scholarly journals Qualitative aspects in dual-phase-lag heat conduction

2006 ◽  
Vol 463 (2079) ◽  
pp. 659-674 ◽  
Author(s):  
Ramón Quintanilla ◽  
Reinhard Racke
Keyword(s):  
Heat Flux ◽  
Heat Conduction ◽  
Uniqueness Result ◽  
Phase Lag ◽  
Dual Phase ◽  
Natural Property ◽  
Unbounded Solutions ◽  
Decay Of Solutions ◽  
Exponentially Stable ◽  
Well Posed

We investigate the equation of the dual-phase-lag heat conduction proposed by Tzou. To describe this equation, we use the phase lag of the heat flux and the phase lag of the gradient of the temperature. We analyse the basic properties of the solutions of this problem. First, we prove that when both parameters are positive, the problem is well posed and the spatial decay of solutions is controlled by an exponential of the distance. When the phase lag of the gradient of the temperature is bigger than the phase lag of the heat flux, the problem is exponentially stable (which is a natural property to expect for a heat equation) and the spatial behaviour is controlled by an exponential of the square of the distance. Also, a uniqueness result for unbounded solutions is proved in this case.

Nonhomogeneous Dual-Phase-Lag Heat Conduction Problem: Analytical Solution and Select Case Studies

2017 ◽  
Vol 140 (3) ◽  
Author(s):  
Simon Julius ◽  
Boris Leizeronok ◽  
Beni Cukurel
Keyword(s):  
Heat Flux ◽  
Heat Conduction ◽  
Analytical Solution ◽  
Heat Generation ◽  
Integral Transform ◽  
Heat Conduction Problem ◽  
Relaxation Times ◽  
Hyperbolic Heat Conduction ◽  
Phase Lag ◽  
Dual Phase

Finite integral transform techniques are applied to solve the one-dimensional (1D) dual-phase heat conduction problem, and a comprehensive analysis is provided for general time-dependent heat generation and arbitrary combinations of various boundary conditions (Dirichlet, Neumann, and Robin). Through the dependence on the relative differences in heat flux and temperature relaxation times, this analytical solution effectively models both parabolic and hyperbolic heat conduction. In order to demonstrate several exemplary physical phenomena, four distinct cases that illustrate the wavelike heat conduction behavior are presented. In the first model, following an initial temperature spike in a slab, the thermal evolution portrays immediate dissipation in parabolic systems, whereas the dual-phase solution depicts wavelike temperature propagation—the intensity of which depends on the relaxation times. Next, the analysis of periodic surface heat flux at the slab boundaries provides evidence of interference patterns formed by temperature waves. In following, the study of Joule heating driven periodic generation inside the slab demonstrates that the steady-periodic parabolic temperature response depends on the ratio of pulsatile electrical excitation and the electrical resistivity of the slab. As for the dual-phase model, thermal resonance conditions are observed at distinct excitation frequencies. Building on findings of the other models, the case of moving constant-amplitude heat generation is considered, and the occurrences of thermal shock and thermal expansion waves are demonstrated at particular conditions.

Dual-phase-lag thermoelastic problem in finite cylindrical domain with relaxation time

2018 ◽  
Vol 14 (5) ◽  
pp. 837-856 ◽  
Author(s):  
Gaurav Mittal ◽  
Vinayak Kulkarni
Keyword(s):  
Heat Flux ◽  
Relaxation Time ◽  
Heat Conduction ◽  
Generalized Thermoelasticity ◽  
Phase Lag ◽  
Dual Phase ◽  
Cylindrical Domain ◽  
Content Type ◽  
Thick Circular Plate ◽  
Lag Model

Purpose The purpose of this paper is to frame a dual-phase-lag model using the fractional theory of thermoelasticity with relaxation time. The generalized Fourier law of heat conduction based upon Tzou model that includes temperature gradient, the thermal displacement and two different translations of heat flux vector and temperature gradient has been used to formulate the heat conduction model. The microstructural interactions and corresponding thermal changes have been studied due to the involvement of relaxation time and delay time translations. This results in achieving the finite speed of thermal wave. Classical coupled and generalized thermoelasticity theories are recovered by considering the various special cases for different order of fractional derivatives and two different translations under consideration. Design/methodology/approach The work presented in this manuscript proposes a dual-phase-lag mathematical model of a thick circular plate in a finite cylindrical domain subjected to axis-symmetric heat flux. The model has been designed in the context of fractional thermoelasticity by considering two successive terms in Taylor’s series expansion of fractional Fourier law of heat conduction in the two different translations of heat flux vector and temperature gradient. The analytical results have been obtained in Laplace transform domain by transforming the original problem into eigenvalue problem using Hankel and Laplace transforms. The numerical inversions of Laplace transforms have been achieved using the Gaver−Stehfast algorithm, and convergence criterion has been discussed. For illustrative purpose, the dual-phase-lag model proposed in this manuscript has been applied to a periodically varying heat source. The numerical results have been depicted graphically and compared with classical, fractional and generalized thermoelasticity for various fractional orders under consideration. Findings The microstructural interactions and corresponding thermal changes have been studied due to the involvement of relaxation time and delay time translations. This results in achieving the finite speed of thermal wave. Classical coupled and generalized thermoelasticity theories are recovered by considering the various special cases for different order of fractional derivatives and two different translations under consideration. This model has been applied to study the thermal effects in a thick circular plate subjected to a periodically varying heat source. Practical implications A dual-phase-lag model can effectively be incorporated to study the transient heat conduction problems for an exponentially decaying pulse boundary heat flux and/or for a short-pulse boundary heat flux in long solid tubes and cylinders. This model is also applicable to study the various effects of the thermal lag ratio and the shift time. These dual-phase-lag models are also practically applicable in the problems of modeling of nanoscale heat transport problems of semiconductor devices and accordingly semiconductors can be classified as per their ability of heat conduction. Originality/value To the authors’ knowledge, no one has discussed fractional thermoelastic dual-phase-lag problem associated with relaxation time in a finite cylindrical domain for a thick circular plate subjected to an axis-symmetric heat source. This is the latest and novel contribution to the field of thermal mechanics.

scholarly journals Dual-phase-lag one-dimensional thermo-porous-elasticity with microtemperatures

2021 ◽  
Vol 40 (6) ◽  
Author(s):  
Z. Liu ◽  
R. Quintanilla
Keyword(s):  
Heat Conduction ◽  
Elastic Problem ◽  
Linear Operators ◽  
Phase Lag ◽  
Dual Phase ◽  
Polynomial Stability ◽  
One Dimensional ◽  
Semigroups Of Linear Operators ◽  
Relaxation Parameters ◽  
Decay Of Solutions

AbstractThis paper is devoted to studying the linear system of partial differential equations modelling a one-dimensional thermo-porous-elastic problem with microtemperatures in the context of the dual-phase-lag heat conduction. Existence, uniqueness, and exponential decay of solutions are proved. Polynomial stability is also obtained in the case that the relaxation parameters satisfy a certain equality. Our arguments are based on the theory of semigroups of linear operators.

On the Analysis of Short-Pulse Laser Heating of Metals Using the Dual Phase Lag Heat Conduction Model

2009 ◽  
Vol 131 (11) ◽  
Author(s):  
K. Ramadan ◽  
W. R. Tyfour ◽  
M. A. Al-Nimr
Keyword(s):  
Heat Flux ◽  
Heat Conduction ◽  
Laser Heating ◽  
Flux Distribution ◽  
Heat Losses ◽  
Phase Lag ◽  
Dual Phase ◽  
Heat Flux Distribution ◽  
Conduction Model ◽  
Initial Heat

Transient heat conduction in a thin metal film exposed to short-pulse laser heating is studied using the dual phase lag heat conduction model. The initial heat flux distribution in the film, resulting from the temporal distribution function of the laser pulse, together with the zero temperature gradients at the boundaries normally used in literature with the presumption that they are equivalent to negligible boundary heat losses is analyzed in detail in this paper. The analysis presented here shows that using zero temperature gradients at the boundaries within the framework of the dual phase lag heat conduction model does not guarantee negligible boundary heat losses unless the initial heat flux distribution is negligibly small. Depending on the value of the initial heat flux distribution, the presumed negligible heat losses from the boundaries can be even way larger than the heat flux at any location within the film during the picosecond laser heating process. Predictions of the reflectivity change of thin gold films due to a laser short heat pulse using the dual phase lag model with constant phase lags are found to deviate considerably from the experimental data. The dual phase lag model is found to overestimate the transient temperature in the thermalization stage of the laser heating process of metal films, although it is still superior to the parabolic and hyperbolic one-step models.

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