This book presents a broad and well-structured overview of various non-Fourier heat conduction models. The classical Fourier heat conduction model is valid for most macroscopic problems. However, it fails when the wave nature of the heat propagation becomes dominant and memory or non-local spatial effects become significant; e.g., during ultrafast heating, heat transfer at the nanoscale, in granular and porous materials, at extremely high values of the heat flux, or in heat transfer in biological tissues. The book looks at numerous non-Fourier heat conduction models that incorporate time non-locality for materials with memory, such as hereditary materials, including fractional hereditary materials, and/or spatial non-locality, i.e. materials with a non-homogeneous inner structure. Beginning with an introduction to classical transport theory, including phase-lag, phonon, and thermomass models, the book then looks at various aspects of relativistic and quantum transport, including approaches based on the Landauer formalism as well as the Green-Kubo theory of linear response. Featuring an appendix that provides an introduction to methods in fractional calculus, this book is a valuable resource for any researcher interested in theoretical and numerical aspects of complex, non-trivial heat conduction problems.
Author(s): Alexander I. Zhmakin
Publisher: Springer
Year: 2023
Language: English
Pages: 418
City: Cham
Preface
Contents
Acronyms
1 Introduction
References
Part I Classical Transport
2 Phase-Lag Models
2.1 Maxwell–Cattaneo–Vernotte Equation
2.1.1 ``Relativistic'' Heat Conduction
2.2 Dual-Phase-Lag Model
2.2.1 Non-local Dual-Phase-Lag Model
2.3 Triple-Phase-Lag Model
2.3.1 Non-local Triple-Phase-Lag Model
References
3 Phonon Models
3.1 Phonon Transport Regimes
3.2 Guyer–Krumhansl (GK) Equation
3.3 Two-Fluid Models
3.3.1 Ballistic–Diffusive Model
3.3.2 Extended Ballistic–Diffusive Model
3.3.3 Unified Non-diffusive-Diffusive Model
3.3.4 Enhanced Fourier Law
3.3.5 Two-fluid Model
3.4 Generalized Fourier Law by Hua et al.
3.5 Phonon Hydrodynamics
3.5.1 Nonequilibrium Thermodynamics of Phonon Hydrodynamic Model
3.5.2 Flux-Limited Behaviour
3.6 Relaxon Model
References
4 Thermomass Model
4.1 Equation of State (EOS) of the Thermon Gas
4.1.1 EOS of Thermon Gas in Ideal Gas
4.1.2 EOS of Thermon Gas in Dielectrics
4.1.3 EOS of Thermon Gas in Metals
4.2 Equations of Motion of Thermon Gas
4.3 Heat Flow Choking Phenomenon
4.4 Dispersion of Thermal Waves
References
5 Mesoscopic Moment Equations
References
6 Microtemperature and Micromorphic Temperature Models
6.1 Microtemperature Models
6.2 Micromorphic Approach
References
7 Thermodynamic Models
7.1 Jou and Cimmelli Model
7.1.1 Heat Conduction in Thermoelectric Systems
7.2 Sellitto and Cimmelli Model
7.3 Kovács and Ván Model
7.4 Famá et al. Model
7.5 Rogolino et al. Models
7.6 Two-Temperature Model by Sellitto et al.
7.7 EIT Ballistic–Diffusive Model
References
8 Fractional Derivative Models
8.1 Fractional Fourier Model
8.1.1 Nonlinear Diffusivity
8.1.2 Fractional Pennes Model
8.2 Zingales's Fractional-Order Model
8.3 Fractional Cattaneo and SPL Models
8.4 Fractional DPL Model
8.5 Fractional TPL Model
8.5.1 Non-local Fractional TPL Model
References
9 Fractional Boltzmann and Fokker–Planck Equations
9.1 Continuous-Time Random Walks
9.1.1 Lévy (Khintchine–Lévy) Walks
9.2 Kramers–Fokker–Planck Equation
9.3 Li and Cao Model
References
10 Elasticity and Thermal Expansion Coupling
10.1 Non-Fourier Thermoelasticity
10.1.1 Fractional Thermoelasticity
References
11 Some Exact Solutions
11.1 Phase-Lag Models
11.2 Phonon Models
11.3 Fractional Models
References
Part II Relativistic Transport
12 Relativistic Brownian Motion
References
13 Relativistic Boltzmann Equation
13.1 General Relativistic Boltzmann Equation
13.2 Particles in External Electromagnetic Fields
13.3 Relativistic Gas in Gravitational Field
13.4 Grad's Moment Method
13.5 Chapman–Enskog Expansion
13.5.1 Anderson–Witting Transport Coefficients in General Relativity
References
14 Variational Models
14.1 Márkus and Gambár Model
14.2 Multifluid Model
References
15 Relativistic Thermodynamics
References
Part III Quantum Transport
16 Landauer Approach
References
17 Green–Kubo Approach
References
18 Coherent Phonon Transport
References
19 Conclusions
References
Appendix An Introduction to Fractional Calculus
A.1 Fractional Derivatives
A.1.1 Riemann–Liouville Fractional Integral
A.1.2 Riemann–Liouville Fractional Derivative
A.1.2.1 Leibniz' Formula
A.1.2.2 Faá di Bruno Formula (The Chain Rule)
A.1.2.3 Fractional Taylor Expansion
A.1.2.4 Symmetrized Space Derivative
A.1.3 Caputo Fractional Derivative
A.1.4 Matrix Approach
A.1.5 Caputo and Fabrizio Fractional Derivatives
A.1.6 GC and GRL Derivatives
A.1.6.1 GC Derivatives
A.1.6.2 GRL Derivatives
A.1.7 Marchaud–Hadamard Fractional Derivatives
A.1.8 Grünwald–Letnikov Derivative
A.1.9 Riesz Fractional Operators
A.1.10 Weyl Fractional Derivative
A.1.11 Erdélye–Kober Fractional Operators
A.1.12 Interpretation of Fractional Integral and Derivatives
A.1.13 Local Fractional Derivatives
A.1.13.1 ``Conformable'' Fractional Derivative
A.2 Tempered Fractional Calculus
A.3 Fractional Differential Equations
A.3.1 Distributed Order Differential Equations
A.3.2 One-Dimensional Fractional Heat Conduction Equation
A.3.3 Special Functions
A.3.3.1 Mittag-Leffler Functions
A.3.3.2 H Functions
A.3.3.3 Wright Functions
A.4 Solution of Fractional Differential Equations
A.4.1 Analytical Methods
A.4.2 Numerical Methods
References
