THE NON-LOCAL FORMULATION OF ELECTROSTATIC PROBLEMS FOR SENSORS HETEROGENEOUS TWO- OR THREE PHASE MEDIA, THE TWO-SCALE SOLUTIONS AND MEASUREMENT APPLICATIONS Текст научной статьи по специальности «Физика»

ВОДОРОДНАЯ ЭНЕРГЕТИКА И ТРАНСПОРТ

Газоаналитические системы и сенсоры водорода

HYDROGEN ENERGY AND TRANSPORT

Gas analytical systems and hydrogen sensors

THE NON-LOCAL FORMULATION OF ELECTROSTATIC PROBLEMS FOR SENSORS HETEROGENEOUS TWO- OR THREE PHASE MEDIA, THE TWO-SCALE SOLUTIONS AND MEASUREMENT APPLICATIONS

V. S. Travkin , A. T. Ponomarenko* It11

1 Member of International Editorial Board -11 Member of International Editorial Advisory Board

Hierarchical Scaled Physics and Technologies (HSPT) 10431 Larwin Ave., Chatsworth, CA 91311, USA E-mail: travkin@iname.com

Enikolopov Institute of Synthetic Polymeric Materials, Russian Academy of Sciences Profsoyuznaya, 70, Moscow, 117393, Russia

The main idea of performing transport and field description of electrostatic problems in heterogeneous media of sensors on the two scales through the spatial non-local theorems of the volume averaging theory (VAT) is to provide the means to account for multiple description scale characteristics and requirements. At present time it is the only consistent and reliable theory available for multiscale either 1D or 2D, 3D, linear or nonlinear statement tasks. The VAT approach has been successfully applied in the last two decades to a number of difficult problems in fluid mechanics, thermal physics, environmental science in heterogeneous media and in porous media. The present work suggests the mathematical formulation and few solutions for heterogeneous media electrostatic problems based on the VAT averaged homogeneous and inhomogeneous Maxwell's equations along with the potential field equations.

Three types of heterogeneous media are addressed for solution: 1D layered, capillary media and globular media. The results for capillary type of media were for media made up of random size capillaries in both dilute approximation and with conjugate interacting particulate fields. New types of mathematical models and governing equations for conductive dielectric composites are suggested. Various simplifications have been analyzed and some applications and comparisons with available theoretical approaches are made.

Nomenclature

» C — mass fraction concentration;

<

<c B — magnetic flux density [Wb/m2]

^ cd — mean drag resistance coefficient in the REV;

| cd — mean skin friction coefficient over the turbu-

| lent area of dSw;

| cp — specific heat, J/(kgK);

* ds — interface differential area in porous medium,

° m2;

i dS12 — internal surface in the REV, m2;

S D — molecular diffusion coefficient, m2/s;

o 2

q D — electric flux density, C/m2; E — electric field, V/m;

f ={fi} — VAT intrinsic phase averaged over AQ; value f ;

(f)f — VAT phase averaged value f, averaged over AQ; in a REV;

f — VAT morpho-fluctuation value of f in a

— time averaged value f ; j — current density, A/m2;

k1 = kf — fluid phase thermal conductivity, W/(mK);

k2 = ks — homogeneous thermal conductivity of solid

phase, W/(mK);

H — magnetic field, A/m;

m — porosity;

(m) — averaged porosity;

n — refraction index;

p — pressure (Pa) and phase function;

Q — electrical power from heater dissipated through

the specimen;

Статья поступила в редакцию 04.01.2005 г. The artisle has entered in publishing office 04.01.2005

— solid phase fraction;

12 — specific surface of a porous medium dS12/ÀO,

1/m;

T — temperature, K;

T2 = Ts — solid phase temperature, K;

T — interface surface temperature when i is in

upward direction, K; U — vector component in x-direction; V — vector component in y-direction; W — vector component in z-direction;

Subscripts

c — charge; e — effective; eff — effective; ex — experimental; f = 1 — fluid phase; 1 — first phase;

i — component of vector variable; L — laminar; r — roughness; s = 2 — solid phase; T — turbulent; w — wall.

Superscripts

--value in fluid phase averaged over the

phase REV AQ„;

* — equilibrium values at the assigned surface and complex conjugate variable; — fluctuation value in a phase.

Greek letters

12

12

a 21 — averaged heat transfer coefficient over dS W/(m2K);

a C — averaged mass transfer coefficient over dS m/s;

ed, em — dielectric permittivity, Fr/m; (xm — magnetic permeability, H/m; v — frequency (Hz) and kinematic viscosity, m2/s; pc — electric charge density, C/m3; p — density, kg/m3;

o — medium specific electric conductivity, A/V/m; O — electric scalar potential, V; ra — angular frequency, rad/s; AQ — representative elementary volume (REV), m3; AQ1 = AQf — pore or phase 1 volume in a REV, m3; AQ2 = AQQ — second or phase 2 volume in a REV, m3; Tw — wall shear stress, N/m2.

Introduction

There are quite a few methods used to describe the field modeling in electrostatic and general electrodynamics problems. In the literature starting from fifties with significant rise in research activity in seventies — eightieths actually given a number of analysis and approaches as by, for example, [1—3] etc. to name just a few. The

initial theoretical efforts were suffering from simplifications of physical as well as of mathematical nature and mostly focused on two or three techniques: 1) power series, perturbation expansions; 2) effective medium theories; and 3) geometrical consideration to separate and describe transport phenomena. Sufficient number of studies review the subject as, for example [1, 4—8]. Dul'nev and Zarichnyak [9-10] reviewed and analyzed different approaches to calculate the generalized (meaning thermal and electrical conductivities, electrical and magnetic inductions, dielectric permittivity, viscosity and diffusion) effective conductivity coefficient in heterogeneous systems making great use of geometrical considerations.

Because of the greatly increased computing power in the last two decades the interest of researches turned to different techniques in heterogeneous sciences that use this power.

The most common way became to treat such problems has been to seek a solution by doing numerical experiments over more or less the exact morphology of interest — what can be called the Detailed Micro-Modeling (DMM) which is often conducted via Direct Numerical Modeling (DNM).

This leads to heavy use of large computers to solve large algebraic statements. The treatment and analysis of the results of such a conventional Direct Numerical Modeling (DNM) is both difficult and limited in analysis. Performing DNM without proper theory is like performing experiments, often very challenging and expensive without data analysis and modeling tools — gives the data, but not the all needed results.

A good example of DMM-DNM for the linear problem of electrical field distribution in two-phase dielectric composites was demonstrated by Cheng and Torquato [11]. Their detailed analysis of numerical simulation results — "show that in general the probability density function for disks and squares exhibits a double-peak character, ... Not surprisingly, therefore, the variance or second moment of the field is generally inadequate in characterizing the field fluctuations in the composite". So, the pure statistical analysis of the DMM-DNM data revealed a clear demand for non-statistical tools for description of phenomena.

It is obvious that DMM-DNM can not meet the entire needs of Heterogeneous Media description and Modeling (HMM). What is the difference between DMM-DNM and Heterogeneous Media Modeling, why can't DMM be self-sufficient in heterogeneous media transport phenomena description? The answers arise via analysis of the issues:

1) There is a basic principal mismatching at the boundaries — boundary conditions problems. Meaning that for the DMM and for the bulk (averaged characteristics) materials' fields the boundary conditions are principally different.

2) The spatial scaling of heterogeneous problems with the chosen Representative Elementary

5

2

Volumes (REV) (for DMM) arises if one needs to address large or small deviation in the elements considered, with different underlying physics for some of them. Which denotes, that when the spatial heterogeneities of characteristics or morpholo-» gy are evolving along the coordinates then there is <c no chance to use DMM for that problem. I 3) Random morphologies treatment — as long 1 as numerical experiments provided with DMM-DNM -g need to be translated to the form which implies

I that the overall spatially bulk characteristics moda;

y eled. Which is not so simple, because of the ques-| tion — what kind of equations are considered as ^ governing equations? And what are the variables | being compared? As in the case of the local poros-

0 ity theory [12, 13], for example, when the results of real porous medium digitized images morphological analysis are used for calculation of effective dielectric constant, assuming applicability of homogeneous medium governing equations.

4) Discrete — continuum gap closure or matching, as long as DMM-DNM actually unrealistically claims on the description and simulation of continuum phenomena. This is the most fundamental drawback when DMM is being used to apply as the most exact reflection of the real physical phenomena. That means, that as soon as the developed solution of the DMM problem exists it needs to be matched to the correspondent HMM. Otherwise, DMM is only valid for the scales in which the problem was stated and in which the experimental confirmation only to be sought. So, no conclusion or generalization could be developed for the next or higher levels of hierarchy of the matter description. Meaning, there is no co-junction in the description of the different physical scales.

5) Interpretation of the results is always a problem. If results are presented for a heterogeneous continuum, then see the above point. If the results are being used as a solution for some discrete problem — the question is how to relate that one to the continuum problem of interest or even to a slightly different problem. Usually, in both ways, at some

^ point there appears a continuum model in the form

1 of some differential equation or equations. And g- those models, equations in all, but VAT models are S for the one scale homogeneous media. If the results i obtained then fit into a statistical model — most

u

I often assumed to be the statistical averaging, which is is the scheme to fit and explain the place of achieved ! characteristics — that means the phenomena are £ being squeezed into the usual statistical averaging § procedure which is in most cases is only correct for © the circumstances when the group of facts are taken separately as independent events.

6) The most sought after characteristics in heterogeneous media transport which are the effective transport coefficients can be correctly determined using the conventional definition as for the effective (steady-state) conductivity, for example

-( j = aV<0) = a2V(0) + (a1 -a2 j VOrf®

AHj

but in only the fraction of problems, while employing the DMM-DNM exact solution (this issue will be discussed in more detail later in the text). The issue is that in majority of problems, as for inhomogeneous, nonlinear coefficients, for example, and in many transient problems having the two-field DMM-DNM exact solution is not enough to find effective coefficients.

7) Multiscale phenomena description problems. Homogenization and fractal methods are inapplicable in most of the situations. Fractal approach is not relevant to most of the morphologies, and the fractal phenomena description is generally too morphological, lacking many physical features presented — as, for example, descriptions in both phases, or description of the phase interchange, etc.

Let's further consider the simplest case of superlattice or multilayer medium, for example:

1) Let's take a situation, when the boundaries are not evenly located. Boundaries routinely unevenly cross the regular boundary cells of the medium. Then for each uneven situation one needs to solve the problem again and again. For example, for angles 5, 10, 20 degrees or when mathematical surface of a boundary crosses the boundary cells straight but at any variable point — endless series of situations.

2) Then one can consider another situation — coefficients are not so nicely constant? They are space-dependent, because of layer's or grain boundaries. The grain boundaries are not perfect and are not just mathematical surfaces without thickness or physical properties. So, the grain boundaries scatter themselves or at least, should be considered as influencing a scattering — but not as mathematical surfaces without any (physical) properties.

3) Then, of course, imperfectness of internal spatial structures — the domain morphologies are not perfect, on any spatial level of consideration.

4) Those are great difficulties for contemporary methods treating heterogeneous medium problems. They've been around for many decades and still are — the series expansion, perturbation methods, Green's function techniques.

Gradually, the formulation of heterogeneous medium transport equations has evolved a great deal since the fifties. Even so, the proper form of the governing equations for a heterogeneous media is still a source of discussion.

Determination of the flow-variables and the magnitude of the scalar transport for problems

involving heterogeneous media is difficult, even when medium is periodical or regular.

Linear or linearized models fail to intrinsically account for many transport phenomena. Allowing inhomogeneities to be of random or stochastic character further compounds the task of properly identifying transport models.

Mathematical simulation of physical processes in heterogeneous media, in general, calls for obtaining averaged characteristics of the medium and, consequently, the averaged equations. The averaging of processes in a regular or randomly organized heterogeneous media can be performed in few different ways. 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 [14]). The hierarchical principle of simulation consists of successively studying the processes at a number of structural levels.

At the beginning one first needs to deal with the smallest scale element, for example a small smooth capillary in a capillary medium or particle in a globular medium. 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 underway 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. The resulting usable modeling 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 physical field of interest as electrical potential, energy, mass or momentum transfer between the phases, in part because of their different boundary conditions.

The first theoretical efforts in this area were published in sixties almost simultaneously by Whi-taker [15], Slattery [16], Anderson and Jackson [17], Marle [18], Zolotarev and Radushkevich [19]. All those works concerned the transport of heat, mass and momentum in porous media. Since then the number of studies were done in those areas.

Up to now the most applications of this theory found and historically has being developed in fields of thermal physics, chemical engineering and fluid mechanics, particularly in porous media transport. The books existing on the subject with problem treatment in these fields [20-23] address primarily the linear heat and momentum transport in porous media modeling theory.

Meanwhile, this kind of treatment for heterogeneous media until recent did not exist in electrodynamics. Numerous works approaching this area use various techniques based on the classical homogeneous medium problem methods, as power series presentation [2], Green's function [24], effective media [1] aiming to primarily to evaluate effective medium coefficients.

At the same time, there is the direction in descriptive fundamental electrodynamics in which the heterogeneous with spacial dispersion media are taken onto the more strict level od consideration see, for

example, [25, 26] and referred therein works. In all the discussions provided around these theories, one of the critical points is that at least for the two theories suggested for heterogeneous media by Landau and Lifshits [27] and with so-called Casimir governing equation forms the modified Maxwell's s equations suggested. In these and others like de- ^ scriptive theories the physical fields are determined * in such a way that they are actually should be ac- f cepted and justified as like averaged somehow fields. ^

The fundamental problem with these theories I is that the averaging procedure there considered ^ as the one by using the known Ostrogradsky-Gauss f theorem developed and justified for homogeneous S medium only. It is also assumed that one can use g the known homogenization method [26]. Mean- 0 while, the "averaging" method of homogenization as shown in [28] is actually the one scale mathematical procedure. The method tied directly to the morphology of the medium, and this is the beneficial factor, still the problem is being solved and analyzed as the one scale homogeneous problem with the sharp change of properties.

That all mean, there is no real averaging provided in the mentioned above theories, as we will show below. The more strict analysis of these and other theories used for descriptive mathematical explanation of heterogeneous media processes and fields is the subject of a separate study some features of that one are given in [29, 30].

Among lower one scale solutions for heterogeneous media obtained on the strict basis, the study by Avellaneda and Torquato [31] shown the peculiarities of the mathematical clarification for the transient Stokes equations while connecting the fluid permeability kdc and formation factor F results using the employment of solution decomposition for eigenfunctions ¥ and using decomposition of the constant unit vector e to seek the electrical field solution E. By presenting the fluid permeability kdc through the series distributed solution in the Of the formula for permeability in porous media was established. As it is obvious from the algorithm and description of the found dependency for permeability kdc and two suggest- ¡1 ed parameters L (or A) and F that the porous me- t dium flow problem was brought down with the £

a;

help of the two following techniques: ^

1) eigenfunction solutions which employ the | Poisson type equation infinite series solution; ¡

2) full scale Laplace equation solution for elec- j? tric field E in the fluid phase Of |

It is worth to note here that the full problem * solution of either one of these tasks only once § does provide an actual complete solution for the 0 entire closure problem together with solution for the VAT based porous medium fluid and electrostatic transport problem (see, for example, [32, 33] ) and gives much more information in terms of the overall behavior of a problem than the DMM-DNM solution.

Another reality, as was shown by Achdou and Avellaneda [34] is that if in the medium which posses the real morphological features — as angles and steep curves of an interface surface — the problem of correspondence of kdc to L (or A) and F falls apart, and primarily due to physics of phenomena. They studied the influence of pore roughness on the dynamic and static medium permeability of single pore and found out, that "even if, in reality, wedgelike singularities are rounded off microscopically, large curvatures on the pore surface will produce strong surface concentration of the electric field". As we will see below exactly these features are fully incorporated into the scaled VAT modeling and simulation equations.

Also, they concluded that "with regard to pore-size dispersion, it is clear that a theory based on a single-tube response function (as for the dynamic permeability kac see [35, 36]) will be inadequate if the dispersion is wide enough."

Nevertheless, the exact flow resistance results have been obtained on the basis of VAT based governing equations in [37] for the random pore diameter distribution for almost the same morphology as was used by Achdou and Avellaneda [34]. The distribution of flow resistance exposed the wide departure from the Darcy law based treatments with constant coefficients. That was shown even for the morphology where a single pore exists with diameter different from the all others. Meanwhile, using the VAT based procedures [33, 37] one can develop the needed variable, nonlinear coefficient of permeability for a Darcy dependency when fluid flow is Stocksian

kdc =

SL

Л

-1

(m2 )2

U 2v

(1)

ing 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.

At present time only the VAT has developed techniques to deal with all those complicated phenomena [33, 37]. The short description and governing equations of heterogeneous media electrodynamics are given in [30].

Meanwhile, prevailing number of published studies that model and simulate the heterogeneous media (including sensor's materials) are concerned with the simple in-series composite formulations for bulk heterogeneous coefficients (properties) see, for example, [38, 39], thus in [39] used the dc in-series model for the bulk conductivity for dc applied field in polymer composites

Oe =

i

i=1

m

(2)

while the ac bulk heterogeneous conductivity was taken in the form — complex frequency dependent

1

m

1

? (ю) Oi (ю)

m

Ы

K en

where K is the dielectric constant of "capacitive barrier", e0 is the permittivity of free space, and the Drude complex conductivity OJ (œ) of the highly conducting material

Oi (ю) =

On

where the coefficient of overall resistanse to momentum transport cd derived for this particular morphology on the basis of exact analytical (in laminar regime) or well established correlations for Fanning friction factor in pore for other flow regimes. In this exact formula Sw is the specific surface of the porous medium and U is the intraphase averaged velocity of the fluid, which can be of any regime — laminar, turbulent or intermediate.

At this time, 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 additional terms that are not usually seen in equations resulting from other (the one scale) methods. In fact, they are of the same order of magnitude as the terms that are normally kept [32, 33].

There are many disagreements about the applicability of models based on conventional diffusivity type models of transport phenomena in porous or any heterogeneous media to media with the follow-

1 - ion,

where o0 is the dc conductivity and Tj is the scattering time for highly conducting region. The more realistic and complicated models like in [40] using the few scale modeling and simulation computation procedures with the scale models those communicated one scale to another scale via heuristic techniques.

We produce the scaled VAT fundamentals for modeling and describing the namo-micromeso-scale phenomena in heterogeneous (and porous) media used in sensors to demonstrate advantageous of strict application of the heterogeneous WSAM theorem, which is the heterogeneous analog of the homogeneous Ostrogradsky-Gauss theorem [33, 37].

1. Electrostatic inhomogeneous single phase medium Maxwell's equations

The set of Maxwell's equations for homogeneous, inhomogeneous (space dependent with relatively smooth characteristics) and nonlinear electrostatic problems

V(eE) = p, VxE = 0, V(MH ) = 0,

(3)

(4)

(5)

VxH = oE + j(e), (6)

with the presence of external current density j(e) will be addressed further for application to heterogeneous media with grain boundaries and abrupt interface phase changes. Also the current conservation equation is the part of description

Vj = 0, j = oE. (7)

The wave formulation of the magnetic field equation with constant phase coefficient also can be covered through the current development

V2H = 0, (8)

and time-invariant potential equations when potentials ®(r) and A determined through

E(r) = -VO(r), VxA = B = |jH, (9)

are

V2 A = -w = -|ioE,

V 2Ф(г ) =

P

(10)

(11)

More complex are potential equations with in-homogeneous or nonlinear coefficients as

V(o(r)VO(r)) = 0. (12)

The second order partial differential equations are describing different fields, nevertheless their VAT versions will be analogous and can be derived in the same way.

If current conductivity o = 0 in a substance, then the Maxwell's equations split in the two subset electrostatic equations

V • (eE) = p, (13)

VxE = 0, (14)

and magnetostatic equations

V-(nH) = 0,

(15)

(16)

VxH = j(e).

2. Volume averaging theory derivations and theorems

The theoretical approach which enables one of further treatments of the electrodynamic equations in multiphase media needs to be introduced in grater detail. The subvolume AQ inside of the domain where the problem is stated and over which functions are averaged called representative elementary volume (REV), see for more detail [22, 23, 33, 41, 42].

Let's assume that the heterogeneous medium consists of two phase with spacial morphology known or at least some features of that morphology already known and the generally random volume fraction of the phase 1 function into additive

m

i (x ) = (mi (x )) + mi (x Mmi) =

AQ

AQ

J f (t, x)dю=(m) f,

1 АП,

f 2 = m

i

AQ,

2 AQ2

while intraphase averages are

m,=f =aq j f (t, x

AQ,

Ш2 = f2 =aq" J f (t, x )d«*

2 AQ2

= fg,

Fig. 1. Representative elementary avaeraging volumes (REV) with the fixed points of representations.

components: the average value of (m1 (X) in the REV and its fluctuations in various directions

AQj

Five types of averaging over the REV function f are defined by the following averaging operators [30, 26]

f = ( f \ + <f) 2 = (m) f +(1 - (m) )/2 phase averages with respect to the

f: = (m)—

(17)

(18)

j f (t, x)dю = (m2)f,, (19)

(20)

(21)

where f is an average over the space of the phase one AQ1 in the REV, f2 is an average over the second phase volume AQ2 = AQ - AQ1, and (f) is an average over the whole REV. When the interface is fixed in space averaged functions for the first and second phase (as liquid and solid) within the REV and over the entire REV fulfill the following conditions, namely

{f + g>! >! +{g}1, H = (22)

for the conditions of steady state phases (with no interface movements)

f =d{ f }l dt1 dt

e

where a — is the constant value, except for the differentiation condition,

{vf1 = vf + at j

AO1

(24)

dSw

f = f - f, /VAQ f,

where dSw is the inner surface in the REV, ds is the solid-phase, inward-directed differential area

in the REV (ds = ndS). The fourth condition implies an unchanging porous medium morphology.

There is an important difference in the definitions of averaged and fluctuation values in regards of their meaning and values in the REV comparing to definitions supported by Whitaker and Quintard see, for example, [43, 44]. The treatment and interpretation of the averaged values inside of the REV are supported in the classical interpretation when a value considered as an averaged inside of the volume is still the constant value in the initial ground scale description space. The more detail on that problem are given in [30, 33].

The three types of averaging fulfill all four of the above conditions as well as the following four consequences

(Д = f. f ={f - Д = 0.

= M,

= fg = 0,

(25)

(26)

At the same time, (f) j and \ f) 2 do not fulfill neither the third of the conditions,

а 1 * а

W1 = \mn

(vf) 1 = v(f) 1 +att j a-

(28)

The above expressions fail to describe some important features of the averaging of differential and commutational operators that are frequently found in the transfer equations. It is possible, however, to improve the equation set by developing further averaging rules. A second averaging of the last expression in (24) yields

{{Vf 11 =jV/+ AO I *

-vfj fdr1-

Aß1 3S12

Utilizing (25) yields

{{Vf I1I1 ={Vf I1

(30)

{f = Vf îa^J d^

[AO1 3S12 J1 AO1 3S12

By using (24) for averaging, one finds what might be called the first return rule, as soon as

1

(31)

(27)

nor all the consequences of the other averaging conditions.

Also the differential condition in accordance with one of the major averaging theorems — theorem of averaging V operator [20, 45] for the heterogeneous medium, the WSAM theorem (after Whitaker-Slattery-Anderson-Marle) the averaged operator V becomes

{Vf} =V{/} J fa =vf

3S12

it follows that

{VfVg\ = {{Vf }1 {Vg}} = VfVg. (32)

Similarly, return rules for averaging operators need to be found for averaged quantities of the gradients,

V f U. W1.

For the first type of gradient, one begins with

<w ,>,=(V</>,*

and from this

V W = (m) (Vf) 1.

1

(33)

as long as

This theorem as long as the statistical characteristics of the REV's morphology, and the averaging conditions with their consequences, lead to the following special ergodic hypothesis: the spa-

cial averages, (f )1, f, and f, converge with increase in the averaging volume to the appropriate probability (statistical) average of the function f of a random value with probability density distribution p . This hypothesis is stated mathematically as follows

fb (x )= j f (x, a) pa {a, x}d a, lim f = fb. (29)

There is a constant discussion in the literature concerning this issue.

a 1 = mа

(34)

(35)

Using (33) and (35) leads to the second return rule

V f >1)1 = Ы (V< f >1 ),

a^ j fds1) = {m) a1t j fds1

dSw

AO

3S.2

(36)

j

Using a combination of the differential condition given by (28)

{vfu =(vf + j fa

AO1 av

with

1

and

{Vf }Д = (m) {Vf},

i

m.

1

AQ

j fdsi

\AQi i, f3si =' , ,

and the second return rule results in the third return rule

f = ({VfU = (rn) {Vf}1 = (rn) (Vf). (37)

Going in for the same kind of operator as rot that will result in the following averaging theorem

(Vxf>1 =VXf >1 +AQ j dsi xf,

AQ as12

also as its consequence the another theorem for intraphase average of Vxf

{Vxf} =Vx{f}1 +aQ j ds1 xf. (39)

Vf))i = V(E1 +AQ j • dsi =

3S12

[< m) {f }i + < m) f }

+ j fE• dsi. AQ d i

= V

(41)

= o.V Ф

+ Ф + AQ J(<& + <fe)

Using this presentation the two forms of potential diffusive flux in gradient form could be taken independently

(°1v°>1 =°1v(®>1 + aq j *ds1>

(43)

3Si-

or

(с^Ф), = o < m + —L J Фds1. (44) x 1 n 1 1 AQ- 1

■dSv

(38)

Averaging of Commutative Operators

Two terms or operators multiplication averaging will result in the following nonlinear formulation after using decomposition f = {f }1 + f, E =

= {} + E

fE = {m) {f\ {E} + {m) {M}. (40)

The averaged divergence operator term with operand as a product of two functions (in this case a scalar and a vector functions) usually becomes after phase averaging

Decomposition of the first term on the right hand side yields fluctuation types of terms that needs to be treated in some way [30].

VAT Averaging of Diffusive Fluxes and Diffusion Operators

The following transformations when applied to the flux gradient term with the phase constant dielectric or conductivity coefficients will better display the source of possible variations of the VAT transport equations. First one can make the transformations of the product of the constant coefficient function o1 times gradient term like

(C1V®\ =G1V(®\ + AQ j®dS =

dSw

(42)

^•(а^Ф) =OiV.

Wi AQ J i

+ AQ J VФ• dsi =oi [V^O VФ + ФV((m1})) +

AQ J (Ф + Ф)

AQ 3S12

= oi [V • ((m1) VФ) + V • (V ((mx))) -

+ _°L Г VФ• ds. =

AQ J

dsi2

f \ - J Ф dsy

AQ J

dS 12

-V-(V((m )))

+ ^ J V<b • ds1 =o1 V• ((m1 )VФ) + AQ as12

+V •

J Ф dsx

AQ J

dSs 12

at j dsi = -vk

aq 3S12

V^o^))^2 (m )Ф)+

+o 1 V^

—— J Фds1

AQ J

»S12

г VФ• ds1,

AQ J

»S12

or

(V • (o i VФ))1=Ol V • (mi )VФ) + +o1 V^

AQ J

or

V^o^)) = (m) о^2Ф-

Further it is turned out to be used for development of the following two forms for averaged diffusive terms in transport equations (with constant coefficient of conductivity or diffusion)

+ — Г o1V<b• ds. (45)

AQ J

ds12

This chain is sustainable because the equality

(46)

is justified in VAT (see, for example, [23]). One's come to the choice of the three forms available for the constant conductivity coefficient second derivative term

(47)

+ —^ r VФds1, (48) AQ J

12

+OjV-

АЦ **

+ _<L J V<p• dsv (49) AQ;J 1

V • (eE) = p, VxE = 0, V(|H ) = 0,

3Sn

or

= V-((e +êi)(Ei + E)) +aq j (eiEi)-ds = = v- [( m) êE 1 +( m) {ел }1 ]+ +aq j (eiEi)- dsi =(рд

V- [( m) êiE i ] + v- [( m) {a} ] +

+AQ j (eiEi )-ds1 = (p>l,

ЭЯ,

V-

((m) l hi i ) + v • [( m) {fliihi}i ]

+aq j №)-dsi =

AQ 3S12

and

(VxHJi = Vx((m,)IIi ) + aQ j dsi xHi =

AQ 3S12

= [( m) <1E 1 + Ы {<^iEEi}1 ].

Quite analogous are the averaged equations for the 2nd phase

3. Non-local two scale two-phase heterogeneous media electrostatic equations

Electrostatic inhomogeneous medium Maxwell's equations

s Assuming here that the coefficients are might

& be variable and even nonlinear functions in equa-

ha tions

(50)

(51)

(52)

VxH = cE, (53)

then averaging of the first, third (50), (52) and forth equations will be done with accounting for possible nonlinear operators. After averaging the equation becomes

(v' m1 »1 = v' (№ »1 +an j (£1E1 )• dS1 =

(54)

and the second electric field equation averaged as

(vxe^ =vx((m)) + an j ds1 xe1 = 0. (55)

Meanwhile, the equations of magnetic field are averaged as following

(56)

(57)

V- [( m2) e 2 E 2 ]+v- (e212)2 +

+ aq j (e2E2 )- ds2 = (p)2 ,

3S|2

Vx(m^)IE2 ) + aq j ds2 XE2 = 0,

(58)

(59)

V- [( m2) 12H 2 ] + V- (Î2 H

+ AQ j (l2H2)• ds2 = 0,

(60)

Vx[(m^ÎÎ2] + aq j ds2 xH

AQ 3S12 = [( m2) ô 2 E 2 +(<<2 E2)2 ].

2

(61)

VAT homogeneous phase electrostatic fields Maxwell equations

More straightforward and simpler are the VAT homogeneous electrostatic equations, like for the phase 1 when the medium (constitutive) coefficients — magnetic permeability dielectric permittivity e1 and its conductivity o1 are constant values, then

V-E =

VxE = 0,

V-H = 0,

VxH = cE, after averaging became (in the 1st phase)

v- (mi>ei ) + aq j ei- = ^,

3S|-

(62)

(63)

(64)

(65)

(66)

(67)

Vx((mj) Ej ) + —- j dsj xEj- = 0,

v- [(mj)hj] + —- j hj-dsj =0, (68)

V x ((m) H ) + j dsj xHj- = (mj) OjEj. (69)

The boundary conditions on the interface surface dS12 give an understanding of inseparability of influence in VAT governing equations of phenomena "in phase" and "on interface" between phases. As to the electrostatic equations derived above then the boundary conditions on interface dS12 between the two homogeneous media are for the electrical field

2

12

п -(е1Е1 -в^ )| =Ps,

(70)

where ps is the interface surface density of electrical charge. Equality of tangential components of electrical fields

E11 = E2 L , (71)

or

" x(E1 - E2 1^12 = 0,

where E1 \t means the tangential component of electrical field vector on the dS12. For the normal components of magnetic field

n .(1H1 -^H 2 = 0, (72)

while the tangential components are connected through

(73)

(Hi - H2 )) = j

S ldSi2

or

n x(Hi - H2)

dSv

= j

s \asv.

where vector js is the interface surface electric current density.

4. Non-local volume averaged equations via electrostatic potential

Homogeneous phase medium (e;- = const, i = 1, 2) non local electrostatic governing equation using electric potential function O can be expressed as

V2 (fa) Ф, ) + V-

— í фА

AO J ' '

1 г - (m /

+— Í Ф, - ds, = AO¿ г г е

P f

(74)

Meanwhile, the inhomogeneous or nonlinear conductivity electric potential equation can take one of the possible 3 forms as, for example in the first phase with no charge

V - ((V((»2i} Ф1))

+ V-

1— í фа

1 AO J 1 1

+v-(<<j1v<p1 >1 ) + aq j o1vo1 • ds, = 0. (75)

aq ds12

The same VAT non-local forms are ready available for the steady magnetic field equations.

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To be continued Продолжение в следующем номере