Neutral inhomogeneities in a two-dimensional steady-state heat

conduction problem

ISTVÁN ECSEDI1, ÁKOS JÓZSEF LENGYEL2*

1,2Institute of Applied Mechanics

1,2University of Miskolc

1,2H-3515, Miskolc-Egyetemváros, Miskolc

1,2HUNGARY

Abstract: A steady-state heat conduction problem is considered in a two-dimensional solid body which is filled

up composite circular inclusions. The composite circular inclusions consist of a core and a coating both of which

are cylindrically orthotropic. In this paper the neutral inhomogeneity is defined as a foreign body (inclusion)

which can be introduced into the host body (matrix) without distributing the temperature field in the originally

homogeneous body. The perfect thermal contacts are assumed to be between the different components of non-

homogeneous bodies.

Key-Words: heat conduction, steady-state, neutral inhomogeneity, two-dimensional

Received: July 28, 2021. Revised: March 19, 2022. Accepted: April 23, 2022. Published: June 7, 2022.

1 Introduction

The existence of the neutral inhomogeneities in a two-

dimensional rectangular body in the case of steady-

state heat conductance is proven. It is assumed that

the original body which has no inclusions is sub-

jected to constant temperature gradient. The consid-

ered two-dimensional body is shown in Fig. 1. The

temperature field of the rectangle in the Cartesian co-

ordinate x,yis prescribed as

T0(y) = −t1

Ly+t1

L,(1)

where t1is a given temperature. It is evident that T0=

T0(y)satisfies the boundary conditions

T0(−L1) = t1, T0(L2) = 0 (2)

and the heat flux vector q0is as follows

q0=k0

Ley,(3)

where k0is the thermal conductance of the homo-

geneous isotropic rectangle. The unit vectors of

the Cartesian coordinate system Oxy are exand ey

(Fig. 1).

The composite cylindrical inhomogeneity is intro-

duced to the homogeneous isotropic rectangular body

as shown in Fig. 2. The inclusion is placed to the

origin of the Cartesian coordinate system Oxy and it

contains two different material components. The first

component is a coating occupying the hollow circu-

lar domain A1and the second component is the core

occupying the solid circular domain A2. In the polar

coordinate system Orϕ the domains A1and A2are

defined as

A1={(r, ϕ)|R2≤r≤R1,0≤ϕ≤2π},(4)

a1a2

Figure 1: Rectangular body subjected to constant heat

flux vector.

A2={(r, ϕ)|0≤r≤R2,0≤ϕ≤2π}.(5)

The materials of the coating and core are homoge-

neous and cylindrically orthotropic with the thermal

conductivities k1r,k1ϕand k2r,k2ϕ. In the polar co-

ordinate system Orϕ

T0(r, ϕ) = −t1

Lsin ϕ+t1

L,(6)

q0(r) = kt1

L(ersin ϕ+eϕcos ϕ).(7)

The unit vectors of the polar coordinate system Orϕ

are er(ϕ)and eϕ(ϕ)(Fig. 2).

WSEAS TRANSACTIONS on HEAT and MASS TRANSFER

DOI: 10.37394/232012.2022.17.15

István Ecsedi, Ákos József Lengyel

E-ISSN: 2224-3461

136

Volume 17, 2022

a1a2

eϕ

T0= 0

OP =r

Figure 2: A coated circular inhomogeneity in homo-

geneous rectangular body.

The existence of neutral inhomogeneity for imper-

fect thermal contact was analysed by Benveniste and

Miloh [1].

For the three-dimensional steady-state heat con-

duction problem the neutral inhomogeneity with

spherical orthotropic inclusions was studied in paper

[2], where the volume fraction of the core in all in-

homogeneities is the same. The functionally graded

material properties of inclusions are also discussed in

paper by Ecsedi and Baksa [2].

The existence of the neutral inhomogeneities in

different boundary-value problems of elasticity was

studied by Ru [3], Ru et al. [4], Benveniste and Chen

[5], Ecsedi and Baksa [6].

The steady-state temperature field in a two-

dimensional cylindrically anisotropic homogeneous

body is described by the following partial differential

equation in cylindrical coordinate system Orϕ [7]

kir

∂2Ti

∂r2+kir

∂Ti

∂r +kiϕ

∂2Ti

∂ϕ2= 0,

(r, ϕ)∈Ai,(i= 1,2).

(8)

The expression of the heat flux in radial direction is

qir(r, ϕ) = −kir

∂Ti

∂r ,(r, ϕ)∈Ai,(i= 1,2).(9)

2 Governing equations

According to equation (6) and

q0r=q0·er=k0

Lsin ϕ, (10)

where the dot between two vectors denotes their

scalar product we assume that

T1(r, ϕ) = F1(r)sin ϕ+t1

L,(r, ϕ)∈A1,(11)

a1a2

Figure 3: Several circular inhomogeneities in rectan-

gular homogeneous two-dimensional body.

T2(r, ϕ) = F2(r)sin ϕ+t1

L,(r, ϕ)∈A2,(12)

From the partial differential equation (8) it follows

that

d2Fi

dr2+1

dFi

dr−k2

r2Fi= 0, ki=skiϕ

kir

for i= 1, R2≤r≤R1,

for i= 2,0≤r≤R2.

(13)

The solution of the ordinary differential equation (13)

for F1=F1(r)and F2=F2(r)are

F1(r) = C1rk1+C2r−k1, R2≤r≤R1,(14)

F2(r) = C3rk2+C4r−k2,0≤r≤R2.(15)

The function F2=F2(r)is bounded at r= 0, from

this fact it follows that

C4= 0.(16)

The temperature fields in inhomogeneity are

T1(r, ϕ) = C1rk1+C2r−k1sin ϕ+t1

(r, ϕ)∈A1,

(17)

T2(r, ϕ) = C3rk2sin ϕ+t1

(r, ϕ)∈A2.

(18)

WSEAS TRANSACTIONS on HEAT and MASS TRANSFER

DOI: 10.37394/232012.2022.17.15

István Ecsedi, Ákos József Lengyel

E-ISSN: 2224-3461

137

Volume 17, 2022

The whole temperature field of the rectangular body

with cylindrically anisotropic inclusion is as follows

T(r, ϕ) = (H(r)−H(r−R2)) T2(r, ϕ)+

+ (H(r−R2)−H(r−R1)) T1(r, ϕ)+

+H(r−R1)T0(r, ϕ),(r, ϕ)∈A=

={(x, y)| − a1≤x≤a2,−L1≤y≤L2}.

(19)

Here, H=H(r)is the Heaviside function [8]. In the

case of perfect thermal contact the following equa-

tions are valid

T1(R1, ϕ) = T0(R1, ϕ),0≤ϕ≤2π, (20)

q1r(R1, ϕ) = q0r(R1, ϕ),0≤ϕ≤2π, (21)

T1(R2, ϕ) = T2(R2, ϕ),0≤ϕ≤2π, (22)

q1r(R2, ϕ) = q2r(R2, ϕ),0≤ϕ≤2π. (23)

The system of equations (20–23) contains only three

unknown C1,C2and C3which has unique solution

if the geometrical and material properties satisfy cer-

tain conditions. Section 3 of this paper deals with the

answering the question what kind of connection must

exist between R1,R2and k0,k1r,k1ϕ,k2r,k2ϕto

compute the constants C1,C2and C3from the sys-

tem of equations (20–23).

3 Formulation of the conditions of

neutral inhomogeneity

The detailed form of system of equations (20–23) is

C1Rk1

1+C2R−k1

1+t1

L= 0,(24)

κ1C1Rk1

1−κ1C2R−k1

1+k0t1

L= 0,(25)

C1Rk1

2+C2R−k1

2−C3Rk2

2= 0,(26)

κ1C1Rk1

2−κ1C2R−k1

2−κ2C3Rk2

2= 0,(27)

where

κ1=pk1rk1ϕ, κ2=pk2rk2ϕ.(28)

System of equations (24–27) generates a homoge-

neous system of linear equations for the C1,C2,C3

and

X=t1

L.(29)

Figure 4: The contour lines of the temperature func-

tion.

This system of equations has only non-trivial solu-

tions for C1,C2,C3and Xif its determinant D0van-

ishes, that is

D0=

Rk1

1R−k1

10 1

κ1Rk1

1−κ1R−k1

10k0

Rk1

2R−k1

2−Rk2

κ1Rk1

2−κ1R−k1

2−κ2Rk2



= 0.

(30)

After some manipulations the determinant D0can be

written in the form

D0=

Rk1

1R−k1

10 1

d1Rk1

1−d2R−k1

10 0

Rk1

2R−k1

2−Rk2

d3Rk1

2−d4R−k1

20 0



= 0,(31)

where

d1=1−k0

κ1, d2=1 + k0

κ1,

d3=1−κ2

κ1, d4=1 + κ2

κ1.

(32)

The condition of the neutral inhomogeneity which as-

sures that there is a non-trivial solution of homoge-

neous system of linear equations for C1,C2,C3and

Xcan be formulated as

1 + k0

κ11−κ2

κ1R−k1

1Rk1

2−

−1−k0

κ11 + κ2

κ1Rk1

1R−k1

2= 0.

(33)

WSEAS TRANSACTIONS on HEAT and MASS TRANSFER

DOI: 10.37394/232012.2022.17.15

István Ecsedi, Ákos József Lengyel

E-ISSN: 2224-3461

138

Volume 17, 2022

Figure 5: The contour lines of the radial component

of heat flux vector.

Independently of the size and the position of neutral

inhomogeneity equation (33) satisfies if

k0=κ1=κ2.(34)

It must be noted that there is no restriction to the po-

sition of the origin of the coordinate system Oxy.

From equations (24–27) the following formulae

can be derived for C1,C2and C3

C1=−t1

R1−k1

L, C2= 0,

C3=−t1

R1−k1

LRk1

2R−k2

(35)

The expression of radial component of the heat flux

vector in the domain A1∪A2is

q2r(r, ϕ) = t1k2r

R1−k1

LRk1

2R−k2

2rk2−1sin ϕ,

0≤r≤R1,0≤ϕ≤2π,

(36)

q1r(r, ϕ) = t1k1r

R1−k1

Lrk1−1sin ϕ,

R2≤r≤R1,0≤ϕ≤2π.

(37)

On the whole rectangular domain the radial compo-

nent of the heat flux vector is as follows

qr(r, ϕ) = (H(r)−H(r−R1)) q1r(r, ϕ)+

+ (H(r−R1)−H(r−R2)) q2r(r, ϕ)+

+H(r−R2)q0r(r, ϕ).

(38)

Figure 6: The plots of temperature function as a func-

tion of r.

Figure 7: The graphs of the radial component of heat

flux vector as a function of r.

WSEAS TRANSACTIONS on HEAT and MASS TRANSFER

DOI: 10.37394/232012.2022.17.15

István Ecsedi, Ákos József Lengyel

E-ISSN: 2224-3461

139

Volume 17, 2022

It is evident that in the case of several circular cylin-

drically anisotropic inclusions (Fig. 3) when the tem-

perature field of the host body is given by equation

(1), if

k0=pkirkiϕ,(i= 1,2. . . , N)(39)

then the temperature field does not disturb outside of

the inclusions.

4 Numerical example

The numerical example uses the following data: t1=

200 K, L1= 0.8m, L2= 0.8m, a1= 0.8m,

a2= 0.8m, R1= 0.25 m, R2= 0.15 m, k1r= 45

W/mK, k1ϕ= 62 W/mK, k2r= 67.5W/mK, k2ϕ=

41.33333 W/mK, k0= 52.82045058 W/mK.

The contour lines of the temperature function T=

T(r, ϕ)is shown in Fig. 4. The contour lines of radial

component of the heat flux vector is given in Fig. 5.

The plots of function T=T(r, ϕ)for five different

values of ϕ(ϕ= 0,ϕ=π

6,ϕ=π

4,ϕ=π

3,ϕ=π

as a function of rfor 0≤r≤5R1are shown in

Fig. 6. The graphs of the radial component of heat

flux vector are presented for five different values of

ϕ(ϕ= 0,ϕ=π

6,ϕ=π

4,ϕ=π

3,ϕ=π

2) as a

function of radial coordinate rfor 0≤r≤5R1in

Fig. 7.

5 Conclusion

Paper gives the existence conditions of neutral inho-

mogeneities in a rectangular domain for a one dimen-

sional steady-state heat flow problem. The compos-

ite inclusions consist of a core and coating which are

cylindrically orthotropic. A numerical example illus-

trates the validity of the presented theory. The main

result of the paper is a contribution to the existing ex-

act benchmark solution for heat conduction in com-

posite solid bodies.

References:

[1] Y. Benveniste, T. Miloh, Neutral inhomo-

geneities in condition phenomena, Journal of the

Mechanics and Physics of Solids, Vol.47, No.9,

1999, pp. 1873-1892.

[2] I. Ecsedi, A. Baksa, Neutral inhomogeneities in

a steady-state heat conduction problem, Interna-

tional Review of Mechanical Engineering, Vol.6,

No.2, 2012, pp. 245-250.

[3] C.-Q. Ru, Interface design of neutral elastic inclu-

sions, International Journal of Solids and Struc-

tures, Vol.35, No.7-8, 1998, pp. 559-572.

[4] C.-Q. Ru, P. Schiavone, A. Mioduchowski, Uni-

formity of Stresses Within a Three-Phase Elliptic

Inclusion in Anti-Plane Shear, Journal of Elastic-

ity, Vol.52, No.2, 1998, pp. 121-128.

[5] Y. Benveniste, T. Chen, The Saint-Venant tor-

sion of a circular bar consisting of a compos-

ite cylinder assemblage with cylindrically or-

thotropic constituents, International Journal of

Solids and Structures, Vol.40, No.25, 2003, pp.

7093-7107.

[6] I. Ecsedi, A. Baksa, Neutral spherical inhomo-

geneities in a twisted circular cylindrical bar, Acta

Mechanica, Vol.224, No.12, 2013, pp. 2929-

2935.

[7] H. S. Carslaw, J. C. Jaeger, Conduction of Heat

in Solids, Oxford University Press, UK, 1959.

[8] G. James, Advanced Modern Engineering Math-

ematics, Pearson Education Limited, Essex, Eng-

land, Third edition, 2004.

Creative Commons Attribution

License 4.0 (Attribution 4.0

International , CC BY 4.0)

This article is published under the terms of the

Creative Commons Attribution License 4.0

https://creativecommons.org/li-

censes/by/4.0/deed.en_US

WSEAS TRANSACTIONS on HEAT and MASS TRANSFER

DOI: 10.37394/232012.2022.17.15

István Ecsedi, Ákos József Lengyel

E-ISSN: 2224-3461

140

Volume 17, 2022