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On the thermomechanical consistency of the time differential dual-phase-lag models of heat conduction
International Journal of Heat and Mass Transfer 114 (2017) 277–285
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International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
On the thermomechanical consistency of the time differential dual-phase-lag models of heat conduction Stan Chiriţa˘ a,b,⇑, Michele Ciarletta c, Vincenzo Tibullo c a
Faculty of Mathematics, Al. I. Cuza University of Iasi, 700506 Iasi, Romania Octav Mayer Mathematics Institute, Romanian Academy, 700505 Iasi, Romania c University of Salerno, via Giovanni Paolo II n. 132, 84084 Fisciano, SA, Italy b
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 1 March 2017 Received in revised form 6 June 2017 Accepted 15 June 2017
This paper deals with the time differential dual-phase-lag heat transfer models aiming, at first, to identify the eventually restrictions that make them thermodynamically consistent. At a first glance it can be observed that the capability of a time differential dual-phase-lag model of heat conduction to describe real phenomena depends on the properties of the differential operators involved in the related constitutive equation. In fact, the constitutive equation is viewed as an ordinary differential equation in terms of the heat flux components (or in terms of the temperature gradient) and it results that, for approximation orders greater than or equal to five, the corresponding characteristic equation has at least a complex root having a positive real part. That leads to a heat flux component (or temperature gradient) that grows to infinity when the time tends to infinity and so there occur some instabilities. Instead, when the approximation orders are lower than or equal to four, this is not the case and there is the need to study the compatibility with the Second Law of Thermodynamics. To this aim the related constitutive equation is reformulated within the system of the fading memory theory, and thus the heat flux vector is written in terms of the history of the temperature gradient and on this basis the compatibility of the model with the thermodynamical principles is analyzed. Ó 2017 Elsevier Ltd. All rights reserved.
2010 MSC: 74F05 80A20 Keywords: Time differential dual-phase-lag model Heat conduction Delay times Stability systems Thermodynamic compatibility
1. Introduction The dual-phase-lag model of heat conduction proposed in [1–3] distinguishes the time instant t þ sq , at which the heat flux flows through a material volume and the time instant t þ sT , at which the temperature gradient establishes across the same material volume:
qi ðx; t þ sq Þ ¼ kij ðxÞT ;j ðx; t þ sT Þ;
with
sq ; sT P 0:
ð1Þ
The above constitutive equation states, synthesizing its meaning, that the temperature gradient T ;j at a certain time t þ sT results in a heat flux vector qi at a different time t þ sq . In the above constitutive Eq. (1), besides the explicit dependence upon the spatial variable, we point out that qi are the components of the heat flux vector, T represents the temperature variation from the constant reference temperature T 0 > 0 and kij are the components of the conductivity tensor; moreover, t is the time variable while sq and sT are ⇑ Corresponding author. E-mail addresses: [email protected] (S. Chiriţa˘), [email protected] (M. Ciarletta), [email protected] (V. Tibullo). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2017.06.071 0017-9310/Ó 2017 Elsevier Ltd. All rights reserved.
the phase lags (or delay times) of the heat flux and of the temperature gradient, respectively. In particular, sq is a relaxation time connected to the fast-transient effects of thermal inertia, while sT is caused by microstructural interactions, such as phonon scattering or phonon-electron interactions [4]. In addition to the thermal conductivity, the phase lags sT and sq are treated as two additional intrinsic thermal properties characterizing the energy-bearing capacity of the material. Eq. (1) describing the lagging behavior in heat transport, when coupled with the energy equation
qi;i ðx; tÞ þ .ðxÞrðx; tÞ ¼ aðxÞ
@T ðx; tÞ; @t
ð2Þ
displays two coupled differential equations of a delayed type. Due to the general time shifts at different scales, sT and sq , no general solution has been known yet. The refined structure of the lagging response depicted by equations (1) and (2), however, has been illustrated by Tzou [3] by expanding Eq. (1) in terms of the Taylor’s series with respect to time:
S. Chiriţa˘ et al. / International Journal of Heat and Mass Transfer 114 (2017) 277–285
278
qi ðx;tÞ þ
sq @qi 1! @t "
ðx;t Þ þ
¼ kij ðxÞ T ;j ðx;tÞ þ
s2q @ 2 qi 2! @t
2
sT @T ;j 1! @t
ðx;t Þ þ þ
ðx;t Þ þ
snq @ n qi n! @t n
2
s @ T ;j 2 T
2! @t 2
dynamic consistency of the model in concern is established when
ðx;t Þ
ðx;t Þ þ þ
m
s @ T ;j m T
m! @tm
# ðx;tÞ : ð3Þ
An interesting discussion concerning this expansion has been developed by Tzou [3] when n m ¼ 0 or n m ¼ 1, relating the progressive interchange between the diffusive and wave behaviors. We emphasize that the related time differential models obtained considering the Taylor series expansions of both sides of the Eq. (1) and retaining terms up to suitable orders in sq and sT (namely, first or second orders in sq and sT ) have been widely investigated with respect to their thermodynamic consistency as well as to interesting stability issues and wave propagation (see, for example, [5–9]). However, the general form of the time differential dual-phase-lag model as given by (3) wasn’t treated up to now, except for the paper by Quintanilla and Racke [10], where the spatial behavior is studied for solutions of the equation obtained by eliminating the heat flux vector between the constitutive Eq. (3) and the energy Eq. (2), provided n ¼ m or n ¼ m þ 1. The main purpose of this paper is to study the thermodynamical and mechanical consistency of the constitutive Eq. (3). We infer that the feasibility study of this constitutive equation greatly depends on the structure of the differential operators involved in its mathematical expression. In fact, if we consider the constitutive Eq. (3) as an ordinary linear differential equation in terms of the unknown function qi ðtÞ (or, equivalently, in terms of the unknown function T ;i ðtÞ) then we can observe that for n P 5 (or m P 5) it admits at least a complex root having a positive real part. That implies that qi ðtÞ (or T ;i ðtÞ) can tends to infinity when the time tends to infinity and so we are led to instability situations. On this way we conclude that the time differential dual-phase-lag model based on a constitutive equation of type (3) with n P 5 or m P 5 cannot be considered able to describe real mechanical situations. Instead, when n ¼ 0; 1; 2; 3; 4 and m ¼ 0; 1; 2; 3; 4 this is not the case and we have to study the thermodynamic consistency of the corresponding model. To this aim we follow [6,11] and we reformulate the constitutive Eq. (3) in such a way that the heat flux vector qi depends on the history of the temperature gradient. In this sense we rewrite the Eq. (3) in the framework of Gurtin and Pipkin [12] and Coleman and Gurtin [13] fading memory theory, and on this basis we analyze the compatibility of the model with the thermodynamical principles. Precisely, the thermo-
ðm; nÞ 2 fð0; 0Þ; ð1; 0Þ; ð0; 1Þ; ð2; 1Þ; ð1; 2Þ; ð2; 2Þ; ð3; 2Þ; ð2; 3Þ; ð3; 3Þ; ð3; 4Þ; ð4; 3Þ; ð4; 4Þg, provided appropriate restrictions are placed on the delay times.
2. Thermomechanical consistency of the model In this Section we consider the Eq. (3) as an ordinary linear nonhomogeneous differential (in time variable) equation in terms of the heat flux vector components and observe that its characteristic equation is
1 n n 1 1 1 s k þ sn1 kn1 þ þ s2q k2 þ sq k þ 1 ¼ 0: n! q ðn 1Þ! q 2! 1!
ð4Þ
This equation is connected with the partial sums of the Maclaurin series for the exponential function e z and with the incomplete gamma function and its roots have been intensively studied in literature (see e. g. Eneström [14–17]). On the basis of the EneströmKakeya theorem it follows that all the roots of the Eq. (4) lie outside of the disk of radius s1q . Moreover, the Eq. (4) has no real root if n is even, while when n is odd, it has only one real root. However, here we are interested if this equation has at least a complex root with a positive real part. To this aim we outline the results obtained by Gábor Szegö [18] and Jean Dieudonné [19] who showed that the roots of the scaled exponential sum function approach the portion of the Szegö curve: jz expð1 zÞj ¼ 1 within the unit disk as n ! 1. Moreover, with the aim to visualize this result for n P 5 we recommend the simulation for the software package Wolfram Mathematica 11 presented in the Appendix (see also the Fig. 1).
Table 1 The values of x ¼ sq k with k solution of the characteristic Eq. (4) for n ¼ 1; 2; 3; 4. n
x ¼ sq k
1 2
1 1:0 1:0 i
3
1:5961 0:70196 1:8073 i
4
0:27056 2:5048 i 1:7294 0:88897 i
Fig. 1. Roots of the exponential sum for n ¼ 4 and for n ¼ 5.
S. Chiriţa˘ et al. / International Journal of Heat and Mass Transfer 114 (2017) 277–285
In view of the above discussion we can conclude that, for values of n greater than or equal to five, the characteristic Eq. (4) admits at least one complex root having a positive real part and this leads to a couple fT; qi g going to infinity as time tends to infinity, that is we are led to instability of the system. In conclusion, for n P 5 the corresponding models of dual-phase-lag of heat conduction are not suitable to describe the lagging behavior. Furthermore, we write the constitutive Eq. (3) in the following form
T ;i ðx;t Þ þ
sT @T ;i 1! @t "
ðx;t Þ þ
¼ K ij ðxÞ qj ðx;t Þ þ
s2T @ 2 T ;i 2! @t 2
sq @qj 1! @t
ðx;tÞ þ þ
ðx;tÞ þ
m! @t m
2! @t 2
ðx;tÞ þ þ
n
s @ qj n q
n! @t n
ðx;tÞ ;
where K ij are so that
ð6Þ
Following an argument similar to that in the above discussion, this time the unknown function being considered T ;i , we can conclude that, for values of m greater than or equal to five, the corresponding constitutive Eq. (3) leads to models of dual-phase-lag of heat conduction that are not suitable to describe the lagging behavior. On the other side, as it can be seen from the Table 1, it follows that for n ¼ 1; 2; 3; 4 all the solutions of the Eq. (4) have a negative real part and hence the corresponding contribution in the heat flux vector components can lead to an asymptotic stable behavior in time couple fT; qi g. In such cases we have to study further the consistency of the constitutive Eq. (3) with the Second Law of Thermodynamics. In conclusion, the constitutive Eq. (3) can be thermodynamically consistent only for n 2 f0; 1; 2; 3; 4g and m 2 f0; 1; 2; 3; 4g. Thus, in what follows we will study the thermodynamical consistency of the constitutive Eq. (3) for ðn; mÞ 2 f0; 1; 2; 3; 4g f0; 1; 2; 3; 4g. For future convenience, we will discuss separately each of the above cases. In order to study the cases involved in the Table 1, we consider the constitutive Eq. (3) as a memory constitutive equation of the type described in Gurtin and Pipkin [12] and Coleman and Gurtin [13]. 3. Case ðn; mÞ ¼ ð0; 0Þ This case yields the classical Fourier law of heat flux which is compatible with the Second Law of Thermodynamics.
The case ðn; mÞ ¼ ð1; 0Þ yields the Cattaneo-Maxwell equation of the heat flux vector which is consistent with thermodynamics for all sq > 0 and it is equivalent to the constitutive equation with fading memory
qi ðx; tÞ ¼
sq
Z
1
s=sq
e 0
kij ðxÞT ;j ðx; t sÞds:
ð7Þ
1
sT
Z 0
1
es=sT K ij ðxÞqj ðx; t sÞds:
Z
1
sq
1
0
h i es=sq kij ðxÞ T ;j ðx; t sÞ þ sT T_ ;j ðx; t sÞ ds;
ð9Þ
which integrated by parts gives
sT k ðxÞT ;j ðx; tÞ sq ij Z 1 1 sT 1 es=sq kij ðxÞT ;j ðx; t sÞds: sq sq 0
qi ðx; tÞ ¼
ð10Þ
To determine the restrictions imposed by thermodynamics on the constitutive equation in concern, we postulate the Second Law of Thermodynamics in terms of a Clausius-Duhem inequality formulated on cyclic histories of period p, that is (see, e.g. Amendola et al. [20], Chapter 8, Section 8.2: Thermodynamic Constraints for Rigid Heat Conductors, page 216):
I
qi ðtÞT ;i ðtÞdt 6 0;
or; equiv alently;
Z
0
p
qi ðtÞT ;i ðtÞdt 6 0;
ð11Þ
where the equality occurs only for the null cycle and having omitted everywhere the explicit dependence upon the space variable. Consequently, any cycle characterized by the history
T ;i ðt sÞ ¼ f i cos xðt sÞ þ g i sin xðt sÞ; x > 0; f i f i þ g i g i > 0; ð12Þ has to fulfil (11) as an inequality. In view of the relations (10) and (12), we have sT qi ðtÞT ;i ðtÞ ¼ kij f i f j þ kij g i g j þ kij f i f j kij g i g j cos2xt 2 sq Z 1 1 sT þ2kij f i g j sin2xt 1 es=sq sq sq 0 1 1 kij f i f j þ kij g i g j cos xsþ kij f i f j kij g i g j cos xð2t sÞ 2 2 þkij f i g j sin xð2t sÞ ds; ð13Þ so that, by replacing into the inequality (11) and recalling that now the period is p ¼ 2xp, we get
Z 2p=x 0
qi ðtÞT ;i ðtÞdt ¼
p kij f i f j þ kij g i g j
s T s q x2 þ 1 ; x 1 þ s2q x2
ð14Þ
which is negative for all x > 0. Concluding, the dual-phase-lag model of heat conduction for ðn; mÞ ¼ ð1; 1Þ is compatible with thermodynamics for all sq > 0 and sT > 0.
Let us first consider the case ðn; mÞ ¼ ð2; 0Þ, that is we study the constitutive equation
qi ðtÞ þ sq q_ i ðtÞ þ
ð8Þ
Hence we conclude that the corresponding model is thermodynamically consistent for all sT > 0.
1 2 s q€i ðtÞ ¼ kij T ;j ðtÞ; 2 q
ð15Þ
where a superposed dot denotes the time differentiation. Then it can be written in the fading memory theory as
qi ðtÞ ¼
The case ðn; mÞ ¼ ð0; 1Þ can be treated by a similar way in view of the form (5) of the constitutive equation and we will have
T ;i ðx; tÞ ¼
qi ðx; tÞ ¼
6. Case ðn; mÞ‰fð2; 0Þ; ð0; 2Þg
4. Case ðn; mÞ‰fð1; 0Þ; ð0; 1Þg
1
The constitutive Eq. (3), for ðn; mÞ ¼ ð1; 1Þ, is equivalent to the constitutive equation with fading memory
#
ð5Þ
K ij kjs ¼ kij K js ¼ dis :
5. Case ðn; mÞ ¼ ð1; 1Þ
ðx;t Þ
2
s @ qj 2 q
smT @ m T ;i
279
2
sq
Z
0
1
s kij T ;j ðt sÞds; es=sq sin
sq
ð16Þ
and hence, by taking into account the relation (12), we have Z 1 1 s=sq s qi ðtÞT ;i ðtÞ ¼ e sin kij f i f j þ kij g i g j cos xs sq 0 sq þ kij f i f j kij g i g j cos xð2t sÞ þ 2kij f i g j sin xð2t sÞ ds:
ð17Þ
S. Chiriţa˘ et al. / International Journal of Heat and Mass Transfer 114 (2017) 277–285
280
Therefore, we obtain
Z 2p=x 0
qi ðtÞT ;i ðtÞdt ¼
2p
k f f þk g g
Z
1
s cos xsds es=sq sin
sq x ij i j ij i j 0
2p kij f i f j þkij g i g j
2 s2q x2 ; ¼ 4 4 x sq x þ4
sq
qi ðtÞ þ sq q_ i ðtÞ þ
1 2 1 sq q€i ðtÞ ¼ kij T ;j ðtÞ þ sT T_ ;j ðtÞ þ s2T T€ ;j ðtÞ ; 2 2
ð26Þ
have been established by Fabrizio and Lazzari in [6] to be
pffiffiffi
pffiffiffi
2 3 sT < sq < 2 þ 3 sT ;
ð27Þ
which are equivalent with
ð18Þ which cannot conserve a constant sign for all x > 0. Thus, the Second Law of Thermodynamics cannot be satisfied and the corresponding dual-phase-lag model is inconsistently from thermodynamic point of view. In view of the form (5) of the constitutive equation and by using an argument similar with that into above we conclude that the corresponding model for ðn; mÞ ¼ ð0; 2Þ is incompatible with thermodynamics.
pffiffiffi
pffiffiffi
2 3 sq < sT < 2 þ 3 sq :
ð28Þ
9. Case ðn; mÞ‰fð3; 0Þ; ð0; 3Þg Let us first consider the constitutive equation
1 1 v qi ðtÞ ¼ kij T ;j ðtÞ þ sT T_ ;j ðtÞ þ s2T T€ ;j ðtÞ þ s3T T ;j ðtÞ ; 2 6
ð29Þ
and note that for any cycle as described by (12) we have
Z 2p=x
7. Case ðn; mÞ‰fð2; 1Þ; ð1; 2Þg We outline here that the case ðn; mÞ ¼ ð2; 1Þ was studied by Fabrizio and Lazzari in [6] and it was shown that the corresponding model, based on the constitutive equation
qi ðtÞ þ sq q_ i ðtÞ þ
1 2 s q€i ðtÞ ¼ kij T ;j ðtÞ þ sT T_ ;j ðtÞ ; 2 q
ð19Þ
is compatible with the thermodynamics if and only if the delay times satisfy the inequality
0 < sq 6 2sT :
ð20Þ
In what follows we consider the case ðn; mÞ ¼ ð1; 2Þ, that is we study the constitutive equation
1 qi ðtÞ þ sq q_ i ðtÞ ¼ kij T ;j ðtÞ þ sT T_ ;j ðtÞ þ s2T T€ ;j ðtÞ : 2
ð21Þ
Considering (21) as a differential equation with the unknown function qi , we obtain the following equivalent representation
qi ðtÞ ¼
1
Z
sq
0
1
1 es=sq kij T ;j ðt sÞ þ sT T_ ;j ðt sÞ þ s2T T€ ;j ðt sÞ ds; 2
ð22Þ
qi ðtÞ ¼
s2T
2sq sq 1 sT 1 1
sq
sq
sT 2sq
sT
kij T ;j ðtÞ
2sq Z 1 0
es=sq kij T ;j ðt sÞds:
ð23Þ
Thus, with the choice (12), from (23) we deduce
Z 2p=x 0
p kij f i f j þ kij g i g j s
s T s q 1 T x2 þ 1 ; qi ðtÞT ;i ðtÞdt ¼ 2sq x 1 þ s2q x2 ð24Þ
and it will be negative for all x > 0 if and only if the following inequality
0 < sT 6 2sq ;
p 2x
kij f i f j þ kij g i g j 2 s2T x2 ;
ð30Þ
which cannot conserve a constant sign for all x > 0. Thus, the Second Law of Thermodynamics cannot be satisfied and the corresponding dual-phase-lag model is inconsistently from thermodynamic point of view. Let us now consider the constitutive equation
qi ðtÞ þ sq q_ i ðtÞ þ
1 2 1 v s q€i ðtÞ þ s3q qi ðtÞ ¼ kij T ;j ðtÞ; 2 q 6
ð31Þ
which can be written as
1 1 v €j ðtÞ þ s3q qj ðtÞ : T ;i ðtÞ ¼ K ij qj ðtÞ þ sq q_ j ðtÞ þ s2q q 2 6
ð32Þ
Thus, for any cycle characterized by
qi ðt sÞ ¼ hi cos xðt sÞ þ ‘i sin xðt sÞ; x > 0; hi hi þ ‘i ‘i > 0; ð33Þ we have
Z 2p=x 0
qi ðtÞT ;i ðtÞdt ¼
p K ij hi hj þ K ij ‘i ‘j 2 s2q x2 ; 2x
ð34Þ
which cannot conserve a constant sign for all x > 0. Thus, the Second Law of Thermodynamics cannot be satisfied and the corresponding dual-phase-lag model is inconsistently from thermodynamic point of view.
and furthermore, by successive integration by parts, we have
sT kij T_ ;j ðtÞ 1
0
qi ðtÞT ;i ðtÞdt ¼
ð25Þ
is fulfilled. Concluding, the dual-phase-lag models based upon the constitutive Eqs. (19) and (21) are compatible with the thermodynamics, provided the corresponding inequalities (20) and (25) are fulfilled. 8. Case ðn; mÞ ¼ ð2; 2Þ The restrictions upon the delay times which follow from Second Law of Thermodynamics for the constitutive equation
10. Case ðn; mÞ‰fð3; 1Þ; ð1; 3Þg We consider first the constitutive equation
1 1 v qi ðtÞ þ sq q_ i ðtÞ ¼ kij T ;j ðtÞ þ sT T_ ;j ðtÞ þ s2T T€ ;j ðtÞ þ s3T T ;j ðtÞ ; ð35Þ 2 6 from which we deduce the following representation
1 es=sq kij T ;j ðt sÞ þ sT T_ ;j ðt sÞ þ s2T T€ ;j ðt sÞ 2 sq 0 1 3v ð36Þ þ sT T ;j ðt sÞ ds: 6
qi ðtÞ ¼
1
Z
1
Further, for any cycle as described by (12), we obtain Z 2p=x p kij f i f j þ kij g i g j 1 3 4 1 2
qi ðtÞT ;i ðtÞdt ¼ sq sT x þ sT sq sT x2 1 ; 6 2 0 x 1 þ s2q x2 ð37Þ
an expression which cannot be negative for all positive values of x. Thus, the dual-phase-lag model based on the constitutive Eq. (35) cannot be compatible with thermodynamics.
S. Chiriţa˘ et al. / International Journal of Heat and Mass Transfer 114 (2017) 277–285
When the case ðn; mÞ ¼ ð3; 1Þ is addressed we write the constitutive equation in the form
T ;i ðtÞ þ sT T_ ;i ðtÞ ¼ K ij
1 1 v €j ðtÞ þ s3q qj ðtÞ ; qj ðtÞ þ sq q_ j ðtÞ þ s2q q 2 6
ð38Þ
where, for convenience, we have set
0
ð40Þ
ð41Þ
ð42Þ
In the other case, that is for ðn; mÞ ¼ ð3; 2Þ, we write the constitutive equation in the following form
1 T ;i ðtÞ þ sT T_ ;i ðtÞ þ s2T T€ ;i ðtÞ 2 1 1 v €j ðtÞ þ s3q qj ðtÞ ; ¼ K ij qj ðtÞ þ sq q_ j ðtÞ þ s2q q 2 6
jðsÞ cos xsds; js ¼
0
js
6 ¼ s2q x4
ð43Þ
ð44Þ
12. Case ðn; mÞ ¼ ð3; 3Þ
s3q
sq
"
12:1972
from which we deduce
i
Z 0
1
ð51Þ
sT 1 sq
2
sq
#
þ 0:0003
sq
sq
s2q x2 þ 24:3944 P 0;
ð52Þ
ð53Þ
where
s3T sT s2 sT ; b1 ¼ 4:06574 T2 6:09877 þ 4:06582 s3q sq sq sq sT 1 sq
2
þ 0:0003;
! ð54Þ
d1 ¼ 24:3944:
Furthermore, we see that the derivative
df ¼ 3a1 z2 2b1 z c1 ; dz
ð55Þ
b1
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 b1 þ 3a1 c1 3a1
< 0;
z2 ¼
b1 þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 b1 þ 3a1 c1 3a1
> 0:
ð56Þ
Thus, f ðzÞ P 0 for all z > 0 if we have f ðz2 Þ > 0, that is
ð45Þ
h
jðsÞkij T ;j ðt sÞ þ sT T_ ;j ðt sÞ
s2T € s3 v T ;j ðt sÞ þ T T ;j ðt sÞ ds; 6
q
for all x P 0. We associate with this inequality the function
z1 ¼
v
€i ðtÞ þ qi ðtÞ qi ðtÞ þ sq q_ i ðtÞ þ q 2 6 s2 s3 v ¼ kij T ;j ðtÞ þ sT T_ ;j ðtÞ þ T T€ ;j ðtÞ þ T T ;j ðtÞ ; 2 6
2
3
admits the roots
We consider here the constitutive equation
þ
ð50Þ
f ðzÞ a1 z3 b1 z2 c1 z þ d1 ;
Thus, we can conclude that the restrictions to be fulfilled by the delay times in order to have the thermodynamic consistency are described by the relations (42) and (44), respectively.
sq ða cÞ2 þ d2
jðsÞ sin xsds:
In view of the values for a; c and d given in (48), it follows that the constitutive Eq. (45) is compatible with thermodynamics if the following condition is fulfilled !
3 s
2 s3 s2 sT T 0:677637 T3 s2q x2 4:0657 T2 6:09877 þ 4:06582 s2q x2
0:28441 sT < sq 6 1:4902 sT :
6
0
3 d2 3c2 þ 2ca s2q x2 1 7 þ 5: a2 þ s2q x2 c2 þ d2 þ s2 x2 2 4d2 s2 x2 q q
c1 ¼ 12:1972
h
ð48Þ
q
and use the same procedure like that in the above to obtain the following restriction
qi ðtÞ ¼
1
a1 ¼ 0:677637
s2q
ð47Þ
ða 2cÞ c2 þ d2 as2q x2 7 a jc ¼ sq 6 þ 4 2 5;
2 a þ s2q x2 c2 þ d2 þ s2 x2 4d2 s2 x2 2
which is negative for all positive x if the delay times satisfy the inequality
0:28441 sq < sT 6 1:4902 sq :
Z
1
2
" 2p kij f i f j þ kij g i g j 1 s2T 2 sT 4 4 qi ðtÞT ;i ðtÞdt ¼ 1 sx 4 4 2 2 sq 3 sq q xðsq x þ 4Þ # !
sT s2 1 T2 s2q x2 þ 2 ; sq sq
sq
;
Furthermore, from the relations (47) and (50), we have
1
þ 2
ds
6p kij f i f j þ kij g i g j h i sq x ða cÞ2 þ d2 1 1 1 s2T x2 jc þ sT x s3T x3 js ; ð49Þ 2 6
Z
jc ¼
and therefore, for any cycle as described by (12), we obtain
Z 2p=x
sq
cos
qi ðtÞT ;i ðtÞdt ¼
where
It is equivalent with the following representation
h s k T ðt sÞ þ sT T_ ;j ðt sÞ es=sq sin sq 0 sq ij ;j 1 1 v þ s2T T€ ;j ðt sÞ þ s3T T ;j ðt sÞ ds; 2 6
ds
Then, for any cycle characterized by (12), we have
0
Let us first consider the case ðn; mÞ ¼ ð2; 3Þ, that is we study the constitutive equation 1 1 1 v €i ðtÞ ¼ kij T ;j ðtÞ þ sT T_ ;j ðtÞ þ s2T T€ ;j ðtÞ þ s3T T ;j ðtÞ : qi ðtÞ þ sq q_ i ðtÞ þ s2q q 2 2 6 ð39Þ
Z
d
sin
a ¼ 1:5961; c ¼ 0:70196; d ¼ 1:8073: Z 2p=x
11. Case ðn; mÞ‰fð3; 2Þ; ð2; 3Þg
2
ac
jðsÞ ¼ eas=sq þ ecs=sq and
and use the above procedure to establish the incompatibility with thermodynamics of the corresponding dual-phase-lag model of heat conduction.
qi ðtÞ ¼
281
ð46Þ
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
3 1
2 3 b1 þ 3a1 c1 6 27a21 d1 2b1 9a1 b1 c1 : 2
ð57Þ
Thus, the constitutive Eq. (45) is compatible with the thermodynamics if the delay times satisfy the inequality (57). It can be verified that the set of delay times fulfilling (57) is non-empty (in fact it contains the case when sT ¼ sq ). Moreover, we have to outline that the inequality (52) has an approximately character and it depends on the approximation order of the roots a; c and d as defined in (48) and obtained in the Table 1.
S. Chiriţa˘ et al. / International Journal of Heat and Mass Transfer 114 (2017) 277–285
282
13. Case ðn; mÞ‰fð4; 0Þ; ð0; 4Þg Let us first consider the constitutive equation
"
qi ðtÞ ¼ kij T ;j ðtÞ þ sT T_ ;j ðtÞ þ
# @4T 1 2€ 1 v 1 sT T ;j ðtÞ þ s3T T ;j ðtÞ þ s4T 4;j ðtÞ ; 2 6 24 @t ð58Þ
and note that for any cycle as described by (12) we have
Z 2p=x 0
qi ðtÞT ;i ðtÞdt ¼
i p kij f i f j þ kij g i g j h 2 2 2 sT x 6 12 ; ð59Þ 24x
which cannot conserve a constant sign for all x > 0. Thus, the Second Law of Thermodynamics cannot be satisfied and the corresponding dual-phase-lag model is inconsistently from thermodynamic point of view. Let us now consider the constitutive equation
qi ðtÞ þ sq q_ i ðtÞ þ
4
1 2 1 v 1 @ q s q€i ðtÞ þ s3q qi ðtÞ þ s4q 4i ðtÞ ¼ kij T ;j ðtÞ; 2 q 6 24 @t ð60Þ
4
T ;i ðtÞ ¼ K ij qj ðtÞ þ sq q_ j ðtÞ þ
@ q 1 2 1 v 1 s q€j ðtÞ þ s3q qj ðtÞ þ s4q 4j ðtÞ : 2 q 6 24 @t
Thus, for any cycle characterized by (33), we obtain
0
qi ðtÞT ;i ðtÞdt ¼
"
T ;i ðtÞ þ sT T_ ;i ðtÞ ¼ K ij qj ðtÞ þ sq q_ j ðtÞ þ
# 4 1 2 1 v 1 @ q sq q€j ðtÞ þ s3q qj ðtÞ þ s4q 4j ðtÞ ; 2 6 24 @t
and we obtain Z 2p=x p K ij hi hj þ K ij ‘i ‘j qi ðtÞT ;i ðtÞdt ¼ x 1 þ s2T x2 0 " # s3q sq sq sq 2 2 4 4 1 s x þ 1 s x þ 1 ; T sT 2sT T 6s3T 4sT
ð67Þ
ð68Þ
which cannot conserve a constant sign for all x > 0. Concluding this section, we see that the dual-phase-lag models based on the constitutive Eqs. (63) and (66) are not compatible with the thermodynamics.
#
ð61Þ Z 2p=x
we write it as
ð66Þ
15. Case ðn; mÞ‰fð4; 2Þ; ð2; 4Þg
which can be written as
"
1 1 v 1 4 @ 4 qi €i ðtÞ þ s3q qi ðtÞ þ qi ðtÞ þ sq q_ i ðtÞ þ s2q q s ðtÞ 2 6 24 q @t4 h i ¼ kij T ;j ðtÞ þ sT T_ ;j ðtÞ ;
p K ij hi hj þ K ij ‘i ‘j 2 2 2 sq x 6 12 ; 24x
Let us first consider the constitutive equation
1 €i ðtÞ qi ðtÞ þ sq q_ i ðtÞ þ s2q q 2 # 1 1 v 1 4 @ 4 T ;j sT 4 ðtÞ ; ¼ kij T ;j ðtÞ þ sT T_ ;j ðtÞ þ s2T T€ ;j ðtÞþ s3T T ;j ðtÞ þ 2 6 24 @t ð69Þ
ð62Þ which cannot conserve a constant sign for all x > 0. Thus, the Second Law of Thermodynamics cannot be satisfied and the corresponding dual-phase-lag model is inconsistently from thermodynamic point of view.
which gives
qi ðtÞ ¼
2
sq
Z
1
es=sq sin
0
s
sq
h kij T ;j ðt sÞ þ sT T_ ;j ðt sÞ
# 1 2€ 1 3v 1 4 @ 4 T ;j þ sT T ;j ðt sÞþ sT T ;j ðt sÞ þ s ðt sÞ ds: 2 6 24 T @t 4
14. Case ðn; mÞ‰fð4; 1Þ; ð1; 4Þg
ð70Þ
Further, for any cycle characterized by (12), we obtain We consider now the constitutive equation " # 1 1 v 1 @4T qi ðtÞ þ sq q_ i ðtÞ ¼ kij T ;j ðtÞ þ sT T_ ;j ðtÞ þ s2T T€ ;j ðtÞ þ s3T T ;j ðtÞ þ s4T 4 ðtÞ ; 2 6 24 @t
Z 2p=x 0
qi ðtÞT ;i ðtÞdt ¼
1
! s4T 6 6 s4T 2s3T s2T 4 4
sx þ þ sq x 12s4q q 6s4q 3s3q s2q
sq
! # s2T 2 2
þ sq x þ 4 ; 2 1 sq s2q
1 es=sq kij T ;j ðt sÞ þ sT T_ ;j ðt sÞ þ s2T T€ ;j ðt sÞ 2 # 4 1 v 1 4@ T s ðt sÞ ds: ð64Þ þ s3T T ;j ðt sÞ þ 6 24 T @t 4
qi ðtÞ ¼
1
Z
2sT
0
Therefore, for any cycle characterized by (12), we get Z 2p=x p kij f i f j þ kij g i g j
qi ðtÞT ;i ðtÞdt ¼ 0 x 1 þ s2q x2 " # s3T sT sT sT 2 2 4 4 1 s x þ 1 s x þ 1 ; q 6s3q 4sq sq 2sq q
ð65Þ which cannot conserve a constant sign for all x > 0. Thus, the Second Law of Thermodynamics cannot be satisfied and the corresponding dual-phase-lag model is inconsistently from thermodynamic point of view. When it is considered the constitutive equation
"
ð63Þ
which furnishes
p kij f i f j þ kij g i g j xðs4q x4 þ 4Þ
ð71Þ
which cannot conserve a negative sign for all x > 0. Thus, the Second Law of Thermodynamics cannot be satisfied and the corresponding dual-phase-lag model is inconsistently from thermodynamic point of view. The same conclusion can be obtained for the case ðn; mÞ ¼ ð4; 2Þ. 16. Case ðn; mÞ‰fð4; 3Þ; ð3; 4Þg Let us first consider the constitutive equation
qi ðtÞ þ sq q_ i ðtÞ þ
1 2 1 v s q€i ðtÞ þ s3q qi ðtÞ 2 q 6
# 1 2€ 1 3v 1 4 @ 4 T ;j _ ¼ kij T ;j ðtÞ þ sT T ;j ðtÞ þ sT T ;j ðtÞþ sT T ;j ðtÞ þ s ðtÞ ; 2 6 24 T @t 4
ð72Þ
S. Chiriţa˘ et al. / International Journal of Heat and Mass Transfer 114 (2017) 277–285
which gives
qi ðtÞ ¼
6
h
sq ðc aÞ2 þ d2
i
Z 0
1
h
jðsÞkij T ;j ðt sÞ þ sT T_ ;j ðt sÞ
# 1 2€ 1 3v 1 4 @ 4 T ;j þ sT T ;j ðt sÞ þ sT T ;j ðt sÞ þ s ðt sÞ ds; 2 6 24 T @t4
ð73Þ
where jðsÞ is given by relation (47) and the values of a; d and c are given by (48). For any cycle characterized by (12), we have
Z 2p=x 0
ð74Þ
2 sT s3 s2 sT 1:01643 T3 4:06574 T2 þ6:09877 4:06582 s2q x2 sq sq sq sq "
sT 1 sq
2
#
þ 0:0003
s2q x2 þ 24:3944 P 0;
ð75Þ
for all x P 0. Thus, the constitutive Eq. (72) is compatible with the thermodynamics if the delay times satisfy the inequalities
sT
0< < 1:33332; sq rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
3 1
2 3 b2 þ 3a2 c2 6 27a22 d2 2b2 9a2 b2 c2 ; 2
ð76Þ
where
s3T s4 0:508231 T4 3 sq sq
a2 ¼ 0:677637 b2 ¼
sT s3 s2 sT 1:01643 T3 þ 4:06574 T2 6:09877 þ 4:06582 sq sq sq sq
c2 ¼ 12:1972
sT 1 sq
which can be written as
qi ðtÞ ¼
! s3T s4T 2 2 3 0:677637 3 0:508231 4 sq x sq sq
12:1972
ð79Þ
!
where
"
1
bs
ð81Þ
1 D ¼ 3ab2 3cd2 þ c3 a3 2ðc aÞ h ðc aÞ2 3a2 þ 3c2 b2 d2 þ b2 d2 3c2 3a2 þ b2 d2 ¼ 22:165;
ð82Þ
and now we have set
a ¼ 0:27056; b ¼ 2:5048; c ¼ 1:7294; d ¼ 0:88897:
ð83Þ
Then, for any cycle characterized by (12), we have Z 2p=x 24p kij f i f j þ kij g i g j 1 1 qi ðtÞT ;i ðtÞdt ¼ 1 s2T x2 þ s4T x4 K c 2 24 sq x D 0 1 3 3 þ sT x sT x K s ; 6 ð84Þ
Z 0
d2 ¼ 24:3944:
ð80Þ
# ða cÞ2 þ d2 b2 bs KðsÞ ¼ e cos sin sq 2bðc aÞ sq " # ds ða cÞ2 þ b2 d2 ds ; ecs=sq cos þ sin sq 2dðc aÞ sq as=sq
Kc ¼
þ 0:0003;
Z
1
Ks ¼
KðsÞ cos xsds;
1
ð85Þ
KðsÞ sin xsds:
0
Further, with the aid of (81), we have
ð77Þ It can be verified that it is non-empty the set of values for the delay times as defined by the relation (76), (in fact it contains at least the case when the delay times are equally, sT ¼ sq ). When the case ðn; mÞ ¼ ð4; 3Þ is considered, the constitutive equation is written in the equivalent form
1 1 v T ;i ðtÞ þ sT T_ ;i ðtÞ þ s2T T€ ;i ðtÞ þ s3T T ;i ðtÞ 2 6 # 4 1 1 v 1 4 @ qj €j ðtÞþ s3q qj ðtÞ þ ¼ K ij qj ðtÞ þ sq q_ j ðtÞ þ s2q q sq 4 ðtÞ ; 2 6 24 @t
2
q
c2 þ d2 a2 b2 s2q x2 þ c2 þ d2
2 K s ¼ s x4 2 q
h
s2q x2 þ a2 b2 caa ðc aÞ2 þ d2 b2
s4q x4 þ 2 a2 b2 s2q x2 þ a2 þ b2 h
s2q x2 þ c2 d2 þ cc a ðc aÞ2 þ b2 d2
q
3 3c2 d2 þ a2 þ b2 4ac 7 5
2 s4q x4 þ 2ðc2 d2 Þs2q x2 þ c2 þ d2
and
and then the thermodynamic restrictions can be obtained easily by using the relation (76).
2 2 2 2 2 2 2 2 2 2 2 2 sq 6 c þ d a b sq x a þ b 3a b þ c þ d 4ac Kc ¼ 4
2 2ðc aÞ s4 x4 þ 2ða2 b2 Þs2 x2 þ a2 þ b2
ð78Þ
Let us now consider the constitutive equation
24 sq D
where
2
17. Case ðn; mÞ ¼ ð4; 4Þ
Z
h s2 KðsÞkij T ;j ðt sÞ þ sT T_ ;j ðt sÞþ T T€ ;j ðt sÞ 2 0 # 4 3 v s 1 4 @ T ;j þ T T ;j ðt sÞ þ s ðt sÞ ds; 24 T @t4 6
qi ðtÞT ;i ðtÞdt ¼
where jc and js are defined by the relation (51). In view of relations (48), (51) and (74) it follows that the Second Law of Thermodynamics can be satisfied if the following inequality is fulfilled
þ
1 1 v 1 4 @ 4 qi €i ðtÞ þ s3q qi ðtÞ þ qi ðtÞ þ sq q_ i ðtÞ þ s2q q s ðtÞ 2 6 24 q @t 4 " # 1 1 v 1 4 @ 4 T ;j sT 4 ðtÞ ; ¼ kij T ;j ðtÞ þ sT T_ ;j ðtÞ þ s2T T€ ;j ðtÞ þ s3T T ;j ðtÞ þ 2 6 24 @t
6p kij f i f j þ kij g i g j h i sq x ðc aÞ2 þ d2 1 1 4 4 1 s2T x2 þ sT x jc 2 24 1 3 3 þ sT x sT x js ; 6
283
s4q x4 þ 2 c2 d2 s2q x2 þ c2 þ d2
2
ð86Þ
i
2
i3 5:
ð87Þ
In view of relations (83), (86) and (87) it follows that the Second Law of Thermodynamics can be satisfied if the following inequality is fulfilled
S. Chiriţa˘ et al. / International Journal of Heat and Mass Transfer 114 (2017) 277–285
284
2:69456
!
3 s4T 2 2 4 s2T s2T sT s x 32:3346 43:1121 þ 32:3348 s2q x2 q 4 2 2 sq sq sq sq
þ 64:668
!
2 s4T s3T s2T sT 258:676 þ 388:015 258:673 þ 64:6695 s2q x2 s4q s3q s2q sq
776:016
!
s2T sT 1552:06 þ 776:031 s2q x2 þ 1552:03 P 0; 2 sq sq
ð88Þ
for all x P 0. It is not difficult to see that the above inequality is fulfilled when sT ¼ sq and so we can conclude that it is non-empty the set of couples ðsT ; sq Þ for which the constitutive Eq. (79) is compatible with the thermodynamics. Unfortunately, there seems to be unavailable to get an explicit expression defining this set. 18. Conclusions
ð0; 0Þ ð0; 1Þ ð0; 2Þ ð0; 3Þ
ð0; 4Þ
1
C B B ð1; 0Þ ð1; 1Þ ð1; 2Þ ð1; 3Þ ð1; 4Þ C C B B m n ¼ B ð2; 0Þ ð2; 1Þ ð2; 2Þ ð2; 3Þ ð2; 4Þ C C; C B @ ð3; 0Þ ð3; 1Þ ð3; 2Þ ð3; 3Þ ð3; 4Þ A ð4; 0Þ ð4; 1Þ ð4; 2Þ ð4; 3Þ ð4; 4Þ provided appropriate restrictions are placed upon the delay times. In fact, from a mathematical point of view, the lagging behavior can be equally described by the following equation equivalent with the constitutive Eq. (3)
qi ðx; tÞ ¼ qhi ðx; tÞ þ qpi ðx; tÞ;
ð89Þ
where qhi ðx; tÞ represents the general integral of the homogeneous differential equation
qi ðx; tÞ þ
sq @qi 1! @t
ðx; tÞ þ
Conflict of interest None declared.
We inferred here an opinion about the time differential dualphase-lag models that is based on the information that we have upon the differential operators involved into the related constitutive equations. It is shown that, when the approximation orders are greater than or equal to five, the corresponding models lead to some instable mechanical systems. Instead, when the approximation orders are lower than or equal to four, then the corresponding models can be compatible with the thermodynamics, provided some appropriate restrictions are assumed upon the delay times. More precisely, the thermodynamical consistency of the model in concern is established when ðm; nÞ take the values in boldface of the matrix
0
ables. The extension of this analysis to nonlinear models (when, for example, the thermophysical properties of substances depend strongly on the temperature), there seems to be a complex task and for this it remains an open problem to be subject in a future research. On the other side, the present paper furnishes a new perspective upon the relationship between the thermodynamic aspects and the stability properties, with reference to the time differential dual-phase-lag models of heat conduction. We believe that this research provides useful information to clarify the applicability of these models.
s2q @ 2 qi 2! @t
2
ðx; t Þ þ þ
snq @ n qi n! @t n
ðx; t Þ ¼ 0;
ð90Þ
Acknowledgments The authors are very grateful to the reviewers for their valuable comments, which have led to an improvement of the present work. Appendix A. Szegö’s curve In this appendix we provide the source code that has been created to generate an animation with the software package Wolfram Mathematica 11. The generated animation shows the complex roots of the scaled exponential sum of order n in the complex plane, with n increasing from 1 to 50, and it shows that the roots approach the Szegö curve as n increases. The following code could be entered in a Mathematica session and, once executed, the animation window is shown. e[n_, z_] :¼ Sum[z^k/k!, {k, 0, n}] axes = Graphics[{Line[{{-1, 0}, {1, 0}}], Line[{{0, 1}, {0, 1}}]}]; szegoeCurve = ContourPlot[Evaluate[Abs[z E^(1 z)] == 1/. z -> x + I y], {x, 1, 1}, {y, 1, 1}, PlotPoints -> 50]; rootsGraph[m_] :¼ Module[{roots, n = IntegerPart [m], points,} roots = NSolve[e[n, n z] == 0, z][[All, 1, 2]]; points = Transpose[{Re[roots], Im[roots]}]; Show[{axes, szegoeCurve, ListPlot[points, PlotRange -> {{-1, 1}, {-1, 1}}, AspectRatio -> 1]}]] Animate[rootsGraph[m], {m, 1, 50}, DefaultDuration > 50]
and qpi ðx; tÞ is a particular solution of the differential Eq. (3). The explicit expression of qhi ðx; tÞ represents a concrete combination of the number of time exponentials relating the roots of the characteristic Eq. (4), while the expression of the particular solution qpi ðx; tÞ represents a convolution (memory) term. In view of the results established by Szegö [18], if follows that, for n P 5, the expression of qhi ðx; tÞ involves some exponentials growing in time at infinity and this proves an asymptotic instable mechanical system. Thus, the situation with ðn; mÞ ¼ ð1; 0Þ, which corresponds to rather widely used model with an only delay time of the heat flux relative to the temperature gradient, appears to lead to an instable mechanical system. Instead, when n < 5 this is not the case and we have searched the memory term described by qpi ðx; tÞ by means of the techniques developed for materials with memory by Gurtin and Pipkin [12] and Coleman and Gurtin [13] and Amendola et al. [20]. Our analysis is limited to the constitutive Eq. (3), where the delay times are assumed to be positive constant parameters. It is worth to mention that our analysis remains valid when the delay times are not constant, but they can depend on the spatial vari-
Here we also provide an image of the roots for n ¼ 4 and n ¼ 5: Fig. 1. References [1] D.Y. Tzou, A unified field approach for heat conduction from macro- to microscales, ASME J. Heat Transfer 117 (1) (1995) 8–16, http://dx.doi.org/10.1115/ 1.2822329. [2] D.Y. Tzou, The generalized lagging response in small-scale and high-rate heating, Int. J. Heat Mass Transfer 38 (17) (1995) 3231–3240, http://dx.doi.org/ 10.1016/0017-9310(95)00052-B. [3] D.Y. Tzou, Experimental support for the lagging behavior in heat propagation, AIAA J. Thermophys. Heat Transfer 9 (4) (1995) 686–693, http://dx.doi.org/ 10.2514/3.725. [4] D.Y. Tzou, Macro- to Microscale Heat Transfer: The Lagging Behavior, John Wiley & Sons, Chichester, 2015, ISBN: 978-1-118-81822-0.. [5] M. Fabrizio, F. Franchi, Delayed thermal models: stability and thermodynamics, J. Therm. Stress. 37 (2) (2014) 160–173, http://dx.doi.org/ 10.1080/01495739.2013.839619.
S. Chiriţa˘ et al. / International Journal of Heat and Mass Transfer 114 (2017) 277–285 [6] M. Fabrizio, B. Lazzari, Stability and second law of thermodynamics in dualphase-lag heat conduction, Int. J. Heat Mass Transfer 74 (2014) 484–489, http://dx.doi.org/10.1016/j.ijheatmasstransfer.2014.02.027. [7] R. Quintanilla, Exponential stability in the dual-phase-lag heat conduction theory, J. Non-Equilib. Thermodynam. 27 (3) (2002) 217–227, http://dx.doi. org/10.1515/JNETDY.2002.012. [8] M. Fabrizio, B. Lazzari, V. Tibullo, Stability and thermodynamic restrictions for a dual-phase-lag thermal model, J. Non-Equilib. Thermodynam., in press, doi: http://dx.doi.org/10.1515/jnet-2016-0039. [9] S. Chiriţa˘, M. Ciarletta, V. Tibullo, On the wave propagation in the time differential dual-phase-lag thermoelastic model, Proc. Roy. Soc. A 471: 20150400, doi: http://dx.doi.org/10.1098/rspa.2015.0400. [10] R. Quintanilla, R. Racke, Spatial behavior in phase-lag heat conduction, Differ. Integral Eqs. 28 (3–4) (2015) 291–308, http://projecteuclid.org/euclid.die/ 142305522. [11] S. Chiriţa˘, C. D’Apice, V. Zampoli, The time differential three-phase-lag heat conduction model: thermodynamic compatibility and continuous dependence, Int. J. Heat Mass Transfer 102 (2016) 226–232, http://dx.doi. org/10.1016/j.ijheatmasstransfer.2016.06.019. [12] M.E. Gurtin, A.C. Pipkin, A general theory of heat conduction with finite wave speeds, Archive Rational Mech. Anal. 31 (2) (1968) 113–126, http://dx.doi.org/ 10.1007/BF00281373.
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[13] B.D. Coleman, M.E. Gurtin, Equipresence and constitutive equations for rigid heat conductors, Zeitschrift Angew. Math. Phys. ZAMP 18 (2) (1967) 199–208, http://dx.doi.org/10.1007/BF01596912. [14] G. Eneström, Härledning af en allmän formel för antalet pensionärer, som vid en godtycklig tidpunkt förefinnas inom en sluten pensionskassa, Öfversigt Kongl. Vetenskaps Akad. Förhandl 50 (1893) 405–415. [15] S. Kakeya, On the zero points of a power series with positive coefficients, Tôhoku Math. J. 3 (1913) 23–24. [16] G. Eneström, Remarque sur un théorème relatif aux racines de l’équation an xn þ an1 xn1 þ þ a1 x þ a0 ¼ 0 où tous les coefficients a sont réels et positifs, Tôhoku Math. J. 18 (1920) 34–36. [17] M. Marden, The Geometry of the Zeros of a Polynomial in a Complex Variable, Mathematical Surveys, No. III, Am. Math. Soc., 1949. [18] G. Szegö, Über eine Eigenschaft der Exponentialreihe, Berlin Math. Ges. Sitzunsber. 23 (1924) 50–64. [19] J. Dieudonné, Sur les zéroes des polynomes-sections de e z , Bull. Sci. Math. 70 (1935) 333–351. [20] G. Amendola, M. Fabrizio, J.M. Golden, Thermodynamics of Materials with Memory, Theory and Applications, Springer, New York, Dordrecht, Heidelberg, London, 2012, ISBN: 978-1-4614-1691-3..
Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
On the thermomechanical consistency of the time differential dual-phase-lag models of heat conduction Stan Chiriţa˘ a,b,⇑, Michele Ciarletta c, Vincenzo Tibullo c a
Faculty of Mathematics, Al. I. Cuza University of Iasi, 700506 Iasi, Romania Octav Mayer Mathematics Institute, Romanian Academy, 700505 Iasi, Romania c University of Salerno, via Giovanni Paolo II n. 132, 84084 Fisciano, SA, Italy b
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 1 March 2017 Received in revised form 6 June 2017 Accepted 15 June 2017
This paper deals with the time differential dual-phase-lag heat transfer models aiming, at first, to identify the eventually restrictions that make them thermodynamically consistent. At a first glance it can be observed that the capability of a time differential dual-phase-lag model of heat conduction to describe real phenomena depends on the properties of the differential operators involved in the related constitutive equation. In fact, the constitutive equation is viewed as an ordinary differential equation in terms of the heat flux components (or in terms of the temperature gradient) and it results that, for approximation orders greater than or equal to five, the corresponding characteristic equation has at least a complex root having a positive real part. That leads to a heat flux component (or temperature gradient) that grows to infinity when the time tends to infinity and so there occur some instabilities. Instead, when the approximation orders are lower than or equal to four, this is not the case and there is the need to study the compatibility with the Second Law of Thermodynamics. To this aim the related constitutive equation is reformulated within the system of the fading memory theory, and thus the heat flux vector is written in terms of the history of the temperature gradient and on this basis the compatibility of the model with the thermodynamical principles is analyzed. Ó 2017 Elsevier Ltd. All rights reserved.
2010 MSC: 74F05 80A20 Keywords: Time differential dual-phase-lag model Heat conduction Delay times Stability systems Thermodynamic compatibility
1. Introduction The dual-phase-lag model of heat conduction proposed in [1–3] distinguishes the time instant t þ sq , at which the heat flux flows through a material volume and the time instant t þ sT , at which the temperature gradient establishes across the same material volume:
qi ðx; t þ sq Þ ¼ kij ðxÞT ;j ðx; t þ sT Þ;
with
sq ; sT P 0:
ð1Þ
The above constitutive equation states, synthesizing its meaning, that the temperature gradient T ;j at a certain time t þ sT results in a heat flux vector qi at a different time t þ sq . In the above constitutive Eq. (1), besides the explicit dependence upon the spatial variable, we point out that qi are the components of the heat flux vector, T represents the temperature variation from the constant reference temperature T 0 > 0 and kij are the components of the conductivity tensor; moreover, t is the time variable while sq and sT are ⇑ Corresponding author. E-mail addresses: [email protected] (S. Chiriţa˘), [email protected] (M. Ciarletta), [email protected] (V. Tibullo). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2017.06.071 0017-9310/Ó 2017 Elsevier Ltd. All rights reserved.
the phase lags (or delay times) of the heat flux and of the temperature gradient, respectively. In particular, sq is a relaxation time connected to the fast-transient effects of thermal inertia, while sT is caused by microstructural interactions, such as phonon scattering or phonon-electron interactions [4]. In addition to the thermal conductivity, the phase lags sT and sq are treated as two additional intrinsic thermal properties characterizing the energy-bearing capacity of the material. Eq. (1) describing the lagging behavior in heat transport, when coupled with the energy equation
qi;i ðx; tÞ þ .ðxÞrðx; tÞ ¼ aðxÞ
@T ðx; tÞ; @t
ð2Þ
displays two coupled differential equations of a delayed type. Due to the general time shifts at different scales, sT and sq , no general solution has been known yet. The refined structure of the lagging response depicted by equations (1) and (2), however, has been illustrated by Tzou [3] by expanding Eq. (1) in terms of the Taylor’s series with respect to time:
S. Chiriţa˘ et al. / International Journal of Heat and Mass Transfer 114 (2017) 277–285
278
qi ðx;tÞ þ
sq @qi 1! @t "
ðx;t Þ þ
¼ kij ðxÞ T ;j ðx;tÞ þ
s2q @ 2 qi 2! @t
2
sT @T ;j 1! @t
ðx;t Þ þ þ
ðx;t Þ þ
snq @ n qi n! @t n
2
s @ T ;j 2 T
2! @t 2
dynamic consistency of the model in concern is established when
ðx;t Þ
ðx;t Þ þ þ
m
s @ T ;j m T
m! @tm
# ðx;tÞ : ð3Þ
An interesting discussion concerning this expansion has been developed by Tzou [3] when n m ¼ 0 or n m ¼ 1, relating the progressive interchange between the diffusive and wave behaviors. We emphasize that the related time differential models obtained considering the Taylor series expansions of both sides of the Eq. (1) and retaining terms up to suitable orders in sq and sT (namely, first or second orders in sq and sT ) have been widely investigated with respect to their thermodynamic consistency as well as to interesting stability issues and wave propagation (see, for example, [5–9]). However, the general form of the time differential dual-phase-lag model as given by (3) wasn’t treated up to now, except for the paper by Quintanilla and Racke [10], where the spatial behavior is studied for solutions of the equation obtained by eliminating the heat flux vector between the constitutive Eq. (3) and the energy Eq. (2), provided n ¼ m or n ¼ m þ 1. The main purpose of this paper is to study the thermodynamical and mechanical consistency of the constitutive Eq. (3). We infer that the feasibility study of this constitutive equation greatly depends on the structure of the differential operators involved in its mathematical expression. In fact, if we consider the constitutive Eq. (3) as an ordinary linear differential equation in terms of the unknown function qi ðtÞ (or, equivalently, in terms of the unknown function T ;i ðtÞ) then we can observe that for n P 5 (or m P 5) it admits at least a complex root having a positive real part. That implies that qi ðtÞ (or T ;i ðtÞ) can tends to infinity when the time tends to infinity and so we are led to instability situations. On this way we conclude that the time differential dual-phase-lag model based on a constitutive equation of type (3) with n P 5 or m P 5 cannot be considered able to describe real mechanical situations. Instead, when n ¼ 0; 1; 2; 3; 4 and m ¼ 0; 1; 2; 3; 4 this is not the case and we have to study the thermodynamic consistency of the corresponding model. To this aim we follow [6,11] and we reformulate the constitutive Eq. (3) in such a way that the heat flux vector qi depends on the history of the temperature gradient. In this sense we rewrite the Eq. (3) in the framework of Gurtin and Pipkin [12] and Coleman and Gurtin [13] fading memory theory, and on this basis we analyze the compatibility of the model with the thermodynamical principles. Precisely, the thermo-
ðm; nÞ 2 fð0; 0Þ; ð1; 0Þ; ð0; 1Þ; ð2; 1Þ; ð1; 2Þ; ð2; 2Þ; ð3; 2Þ; ð2; 3Þ; ð3; 3Þ; ð3; 4Þ; ð4; 3Þ; ð4; 4Þg, provided appropriate restrictions are placed on the delay times.
2. Thermomechanical consistency of the model In this Section we consider the Eq. (3) as an ordinary linear nonhomogeneous differential (in time variable) equation in terms of the heat flux vector components and observe that its characteristic equation is
1 n n 1 1 1 s k þ sn1 kn1 þ þ s2q k2 þ sq k þ 1 ¼ 0: n! q ðn 1Þ! q 2! 1!
ð4Þ
This equation is connected with the partial sums of the Maclaurin series for the exponential function e z and with the incomplete gamma function and its roots have been intensively studied in literature (see e. g. Eneström [14–17]). On the basis of the EneströmKakeya theorem it follows that all the roots of the Eq. (4) lie outside of the disk of radius s1q . Moreover, the Eq. (4) has no real root if n is even, while when n is odd, it has only one real root. However, here we are interested if this equation has at least a complex root with a positive real part. To this aim we outline the results obtained by Gábor Szegö [18] and Jean Dieudonné [19] who showed that the roots of the scaled exponential sum function approach the portion of the Szegö curve: jz expð1 zÞj ¼ 1 within the unit disk as n ! 1. Moreover, with the aim to visualize this result for n P 5 we recommend the simulation for the software package Wolfram Mathematica 11 presented in the Appendix (see also the Fig. 1).
Table 1 The values of x ¼ sq k with k solution of the characteristic Eq. (4) for n ¼ 1; 2; 3; 4. n
x ¼ sq k
1 2
1 1:0 1:0 i
3
1:5961 0:70196 1:8073 i
4
0:27056 2:5048 i 1:7294 0:88897 i
Fig. 1. Roots of the exponential sum for n ¼ 4 and for n ¼ 5.
S. Chiriţa˘ et al. / International Journal of Heat and Mass Transfer 114 (2017) 277–285
In view of the above discussion we can conclude that, for values of n greater than or equal to five, the characteristic Eq. (4) admits at least one complex root having a positive real part and this leads to a couple fT; qi g going to infinity as time tends to infinity, that is we are led to instability of the system. In conclusion, for n P 5 the corresponding models of dual-phase-lag of heat conduction are not suitable to describe the lagging behavior. Furthermore, we write the constitutive Eq. (3) in the following form
T ;i ðx;t Þ þ
sT @T ;i 1! @t "
ðx;t Þ þ
¼ K ij ðxÞ qj ðx;t Þ þ
s2T @ 2 T ;i 2! @t 2
sq @qj 1! @t
ðx;tÞ þ þ
ðx;tÞ þ
m! @t m
2! @t 2
ðx;tÞ þ þ
n
s @ qj n q
n! @t n
ðx;tÞ ;
where K ij are so that
ð6Þ
Following an argument similar to that in the above discussion, this time the unknown function being considered T ;i , we can conclude that, for values of m greater than or equal to five, the corresponding constitutive Eq. (3) leads to models of dual-phase-lag of heat conduction that are not suitable to describe the lagging behavior. On the other side, as it can be seen from the Table 1, it follows that for n ¼ 1; 2; 3; 4 all the solutions of the Eq. (4) have a negative real part and hence the corresponding contribution in the heat flux vector components can lead to an asymptotic stable behavior in time couple fT; qi g. In such cases we have to study further the consistency of the constitutive Eq. (3) with the Second Law of Thermodynamics. In conclusion, the constitutive Eq. (3) can be thermodynamically consistent only for n 2 f0; 1; 2; 3; 4g and m 2 f0; 1; 2; 3; 4g. Thus, in what follows we will study the thermodynamical consistency of the constitutive Eq. (3) for ðn; mÞ 2 f0; 1; 2; 3; 4g f0; 1; 2; 3; 4g. For future convenience, we will discuss separately each of the above cases. In order to study the cases involved in the Table 1, we consider the constitutive Eq. (3) as a memory constitutive equation of the type described in Gurtin and Pipkin [12] and Coleman and Gurtin [13]. 3. Case ðn; mÞ ¼ ð0; 0Þ This case yields the classical Fourier law of heat flux which is compatible with the Second Law of Thermodynamics.
The case ðn; mÞ ¼ ð1; 0Þ yields the Cattaneo-Maxwell equation of the heat flux vector which is consistent with thermodynamics for all sq > 0 and it is equivalent to the constitutive equation with fading memory
qi ðx; tÞ ¼
sq
Z
1
s=sq
e 0
kij ðxÞT ;j ðx; t sÞds:
ð7Þ
1
sT
Z 0
1
es=sT K ij ðxÞqj ðx; t sÞds:
Z
1
sq
1
0
h i es=sq kij ðxÞ T ;j ðx; t sÞ þ sT T_ ;j ðx; t sÞ ds;
ð9Þ
which integrated by parts gives
sT k ðxÞT ;j ðx; tÞ sq ij Z 1 1 sT 1 es=sq kij ðxÞT ;j ðx; t sÞds: sq sq 0
qi ðx; tÞ ¼
ð10Þ
To determine the restrictions imposed by thermodynamics on the constitutive equation in concern, we postulate the Second Law of Thermodynamics in terms of a Clausius-Duhem inequality formulated on cyclic histories of period p, that is (see, e.g. Amendola et al. [20], Chapter 8, Section 8.2: Thermodynamic Constraints for Rigid Heat Conductors, page 216):
I
qi ðtÞT ;i ðtÞdt 6 0;
or; equiv alently;
Z
0
p
qi ðtÞT ;i ðtÞdt 6 0;
ð11Þ
where the equality occurs only for the null cycle and having omitted everywhere the explicit dependence upon the space variable. Consequently, any cycle characterized by the history
T ;i ðt sÞ ¼ f i cos xðt sÞ þ g i sin xðt sÞ; x > 0; f i f i þ g i g i > 0; ð12Þ has to fulfil (11) as an inequality. In view of the relations (10) and (12), we have sT qi ðtÞT ;i ðtÞ ¼ kij f i f j þ kij g i g j þ kij f i f j kij g i g j cos2xt 2 sq Z 1 1 sT þ2kij f i g j sin2xt 1 es=sq sq sq 0 1 1 kij f i f j þ kij g i g j cos xsþ kij f i f j kij g i g j cos xð2t sÞ 2 2 þkij f i g j sin xð2t sÞ ds; ð13Þ so that, by replacing into the inequality (11) and recalling that now the period is p ¼ 2xp, we get
Z 2p=x 0
qi ðtÞT ;i ðtÞdt ¼
p kij f i f j þ kij g i g j
s T s q x2 þ 1 ; x 1 þ s2q x2
ð14Þ
which is negative for all x > 0. Concluding, the dual-phase-lag model of heat conduction for ðn; mÞ ¼ ð1; 1Þ is compatible with thermodynamics for all sq > 0 and sT > 0.
Let us first consider the case ðn; mÞ ¼ ð2; 0Þ, that is we study the constitutive equation
qi ðtÞ þ sq q_ i ðtÞ þ
ð8Þ
Hence we conclude that the corresponding model is thermodynamically consistent for all sT > 0.
1 2 s q€i ðtÞ ¼ kij T ;j ðtÞ; 2 q
ð15Þ
where a superposed dot denotes the time differentiation. Then it can be written in the fading memory theory as
qi ðtÞ ¼
The case ðn; mÞ ¼ ð0; 1Þ can be treated by a similar way in view of the form (5) of the constitutive equation and we will have
T ;i ðx; tÞ ¼
qi ðx; tÞ ¼
6. Case ðn; mÞ‰fð2; 0Þ; ð0; 2Þg
4. Case ðn; mÞ‰fð1; 0Þ; ð0; 1Þg
1
The constitutive Eq. (3), for ðn; mÞ ¼ ð1; 1Þ, is equivalent to the constitutive equation with fading memory
#
ð5Þ
K ij kjs ¼ kij K js ¼ dis :
5. Case ðn; mÞ ¼ ð1; 1Þ
ðx;t Þ
2
s @ qj 2 q
smT @ m T ;i
279
2
sq
Z
0
1
s kij T ;j ðt sÞds; es=sq sin
sq
ð16Þ
and hence, by taking into account the relation (12), we have Z 1 1 s=sq s qi ðtÞT ;i ðtÞ ¼ e sin kij f i f j þ kij g i g j cos xs sq 0 sq þ kij f i f j kij g i g j cos xð2t sÞ þ 2kij f i g j sin xð2t sÞ ds:
ð17Þ
S. Chiriţa˘ et al. / International Journal of Heat and Mass Transfer 114 (2017) 277–285
280
Therefore, we obtain
Z 2p=x 0
qi ðtÞT ;i ðtÞdt ¼
2p
k f f þk g g
Z
1
s cos xsds es=sq sin
sq x ij i j ij i j 0
2p kij f i f j þkij g i g j
2 s2q x2 ; ¼ 4 4 x sq x þ4
sq
qi ðtÞ þ sq q_ i ðtÞ þ
1 2 1 sq q€i ðtÞ ¼ kij T ;j ðtÞ þ sT T_ ;j ðtÞ þ s2T T€ ;j ðtÞ ; 2 2
ð26Þ
have been established by Fabrizio and Lazzari in [6] to be
pffiffiffi
pffiffiffi
2 3 sT < sq < 2 þ 3 sT ;
ð27Þ
which are equivalent with
ð18Þ which cannot conserve a constant sign for all x > 0. Thus, the Second Law of Thermodynamics cannot be satisfied and the corresponding dual-phase-lag model is inconsistently from thermodynamic point of view. In view of the form (5) of the constitutive equation and by using an argument similar with that into above we conclude that the corresponding model for ðn; mÞ ¼ ð0; 2Þ is incompatible with thermodynamics.
pffiffiffi
pffiffiffi
2 3 sq < sT < 2 þ 3 sq :
ð28Þ
9. Case ðn; mÞ‰fð3; 0Þ; ð0; 3Þg Let us first consider the constitutive equation
1 1 v qi ðtÞ ¼ kij T ;j ðtÞ þ sT T_ ;j ðtÞ þ s2T T€ ;j ðtÞ þ s3T T ;j ðtÞ ; 2 6
ð29Þ
and note that for any cycle as described by (12) we have
Z 2p=x
7. Case ðn; mÞ‰fð2; 1Þ; ð1; 2Þg We outline here that the case ðn; mÞ ¼ ð2; 1Þ was studied by Fabrizio and Lazzari in [6] and it was shown that the corresponding model, based on the constitutive equation
qi ðtÞ þ sq q_ i ðtÞ þ
1 2 s q€i ðtÞ ¼ kij T ;j ðtÞ þ sT T_ ;j ðtÞ ; 2 q
ð19Þ
is compatible with the thermodynamics if and only if the delay times satisfy the inequality
0 < sq 6 2sT :
ð20Þ
In what follows we consider the case ðn; mÞ ¼ ð1; 2Þ, that is we study the constitutive equation
1 qi ðtÞ þ sq q_ i ðtÞ ¼ kij T ;j ðtÞ þ sT T_ ;j ðtÞ þ s2T T€ ;j ðtÞ : 2
ð21Þ
Considering (21) as a differential equation with the unknown function qi , we obtain the following equivalent representation
qi ðtÞ ¼
1
Z
sq
0
1
1 es=sq kij T ;j ðt sÞ þ sT T_ ;j ðt sÞ þ s2T T€ ;j ðt sÞ ds; 2
ð22Þ
qi ðtÞ ¼
s2T
2sq sq 1 sT 1 1
sq
sq
sT 2sq
sT
kij T ;j ðtÞ
2sq Z 1 0
es=sq kij T ;j ðt sÞds:
ð23Þ
Thus, with the choice (12), from (23) we deduce
Z 2p=x 0
p kij f i f j þ kij g i g j s
s T s q 1 T x2 þ 1 ; qi ðtÞT ;i ðtÞdt ¼ 2sq x 1 þ s2q x2 ð24Þ
and it will be negative for all x > 0 if and only if the following inequality
0 < sT 6 2sq ;
p 2x
kij f i f j þ kij g i g j 2 s2T x2 ;
ð30Þ
which cannot conserve a constant sign for all x > 0. Thus, the Second Law of Thermodynamics cannot be satisfied and the corresponding dual-phase-lag model is inconsistently from thermodynamic point of view. Let us now consider the constitutive equation
qi ðtÞ þ sq q_ i ðtÞ þ
1 2 1 v s q€i ðtÞ þ s3q qi ðtÞ ¼ kij T ;j ðtÞ; 2 q 6
ð31Þ
which can be written as
1 1 v €j ðtÞ þ s3q qj ðtÞ : T ;i ðtÞ ¼ K ij qj ðtÞ þ sq q_ j ðtÞ þ s2q q 2 6
ð32Þ
Thus, for any cycle characterized by
qi ðt sÞ ¼ hi cos xðt sÞ þ ‘i sin xðt sÞ; x > 0; hi hi þ ‘i ‘i > 0; ð33Þ we have
Z 2p=x 0
qi ðtÞT ;i ðtÞdt ¼
p K ij hi hj þ K ij ‘i ‘j 2 s2q x2 ; 2x
ð34Þ
which cannot conserve a constant sign for all x > 0. Thus, the Second Law of Thermodynamics cannot be satisfied and the corresponding dual-phase-lag model is inconsistently from thermodynamic point of view.
and furthermore, by successive integration by parts, we have
sT kij T_ ;j ðtÞ 1
0
qi ðtÞT ;i ðtÞdt ¼
ð25Þ
is fulfilled. Concluding, the dual-phase-lag models based upon the constitutive Eqs. (19) and (21) are compatible with the thermodynamics, provided the corresponding inequalities (20) and (25) are fulfilled. 8. Case ðn; mÞ ¼ ð2; 2Þ The restrictions upon the delay times which follow from Second Law of Thermodynamics for the constitutive equation
10. Case ðn; mÞ‰fð3; 1Þ; ð1; 3Þg We consider first the constitutive equation
1 1 v qi ðtÞ þ sq q_ i ðtÞ ¼ kij T ;j ðtÞ þ sT T_ ;j ðtÞ þ s2T T€ ;j ðtÞ þ s3T T ;j ðtÞ ; ð35Þ 2 6 from which we deduce the following representation
1 es=sq kij T ;j ðt sÞ þ sT T_ ;j ðt sÞ þ s2T T€ ;j ðt sÞ 2 sq 0 1 3v ð36Þ þ sT T ;j ðt sÞ ds: 6
qi ðtÞ ¼
1
Z
1
Further, for any cycle as described by (12), we obtain Z 2p=x p kij f i f j þ kij g i g j 1 3 4 1 2
qi ðtÞT ;i ðtÞdt ¼ sq sT x þ sT sq sT x2 1 ; 6 2 0 x 1 þ s2q x2 ð37Þ
an expression which cannot be negative for all positive values of x. Thus, the dual-phase-lag model based on the constitutive Eq. (35) cannot be compatible with thermodynamics.
S. Chiriţa˘ et al. / International Journal of Heat and Mass Transfer 114 (2017) 277–285
When the case ðn; mÞ ¼ ð3; 1Þ is addressed we write the constitutive equation in the form
T ;i ðtÞ þ sT T_ ;i ðtÞ ¼ K ij
1 1 v €j ðtÞ þ s3q qj ðtÞ ; qj ðtÞ þ sq q_ j ðtÞ þ s2q q 2 6
ð38Þ
where, for convenience, we have set
0
ð40Þ
ð41Þ
ð42Þ
In the other case, that is for ðn; mÞ ¼ ð3; 2Þ, we write the constitutive equation in the following form
1 T ;i ðtÞ þ sT T_ ;i ðtÞ þ s2T T€ ;i ðtÞ 2 1 1 v €j ðtÞ þ s3q qj ðtÞ ; ¼ K ij qj ðtÞ þ sq q_ j ðtÞ þ s2q q 2 6
jðsÞ cos xsds; js ¼
0
js
6 ¼ s2q x4
ð43Þ
ð44Þ
12. Case ðn; mÞ ¼ ð3; 3Þ
s3q
sq
"
12:1972
from which we deduce
i
Z 0
1
ð51Þ
sT 1 sq
2
sq
#
þ 0:0003
sq
sq
s2q x2 þ 24:3944 P 0;
ð52Þ
ð53Þ
where
s3T sT s2 sT ; b1 ¼ 4:06574 T2 6:09877 þ 4:06582 s3q sq sq sq sT 1 sq
2
þ 0:0003;
! ð54Þ
d1 ¼ 24:3944:
Furthermore, we see that the derivative
df ¼ 3a1 z2 2b1 z c1 ; dz
ð55Þ
b1
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 b1 þ 3a1 c1 3a1
< 0;
z2 ¼
b1 þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 b1 þ 3a1 c1 3a1
> 0:
ð56Þ
Thus, f ðzÞ P 0 for all z > 0 if we have f ðz2 Þ > 0, that is
ð45Þ
h
jðsÞkij T ;j ðt sÞ þ sT T_ ;j ðt sÞ
s2T € s3 v T ;j ðt sÞ þ T T ;j ðt sÞ ds; 6
q
for all x P 0. We associate with this inequality the function
z1 ¼
v
€i ðtÞ þ qi ðtÞ qi ðtÞ þ sq q_ i ðtÞ þ q 2 6 s2 s3 v ¼ kij T ;j ðtÞ þ sT T_ ;j ðtÞ þ T T€ ;j ðtÞ þ T T ;j ðtÞ ; 2 6
2
3
admits the roots
We consider here the constitutive equation
þ
ð50Þ
f ðzÞ a1 z3 b1 z2 c1 z þ d1 ;
Thus, we can conclude that the restrictions to be fulfilled by the delay times in order to have the thermodynamic consistency are described by the relations (42) and (44), respectively.
sq ða cÞ2 þ d2
jðsÞ sin xsds:
In view of the values for a; c and d given in (48), it follows that the constitutive Eq. (45) is compatible with thermodynamics if the following condition is fulfilled !
3 s
2 s3 s2 sT T 0:677637 T3 s2q x2 4:0657 T2 6:09877 þ 4:06582 s2q x2
0:28441 sT < sq 6 1:4902 sT :
6
0
3 d2 3c2 þ 2ca s2q x2 1 7 þ 5: a2 þ s2q x2 c2 þ d2 þ s2 x2 2 4d2 s2 x2 q q
c1 ¼ 12:1972
h
ð48Þ
q
and use the same procedure like that in the above to obtain the following restriction
qi ðtÞ ¼
1
a1 ¼ 0:677637
s2q
ð47Þ
ða 2cÞ c2 þ d2 as2q x2 7 a jc ¼ sq 6 þ 4 2 5;
2 a þ s2q x2 c2 þ d2 þ s2 x2 4d2 s2 x2 2
which is negative for all positive x if the delay times satisfy the inequality
0:28441 sq < sT 6 1:4902 sq :
Z
1
2
" 2p kij f i f j þ kij g i g j 1 s2T 2 sT 4 4 qi ðtÞT ;i ðtÞdt ¼ 1 sx 4 4 2 2 sq 3 sq q xðsq x þ 4Þ # !
sT s2 1 T2 s2q x2 þ 2 ; sq sq
sq
;
Furthermore, from the relations (47) and (50), we have
1
þ 2
ds
6p kij f i f j þ kij g i g j h i sq x ða cÞ2 þ d2 1 1 1 s2T x2 jc þ sT x s3T x3 js ; ð49Þ 2 6
Z
jc ¼
and therefore, for any cycle as described by (12), we obtain
Z 2p=x
sq
cos
qi ðtÞT ;i ðtÞdt ¼
where
It is equivalent with the following representation
h s k T ðt sÞ þ sT T_ ;j ðt sÞ es=sq sin sq 0 sq ij ;j 1 1 v þ s2T T€ ;j ðt sÞ þ s3T T ;j ðt sÞ ds; 2 6
ds
Then, for any cycle characterized by (12), we have
0
Let us first consider the case ðn; mÞ ¼ ð2; 3Þ, that is we study the constitutive equation 1 1 1 v €i ðtÞ ¼ kij T ;j ðtÞ þ sT T_ ;j ðtÞ þ s2T T€ ;j ðtÞ þ s3T T ;j ðtÞ : qi ðtÞ þ sq q_ i ðtÞ þ s2q q 2 2 6 ð39Þ
Z
d
sin
a ¼ 1:5961; c ¼ 0:70196; d ¼ 1:8073: Z 2p=x
11. Case ðn; mÞ‰fð3; 2Þ; ð2; 3Þg
2
ac
jðsÞ ¼ eas=sq þ ecs=sq and
and use the above procedure to establish the incompatibility with thermodynamics of the corresponding dual-phase-lag model of heat conduction.
qi ðtÞ ¼
281
ð46Þ
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
3 1
2 3 b1 þ 3a1 c1 6 27a21 d1 2b1 9a1 b1 c1 : 2
ð57Þ
Thus, the constitutive Eq. (45) is compatible with the thermodynamics if the delay times satisfy the inequality (57). It can be verified that the set of delay times fulfilling (57) is non-empty (in fact it contains the case when sT ¼ sq ). Moreover, we have to outline that the inequality (52) has an approximately character and it depends on the approximation order of the roots a; c and d as defined in (48) and obtained in the Table 1.
S. Chiriţa˘ et al. / International Journal of Heat and Mass Transfer 114 (2017) 277–285
282
13. Case ðn; mÞ‰fð4; 0Þ; ð0; 4Þg Let us first consider the constitutive equation
"
qi ðtÞ ¼ kij T ;j ðtÞ þ sT T_ ;j ðtÞ þ
# @4T 1 2€ 1 v 1 sT T ;j ðtÞ þ s3T T ;j ðtÞ þ s4T 4;j ðtÞ ; 2 6 24 @t ð58Þ
and note that for any cycle as described by (12) we have
Z 2p=x 0
qi ðtÞT ;i ðtÞdt ¼
i p kij f i f j þ kij g i g j h 2 2 2 sT x 6 12 ; ð59Þ 24x
which cannot conserve a constant sign for all x > 0. Thus, the Second Law of Thermodynamics cannot be satisfied and the corresponding dual-phase-lag model is inconsistently from thermodynamic point of view. Let us now consider the constitutive equation
qi ðtÞ þ sq q_ i ðtÞ þ
4
1 2 1 v 1 @ q s q€i ðtÞ þ s3q qi ðtÞ þ s4q 4i ðtÞ ¼ kij T ;j ðtÞ; 2 q 6 24 @t ð60Þ
4
T ;i ðtÞ ¼ K ij qj ðtÞ þ sq q_ j ðtÞ þ
@ q 1 2 1 v 1 s q€j ðtÞ þ s3q qj ðtÞ þ s4q 4j ðtÞ : 2 q 6 24 @t
Thus, for any cycle characterized by (33), we obtain
0
qi ðtÞT ;i ðtÞdt ¼
"
T ;i ðtÞ þ sT T_ ;i ðtÞ ¼ K ij qj ðtÞ þ sq q_ j ðtÞ þ
# 4 1 2 1 v 1 @ q sq q€j ðtÞ þ s3q qj ðtÞ þ s4q 4j ðtÞ ; 2 6 24 @t
and we obtain Z 2p=x p K ij hi hj þ K ij ‘i ‘j qi ðtÞT ;i ðtÞdt ¼ x 1 þ s2T x2 0 " # s3q sq sq sq 2 2 4 4 1 s x þ 1 s x þ 1 ; T sT 2sT T 6s3T 4sT
ð67Þ
ð68Þ
which cannot conserve a constant sign for all x > 0. Concluding this section, we see that the dual-phase-lag models based on the constitutive Eqs. (63) and (66) are not compatible with the thermodynamics.
#
ð61Þ Z 2p=x
we write it as
ð66Þ
15. Case ðn; mÞ‰fð4; 2Þ; ð2; 4Þg
which can be written as
"
1 1 v 1 4 @ 4 qi €i ðtÞ þ s3q qi ðtÞ þ qi ðtÞ þ sq q_ i ðtÞ þ s2q q s ðtÞ 2 6 24 q @t4 h i ¼ kij T ;j ðtÞ þ sT T_ ;j ðtÞ ;
p K ij hi hj þ K ij ‘i ‘j 2 2 2 sq x 6 12 ; 24x
Let us first consider the constitutive equation
1 €i ðtÞ qi ðtÞ þ sq q_ i ðtÞ þ s2q q 2 # 1 1 v 1 4 @ 4 T ;j sT 4 ðtÞ ; ¼ kij T ;j ðtÞ þ sT T_ ;j ðtÞ þ s2T T€ ;j ðtÞþ s3T T ;j ðtÞ þ 2 6 24 @t ð69Þ
ð62Þ which cannot conserve a constant sign for all x > 0. Thus, the Second Law of Thermodynamics cannot be satisfied and the corresponding dual-phase-lag model is inconsistently from thermodynamic point of view.
which gives
qi ðtÞ ¼
2
sq
Z
1
es=sq sin
0
s
sq
h kij T ;j ðt sÞ þ sT T_ ;j ðt sÞ
# 1 2€ 1 3v 1 4 @ 4 T ;j þ sT T ;j ðt sÞþ sT T ;j ðt sÞ þ s ðt sÞ ds: 2 6 24 T @t 4
14. Case ðn; mÞ‰fð4; 1Þ; ð1; 4Þg
ð70Þ
Further, for any cycle characterized by (12), we obtain We consider now the constitutive equation " # 1 1 v 1 @4T qi ðtÞ þ sq q_ i ðtÞ ¼ kij T ;j ðtÞ þ sT T_ ;j ðtÞ þ s2T T€ ;j ðtÞ þ s3T T ;j ðtÞ þ s4T 4 ðtÞ ; 2 6 24 @t
Z 2p=x 0
qi ðtÞT ;i ðtÞdt ¼
1
! s4T 6 6 s4T 2s3T s2T 4 4
sx þ þ sq x 12s4q q 6s4q 3s3q s2q
sq
! # s2T 2 2
þ sq x þ 4 ; 2 1 sq s2q
1 es=sq kij T ;j ðt sÞ þ sT T_ ;j ðt sÞ þ s2T T€ ;j ðt sÞ 2 # 4 1 v 1 4@ T s ðt sÞ ds: ð64Þ þ s3T T ;j ðt sÞ þ 6 24 T @t 4
qi ðtÞ ¼
1
Z
2sT
0
Therefore, for any cycle characterized by (12), we get Z 2p=x p kij f i f j þ kij g i g j
qi ðtÞT ;i ðtÞdt ¼ 0 x 1 þ s2q x2 " # s3T sT sT sT 2 2 4 4 1 s x þ 1 s x þ 1 ; q 6s3q 4sq sq 2sq q
ð65Þ which cannot conserve a constant sign for all x > 0. Thus, the Second Law of Thermodynamics cannot be satisfied and the corresponding dual-phase-lag model is inconsistently from thermodynamic point of view. When it is considered the constitutive equation
"
ð63Þ
which furnishes
p kij f i f j þ kij g i g j xðs4q x4 þ 4Þ
ð71Þ
which cannot conserve a negative sign for all x > 0. Thus, the Second Law of Thermodynamics cannot be satisfied and the corresponding dual-phase-lag model is inconsistently from thermodynamic point of view. The same conclusion can be obtained for the case ðn; mÞ ¼ ð4; 2Þ. 16. Case ðn; mÞ‰fð4; 3Þ; ð3; 4Þg Let us first consider the constitutive equation
qi ðtÞ þ sq q_ i ðtÞ þ
1 2 1 v s q€i ðtÞ þ s3q qi ðtÞ 2 q 6
# 1 2€ 1 3v 1 4 @ 4 T ;j _ ¼ kij T ;j ðtÞ þ sT T ;j ðtÞ þ sT T ;j ðtÞþ sT T ;j ðtÞ þ s ðtÞ ; 2 6 24 T @t 4
ð72Þ
S. Chiriţa˘ et al. / International Journal of Heat and Mass Transfer 114 (2017) 277–285
which gives
qi ðtÞ ¼
6
h
sq ðc aÞ2 þ d2
i
Z 0
1
h
jðsÞkij T ;j ðt sÞ þ sT T_ ;j ðt sÞ
# 1 2€ 1 3v 1 4 @ 4 T ;j þ sT T ;j ðt sÞ þ sT T ;j ðt sÞ þ s ðt sÞ ds; 2 6 24 T @t4
ð73Þ
where jðsÞ is given by relation (47) and the values of a; d and c are given by (48). For any cycle characterized by (12), we have
Z 2p=x 0
ð74Þ
2 sT s3 s2 sT 1:01643 T3 4:06574 T2 þ6:09877 4:06582 s2q x2 sq sq sq sq "
sT 1 sq
2
#
þ 0:0003
s2q x2 þ 24:3944 P 0;
ð75Þ
for all x P 0. Thus, the constitutive Eq. (72) is compatible with the thermodynamics if the delay times satisfy the inequalities
sT
0< < 1:33332; sq rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
3 1
2 3 b2 þ 3a2 c2 6 27a22 d2 2b2 9a2 b2 c2 ; 2
ð76Þ
where
s3T s4 0:508231 T4 3 sq sq
a2 ¼ 0:677637 b2 ¼
sT s3 s2 sT 1:01643 T3 þ 4:06574 T2 6:09877 þ 4:06582 sq sq sq sq
c2 ¼ 12:1972
sT 1 sq
which can be written as
qi ðtÞ ¼
! s3T s4T 2 2 3 0:677637 3 0:508231 4 sq x sq sq
12:1972
ð79Þ
!
where
"
1
bs
ð81Þ
1 D ¼ 3ab2 3cd2 þ c3 a3 2ðc aÞ h ðc aÞ2 3a2 þ 3c2 b2 d2 þ b2 d2 3c2 3a2 þ b2 d2 ¼ 22:165;
ð82Þ
and now we have set
a ¼ 0:27056; b ¼ 2:5048; c ¼ 1:7294; d ¼ 0:88897:
ð83Þ
Then, for any cycle characterized by (12), we have Z 2p=x 24p kij f i f j þ kij g i g j 1 1 qi ðtÞT ;i ðtÞdt ¼ 1 s2T x2 þ s4T x4 K c 2 24 sq x D 0 1 3 3 þ sT x sT x K s ; 6 ð84Þ
Z 0
d2 ¼ 24:3944:
ð80Þ
# ða cÞ2 þ d2 b2 bs KðsÞ ¼ e cos sin sq 2bðc aÞ sq " # ds ða cÞ2 þ b2 d2 ds ; ecs=sq cos þ sin sq 2dðc aÞ sq as=sq
Kc ¼
þ 0:0003;
Z
1
Ks ¼
KðsÞ cos xsds;
1
ð85Þ
KðsÞ sin xsds:
0
Further, with the aid of (81), we have
ð77Þ It can be verified that it is non-empty the set of values for the delay times as defined by the relation (76), (in fact it contains at least the case when the delay times are equally, sT ¼ sq ). When the case ðn; mÞ ¼ ð4; 3Þ is considered, the constitutive equation is written in the equivalent form
1 1 v T ;i ðtÞ þ sT T_ ;i ðtÞ þ s2T T€ ;i ðtÞ þ s3T T ;i ðtÞ 2 6 # 4 1 1 v 1 4 @ qj €j ðtÞþ s3q qj ðtÞ þ ¼ K ij qj ðtÞ þ sq q_ j ðtÞ þ s2q q sq 4 ðtÞ ; 2 6 24 @t
2
q
c2 þ d2 a2 b2 s2q x2 þ c2 þ d2
2 K s ¼ s x4 2 q
h
s2q x2 þ a2 b2 caa ðc aÞ2 þ d2 b2
s4q x4 þ 2 a2 b2 s2q x2 þ a2 þ b2 h
s2q x2 þ c2 d2 þ cc a ðc aÞ2 þ b2 d2
q
3 3c2 d2 þ a2 þ b2 4ac 7 5
2 s4q x4 þ 2ðc2 d2 Þs2q x2 þ c2 þ d2
and
and then the thermodynamic restrictions can be obtained easily by using the relation (76).
2 2 2 2 2 2 2 2 2 2 2 2 sq 6 c þ d a b sq x a þ b 3a b þ c þ d 4ac Kc ¼ 4
2 2ðc aÞ s4 x4 þ 2ða2 b2 Þs2 x2 þ a2 þ b2
ð78Þ
Let us now consider the constitutive equation
24 sq D
where
2
17. Case ðn; mÞ ¼ ð4; 4Þ
Z
h s2 KðsÞkij T ;j ðt sÞ þ sT T_ ;j ðt sÞþ T T€ ;j ðt sÞ 2 0 # 4 3 v s 1 4 @ T ;j þ T T ;j ðt sÞ þ s ðt sÞ ds; 24 T @t4 6
qi ðtÞT ;i ðtÞdt ¼
where jc and js are defined by the relation (51). In view of relations (48), (51) and (74) it follows that the Second Law of Thermodynamics can be satisfied if the following inequality is fulfilled
þ
1 1 v 1 4 @ 4 qi €i ðtÞ þ s3q qi ðtÞ þ qi ðtÞ þ sq q_ i ðtÞ þ s2q q s ðtÞ 2 6 24 q @t 4 " # 1 1 v 1 4 @ 4 T ;j sT 4 ðtÞ ; ¼ kij T ;j ðtÞ þ sT T_ ;j ðtÞ þ s2T T€ ;j ðtÞ þ s3T T ;j ðtÞ þ 2 6 24 @t
6p kij f i f j þ kij g i g j h i sq x ðc aÞ2 þ d2 1 1 4 4 1 s2T x2 þ sT x jc 2 24 1 3 3 þ sT x sT x js ; 6
283
s4q x4 þ 2 c2 d2 s2q x2 þ c2 þ d2
2
ð86Þ
i
2
i3 5:
ð87Þ
In view of relations (83), (86) and (87) it follows that the Second Law of Thermodynamics can be satisfied if the following inequality is fulfilled
S. Chiriţa˘ et al. / International Journal of Heat and Mass Transfer 114 (2017) 277–285
284
2:69456
!
3 s4T 2 2 4 s2T s2T sT s x 32:3346 43:1121 þ 32:3348 s2q x2 q 4 2 2 sq sq sq sq
þ 64:668
!
2 s4T s3T s2T sT 258:676 þ 388:015 258:673 þ 64:6695 s2q x2 s4q s3q s2q sq
776:016
!
s2T sT 1552:06 þ 776:031 s2q x2 þ 1552:03 P 0; 2 sq sq
ð88Þ
for all x P 0. It is not difficult to see that the above inequality is fulfilled when sT ¼ sq and so we can conclude that it is non-empty the set of couples ðsT ; sq Þ for which the constitutive Eq. (79) is compatible with the thermodynamics. Unfortunately, there seems to be unavailable to get an explicit expression defining this set. 18. Conclusions
ð0; 0Þ ð0; 1Þ ð0; 2Þ ð0; 3Þ
ð0; 4Þ
1
C B B ð1; 0Þ ð1; 1Þ ð1; 2Þ ð1; 3Þ ð1; 4Þ C C B B m n ¼ B ð2; 0Þ ð2; 1Þ ð2; 2Þ ð2; 3Þ ð2; 4Þ C C; C B @ ð3; 0Þ ð3; 1Þ ð3; 2Þ ð3; 3Þ ð3; 4Þ A ð4; 0Þ ð4; 1Þ ð4; 2Þ ð4; 3Þ ð4; 4Þ provided appropriate restrictions are placed upon the delay times. In fact, from a mathematical point of view, the lagging behavior can be equally described by the following equation equivalent with the constitutive Eq. (3)
qi ðx; tÞ ¼ qhi ðx; tÞ þ qpi ðx; tÞ;
ð89Þ
where qhi ðx; tÞ represents the general integral of the homogeneous differential equation
qi ðx; tÞ þ
sq @qi 1! @t
ðx; tÞ þ
Conflict of interest None declared.
We inferred here an opinion about the time differential dualphase-lag models that is based on the information that we have upon the differential operators involved into the related constitutive equations. It is shown that, when the approximation orders are greater than or equal to five, the corresponding models lead to some instable mechanical systems. Instead, when the approximation orders are lower than or equal to four, then the corresponding models can be compatible with the thermodynamics, provided some appropriate restrictions are assumed upon the delay times. More precisely, the thermodynamical consistency of the model in concern is established when ðm; nÞ take the values in boldface of the matrix
0
ables. The extension of this analysis to nonlinear models (when, for example, the thermophysical properties of substances depend strongly on the temperature), there seems to be a complex task and for this it remains an open problem to be subject in a future research. On the other side, the present paper furnishes a new perspective upon the relationship between the thermodynamic aspects and the stability properties, with reference to the time differential dual-phase-lag models of heat conduction. We believe that this research provides useful information to clarify the applicability of these models.
s2q @ 2 qi 2! @t
2
ðx; t Þ þ þ
snq @ n qi n! @t n
ðx; t Þ ¼ 0;
ð90Þ
Acknowledgments The authors are very grateful to the reviewers for their valuable comments, which have led to an improvement of the present work. Appendix A. Szegö’s curve In this appendix we provide the source code that has been created to generate an animation with the software package Wolfram Mathematica 11. The generated animation shows the complex roots of the scaled exponential sum of order n in the complex plane, with n increasing from 1 to 50, and it shows that the roots approach the Szegö curve as n increases. The following code could be entered in a Mathematica session and, once executed, the animation window is shown. e[n_, z_] :¼ Sum[z^k/k!, {k, 0, n}] axes = Graphics[{Line[{{-1, 0}, {1, 0}}], Line[{{0, 1}, {0, 1}}]}]; szegoeCurve = ContourPlot[Evaluate[Abs[z E^(1 z)] == 1/. z -> x + I y], {x, 1, 1}, {y, 1, 1}, PlotPoints -> 50]; rootsGraph[m_] :¼ Module[{roots, n = IntegerPart [m], points,} roots = NSolve[e[n, n z] == 0, z][[All, 1, 2]]; points = Transpose[{Re[roots], Im[roots]}]; Show[{axes, szegoeCurve, ListPlot[points, PlotRange -> {{-1, 1}, {-1, 1}}, AspectRatio -> 1]}]] Animate[rootsGraph[m], {m, 1, 50}, DefaultDuration > 50]
and qpi ðx; tÞ is a particular solution of the differential Eq. (3). The explicit expression of qhi ðx; tÞ represents a concrete combination of the number of time exponentials relating the roots of the characteristic Eq. (4), while the expression of the particular solution qpi ðx; tÞ represents a convolution (memory) term. In view of the results established by Szegö [18], if follows that, for n P 5, the expression of qhi ðx; tÞ involves some exponentials growing in time at infinity and this proves an asymptotic instable mechanical system. Thus, the situation with ðn; mÞ ¼ ð1; 0Þ, which corresponds to rather widely used model with an only delay time of the heat flux relative to the temperature gradient, appears to lead to an instable mechanical system. Instead, when n < 5 this is not the case and we have searched the memory term described by qpi ðx; tÞ by means of the techniques developed for materials with memory by Gurtin and Pipkin [12] and Coleman and Gurtin [13] and Amendola et al. [20]. Our analysis is limited to the constitutive Eq. (3), where the delay times are assumed to be positive constant parameters. It is worth to mention that our analysis remains valid when the delay times are not constant, but they can depend on the spatial vari-
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