On high-order approximations for describing the lagging behavior of heat conduction

Abstract

This paper is dedicated to high-order effects of thermal lagging in correlation with heat transfer models in micro- or nanoscale, relating the number of energy carriers and the associated resonance phenomenon under high-frequency excitations. Thus, a class of constitutive equations is considered for the heat flux describing high-order effects in the lagging behavior of heat transport. Tzou’s model, which is based on time-differential dual-phase-lag approximations of heat conduction, is generalized, incorporating the microstructural interaction effect in the fast-transient process of heat transport. More precisely, polynomial approximations of order n for the heat flux vector and of order m for the gradient of the temperature variation are considered. Further, well-posedness is established for solutions of the specific initial boundary value problems for the mathematical model when: (i) $m = n$ ; (ii) $m = n - 1$ ; and (iii) $m = n + 1$ . This means that uniqueness and continuous dependences of solutions are established with respect to the given prescribed data. With this aim, some modified initial boundary value problems related to the operators of the constitutive equation are introduced and suitable time-weighted measures of the solution are presented, and are used to establish the uniqueness theorems and to generate some estimates describing the continuous dependence of solutions with respect to the given data for each of the three cases specified. Moreover, the spatial behavior of the transient solutions is studied and an influence domain result is established for $m = n \pm 1$ , while for $m = n$ some exponential decay estimates of Saint-Venant type are established. All results are established without restrictions on the characteristic parameters of the constitutive equation other than that the product of the coefficients of the time derivatives of greatest order of the two operators involved in the constitutive equation is positive.

Keywords

Lagging behavior heat conduction models uniqueness continuous dependence spatial behavior time-weighted measure

Get full access to this article

View all access options for this article.

References

Tzou

. A unified approach for heat conduction from macro to micro-scales. J Heat Transfer 1995; 117: 8–16.

Tzou

. The generalized lagging response in small-scale and high-rate heating. Int J Heat Mass Transfer 1995; 38: 3231–3234.

Tzou

. Experimental support for the lagging behavior in heat propagation. J Thermophys Heat Transfer 1995; 9: 686–693.

Tzou

. Macro- to micro-scale heat transfer: The lagging behavior. Chichester, UK: John Wiley & Sons, 2015.

Chandrasekharaiah

. Hyperbolic thermoelasticity: A review of recent literature. Appl Mech Rev 1998; 51: 705–729.

Fabrizio

Lazzari

. Stability and second law of thermodynamics in dual-phase-lag heat conduction. Int J Heat Mass Transfer 2014; 74: 484–489.

Wang

Zhou

. Well-posedness and solution structure of dual-phase-lagging heat conduction. Int J Heat Mass Transfer 2001; 44: 1659–1669.

Wang

. Well-posedness of dual-phase-lagging heat equation: Higher dimensions. Int J Heat Mass Transfer 2002; 45: 1165–1171.

Wang

. Thermal oscillation and resonance in dual-phase-lagging heat conduction. Int J Heat Mass Transfer 2002; 45: 1055–1061.

10.

Quintanilla

. Exponential stability in the dual-phase-lag heat conduction theory. J Non-Eq123 Thermodyn 2002; 27: 217–227.

11.

Horgan

Quintanilla

. Spatial behaviour of solutions of the dual-phase-lag heat equation. Math Methods Appl Sci 2005; 28: 43–57.

12.

Quintanilla

Racke

. A note on stability in dual-phase-lag heat conduction. Int J Heat Mass Transfer 2006; 49: 1209–1213.

13.

Quintanilla

Racke

. Qualitative aspects in dual-phase-lag thermoelasticity. SIAM J Appl Math 2006; 66: 977–1001.

14.

Quintanilla

Racke

. Qualitative aspects in dual-phase-lag heat conduction. Proc R Soc London, Ser A 2007; 463: 659–674.

15.

Liu

Chang

. Analysis of dual-phase-lag heat conduction in cylindrical system with a hybrid method. Appl Math Modell 2007; 31: 369–380.

16.

Fabrizio

Franchi

. Delayed thermal models: Stability and thermodynamics. J Therm Stresses 2014; 37: 160–173.

17.

Quintanilla

Racke

. Spatial behavior in phase-lag heat conduction. Differ Integral Equ 2015; 28: 291–308.

18.

Abouelregal

Abo-Dahab

. Study of the dual-phase-lag model of thermoelasticity for a half-space problem with rigidly fixed surface in the presence of a thermal shock. J Comput Theor Nanosci 2015; 12: 38–45.

19.

Fabrizio

Lazzari

Tibullo

. Stability and thermodynamic restrictions for a dual-phase-lag thermal model. J Non-Eq124 Thermodyn 2017; 42: 243–252.

20.

Chiriţă

. On the time differential dual-phase-lag thermoelastic model. Meccanica 2017; 52: 349–361.

21.

Quintanilla

. A well posed problem for the dual-phase-lag heat conduction. J Therm Stresses 2008; 31: 260–269.

22.

Quintanilla

. A well-posed problem for the three-dual-phase-lag heat conduction. J Therm Stresses 2009; 32: 1270–1278.

23.

Quintanilla

. On uniqueness and stability for a thermoelastic theory. Math Mech Solids 2016; 22: 1387–1396.

24.

Amendola

Fabrizio

Golden

et al . Second-order approximation for heat conduction: Dissipation principle and free energies. Proc R Soc London, Ser A 2016; 472: 20150707.

25.

Chiriţă

Ciarletta

Tibullo

. Qualitative properties of solutions in the time differential dual-phase-lag model of heat conduction. Appl Math Modell 2017; 50: 380–393.

26.

Chiriţă

Ciarletta

Tibullo

. On the wave propagation in the time differential dual-phase-lag thermoelastic model. Proc R Soc London, Ser A 2015; 471: 20150400.

27.

Chiriţă

Ciarletta

Tibullo

. On the thermomechanical consistency of the time differential dual-phase-lag models of heat conduction. Int J Heat and Mass Transfer 2017; 114: 277–285.

28.

Chiriţă

Ciarletta

. Time-weighted surface power function method for the study of spatial behaviour in dynamics of continua. Eur J Mech A Solids 1999; 18: 915–933.