Jtam.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 4, 39, 2001 ADMIXTURE DIFFUSION IN A TWO-PHASE RANDOM NONHOMOGENEOUS STRATIFIED LAYER Yevgen Chaplia Department of Environmental Mechanics, Bydgoszcz Academy e-mail: czapla@wsp.bydgoszcz.pl Institute of Applied Problems of Mechanics and Mathematics, Ukrainian National Academy of Sciences, Lviv, Ukraine Olha Chernukha Institute of Applied Problems of Mechanics and Mathematics, Ukrainian National Academy of Sciences, Lviv, Ukraine e-mail: cher@cmm.lviv.ua Vertical admixture diffusion has been considered in a layerwith random nonhomogeneous two-phase stratified structure of the material. Diffe- rent phase diffusion coefficients andphase densities have been taken into account as well as jump discontinuities of the diffusion coefficient at interphase boundaries. Averaging the obtained expressions for the ad- mixture concentration has been done over the ensemble of sublayer con- figurations with equally probable distribution, and two particular cases of beta-distribution of phases in the body. Keywords:admixturediffusion, randomnonhomogeneousstratified layer 1. Introduction In practice, often the necessity occurs to describe the process of admi- xture mass transfer in nonhomogeneous stratified structures. Admixture and behaviour of its distribution in a body have an essential influence on its phy- sical and mechanical properties. The rigorous geometric composition of such structures is unknown, i.e. position and thickness of the sublayers in different materials are randommagnitudes. However, their corresponding densities and diffusion coefficients of admixture particles are determined accurately enough. 930 Admixture diffusion in a two-phase random... In certain cases the diffusion coefficients values can differ by some orders of magnitude in different sublayers. To evaluate the influence of such a structure with substantially different diffusive properties of sublayers on the mass transfer in a body, the methods of homogenisation (Lydzba, 1998; Matysiak and Mieszkowski, 1999) and in- troduction of effective diffusion coefficients (Kanovsky and Tkachenko, 1991; Lyubov, 1981; Shatinsky and Nesterenko, 1988) has been proposed. At the study of transfer processes in regular structures, the methods of solving the initial-boundary value problems developed in (Podstrigach et al., 1984) concerning the heat processes can be used. If the body structure is such that there aremacroscopic quantities of par- ticles of different kind sublayers and admixture within an arbitrarily chosen physically small body element, then the continuum-thermodynamical models for description of the diffusion processes (Burak and Chaplia, 1993; Burak et al., 1995) can be also used. However, the cases have been described in literature (Lyubov, 1981; Ka- novsky and Tkachenko, 1991) when introduction of an effective diffusion co- efficient and experimental data interpretation on this basis are not always physically justified. But we canmake certain reliable assumptions concerning the stochastic distribution of sublayers in the body. 2. Problem formulation Let the admixture particles migrate in a dispersed layer of thickness z0 with randomly nonhomogeneous stratified structure of material. The body is composed of two solid phases with different densities (Fig.1), and admixture diffusion coefficients can differ essentialy in these phases. The discussion is restricted to the case when the volume fraction v0 of one phase (the basic phase, marked by the index 0) is much greater than that of another phase v0 ≫ v1. If an arbitraryvertical bodyvolume is denotedby V then V =V (0)+V (1), where V (j) is the volume of the j-phase, and V (j) = nj ⋃ i=1 V (j) i j=0;1 Here V (j) i is thevolumeof sublayer iof the j-phase, i is the sublayer number, i= 1,2, ...,nj, nj is the number of sublayers of kind j. And we assume that Ye.Chaplia, O.Chernukha 931 Fig. 1. One of possible realizations of the body structure the body density ρ(z) and diffusion coefficient D(z) are constant in the volume of each phase. At the same time, the phase configuration is a random magnitude. Let us introduce into consideration the random operator ηij(z) that de- pends on the phase configuration and doesn’t depend on their physical cha- racteristics. It is defined by the formula (Lydzba, 1998) ηij(z)=    1 z∈V (j) i 0 z 6∈V (j) i (2.1) Note that 1 ∑ j=0 nj ∑ i=1 ηij(z)= 1 (2.2) Relationship (2.2) represents the body continuity. Then the diffusion coefficient D(z) and density of the body ρ(z) are presented by the random operator (2.1) as follows D(z)= 1 ∑ j=0 nj ∑ i=1 Djηij(z) ρ(z)= 1 ∑ j=0 nj ∑ i=1 ρjηij(z) (2.3) where Dj, ρj are values of the corresponding coefficients in j-phase. Using theapproachof generalized functions (Vladimirov, 1976;Podstrigach et al., 1984), diffusion of admixture particles in such a body is described in the form L(z,t)c(z,t)≡ ρ(z) ∂c(z,t) ∂t −∇[D(z)∇c(z,t)] = 0 (2.4) 932 Admixture diffusion in a two-phase random... where c(z,t) denotes the field of admixture concentration in the body; ρ(z)= ρ(z)/ρ0 is the normalized random density and ρ(z) is the body density, ρ0 is the density of the phase 0; D(z) is a random admixture diffusion coefficient, D(z)= d(z)/ρ0, and here d(z) is a randomkinetic coefficient; ∇= ∂/∂z, t is time. Let a constantmass source act on the upper boundary of the layer referred to rectangular coordinates so that the Oz-axis is perpendicular to its surface z=0 c(z,t)|z=0 = c ∗ (c∗ = const) Another boundary condition and the initial one are also given c(z,t)|z=z0 =0 c(z,t)|t=0 =0 (2.5) Substitute the coefficient (2.3) intoEq. (2.4) and assume that (Vladimirov, 1974) 1 ∑ j=0 nj ∑ i=1 ∇ ( Djηij(z) ) = 1 ∑ j=0 nj ∑ i=1 [Dj]Γδ(z−z Γ ij) where [Dj]Γ denotes a jump of the diffusion coefficient on the boundaries of the i-layer of the j-phase (V (j) i ), δ(z) is the Dirac delta-function, z Γ ij is the boundary of subregion V (j) i (henceforth zij denotes the upper boundary of V (j) i (random magnitude); zij + δzj is the lower boundary of this sublayer, δzj is the width of the j-phase layer). Then we obtain L(z,t)c(z,t) = 1 ∑ j=0 nj ∑ i=1 Lij(z,t)c(z,t) = 0 (2.6) where the random operator Lij is Lij(z,t)= ρjηij(z) ∂ ∂t −Djηij(z) ∂2 ∂z2 − − [ [Dj] h Γδ(z−zij)+ [Dj] l Γδ(z− (zij + δzj)) ] ∂ ∂z Here [Dj] h Γ , [Dj] l Γ are jumps of the diffusion coefficient on the upper and lower boundaries of the i-sublayer of the j-phase (randommagnitude). Ye.Chaplia, O.Chernukha 933 3. Neyman series for the diffusion problem In Eq. (2.6) add and subtract deterministic operator L0(z,t) defined in the entire interval (t∈ [0;∞[, z∈ [0;z0]) L0(z,t)= ρ0 ∂ ∂t −D0 ∂2 ∂z2 the coefficients of which are characteritics of the basic phase. Then using con- ditions (2.2) we have L0(z,t)c(z,t) =Ls(z,t)c(z,t) (3.1) where Ls(z,t)≡L0−L= ρ∗ n1 ∑ i=1 ηi1(z) ∂ ∂t −D∗ n1 ∑ i=1 ηi1(z) ∂2 ∂z2 + (3.2) +D∗ n1 ∑ i=1 [ δ(z−zi1)− δ(z− (zi1+ δz1)) ] ∂ ∂z Here ρ ∗ = ρ0 −ρ1 and D∗ = D0 −D1. We consider the right-hand side of Eq. (3.1) as a source, i.e. the medium nonhomogeneity is treated as internal source. The solution of initial-boundary value problem (3.1), (2.5) is found in the form of Neyman series (Rytov et al., 1978). Let c0(z,t) by adeterministic field of admixture concentration in thebody with characteristics ρ0,D0. It satisfies the following homogeneous equation L0(z,t)c0(z,t) = 0 and the initial boundary conditions (2.5), i.e. (Crank, 1975) c0(z,t)= c ∗ { 1− z z0 − ∞ ∑ n=1 2 nπ exp ( − D0 ρ0 y2nt ) sin(ynz) } (3.3) where yn =nπ/z0. Write G(z,z′, t,t′) for the unperturbedGreen function satisfying a diffu- sion equation for a point source ρ0 ∂G ∂t −D0 ∂2G ∂z2 = δ(t− t′)δ(z−z′) 934 Admixture diffusion in a two-phase random... and the initial and boundary conditions G(z,z′, t,t′)|t=0 =0 G(z,z′, t,t′)|z=0 =G(z,z ′, t,t′)|z=z0 =0 Then the initial-boundary value problem (3.1), (2.5) is equivalent to the in- tegral equation for the random field of admixture concentration c(z,t) in a two-phase stratified layer c(z,t) = c0(z,t)+ t ∫ 0 z0 ∫ 0 G(z,z′, t,t′)Ls(z ′, t′)c(z′, t′) dz′dt′ (3.4) where the Green function is G(z,z′, t,t′)= 1 2ρ0 ∞ ∑ n=1 exp [ − D0 ρ0 y2n(t− t ′) ][ cos(yn(z−z ′))− cos(yn(z+z ′)) ] (3.5) TheNeyman series for the problem (3.1), (2.5) is built by iterating (Rytov at al., 1978) the integral equation (3.4). Let us restrict the expression to the first two terms in the Neyman series. Then we obtain c(z,t)≈ c0(z,t)+ t ∫ 0 z0 ∫ 0 G(z,z′, t,t′)Ls(z ′, t′)c0(z ′, t′) dz′dt′ (3.6) If we substitute the operator Ls(z ′, t′) defined by (3.2) into Eq. (3.6), we have c(z,t)≈ c0(z,t)+ t ∫ 0 z0 ∫ 0 G(z,z′, t,t′) ∞ ∑ n=1 [ ρ ∗ ∂c0 ∂t′ −D∗ ∂2c0 ∂z ′2 ] ηi1(z ′) dz′dt′+ (3.7) +D∗ t ∫ 0 z0 ∫ 0 G(z,z′, t,t′) ∞ ∑ n=1 [ δ(z′−zi1)−δ(z ′− (zi1+ δz1)) ]∂c0 ∂z′ dz′ 4. Averaging approximate solution Let us consider averaging of the concentration field over the ensemble of sublayer configurations with different distributions of phases in the body. Ye.Chaplia, O.Chernukha 935 I. Let the phases be distributed with equal probabilities. As c0(z,t) is a deterministic field, then 〈c0(z,t)〉conf = c0(z,t). Consider the first integral in (3.7). As long as ηi1(z ′)= { 1 z′ ∈ [zi1;zi1+ δz1] 0 z 6∈ [zi1;zi1+ δz1] = { 1 z′−zi1 ∈ [0;δz1] 0 z′−zi1 6∈ [0;δz1] = ηi1(z ′−zi1) (4.1) only the function ηi1(z ′ − zi1) depends on zi1 under the integral and there are not other terms with index i, then 〈I1〉conf = t ∫ 0 z0 ∫ 0 G(z,z′, t,t′)L∗(z ′, t′)c0(z ′, t′) 1 V nj ∑ i=1 ∫ V ηi1(z ′−zi1) dzi1dz ′dt′ L∗(z ′, t′)= ρ ∗ ∂ ∂t′ −D∗ ∂2 ∂z ′2 Taking into account the properties of function ηi1(z ′−zi1), we can write 1 V nj ∑ i=1 ∫ V ηi1(z ′−zi1)dzi1 =    v1 z′ δz1 z′ <δz1 v1 z ′ ­ δz1 Then we obtain 〈I1〉conf = v1 δz1 t ∫ 0 δz1 ∫ 0 GL∗c0(z ′, t′)z′ dz′dt′+v1 t ∫ 0 z0 ∫ δz1 GL∗c0(z ′, t′) dz′dt′ (4.2) Consider averaging of the second integral in (3.7). Since the δ-function is even, we have (Abramowitz and Stegun, 1979) z0−δz1 ∫ 0 δ(zi1−z ′) dzi1 =          1 2 z′ =0 or z′ = z0−dz1 1 z′ ∈]0;z0− δz1[ 0 for other z′ and 1 V nj ∑ i=1 ∫ V δ(zi1−z ′) dzi1 =          v1 2δz1 z′ =0 or z′ = z0− δz1 v1 δz1 z′ ∈]0;z0− δz1[ 0 for other z′ (4.3) 936 Admixture diffusion in a two-phase random... We find the internal integral of the second δ-function in the same way 1 V nj ∑ i=1 ∫ V δ(zi1+ δz1−z ′) dzi1 =          v1 2δz1 z′ = δz1 or z ′ = z0 v1 δz1 z′ ∈]δz1;z0[ 0 for other z′ (4.4) Then, allowing for (4.3), (4.4), the definition of an improper integral, the boundary conditions for the Green function, we obtain 〈I2〉conf =D∗ v1 δz1 t ∫ 0 {1 2 ( G ∂c0 ∂z′ ∣ ∣ ∣ z′=z0−δz1 −G ∂c0 ∂z′ ∣ ∣ ∣ z′=δz1 ) + (4.5) + δz1+0 ∫ +0 G ∂c0 ∂z′ dz′− z0−0 ∫ z0−δz1+0 G ∂c0 ∂z′ dz′ } dt′ As longas (4.2) and (4.5) takeplace,wecanwrite the expression for calculating the approximate concentration field averaged over the ensemble of sublayer configurations 〈c〉conf = c0(z,t)+ v1 δz1 t ∫ 0 { δz1 ∫ 0 GL∗c0(z ′, t′)z′ dz′+ +δz1 z0 ∫ δz1 GL∗c0(z ′, t′) dz′+D∗ [1 2 ( G ∂c0 ∂z′ ∣ ∣ ∣ z′=z0−δz1 −G ∂c0 ∂z′ ∣ ∣ ∣ z′=δz1 ) +(4.6) + δz1+0 ∫ +0 G ∂c0 ∂z′ dz′− z0−0 ∫ z0−δz1+0 G ∂c0 ∂z′ dz′ ]} dt′ II. Let the phase j=1 have the beta-distribution in the layer. Note that the density of the beta-distribution in a layer with thickness z0 is f(z)=    Γ(α+β) Γ(α)Γ(β) ( z z0 )α−1( 1− z z0 )β−1 z∈ [0;z0] 0 z 6∈ [0;z0] (α> 0, β > 0) Below we consider two special cases: (i) α> 1, β = 1; (ii) α= 1, β > 1 (Fig.2). Ye.Chaplia, O.Chernukha 937 Fig. 2. Density of beta-distribution Let us average the concentration field over the ensemble of sublayer confi- gurations (3.7)with the beta-distribution of the phase j=1.For this purpose consider the averaging of the first integral in (3.7) 〈I1〉conf = t ∫ 0 z0 ∫ 0 GL∗c0(z ′, t′) nj ∑ i=1 ∫ V ηi1(z ′)f(zi1) dzi1dz ′dt′ (i)Taking into account the expression for f(z), in this case we have nj ∑ i=1 ∫ V ηi1(z ′)f(zi1) dzi1 = Γ(1+α) Γ(α) nj ∑ i=1 z0−δz1 ∫ 0 ηi1(z ′−zi1) ( zi1 z0− δz1 )α−1 dzi1 Using (4.1) we obtain two cases: if z′ <δz1 then nj ∑ i=1 ∫ V ηi1(z ′)f(zi1) dzi1 = Γ(1+α) αΓ(α) v1(z ′)α δz1(z0− δz1)α−2 When z′ ­ δz1 nj ∑ i=1 ∫ V ηi1(z ′)f(zi1) dzi1 = Γ(1+α) αΓ(α) v1[(z ′)α− (z′− δz1) α] δz1(z0− δz1)α−2 In consequence, we obtain 〈I1〉conf = Γ(1+α) αΓ(α) v1 δz1 (z0− δz1) 2−α · · t ∫ 0 { δz1 ∫ 0 z ′αGL∗c0(z ′, t′) dz′− z0 ∫ δz1 (z′− δz1) αGL∗c0(z ′, t′) dz′ } dt′ 938 Admixture diffusion in a two-phase random... (ii) Using the expression of the beta-distribution density when α = 1, β> 1, we have nj ∑ i=1 ∫ V ηi1(z ′)f(zi1) dzi1 = Γ(1+β) Γ(β) nj ∑ i=1 z0−δz1 ∫ 0 ηi1(z ′−zi1) ( 1− zi1 z0− δz1 )β−1 dzi1 Integrating the last expression we obtain 〈I1〉conf = Γ(1+β) βΓ(β) v1 δz1 (z0− δz1) 2−β t ∫ 0 { z0 ∫ δz1 (z0−z ′)βGL∗c0(z ′, t′) dz′− − z0 ∫ 0 (z0− δz1−z ′)βGL∗c0(z ′, t′) dz′+(z0−δz1) β δz1 ∫ 0 GL∗c0(z ′, t′) dz′ } dt′ Since the Dirac function is an even one, the averaged second integral in (3.7) can be written in the form: (i) 〈I2〉conf =D∗ Γ(1+α) Γ(α) v1 δz1 (z0− δz1) 2−α t ∫ 0 z0 ∫ 0 GL∗c0(z ′, t′) · · ( z ′α−1− (z′− δz1) α−1 ) dz′dt′ (ii) 〈I2〉conf =D∗ Γ(1+β) Γ(β) v1 δz1 (z0− δz1) 2−β t ∫ 0 z0 ∫ 0 GL∗c0(z ′, t′) · · ( (z0− δz1−z ′)β−1− (z0− δz1) β−1 ) dz′dt′ Asa resultwe obtain the formulae for the admixture concentration field avera- ged over the ensemble of sublayer configurations with their beta-distribution: (i) 〈c〉conf = c0(z,t)+ Γ(1+α) Γ(α) v1 δz1 ( z0− δz1 )2−α t ∫ 0 {1 α δz1 ∫ 0 z ′αGL∗c0(z ′, t′)dz′− (4.7) − 1 α z0 ∫ δz1 (z′− δz1) αGL∗c0(z ′, t′)dz′+D∗ z0 ∫ 0 G ∂c0 ∂z′ [ z ′α−1− (z′− δz1) α−1 ] dz′ } dt′ Ye.Chaplia, O.Chernukha 939 (ii) 〈c〉conf = c0+ Γ(1+β) Γ(β) v1 δz1 ( z0− δz1 )2−β t ∫ 0 {1 β z0 ∫ δz1 (z0−z ′)βGL∗c0(z ′, t′)dz′− − 1 β z0 ∫ 0 (z0−z ′−δz1) βGL∗c0(z ′, t′)dz′+ 1 β (z0− δz1) β δz1 ∫ 0 GL∗c0(z ′, t′)dz′+ (4.8) +D∗ z0 ∫ 0 G ∂c0 ∂z′ [ (z0−z ′− δz1) β−1− (z0− δz1) β−1 ] dz′ } dt′ 5. Analysis of the obtained solutions The final expression for the averaged field of the admixture concentration for different distributions of sublayers in the two-phase stratified layer is ob- tained by substituting the formulae for theGreen function and the admixture concentration in the homogeneous medium with characteristics of the phase j=0 into the respective expressions for the averaged concentration fields. I.The equally probable distribution of the phases. Substituting Eqs (3.3) and (3.5) into (4.6) we have 1 c∗ 〈c(z,t)〉conf =1− z z0 − ∞ ∑ n=1 2 nπ exp ( − D0 ρ0 y2nt ) sin(ynz)+ + v1 δz1 D∗ 2z0D0 { δz1(z0−2z)+ ∞ ∑ k=1 1 yk exp ( − D0 ρ0 y2kt )[( B1+ 1 yk B2 ) cos(ykδz1)− −B2z0 sin(ykδz1) ] +sin(ykz) [D∗ y3 k (1− (−1)k)(1− cos(ykδz1))+ (5.1) + ∞ ∑ n=1 1 y2n−y 2 k [ exp ( − D0 ρ0 y2kt ) −exp ( − D0 ρ0 y2nt )] · · ( 2 Dρ D∗ yn(A−−A+)− (1+(−1) k+n)Akn )]} where B1 =2sin(ykδz1)cos(ykz) B2 =2cos(ykδz1)cos(ykz) 940 Admixture diffusion in a two-phase random... Dρ = D1ρ0−D0ρ1 ρ0 A± = 1 (yk±yn) 2 [cos((yk ±yn)δzj)−1] Akn = 2yk y2 k −y2n − cos[(yk −yn)δz1] yk−yn − cos[(yk+yn)δz1] yk+yn Illustration of the influence of the material structure nonhomogeneity on the distribution of the admixture concentration in a layer under the action of a constant source on the upper boundary is given in Fig.3 and Fig.4. Nu- merical calculation was done for the dimensionless quantities ξ = z/z0 and Fo = D0t/z 2 0. It is assumed that D1 = D1/D0 = 0.5, δξ1 = δz1/z0 = 0.01, Fo= 10−2, v1 =0.1. The solid line marks the respective function for the ad- mixture concentration averaged over the ensemble of sublayer configurations and calculated by (5.1). The dashed line identifies the admixture concentra- tion in the homogeneous medium with the basic phase characteristics. The dimensionless coordinate ξ has been assumed as abscissa, the ratio of the concentration to its value on the upper body boundary c∗ has been taken as ordinate. The distributions of the admixture concentration are compared in Fig.3a for different values of the reduced diffusion coefficient D1 = 0.2, 0.5, 0.8, 1.2, 1.5, curves 1-5, respectively. The concentration distributions are presented for different values of the Fourier number Fo = 10−2, 10−3, 10−4, curves 1-3 (1a−3a) respectively, in Fig.3b. Fig. 3. Fig.4a illustrates the behaviour of the concentration field in dependence on the quantity of the volume fraction of sublayers v1 = 0.2, 0.15, 0.1, 0.05, 0.01, curves 1-5, respectively. Dependence of the admixture concentration on the sublayer thickness δξ1 =0.05, 0.02, 0.01, 0.008, 0.007, curves 1-5, is shown in Fig.4b. The performed analysis of the obtained results shows that distinctions in diffusive properties of the randomly distributed phases can cause essential Ye.Chaplia, O.Chernukha 941 Fig. 4. changes of the character of the admixture concentration field in the body. Thus in quantitative description of the mass transfer it is necessary to take into account explicitly both different values of the diffusion coefficient and its jump discontinuities at phase boundaries. In the case when the diffusion coefficient in thin layers is greater than the one in the matrix, it leads the admixture concentration decrease in the body. And occurence of sublayers with the diffusion coefficient smaller than one in thematrix causes its essential increase (Fig.3a). Change of the other material parameters affects also substantially the va- lues of the averaged concentration field in a nonhomogeneous medium. Thus, in the case of the admixture diffusion in bodieswith D1 1 1 c∗ 〈c(z,t)〉conf =1− z z0 − ∞ ∑ n=1 2 nπ exp ( − D0 ρ0 y2nt ) sin(ynz)+ + Γ(1+β) Γ(β) v1 δz1 (z0− δz1) 2−β z0D0 { D∗ zz0 8 ( 1− z z0 ) (z0−δz1) β−1+ + ∞ ∑ k=1 sin(ykz) {D∗ 2 [ 1 y2 k fβ−1s (−z0,0,δz1 −z0,yk) [ 1− exp ( − D0 ρ0 y2kt )] − −(z0− δz1) β−1(1− (−1)k) ( 1 y2 k −1 ) 1 yk exp ( − D0 ρ0 y2kt )] + + ∞ ∑ n=1 1 y2n−y 2 k [ exp ( − D0 ρ0 y2kt ) − exp ( − D0 ρ0 y2nt )][Dρ 2β ynA(z0− δz1) β − (5.3) − Dρ β yn [ fαc (−z0,0,δz1 −z0,yk−yn)−f α c (−z0,0,δz1 −z0,yk+yn)+ Ye.Chaplia, O.Chernukha 943 +fαc (−z0,−δz1,−z0,yk−yn)−f α c (−z0,−δz1,−z0,yk+yn) ] − − 1 2 D∗ [ fβ−1s (−z0,0,δz1 −z0,yk−yn)+f β−1 s (−z0,0,δz1 −z0,yk+yn)] ] − −D∗yk ∞ ∑ n=1 1− (−1)k+n y2n−y 2 k exp ( − D0 ρ0 y2nt )}} where A= sin(yk−yn)δz1 yk−yn − sin(yk+yn)δz1 yk+yn The distributions of the admixture concentration field in a stratified layer is given in Fig.5 and Fig.6 for the particular cases of the probable beta- distribution of sublayers. Numerical calculation was also done for the di- mensionless quantities ξ = z/z0 and Fo = D0t/z 2 0. Then we assume D1 = D1/D0 = 0.5, δξ1 = δz1/z0 = 0.01, Fo = 10 −1, v1 = 0.1. The dashed line (curves a)marks the respective function for admixture concentra- tion averaged over the ensemble of sublayers configurations with their beta- distribution in thebody for the case α> 1,β=1andcalculated by (5.2). The solid line (curves b) identifies the admixture concentration for the case α=1, β> 1 and calculated by the expression (5.3). The dimensionless coordinate ξ has been assumed as abscissa, the ratio of the concentration to its value on the upper body boundary c∗ has been assumed as ordinate. The distributions of the admixture concentration are compared in Fig.5a for different values of the reduced diffusion coefficient D1 = 0.2, 0.5, 0.8, 1.2, 1.5, curves 1-5, re- spectively. The concentration distributions are presented for different values of Fourier number Fo=10−1, 5 ·10−2, 10−2, curves 1-3 respectively, in Fig.5b. Fig. 5. Fig.6a illustrates the behaviour of the concentration field in dependence 944 Admixture diffusion in a two-phase random... on the quantity of the volume fraction of sublayers v1 = 0.2, 0.15, 0.1, 0.05, 0.01, curves 1-5, respectively. Dependence of the admixture concentration on the sublayer thickness δξ1 = 0.05, 0.02, 0.01, 0.008, curves 1-4, is shown in Fig.6b. Fig. 6. Numerical calculations show that for the case α> 1,β=1of the sublayer beta-distribution, i.e. it is known a priori that there is the matrix near the surface z = 0, and sublayers position is most probable near another layer boundary z= z0 (see Fig.2), and the procedure ofmodel homogenisation can beused effectively.We also note that for such aprobable sublayer distribution, changes of the model parameters do not produce behaviour changes of the admixture concentration field. And only the change of the diffusion coefficient influences the quantitativemagnitude of the admixtureparticles concentration in the body (Fig.5a). An altogether different picture emerges in the case α = 1, β > 1 of the sublayer beta-distribution, i.e. the matrix is a priori on the boundary z= z0 and sublayers position is themost probable near the boundary z=0 (Fig.2). In this case, using of the homogenisation procedure is inefficient. Change of the model parameters can essentially affect the behaviour of the concentra- tion field. Thus, for example, if the diffusion coefficient of thematrix is larger than one in the sublayer material then increase of the admixture particles concentration occurs near the surface z = 0. And when the matrix diffusion coefficient is less than one in sublayers, accumulation of the admixture partic- les concentration occurs near another layer boundary z = z0 (Fig.5a). The value of the sublayer volume fraction (Fig.6a) and its thickness (Fig.6b) affect essentially the concentration values, without changing the function behaviour. Remark that the obtained expressions for the admixture concentration Ye.Chaplia, O.Chernukha 945 field averaged over the ensemble of phase configurations give the possibility to determine also thedispersionof the concentration fieldbyusing theknown for- mula (Rytov et al., 1978). It is important, in particular, to verify the obtained values of the averaged concentration. So we can obtain the practically important information on the character of the admixture distribution in a body using some a priori data concerning their structure and physical properties. References 1. Abramowitz M., Stegun I.A., 1979,Handbook of Mathematical Functions, Nauka, Moscow 2. Burak Y.Y., Chaplia Y.Y., 1993, Initial Principles of the Mathematical Model of Heterodiffusion Transfer,Reports of Ukrainian National Academy of Sciences, 10, 59-63 3. Burak Y.Y., Chaplia Y.Y., Chernukha O.Y., 1995, On Radionuclides VerticalMigration in Soil,Reports of Ukrainian National Academy of Sciences, 10, 34-37 4. Crank J.C., 1975,Mathematics of Diffusion, Clarendon Press, Oxford 5. Kanovsky I.Y., Tkachenko I.V., 1991, An Effective Diffusion Coefficient in a NonhomogeneousMedium,Ukrainian Physical Journal, 36, 3, 432-434 6. Lyubov B.Y., 1981, Diffusion Processes in Nonhomogeneous Media, Nauka, Moscow 7. Lydzba D., 1998, Homogenisation Theories Applied to PorousMedia Mecha- nics, Journal of Theoretical and Applied Mechanics, 36, 3, 657-679 8. Matysiak S.J.,MieszkowskiR., 1999,OnHomogenisation ofDiffusionPro- cesses inMicroperiodic Stratified Bodies, Int. J. Heat Mass Transfer, 26, 539- 547 9. PodstrigachY.S., LomakinV.A.,KolyanoY.M., 1984,Thermoelasticity of Bodies of Nonhomogeneous Structure, Nauka,Moscow 10. Rytov S.M., KravtsovY.A., Tatarskiy B.I., 1978, Introduction into Sta- tistical Radio Physics. II. Random Fields, Nauka, Moscow 11. Shatinsky V.F., Nesterenko A.I., 1988, Protective Diffusive Coverages, Naukova Dumka, Kyiv 12. Vladimirov V.S., 1976,Equations of Mathematical Physics, Nauka,Moscow 946 Admixture diffusion in a two-phase random... Dyfuzja substancji domieszkowej w dwufazowej warstwie losowo niejednorodnej Rozważona jest jednowymiarowa (pionowa) dyfuzja substancji domieszkowej w war- stwie tworzonej przez losowo-niejednorodny, dwufazowy materiał warstwowy. Przy konstruowaniu rozwiazańuwzględniono zarówno różnicewspółczynnikówdyfuzji i gę- stości w różnych fazach, jak i nieciągłości współczynnika dyfuzji na granicach faz. Zgodnie z zaproponowanym podejściem przy rozwiązaniu zagadnienia brzegowego dyfuzji wpływniejednorodnościmateriału sprowadzą się do rozpatrywania źródełwe- wnętrznychmasy, a same zagadnienie – do równania całkowego, które z koleji rozwią- zano metodą rozwinięcia w szereg Neymana. Uśrednienie przybliżonego rozwiązania po zbiorze konfiguracji faz, z których złożone jest ciało, wykonano dla równomier- nego losowego rozkładu faz oraz dwóch szczegółowych przypadków rozkładu beta. Porównanie rozkładów uśrednionego pola koncentracji i koncentracji w jednorodnym ośrodku pokazało potrzebę wzięcia pod uwagę zarówno różnych dyfuzyjnych wła- ściwości faz, jak i nieciągłości współczynnika dyfuzji na granicach faz w warunkach doskonałego kontaktu. Oprócz wyznaczono zależność uśrednionej koncentracji skład- nika domieszkowego odwspółczynnikówdyfuzji, gęstości i objętościowychudziału faz dla losowego rozkładu beta podwarstw. Manuscript received February 1, 2001; accepted for print April 24, 2001