discontinuities, etc. for the Upper scale.


  • 2) Because of those facts given above, to the important next question should be given more thorough investigation for problems in acoustics and elasticity theory.

    What is the significance and value of difference in the problems with openly admitted interface boundaries movements and those with application of time harmonic presentation ?

    As we have already studied, the time harmonic presentation can dramatically change the additional terms in VAT upper scale governing equations - those terms which control and determine the movements of interface during the wave propagation process.


  • 3) One of the silent deficiencies of the HT (that's my guess, anyway so far) is that the Upper scale "averaged" HT equations would not sustain the balances of the fields characteristics - as the energy balance, momentum balance, etc.

    The reason I am bold to talk about that - is because we, using the VAT, were able to calculate the Upper scale characteristics and properties. And those calculations were demanding the terms as they exist in the VAT upper scale governing equations!

    References

    Auriault, J.L. and Boutin, C., (1994), "Deformable Porous Media with Double Porosity III: Acoustics", Transport in Porous Media, Vol. 14, pp. 143-162.

    Auriault, J.L., (1991), "Hetrogeneous Medium. Is an Equivalent Macroscopic Description Possible?" Int. J. Engng. Sci., Vol. 29, No. 7, pp. 785-795.

    Bakhvalov, N.S. and Panasenko, G.P., Averaging Processes in Periodoc Media. Mathematical Problems of Mechanics of Composite Materials. Hauka, Moscow, 1984. 352 p. (in Russian).

    Bakhvalov, N.S. and Panasenko, G.P. (1989), Homogenization: Averaging Processes in Periodoc Media. Mathematical Problems in the Mechanics of Composite Materials, Kluwer Acad. Publishers, Dordrecht, 1989. 366 p.

    Bensoussan, A., Lions, J.-L., and Papanicolaou, G.C., Asymptotic Analysis for Periodic Structures, in Studies in Mathematics and its Applications, Vol. 5, North-Holland, Amsterdam, 1978.

    Hornung, U. ed., Homogenization and Porous Media, Vol. 6 in "Interdisciplinary Applied Mathematics" ser., Springer-Verlag, New York, 1997. 279 p.

    Jikov, V.V., Kozlov, S.M., and Oleinik, O.A., Homogenization of Differential Operators and Integral Functions, Springer-Verlag, Berlin, 1994.

    Kröner, E., in Modeling Small Deformations of Polycrystals, Chap. 8, edited by J. Gittus and J. Zarka, Elsevier, London, 1986.

    Oleinik, O.A., Shamaev, A.S., and Yosifian, G.A., Mathematical Problems in Elasticity and Homogenization, Elsevier, Amsterdam, 1992.

    Sanchez-Palencia, E., Non-Homogeneous Media and Vibration Theory, Lecture Notes in Physics, Vol. 127, Springer, Berlin, 1980.

    Zaoui, A., in Homogenization Techniques for Composite Media, Lecture Notes in Physics, Vol. 272, Springer, Berlin, 1987.


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