Three mathematical representations and an improved ADI method for hyperbolic heat conduction

Nie, Ben-Dian; Cao, Bing‐Yang

doi:10.1016/j.ijheatmasstransfer.2019.02.026

Cited by 21 publications

(9 citation statements)

References 54 publications

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“…The choices of these coefficients are important, of course. However, it does not influence the qualitative characteristics of the equations, as it has been adopted in the reference [ 59 , 60 , 61 , 62 ]. The propagation patterns of the energy density with and without the second term are displayed in Figure 2 .…”

Section: Numerical Resultsmentioning

confidence: 99%

Thermomass Theory in the Framework of GENERIC

Nie

Cao

Zhang

et al. 2020

Entropy

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Thermomass theory was developed to deal with the non-Fourier heat conduction phenomena involving the influence of heat inertia. However, its structure, derived from an analogy to fluid mechanics, requires further mathematical verification. In this paper, General Equation for Non-Equilibrium Reversible-Irreversible Coupling (GENERIC) framework, which is a geometrical and mathematical structure in nonequilibrium thermodynamics, was employed to verify the thermomass theory. At first, the thermomass theory was introduced briefly; then, the GENERIC framework was applied in the thermomass gas system with state variables, thermomass gas density ρ h and thermomass momentum m h , and the time evolution equations obtained from GENERIC framework were compared with those in thermomass theory. It was demonstrated that the equations generated by GENERIC theory were the same as the continuity and momentum equations in thermomass theory with proper potentials and eta-function. Thermomass theory gives a physical interpretation to the GENERIC theory in non-Fourier heat conduction phenomena. By combining these two theories, it was found that the Hamiltonian energy in reversible process and the dissipation potential in irreversible process could be unified into one formulation, i.e., the thermomass energy. Furthermore, via the framework of GENERIC, thermomass theory could be extended to involve more state variables, such as internal source term and distortion matrix term. Numerical simulations investigated the influences of the convective term and distortion matrix term in the equations. It was found that the convective term changed the shape of thermal energy distribution and enhanced the spreading behaviors of thermal energy. The distortion matrix implies the elasticity and viscosity of the thermomass gas.Entropy 2020, 22, 227 2 of 24 the equation predicted negative temperature at some cases which is unphysical [22]. Afterwards, Phase lag models, such as Dual Phase Lag model [23,24], were proposed and thought that a time lag existed between heat flux and temperature gradient and it could reduce to Cattaneo-Vernotte equation via expansions. However, it was found that the solutions were not continually dependent on the initial condition [19], which resulted in instability, and it could not avoid the problem of negative temperature [25][26][27]. Another series of non-Fourier equations were derived from the Boltzmann equation [28][29][30]. Guyer-Krumhansl equation [28,29], typical for this kinds of equations led to the investigations into phonon hydrodynamics, which was named due to its similar structure with Navier-Stokes equation. However, it also met mathematical difficulties [31,32], such as the violation of maximum principle for stability and the failure in preserving variable non-negativity. Recently, to study the heat transfer in a more intrinsic way, thermomass theory [33-36] was developed. It was found that the heat was conserved during conduction, similar to mass. Then, the mass nature of heat was recognized in heat conducti...

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Section: Numerical Resultsmentioning

confidence: 99%

Thermomass Theory in the Framework of GENERIC

Nie

Cao

Zhang

et al. 2020

Entropy

Self Cite

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“…The complementary function can be obtained by solving (13). Let (19) Solution of particular integral is obtained by applying the differential transform on (14).The transformed form of (14) is given by (32) Using (23), (32), (31) in (27) and (28) one can get displacement function, which is given as:…”

Section: Solution For Stress Functionmentioning

confidence: 99%

“…Noroozi and Goodarzi [12] have studied the effect of laser heat source on a finite tissue under one-dimensional HHC model applying the ADM method. Recently, Nie and Cao [13] have proposed three different mathematical representation of HHC and solved using alternative direction implicit method.…”

Section: Introductionmentioning

confidence: 99%

Thermoelastic Response of a Rectangular Plate under Hyperbolic Heat Conduction Model in Differential Transform Domain

Chaudhari¹,

Sutar²

2019

IJEAT

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In the present work, a semi-analytical solution is presented for the thermoelastic response of a finite plate of rectangular geometry considering hyperbolic heat conduction model. The solution of thermoelastic displacement, thermal stresses and temperature are obtained using differential transform method under hyperbolic, non-Fourier heat conduction theory. For special case, thermal stresses and displacement functions are determined numerically and plotted graphically to analyze the effect of the thermal relaxation time.

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“…These theories are more and more important in nanostructures and are subjects of various challenging physical, mathematical and numerical researches. For example, nonlocal effects and the role of effective temperature are investigated in [1][2][3][4][5], particular special functions were constructed and exact solutions were calculated for both the hyperbolic and Guyer-Krumhansl heat conduction [6][7][8][9], adapted numerical methods were developed in [10,11], the role of internal variables in complex media modelling was investigated in [12][13][14], and the particularities of heat conduction in nanomaterials are discov-ered in [15][16][17]. These investigations are often related to various concepts of non-equilibrium temperature, too.…”

Section: Introductionmentioning

confidence: 99%

Generalized ballistic-conductive heat transport laws in three-dimensional isotropic materials

Famà

Restuccia

Ván

2020

Continuum Mech. Thermodyn.

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General constitutive equations of heat transport with second sound and ballistic propagation in isotropic materials are given using non-equilibrium thermodynamics with internal variables. The consequences of Onsager reciprocity relations between thermodynamic fluxes and forces and positive definiteness of the entropy production are considered. The relation to theories of Extended Thermodynamics is discussed in detail. We provide an explicit expression for all the components of the matrices of the transport coefficients. The expressions are cumbersome but are expected to be useful for computer programming for simulations of the corresponding physical effects.

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Three mathematical representations and an improved ADI method for hyperbolic heat conduction

Cited by 21 publications

References 54 publications

Thermomass Theory in the Framework of GENERIC

Thermomass Theory in the Framework of GENERIC

Thermoelastic Response of a Rectangular Plate under Hyperbolic Heat Conduction Model in Differential Transform Domain

Generalized ballistic-conductive heat transport laws in three-dimensional isotropic materials

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