fundamental and plane wave solution in swelling porous medium

Afr. Mat.DOI 10.1007/s13370-012-0123-5

Fundamental and plane wave solution in swelling porousmedium

Rajneesh Kumar · Divya Taneja · Kuldeep Kumar

Received: 13 June 2012 / Accepted: 6 November 2012© African Mathematical Union and Springer-Verlag Berlin Heidelberg 2012

Abstract In the present paper propagation of plane waves in swelling porous medium(SP) is studied. The phase velocity and attenuation coefficients of these waves are computednumerically and presented graphically. The results so obtained have been compared to withoutswelling porous elastic medium (EL). The fundamental solution of the system of differentialequations in swelling porous medium in case of steady oscillations in terms of elementaryfunctions has been constructed. Some basic properties are established and particular case ofinterest is also deduced.

Keywords Plane waves · Swelling porous medium · Fundamental solution ·Steady oscillations.

Mathematics Subject Classification (2000) 76D33

1 Introduction

The continuum theory of mixtures has extensively been studied in literature. A presentationof the work on the subject can be found in review articles by Atkin and Craine [1,2], andBedford and Drumheller [3]. Eringen [4] pointed out the importance of the theory of mixturesto the applied field of swelling. In this connection, Eringen has developed a continuum theoryof swelling porous elastic soils as a continuum theory of mixtures for porous elastic solidsfilled with fluid and gas. Swelling porous medium (material) is a porous material that swells(shrinks) upon whetting (drying). The intended applications of the theory are in the field of

R. KumarDepartment of Mathematics, Kurukshetra University,Kurukshetra, Haryana 136 119, Indiae-mail: [email protected]

D. Taneja (B) · Kuldeep KumarDepartment of Mathematics, N.I.T.,Kurukshetra, Haryana 136 119, Indiae-mail: [email protected]

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swelling, oil exploration, slurries and consolidation problems. Porous material consisting ofa porous solid with fluid-filled or gas-filled pores appear in a wide variety of applicationsincluding construction, agriculture, food stuffs, drug delivery systems and bio-tissues.

Bofill and Quintanilla [5] studied anti plane shear deformations of swelling porous elasticsoils in case of fluid saturation and gas saturation. Gales [6] investigated some theoreticalproblems concerning waves and vibrations within the context of the isothermal linear the-ory of swelling porous elastic soils with fluid or gas saturation. Tessa and Bennethum [7]derived transport equations for porous swelling materials that undergo finite deformations.Kleinfelter et al. [8] discussed the various aspects of mixture theory applied to unsatu-rated/saturated swelling soils and studied the two and three phase problems. Gales [9] studiedthe asymptotic spatial behavior of solutions in a mixture consisting of two thermo elasticsolids. Gales [10] also studied the spatial behavior of the harmonic vibrations in thermal vis-coelastic mixtures. Zhu et al. [11] presented a model for flow and deformation in context ofunsaturated swelling porous media. Jahanshahi et al. [12] obtained the numerical simulationof a three layered radiant porous heat exchanger. He include lattice boltzmann simulationof fluid flow. Shahnazari et al. [13] studied the comparison of thermal dispersion effects forsingle and two phase analysis of heat transfer in porous media. Merlani et al. [14] presenteda non-linear model of mechanic and chemo-poroelastic interactions between fluids conta-minants and solid matrix. Heider et al. [15] discussed the propagation of dynamic wave ininfinite saturated porous media.

Svanadze [16] obtained the fundamental solution of the system of linear coupled partialdifferential equations of the steady oscillations of the theory of microstretch elastic solids.Boer and Svanadze [17] studied the linear theory of the liquid-saturated porous mediumconsisting of a microscopically incompressible solid skeleton containing microscopicallyincompressible liquid. He obtained the fundamental solution of the system of linear coupledpartial differential equations of the steady oscillations of the porous solids.

Svanadze and Cicco [18] studied the linear theory of thermomicrostretch elastic solids andconstruct the fundamental solution of the system of differential equations in the case of steadyoscillations in terms of elementary functions. Ciarletta et al. [19] studied the linear theory ofmicropolar thermoelasticity for materials with voids. He constructed the fundamental solutionof the system of differential equations in the case of steady oscillations in terms of elementaryfunctions. Svanadze et al. [20] presented the linear theory of micropolar thermoelasticitywithout energy dissipation.

In the present paper the propagation of plane waves in swelling porous medium has beenstudied. The phase velocity and attenuation coefficient of plane wave has been computedand presented graphically for different values of frequency. The fundamental solution ofsystem of equations in the case of steady oscillations has also been constructed in terms ofelementary functions.

2 Basic equations

Let x=(x1, x2, x3) be the point of the Euclidean three-dimensional space E3, |x| =(x2

1 , x22 , x2

3 )12 , and let t denote the time variable. Following [5], the basic equations in swelling

porous medium in absence of body forces and heat sources are:

μ�us + (λ + μ)grad divus − σ f grad div u f + ξ f f (u f − us) = ρs0us, (1)

μv�u f + (λv + μv)grad div u f − σ f grad div us − σ f f grad div u f

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Fundamental and plane wave solution in SP medium

−ξ f f (u f − us) = ρf

0 u f , (2)

where the superscripts s and f denote, the elastic solid and the fluid respectively, uk =(uk

1, uk2, uk

3) is the displacement component, ρk0 is the density, λ,μ, λv, μv, σ

f , σ f f , ξ f f

are constitutive constants, � is the Laplacian operator and a superposed dot denotes timedifferentiation. (k = s, f )

We define dimensionless quantities:

x′ = ω∗

c1x, ui ′ = ω∗

c1ui , t ′ = ω∗t, ω∗ = ξ f f

ρs0

, c21 = λ + 2μ

ρs0

(3)

Upon introducing the quantities (3) in the basic Eqs. (1) and (2), after suppressing the primes,we obtain

δ21�us +(1−δ2

1)grad div us −a1grad div u f +a2(uf −us)= us, (4)

δ22�u f +(1−δ2

2)grad div u f −h1grad div us −h2grad div u f −h3(u f −us)=h4u f ,

(5)

where

δ21 = μ

λ + 2μ, a1 = σ f

λ + 2μ, a2 = ξ f f

ρs0ω

∗ , δ22 = μv

λv + 2μv

, h1 = σ f

(λv + 2μv)ω∗ ,

h2 = σ f f

(λv + 2μv)ω∗ , h3 = ξ f f c21

ω∗2(λv + 2μv), h4 = c2

1ρf

0

ω∗(λv + 2μv).

3 Plane waves

Suppose that plane waves corresponding to the wave number ξ and frequency ω are propa-gating in the x1 direction through the homogeneous isotropic swelling porous medium. Weassume the solution of the form:

(uk(x, t)) = Akei(ξ x1−ωt) (6)

where, Ak = (Ak1, Ak

2, Ak3) is a constant vector and ωis the oscillation frequency (ω > 0).

Using (6) in the Eqs. (4, 5), we obtain

[−ξ2(δ21 + (1 − δ2

1)δ1m) + τ2]Asm + [a1ξ

2δ1m − ia2ω]A fm = 0, (7)

[(δ22 iω + ((1 − δ2

2)iω + h2)δ1m)ξ2) + τ3]A fm + [h1ξ

2δ1m − ih3ω]Asm = 0, (8)

where δ1m is the Kronecker delta function and τ2 = iω(−iω + a2), τ3 = iω(−ih4ω + h3).

From Eqs. (7, 8) for As1, A f

1 we have the following system of equations

[−ξ2 + τ2]As1 + [a1ξ

2 − ia2ω]A f1 = 0, (9)

[h1ξ2 − ih3ω]As

1 + [(iω + h2)ξ2 + τ3]A f

1 = 0, (10)

The non-trivial solution of the system of Eqs. (9, 10) is ensured by a determinate equation∣∣∣∣

−ξ2 + τ2 a1ξ2 − ia2ω

h1ξ2 − ih3ω ξ2(iω + h2) + τ3

∣∣∣∣= 0, (11)

The Eq. (11) yields to the following polynomial equation in ξ as:

F1ξ4 + F2ξ

2 + F3 = 0, (12)

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where

F1 = −(iω + h2), F2 = −τ3 + τ2(iω + h2) + (a2h1 + h3a1)iω − a1h1,

F3 = τ2τ3 + h3ω2a2.

From Eqs. (7, 8) for Akl , l = 2, 3, we have the following system of equations

[−δ21ξ2 + τ2]As

2 − ia2ωA f2 = 0, (13)

−ih3ωAs2 + [δ2

2 iωξ2 + τ3]A f2 = 0, (14)

Equations (13) and (14) will have a non trivial solution if∣∣∣∣∣

−δ21ξ2 + τ2 −ia2ω

−ih3ω δ22 iωξ2 + τ3

∣∣∣∣∣= 0, (15)

From (15), we obtain

− δ21δ2

2 iωξ4 + ξ2(−δ21τ3 + τ2δ

22 iω) + τ2τ3 + h3a2ω

2 = 0, (16)

Equations (12) and (16) will be called the dispersion equations of longitudinal and transversalplane waves respectively. From these equations we obtain eight roots of ξ , in which we areinterested to those roots whose imaginary parts are positive because only those roots givethe negative roots of the decay coefficient, Im(ξ). Corresponding to these roots, there existlongitudinal wave in solid, longitudinal wave in fluid, transversal wave in solid, transversalwave in fluid. We denote the values of ξ associated with these modes by ξ1, ξ2, ξ3 and ξ4

respectively.We now derive the expressions of phase velocity and attenuation coefficient of these type

of waves as:

3.1 Phase velocity

The phase velocities are given by

Vj = ω∣∣Re(ξ j )

∣∣, j = 1, 2, 3, 4 (17)

where V1 and V2 are the longitudinal phase velocities in solid and fluid respectively; and V3

and V4 are the transversal phase velocities in solid and fluid respectively;

3.2 Attenuation coefficient

The attenuation coefficients are defined as

Q j = ∣∣Im(ξ j )

∣∣ , j = 1, 2, 3, 4 (18)

Now, we consider the case of steady oscillations. We assume the displacement vector in solid,displacement vector in fluid and temperature change as

uk(x, t) = Re[(uk(x, t))e−iωt ] (19)

Using Eq. (19) into Eqs. (4, 5), we obtain the system of equations of steady oscillations as

(δ21� + τ2)us + (1 − δ2

1)grad div us − a1grad div u f − ia2ωu f = 0 (20)

(−iωδ22� + τ3)u f + ((1 − δ2

2)(−iω) − h2)grad divu f − h1grad div us − h3iωus = 0

(21)

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We introduce the matrix differential operator

F = [Fmn]m,n=1,2,...,6

where

Fmn = (δ21� + τ2)δmn + (1 − δ2

1)∂2

∂xm∂xn, Fm,n+3 = −a1

∂2

∂xm∂xn− a2iω,

Fm+3,n+3 = (−iωδ22� + τ3)δmn + ((1 − δ2

2)(−iω) − h2)∂2

∂xm∂xn,

Fm+3,n = −h1∂2

∂xm∂xn− h3iω m, n = 1, 2, 3

Here δmn is Kronecker delta function.The system of Eqs. (20, 21) can be written as

F U(x) = 0

where U = (uk) is a six component vector in E3.We assume that

− δ21δ2

2 iω �= 0 (22)

If the condition (22) is satisfied, then F is an elliptic differential operator [21].

Definition The fundamental solution of the system of Eqs. (20, 21) (the fundamental matrixof operator F) is the matrix G(x) = [

Ggh(x)]

6×6 satisfying condition [21]

F G(x) = δ(x)I(x) (23)

where δ is the Dirac delta, I = [

δgh]

6×6 is the unit matrix and x ∈ E3.Now we construct G(x)in terms of elementary functions.

4 Fundamental solution of system of equations of steady oscillations

Consider the system of equations

(δ21� + τ2)us + (1 − δ2

1)grad divus − h1grad divu f − h3iωu f = H′, (24)

− a1grad divus −a2iωus +(− δ22 iω�+τ3)u f +((1 − δ2

2)(− iω)−h2)grad divu f =H′′,

(25)

where H′, H′′ are three component vector function on E3.The system of Eqs. (24, 25) may be written in the form

Ftr U(x) = Q(x) (26)

where, Ftr is the transpose of matrix F, Q = (H′, H′′) and x ∈ E3.Applying the operator div to Eqs. (24) and (25), we obtain

C1divus + C2divu f = divH′ (27)

D1divus + D2divu f = divH′′ (28)

where C1 = τ2 + �, C2 = −h1� − h3iω, D1 = −a1� − a2iω, D2 = τ3 − (iω + h2)�

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R. Kumar et al.

Equations (27) and (28) may be written in the form

N(�)S = Q (29)

where S = (divus, divu f ), Q = (d1, d2) = (divH′, divH′′) and

N (�) = [Nmn(�)]2×2 =[

C1 C2

D1 D2

]

2×2(30)

The system of Eqs. (27) and (28) may also be written as

�1(�)S = � (31)

where,

� = (�1, �2), �n = e∗2

∑

m=1

N∗mndm (32)

�1(�) = e∗ det N (�), e∗ = −1

(iω + h2) − a1h1n = 1, 2 (33)

and N∗mn is the cofactor of the elements Nmn of the matrix N .

From Eqs. (30) and (33) we see that

�1(�) =2

∏

m=1

(� + λ2m)

where λ2m , m = 1, 2 are the roots of the equation �(−k) = 0 (with respect to k).

From (27) and (28) we get

�1(�)divus = φ1, �1(�)divu f = φ2, (34)

where

φ1 = (D2divH′ − C2divH′′), φ2 = (−D1divH′ + C1divH′′). (35)

Operate �1(�)on (24) and (25)

�1(�)((δ21�+τ2)us −h3iωu f ) = f1, �1(�)(−a2iωus +(−δ2

2 iω�+τ3)u f )= f2, (36)

where

f1 = −(1 − δ21)gradφ1 + h1gradφ2 + �1(�)H′,

f2 = a1gradφ1 + ((1 − δ22)(iω) + h2)gradφ2 + �1(�)H′′.

From system (36) we have

�1(�)�2(�)us = � ′ �1(�)�2(�)u f = � ′′, (37)

where

�2(�) = f ∗ det

∥∥∥∥

δ21� + τ2 h3ω

−a2ω −δ22 iω� + τ3

∥∥∥∥

2×2, f ∗ = −1

δ21δ2

2 iω,

123


and

� ′ = f ∗{[(δ22 iω�−τ3)(1−δ2

1)+h3iωa1]gradφ1+[(−δ22 iω�+τ3)h1

+h3iω((1−δ22)iω+h2)]gradφ2+h3iω�1(�)H′′+(−δ2

2 iω�+τ3)�1(�)H′, (38)

� ′′ = f ∗{[(δ21� + τ2)a1 − a2iω(1 − δ2

1)]gradφ1 + [(δ21� + τ2)((1 − δ2

2)iω + h2)

+a2iωh1]gradφ2 + a2iω�1(�)H′ + (δ21� + τ2)�1(�)H′′. (39)

From Eqs. (31) and (37) we obtain

�(�)U(x) = �(x) (40)

where

� = (� ′, � ′′)�(�) = [

�gh(�)]

6×6

�mm(�) = �1(�)�2(�) =4

∏

q=1

(� + λ2q)

�gh(�)=0,�55(�)=�66(�) = �1(�); m =1, 2, 3, 4; g, h =1, 2, 3, 4, 5, 6; g �= h

From Eqs. (35) and (38), we obtain

� ′ = [ f ∗(−δ22 iω�+τ3)�1(�)J+q11(�)grad div]H′+[h3iω�1(�)

+q12(�)grad div]H′′, (41)

� ′′ = [ f ∗a2iω�1(�)+q21(�)grad div]H′+[(δ21�+τ2)�1(�)J f ∗

+q22(�)grad div]H′′ (42)

where J = [

δgh]

3×3 is the unit matrix.In Eqs. (41, 42), we have used the following notations:

q1m(�) = f ∗e∗{[(δ22 iω� − τ3)(1 − δ2

1) + a1h3iω]N∗m1 + [−(δ2

2 iω� − τ3)h1

+((1 − δ22)iω + h2)h3iω]N∗

m2},q2m(�) = f ∗e∗{[a1(δ

21� + τ2) − a2iω(1 − δ2

1)]N∗m1

+[(a2iωh1 + ((1 − δ22)iω + h2))(δ

21� + τ2)]N∗

m2}. m = 1, 2;From Eqs. (41) and (42), we obtain

�(x) = Rtr Q(x) (43)

where

R = [

Rgh]

6×6

Rmn = f ∗(−δ22 iω�+τ3)�1(�)δmn +q11(�)

∂2

∂xm∂xn,

Rm,n+3 = f ∗a2iω�1(�) + q21(�)∂2

∂xm∂xn,

Rm+3,n = h3iω�1(�) + q21(�)∂2

∂xm∂xn,

Rm+3,n+3 = f ∗(δ21� + τ2)�1(�)δmn + q22(�)

∂2

∂xm∂xn, m, n = 1, 2

(44)

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R. Kumar et al.

From Eqs. (26), (40) and (43) we obtain,

�U = Rtr Q(x)

It implies that

� = Rtr Ftr

�(�) = F R (45)

We assume that

λ2m �= λ2

n �= 0 m, n = 1, 2, 3, 4; m �= n

Let

Y(x) = [Yrs(x)]6×6 , Ymn(x) =4

∑

n=1

r1nς(x)

Y55(x) = Y66(x) =2

∑

n=1

r2nς(x)

Yvw(x) = 0, m = 1, 2, 3, 4; v,w = 1, 2, 3, 4, 5, 6; v �= w

where

ςn(x) = −1

4π |x| exp(iλn |x|), n = 1, 2, 3, 4

r1l(x) =4

∏

m=1, m �=l

(λ2m−λ2

l )−1, l = 1, 2, 3, 4

r2w(x) =2

∏

m=1, m �=w

(λ2m−λ2

w)−1, w = 1, 2

We will prove the following lemma:

Lemma The matrix Y defined above is the fundamental matrix of operator�(�), that is

�(�)Y (x) = δ(x)I(x) (46)

Proof To prove the lemma it is sufficient to prove that

�1(�)�2(�)Y11(x) = δ(x), �1(�) Y55(x) = δ(x) (47)

We find that

r21+r22 =0, r22(λ21 − λ2

2)=1, (�+λ2m)ςn(x)=δ(x)+(λ2

m −λ2n)ςn(x); m, n =1, 2 (48)

Now consider,

�1(�)Y55(x) = (� + λ22)

2∑

n=1

r2n[δ + (λ21 − λ2

n)ςn]

= (� + λ22)(λ

21 − λ2

2)ς2r22

= (� + λ22)ς2 = δ

123


Similarly Eq. (47)1 can be proved.We introduce the matrix

G(x) = R Y(x)

F G(x) = F R Y(x)

= �(�)Y(x)

= δ(x) I(x) (49)

Thus G(x) satisfies the condition given by Eq. (23) and hence a proof has been made to thefollowing theorem:

Theorem The matrix G(x) defined by Eq. (49) is the fundamental solution of system of Eqs.(20) and (21).

5 Basic properties of the matrix G(x)

Property Each column of the matrix G(x) is the solution of the system of Eqs. (20, 21) atevery point x ∈ E3 except the origin.

Property The matrix G(x)can be written in the form

G = [

Ggh]

6×6

Gmn(x) = Rmn Y11(x)

Gmp(x) = Rmp Y55(x) m = 1, . . . , 6; n = 1, . . . , 4; p = 5, 6.

5.1 Special cases

If we neglect the effect of swelling porosity i.e. σ f = σ f f = ξ f f = μv = λv = ρf

0 = 0then we obtain corresponding results for elastic medium (EL).

6 Relevant dimensionless quantities

In order to illustrate theoretical results obtained in the preceding sections, we now presentsome numerical results. For numerical computation, the relevant dimensionless quantitiesare taken as:

a1 = 0.00510, a2 = 1.00005, δ21 = 0.36390, δ2

2 = 0.35460, ω∗ = 0.65849,

h1 = 0.00904, h2 = 0.22617, h3 = 1.77106, h4 = 1.28318

The software Matlab 7.0.4 has been used to determine the values of phase velocity andattenuation coefficient of plane waves. The variations of phase velocities and attenuationcoefficients with respect to frequency have been shown in Figs. 1, 2 and 3 and 4, 5 and 6respectively. In all Figures solid line corresponds to SP medium and dash line correspondsto EL medium.

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R. Kumar et al.

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Frequency ( (

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2

2.1P

hase

vel

ocity

(V

1) SPEL

Fig. 1 Variation of phase velocity V1 w.r.t. frequency

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 150

1

2

3

4

5

Pha

se v

eloc

ity (

V2)

SP

Frequency ( (


7 Phase velocity

Figures 1, 2, 3 and 4 depicts the variation in phase velocities (Vi ); i = 1,2,3,4 in SP mediumand EL medium. From Figs. 1 and 3, it is observed that phase velocity for SP mediuminitially oscillates then became stationary; whereas for EL medium, it remains stationary.

123


0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 150.56

0.6

0.64

0.68

0.72P

hase

vel

ocity

(V

3)

SPEL

Frequency ( (


0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 150

2

4

6

Pha

se v

eloc

ity (

V4)

SP

Frequency ( (


At initial stage the phase velocity for SP medium remain greater than the phase velocity forEL medium, but after a finite stage the reverse behavior is observed. The transversal phasevelocity V3 remain less than the longitudinal phase velocity V1 for both SP and EL media.From Figs. 2 and 4 it is noticed, that phase velocity V2 and V4 in SP medium increase withfrequency (ω) in the whole range. The values of phase velocity V4 remain greater than thevalues obtained for V2 in the whole range.

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R. Kumar et al.

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 150.2

0.4

0.6

0.8

1

1.2A

ttenu

atio

n co

effic

ient

(Q

1) SP

Frequency ( (

Fig. 5 Variation of attenuation coefficient Q1 w.r.t. frequency

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 150

0.5

1

1.5

2

2.5

3

3.5

Atte

nuat

ion

coef

ficie

nt (

Q2)

SP

Frequency ( (


7.1 Attenuation coefficient

Figures 5, 6, 7 and 8 shows the variation in attenuation coefficient (Qi ) i = 1, 2, 3, 4 w.r.t ω.From Figs. 5 and 7 it is observed that attenuation coefficient Q1 and Q3 initially oscillatesthen became stationary, whereas attenuation coefficient Q2 and Q4 increase with ω in thewhole range. The values of attenuation coefficient for transversal wave remain less the valuesobtained for longitudinal wave propagation in the whole range.

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0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 150.15

0.2

0.25

0.3

0.35

0.4A

ttenu

atio

n co

effic

ient

(Q

3)

SP

Frequency ( (


0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 150

1

2

3

Atte

nuat

ion

coef

ficie

nt (

Q4)

SP

Frequency ( (


8 Conclusion

From the above figures, we conclude that the phase velocity and attenuation coefficient insolid initially oscillates and became stationary, whereas, for fluid it increases with frequency.The fundamental solution G(x) of the system of Eqs. (20, 21) makes it possible to investigatethree dimensional boundary value problems of swelling porous elastic media with potentialmethod. The main results obtained in the classical theory of elasticity and thermoelasticitywith potential method are given in the book of Kupradze et al. [22].

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R. Kumar et al.

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