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Bounds for the effective heat conduction coefficient
MECHANICS RESEARCH COMMUNICATIONS
Mechanics Research Communications 29 (2002) 189–193 www.elsevier.com/locate/mechrescom
Bounds for the effective heat conduction coefficient Istvan Ecsedi Department of Mechanics, University of Miskolc, Miskolc-Egyetemv aros H-3515 Hungary Received 7 September 2001
Abstract The linear problem of the steady-state heat conduction is studied in isotropic nonhomogeneous hollow rigid bodies. Upper and lower bounds are derived for the effective heat conduction coefficient. It is proven that, the effective heat conduction coefficient of a compound body is between the weighted arithmetic and harmonic means of heat conduction coefficients of the homogeneous body components. Ó 2002 Elsevier Science Ltd. All rights reserved. Keywords: Heat conduction; Nonhomogeneous body; Effective heat conductivity coefficient; Upper and lower bound
1. Introduction The paper deals with the linear problem of steady-state heat conduction in nonhomogeneous hollow rigid bodies. The hollow body considered is bounded by the closed surfaces oB1 and oB2 . The ‘‘outer’’ boundary of the body B is oB2 and the ‘‘inner’’ boundary of body B is oB1 . The set of the inner points of B is denoted by B and the set of the points on the boundary of B is denoted by oB, it is evident oB ¼ oB1 [ oB2 and oB1 \ oB2 ¼ f0g. A point of B is indicated by the vector r ¼ x1 e1 þ x2 e2 þ x3 e3 in a given orthogonal Cartesian coordinate system ð0; x1 ; x2 ; x3 Þ with the unit vectors e1 , e2 , e3 . The temperature in the body is indicated by T ¼ T ðrÞ, r 2 B and the heat flux vector by q ¼ fq1 ; q2 ; q3 g, q ¼ qðrÞ r 2 B. The heat flux through a surface element, whose unit normal n, denoted by q, q ¼ q n. Here, the dot between two vectors denotes their scalar product. There is no distributed heat source in B. The heat conduction coefficient of the nonhomogeneous body B is k ¼ kðrÞ. It is assumed that T ¼ T1 on oB1 ;
T ¼ T2 on oB2 ;
T1 > T2 :
ð1:1Þ
The problem of steady-state conduction in the nonhomogenous body B is ruled by the balance equation r q ¼ 0 in B
ð1:2Þ
and by the Fourier’s law q ¼ krT in B
E-mail address: [email protected] (I. Ecsedi). 0093-6413/02/$ - see front matter Ó 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 9 3 - 6 4 1 3 ( 0 2 ) 0 0 2 3 8 - 0
ð1:3Þ
190
I. Ecsedi / Mechanics Research Communications 29 (2002) 189–193
with the boundary condition (1.1) (Carslaw and Jaeger, 1986; Ozisik, 1993). In the Eqs. (1.2) and (1.3) r is the gradient (del) operator. The total heat flow between the boundary surfaces oB1 and oB2 is Z Z Z q dS ¼ ðT1 T2 Þ kn rF dS ¼ ðT1 T2 Þ k j rF j2 dV ; ð1:4Þ Q¼ oB1
oB1
B
where F ¼ F ðrÞ is the solution to the next boundary-value problem r ðkrF Þ ¼ 0
in B;
ð1:5Þ
F ¼ 1 on oB1 ;
F ¼ 0 on oB2
ð1:6Þ
and dV is the volume element, dS is the surface element, n is outward unit normal vector on oB1 in (1.4). The temperature field T ¼ T ðrÞ of body B can be expressed in the term of F as T ðrÞ ¼ ðT1 T2 ÞF ðrÞ þ T2 :
ð1:7Þ
If the body is homogeneous then we have k ¼ k0 ¼ constant
ð1:8Þ
and F ðrÞ ¼ F0 ðrÞ, where F0 ¼ F0 ðrÞ, is the solution to the boundary value problem formulated in Eqs. (1.9) and (1.10): DF0 ¼ 0 in B; F0 ¼ 1 on oB1 ;
ð1:9Þ F0 ¼ 0 on oB2 :
ð1:10Þ
Here D ¼ r r is the Laplace operator. The heat transferred between the surfaces oB1 , oB2 per unit time in this case will be Z Z 2 n rF0 dS: Q0 ¼ ðT1 T2 Þk0 ðrF0 Þ dV ¼ ðT1 T2 Þk0
ð1:11Þ
oB1
B
2. Effective heat conduction coefficient The definition of the effective heat conduction coefficient (EHCC) is based on the equation Q ¼ Q0 :
ð2:1Þ
From the Eq. (2.1), we obtain the expression of the EHCC ke ¼ k0 , that is R kðrÞ j rF j2 dV ke ¼ BR : j rF0 j2 dV B
ð2:2Þ
If the body B is a composition of n different homogeneous component and the component ‘‘i’’ fills the region Bi 2 Bði ¼ 1; 2; . . . ; nÞ then we have Z Z n X 2 kðrÞ j rF j dV ¼ ki j rF j2 dV : ð2:3Þ B
i¼1 n
Bi
It is evident, that B ¼ [ Bi and Bi \ Bj ¼ f0g if i 6¼ j. In the Eq. (2.3) ki denotes the heat conduction coi¼1 efficient of that homogeneous component which fills the region Bi ði ¼ 1; 2; . . . ; nÞ. We note that, the definition of EHCC given by the formula (2.2) is different from which is used in Hasin and Shtrikman (1962)
I. Ecsedi / Mechanics Research Communications 29 (2002) 189–193
191
and Wojnar (1998). The aim of this paper is to give upper and lower bounds for the EHCC in the terms of F0 ¼ F0 ðx1 ; x2 ; x3 Þ. 3. Bounds for the EHCC Theorem 1. The two side bounding formula R R kðrÞ j rF0 j2 dV j rF0 j2 dV BR P ke P R B 1 2 j rF0 j dV j rF0 j2 dV B B kðrÞ
ð3:1Þ
holds. Theorem 2. For the composite body made of different homogeneous phases of any shape and arrangement of body components we have n X 1 aj kj P ke P Pn aj ; ð3:2Þ j¼1 kj
j¼1
where
R
B aj ¼ R j B
j rF0 j2 dV j rF0 j2 dV
ðj ¼ 1; 2; . . . ; nÞ;
ð3:3Þ
and n X
aj > 0 ðj ¼ 1; 2; . . . ; nÞ;
aj ¼ 1:
ð3:4Þ
j¼1
Proof. Starting from the Schwarz’s inequality Z 2 Z Z pðrÞa b dV 6 pðrÞa2 dV pðrÞb2 dV ; B
B
ð3:5Þ
B
where a ¼ aðrÞ, b ¼ bðrÞ are such vector–vector functions for which the integrals appeared in (3.5) exist and they have finite values with a given nonnegative weight function p ¼ pðrÞ. Putting pðrÞ ¼ kðrÞ;
a ¼ rF ;
b ¼ rF0
in (3.5), we get Z 2 Z Z kðrÞrF rF0 dV 6 kðrÞ j rF j2 dV kðrÞ j rF0 j2 dV : B
B
ð3:6Þ
B
By the application of the Leibnitz’s rule of the derivation of product function and Gauss–Green’s theorem transforming a volume integral to a surface integral we obtain Z Z Z Z kðrÞrF rF0 dV ¼ ðkðrÞrFF0 Þ r dV F0 r ðkðrÞrF Þ dV ¼ kðrÞF0 n rF dS B oB ZB Z B ¼ kðrÞF n rF dS ¼ ðkðrÞF rF Þ r dV oB Z ZB Z 2 ¼ kðrÞ j rF j dV þ F r ðkðrÞrF Þ dV ¼ kðrÞ j rF j2 dV : ð3:7Þ B
B
B
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I. Ecsedi / Mechanics Research Communications 29 (2002) 189–193
In the derivation above given, the Eqs. (1.5), (1.6) and (1.10) have been used. The combination of the Eqs. (2.2) and (3.7) with inequality (3.6) gives the upper bound relation which is formulated in (3.1) for EHCC. In order to prove the lower bound relation of EHCC contained in two side bound (3.1) we substitute 1 ; kðrÞ
pðrÞ ¼
a ¼ kðrÞrF ;
b ¼ rF0
in (3.5). This substitution yields the next relation: Z 2 Z Z j rF0 j2 dV : rF rF0 dV 6 kðrÞ j rF j2 dV kðrÞ B B B
ð3:8Þ
By the same method which was used in the derivation of Eq. (3.7) we obtain Z Z Z Z Z rF rF0 dV ¼ r ðF rF0 Þ dV F DF0 dB ¼ F n rF0 dS ¼ F0 n rF0 dS B B B oB oB Z Z Z Z ¼ r ðF0 rF0 Þ dV ¼ j rF0 j2 dV F0 DF0 dV ¼ j rF0 j2 dV : B
B
B
ð3:9Þ
B
Combining Eq. (2.2) and the inequality relation (3.8) with the Eq. (3.9) we get the lower bound relation asserted in Theorem 1. The application of Theorem 1 to the composite body which consists of n homogeneous components leads to the result formulated in the relation (3.2), since kðrÞ ¼ ki ¼ constant, r 2 Bi ði ¼ 1; 2; . . . ; nÞ.
4. Examples Example 4.1. Let us consider a solid body bounded by two concentric spheres with center at the origin and radii R1 and R2 ð0 < R1 < R2 Þ. Let the heat conduction coefficient be a known function of the spherical coordinates r, u, #, that is k ¼ kðr; u; #Þ. The connection between x1 , x2 , x3 and r, u, # is as follows: x1 ¼ r sin # cos u;
x2 ¼ r sin # sin u;
x3 ¼ r cos #:
The application of the bounding formula (3.1) gives the following lower and upper bounds for EHCC R R2 R 2p R p kðr;u;#Þ sin # d# du dr 2 r¼R1 u¼0 #¼0 r ke 6 ; ð4:1Þ 4p R11 R12
ke P R R 2 r¼R1
R 2p
4p R11 R12 Rp sin #
u¼0
#¼0 r2 kðr;u;#Þ
d# du dr
:
ð4:2Þ
Example 4.2. Let us consider the same body as in Example 4.1, but the heat condition coefficient is specified as p kðr; u; #Þ ¼ k1 for R1 6 r 6 R2 ; 0 6 u 6 2p; 0 6 # < ; 2 p kðr; u; #Þ ¼ k2 for R1 6 r 6 R2 ; 0 6 u 6 2p; < # 6 p: 2 This body is a composite of two homogeneous bodies having different values of heat conduction coefficients. Each of component bodies is bounded by two half spheres and a diameter plane.
I. Ecsedi / Mechanics Research Communications 29 (2002) 189–193
193
By the use of two side bounding relation (3.2) we get 1 ðk1 þ k2 Þ P ke P 2
1 k1
2 : þ k12
ð4:3Þ
Example 4.3. In this example the hollow spherical body consists of n solid spheres, each of radius d. These embedding spheres have the same heat conduction coefficient which is k2 . The centers of the spheres of radius d are spaced uniformly on a spherical surface of radius R3 . The centers of the spherical surfaces of radii R1 , R2 and R3 are the same point. The volume of the base (matrix) material is 4 Vm ¼ pðR32 R31 nd3 Þ; 3 and its heat conduction coefficient is k1 . The application of the relation (3.2) gives the result ð1 aÞk1 þ ak2 P ke P
1 a k1
1 ; þ ka2
ð4:4Þ
where n 4R3
a¼
2dR3 ‘n RR33 d þ þd R2 d2 3
1 R1
R12
:
ð4:5Þ
5. Conclusions A method to obtain upper and lower bounds for the EHCC is presented. The formulation of the steadystate heat conduction problem in nonhomogeneous hollow solid bodies is based on Fourier’s theory. Schwarz’s inequality is used with two different weighted functions to prove the bounding formulae. The lower–upper bound relation of the EHCC for composite bodies has a simple meaning, namely the upper bound expression is a weighted arithmetic mean and the lower bound expression is a weighted harmonic mean of the heat conduction coefficients of the homogeneous body components. The weight factors for the both two means are same they depend on the solution of heat conduction problem in a homogeneous body which is geometrically identical to the considered nonhomogeneous body.
References Carslaw, H.S., Jaeger, J.C., 1986. Conduction of Heat in Solids, second ed. Clarendon Press, Oxford. Ozisik, N., 1993. Heat Conduction. Wiley, Chichester. Hasin, Z., Shtrikman, S., 1962. A variational approach to the theory of effective magnetic permeability of multiphase materials. J. Appl. Phys. 23, 3125–3131. Wojnar, R., 1998. Upper and lower bounds on heat flux. J. Thermal Stresses 21, 381–403.
Mechanics Research Communications 29 (2002) 189–193 www.elsevier.com/locate/mechrescom
Bounds for the effective heat conduction coefficient Istvan Ecsedi Department of Mechanics, University of Miskolc, Miskolc-Egyetemv aros H-3515 Hungary Received 7 September 2001
Abstract The linear problem of the steady-state heat conduction is studied in isotropic nonhomogeneous hollow rigid bodies. Upper and lower bounds are derived for the effective heat conduction coefficient. It is proven that, the effective heat conduction coefficient of a compound body is between the weighted arithmetic and harmonic means of heat conduction coefficients of the homogeneous body components. Ó 2002 Elsevier Science Ltd. All rights reserved. Keywords: Heat conduction; Nonhomogeneous body; Effective heat conductivity coefficient; Upper and lower bound
1. Introduction The paper deals with the linear problem of steady-state heat conduction in nonhomogeneous hollow rigid bodies. The hollow body considered is bounded by the closed surfaces oB1 and oB2 . The ‘‘outer’’ boundary of the body B is oB2 and the ‘‘inner’’ boundary of body B is oB1 . The set of the inner points of B is denoted by B and the set of the points on the boundary of B is denoted by oB, it is evident oB ¼ oB1 [ oB2 and oB1 \ oB2 ¼ f0g. A point of B is indicated by the vector r ¼ x1 e1 þ x2 e2 þ x3 e3 in a given orthogonal Cartesian coordinate system ð0; x1 ; x2 ; x3 Þ with the unit vectors e1 , e2 , e3 . The temperature in the body is indicated by T ¼ T ðrÞ, r 2 B and the heat flux vector by q ¼ fq1 ; q2 ; q3 g, q ¼ qðrÞ r 2 B. The heat flux through a surface element, whose unit normal n, denoted by q, q ¼ q n. Here, the dot between two vectors denotes their scalar product. There is no distributed heat source in B. The heat conduction coefficient of the nonhomogeneous body B is k ¼ kðrÞ. It is assumed that T ¼ T1 on oB1 ;
T ¼ T2 on oB2 ;
T1 > T2 :
ð1:1Þ
The problem of steady-state conduction in the nonhomogenous body B is ruled by the balance equation r q ¼ 0 in B
ð1:2Þ
and by the Fourier’s law q ¼ krT in B
E-mail address: [email protected] (I. Ecsedi). 0093-6413/02/$ - see front matter Ó 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 9 3 - 6 4 1 3 ( 0 2 ) 0 0 2 3 8 - 0
ð1:3Þ
190
I. Ecsedi / Mechanics Research Communications 29 (2002) 189–193
with the boundary condition (1.1) (Carslaw and Jaeger, 1986; Ozisik, 1993). In the Eqs. (1.2) and (1.3) r is the gradient (del) operator. The total heat flow between the boundary surfaces oB1 and oB2 is Z Z Z q dS ¼ ðT1 T2 Þ kn rF dS ¼ ðT1 T2 Þ k j rF j2 dV ; ð1:4Þ Q¼ oB1
oB1
B
where F ¼ F ðrÞ is the solution to the next boundary-value problem r ðkrF Þ ¼ 0
in B;
ð1:5Þ
F ¼ 1 on oB1 ;
F ¼ 0 on oB2
ð1:6Þ
and dV is the volume element, dS is the surface element, n is outward unit normal vector on oB1 in (1.4). The temperature field T ¼ T ðrÞ of body B can be expressed in the term of F as T ðrÞ ¼ ðT1 T2 ÞF ðrÞ þ T2 :
ð1:7Þ
If the body is homogeneous then we have k ¼ k0 ¼ constant
ð1:8Þ
and F ðrÞ ¼ F0 ðrÞ, where F0 ¼ F0 ðrÞ, is the solution to the boundary value problem formulated in Eqs. (1.9) and (1.10): DF0 ¼ 0 in B; F0 ¼ 1 on oB1 ;
ð1:9Þ F0 ¼ 0 on oB2 :
ð1:10Þ
Here D ¼ r r is the Laplace operator. The heat transferred between the surfaces oB1 , oB2 per unit time in this case will be Z Z 2 n rF0 dS: Q0 ¼ ðT1 T2 Þk0 ðrF0 Þ dV ¼ ðT1 T2 Þk0
ð1:11Þ
oB1
B
2. Effective heat conduction coefficient The definition of the effective heat conduction coefficient (EHCC) is based on the equation Q ¼ Q0 :
ð2:1Þ
From the Eq. (2.1), we obtain the expression of the EHCC ke ¼ k0 , that is R kðrÞ j rF j2 dV ke ¼ BR : j rF0 j2 dV B
ð2:2Þ
If the body B is a composition of n different homogeneous component and the component ‘‘i’’ fills the region Bi 2 Bði ¼ 1; 2; . . . ; nÞ then we have Z Z n X 2 kðrÞ j rF j dV ¼ ki j rF j2 dV : ð2:3Þ B
i¼1 n
Bi
It is evident, that B ¼ [ Bi and Bi \ Bj ¼ f0g if i 6¼ j. In the Eq. (2.3) ki denotes the heat conduction coi¼1 efficient of that homogeneous component which fills the region Bi ði ¼ 1; 2; . . . ; nÞ. We note that, the definition of EHCC given by the formula (2.2) is different from which is used in Hasin and Shtrikman (1962)
I. Ecsedi / Mechanics Research Communications 29 (2002) 189–193
191
and Wojnar (1998). The aim of this paper is to give upper and lower bounds for the EHCC in the terms of F0 ¼ F0 ðx1 ; x2 ; x3 Þ. 3. Bounds for the EHCC Theorem 1. The two side bounding formula R R kðrÞ j rF0 j2 dV j rF0 j2 dV BR P ke P R B 1 2 j rF0 j dV j rF0 j2 dV B B kðrÞ
ð3:1Þ
holds. Theorem 2. For the composite body made of different homogeneous phases of any shape and arrangement of body components we have n X 1 aj kj P ke P Pn aj ; ð3:2Þ j¼1 kj
j¼1
where
R
B aj ¼ R j B
j rF0 j2 dV j rF0 j2 dV
ðj ¼ 1; 2; . . . ; nÞ;
ð3:3Þ
and n X
aj > 0 ðj ¼ 1; 2; . . . ; nÞ;
aj ¼ 1:
ð3:4Þ
j¼1
Proof. Starting from the Schwarz’s inequality Z 2 Z Z pðrÞa b dV 6 pðrÞa2 dV pðrÞb2 dV ; B
B
ð3:5Þ
B
where a ¼ aðrÞ, b ¼ bðrÞ are such vector–vector functions for which the integrals appeared in (3.5) exist and they have finite values with a given nonnegative weight function p ¼ pðrÞ. Putting pðrÞ ¼ kðrÞ;
a ¼ rF ;
b ¼ rF0
in (3.5), we get Z 2 Z Z kðrÞrF rF0 dV 6 kðrÞ j rF j2 dV kðrÞ j rF0 j2 dV : B
B
ð3:6Þ
B
By the application of the Leibnitz’s rule of the derivation of product function and Gauss–Green’s theorem transforming a volume integral to a surface integral we obtain Z Z Z Z kðrÞrF rF0 dV ¼ ðkðrÞrFF0 Þ r dV F0 r ðkðrÞrF Þ dV ¼ kðrÞF0 n rF dS B oB ZB Z B ¼ kðrÞF n rF dS ¼ ðkðrÞF rF Þ r dV oB Z ZB Z 2 ¼ kðrÞ j rF j dV þ F r ðkðrÞrF Þ dV ¼ kðrÞ j rF j2 dV : ð3:7Þ B
B
B
192
I. Ecsedi / Mechanics Research Communications 29 (2002) 189–193
In the derivation above given, the Eqs. (1.5), (1.6) and (1.10) have been used. The combination of the Eqs. (2.2) and (3.7) with inequality (3.6) gives the upper bound relation which is formulated in (3.1) for EHCC. In order to prove the lower bound relation of EHCC contained in two side bound (3.1) we substitute 1 ; kðrÞ
pðrÞ ¼
a ¼ kðrÞrF ;
b ¼ rF0
in (3.5). This substitution yields the next relation: Z 2 Z Z j rF0 j2 dV : rF rF0 dV 6 kðrÞ j rF j2 dV kðrÞ B B B
ð3:8Þ
By the same method which was used in the derivation of Eq. (3.7) we obtain Z Z Z Z Z rF rF0 dV ¼ r ðF rF0 Þ dV F DF0 dB ¼ F n rF0 dS ¼ F0 n rF0 dS B B B oB oB Z Z Z Z ¼ r ðF0 rF0 Þ dV ¼ j rF0 j2 dV F0 DF0 dV ¼ j rF0 j2 dV : B
B
B
ð3:9Þ
B
Combining Eq. (2.2) and the inequality relation (3.8) with the Eq. (3.9) we get the lower bound relation asserted in Theorem 1. The application of Theorem 1 to the composite body which consists of n homogeneous components leads to the result formulated in the relation (3.2), since kðrÞ ¼ ki ¼ constant, r 2 Bi ði ¼ 1; 2; . . . ; nÞ.
4. Examples Example 4.1. Let us consider a solid body bounded by two concentric spheres with center at the origin and radii R1 and R2 ð0 < R1 < R2 Þ. Let the heat conduction coefficient be a known function of the spherical coordinates r, u, #, that is k ¼ kðr; u; #Þ. The connection between x1 , x2 , x3 and r, u, # is as follows: x1 ¼ r sin # cos u;
x2 ¼ r sin # sin u;
x3 ¼ r cos #:
The application of the bounding formula (3.1) gives the following lower and upper bounds for EHCC R R2 R 2p R p kðr;u;#Þ sin # d# du dr 2 r¼R1 u¼0 #¼0 r ke 6 ; ð4:1Þ 4p R11 R12
ke P R R 2 r¼R1
R 2p
4p R11 R12 Rp sin #
u¼0
#¼0 r2 kðr;u;#Þ
d# du dr
:
ð4:2Þ
Example 4.2. Let us consider the same body as in Example 4.1, but the heat condition coefficient is specified as p kðr; u; #Þ ¼ k1 for R1 6 r 6 R2 ; 0 6 u 6 2p; 0 6 # < ; 2 p kðr; u; #Þ ¼ k2 for R1 6 r 6 R2 ; 0 6 u 6 2p; < # 6 p: 2 This body is a composite of two homogeneous bodies having different values of heat conduction coefficients. Each of component bodies is bounded by two half spheres and a diameter plane.
I. Ecsedi / Mechanics Research Communications 29 (2002) 189–193
193
By the use of two side bounding relation (3.2) we get 1 ðk1 þ k2 Þ P ke P 2
1 k1
2 : þ k12
ð4:3Þ
Example 4.3. In this example the hollow spherical body consists of n solid spheres, each of radius d. These embedding spheres have the same heat conduction coefficient which is k2 . The centers of the spheres of radius d are spaced uniformly on a spherical surface of radius R3 . The centers of the spherical surfaces of radii R1 , R2 and R3 are the same point. The volume of the base (matrix) material is 4 Vm ¼ pðR32 R31 nd3 Þ; 3 and its heat conduction coefficient is k1 . The application of the relation (3.2) gives the result ð1 aÞk1 þ ak2 P ke P
1 a k1
1 ; þ ka2
ð4:4Þ
where n 4R3
a¼
2dR3 ‘n RR33 d þ þd R2 d2 3
1 R1
R12
:
ð4:5Þ
5. Conclusions A method to obtain upper and lower bounds for the EHCC is presented. The formulation of the steadystate heat conduction problem in nonhomogeneous hollow solid bodies is based on Fourier’s theory. Schwarz’s inequality is used with two different weighted functions to prove the bounding formulae. The lower–upper bound relation of the EHCC for composite bodies has a simple meaning, namely the upper bound expression is a weighted arithmetic mean and the lower bound expression is a weighted harmonic mean of the heat conduction coefficients of the homogeneous body components. The weight factors for the both two means are same they depend on the solution of heat conduction problem in a homogeneous body which is geometrically identical to the considered nonhomogeneous body.
References Carslaw, H.S., Jaeger, J.C., 1986. Conduction of Heat in Solids, second ed. Clarendon Press, Oxford. Ozisik, N., 1993. Heat Conduction. Wiley, Chichester. Hasin, Z., Shtrikman, S., 1962. A variational approach to the theory of effective magnetic permeability of multiphase materials. J. Appl. Phys. 23, 3125–3131. Wojnar, R., 1998. Upper and lower bounds on heat flux. J. Thermal Stresses 21, 381–403.