One-Dimensional Heat Conduction and Entropy Production

Кривцов, А. М.; Sokolov, A. A.; Müller, Wolfgang; Freidin, A. B.

doi:10.1007/978-3-319-73694-5_12

Cited by 11 publications

(8 citation statements)

References 13 publications

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“…In particular, a heat transfer equation was obtained that differs from the extended heat transfer equations suggested earlier [40][41][42][43]; however, it is in an excellent agreement with molecular dynamics simulations and previous analytical estimates [31]. The properties of the solutions describing heat transfer in a one-dimensional harmonic crystal were discussed in [44][45][46]. Later this approach was generalized [37,38,[47][48][49][50][51]] to a number of systems, namely, to an infinite one-dimensional crystal on an elastic substrate [47], to an infinite one-dimensional diatomic harmonic crystal [52], to a finite one-dimensional crystal [51], and to two and three-dimensional infinite harmonic lattices [48][49][50].…”

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confidence: 66%

Steady-state kinetic temperature distribution in a two-dimensional square harmonic scalar lattice lying in a viscous environment and subjected to a point heat source

Гаврилов

Кривцов

2019

Continuum Mech. Thermodyn.

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We consider heat transfer in an infinite two-dimensional square harmonic scalar lattice lying in a viscous environment and subjected to a heat source. The basic equations for the particles of the lattice are stated in the form of a system of stochastic ordinary differential equations. We perform a continualization procedure and derive an infinite system of linear partial differential equations for covariance variables. The most important results of the paper are the deterministic differential-difference equation describing non-stationary heat propagation in the lattice and the analytical formula in the integral form for its steady-state solution describing kinetic temperature distribution caused by a point heat source of a constant intensity. The comparison between numerical solution of stochastic equations and obtained analytical solution demonstrates a very good agreement everywhere except for the main diagonals of the lattice (with respect to the point source position), where the analytical solution is singular.Keywords ballistic heat transfer · 2D harmonic scalar lattice · kinetic temperature 1 Introduction At the macroscale, Fourier's law of heat conduction is widely and successfully used to describe heat transfer processes. However, recent experimental observations demonstrate that Fourier's law is violated at the microscale and nanoscale, in particular, in low-dimensional nanostructures [1][2][3][4][5][6], where the ballistic heat transfer is realized. The anomalous heat transfer also may be related with the spontaneous emergence of long-range correlations; the latter is typical for momentum-conserving systems [7,8]. The simplest theoretical approach to describe the ballistic heat propagation is to use harmonic lattice models. In some cases such models allow one to obtain the analytical description of thermomechanical processes in solids [9][10][11][12][13][14][15]. In the literature, the problems concerning heat transfer in harmonic lattices are mostly considered

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confidence: 66%

Steady-state kinetic temperature distribution in a two-dimensional square harmonic scalar lattice lying in a viscous environment and subjected to a point heat source

Гаврилов

Кривцов

2019

Continuum Mech. Thermodyn.

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“…However, it is criticized for possessing an infinite speed of signal propagations and an absence of thermal fronts. Much efforts have been made to amend this problem, including the Cattaneo's law, ballistic heat propagation and many others [5,67]. The generalization of Fourier's law based on CDF was first proposed by Zhu et al [26].…”

Section: Non-fourier Heat Conductionmentioning

confidence: 99%

“…with the subscript V, E, L denoting the homogeneous viscous part, Ericksen part for the static state, and Leslie part for the non-equilibrium state, respectively. Now we introduce a strictly concave mathematical entropy function η = ρs(ν, u, d, ∇d, C, K, l, h), (67) where ν = 1/ρ, (C, K) are tensors with the same size of (σ, π), and (l, h) are vectors with the same size of (g, q). (C, K, l) are used to describe the viscous-elastic effects of nematic liquid crystal flows, and h characterizes the heat conduction induced by temperature gradients.…”

Section: Non-isothermal Flows Of Liquid Crystalsmentioning

confidence: 99%

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Recent Advances in Conservation–Dissipation Formalism for Irreversible Processes

Peng

Hong

2021

Entropy

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The main purpose of this review is to summarize the recent advances of the Conservation–Dissipation Formalism (CDF), a new way for constructing both thermodynamically compatible and mathematically stable and well-posed models for irreversible processes. The contents include but are not restricted to the CDF’s physical motivations, mathematical foundations, formulations of several classical models in mathematical physics from master equations and Fokker–Planck equations to Boltzmann equations and quasi-linear Maxwell equations, as well as novel applications in the fields of non-Fourier heat conduction, non-Newtonian viscoelastic fluids, wave propagation/transportation in geophysics and neural science, soft matter physics, etc. Connections with other popular theories in the field of non-equilibrium thermodynamics are examined too.

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“…There is an extensive literature on the calculations by each of the methods di fferent problems of discrete mechanics [10][11][12][13][14]. It should be noted that for the kinetic theory (the Boltzmann equation) the law of conservation of angular momentum does not hold.…”

Section: Introductionmentioning

confidence: 99%

The Effect of Angular Momentum and Ostrogradsky-Gauss Theorem in the Equations of Mechanics

Prozorova

2020

Wseas Transactions on Fluid Mechanics

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There are many experimental facts that currently cannot be described theoretically. A possible reason is bad mathematical models and algorithms for calculation, despite the many works in this area of research. The aim of this work is to clarificate the mathematical models of describing for rarefied gas and continuous mechanics and to study the errors that arise when we describe a rarefied gas through distribution function. Writing physical values conservation laws via delta functions, the same classical definition of physical values are obtained as in classical mechanics. Usually the derivation of conservation laws is based using the Ostrogradsky-Gauss theorem for a fixed volume without moving. The theorem is a consequence of the application of the integration in parts at the spatial case. In reality, in mechanics and physics gas and liquid move and not only along a forward path, but also rotate. Discarding the out of integral term means ignoring the velocity circulation over the surface of the selected volume. When taking into account the motion of a gas, this term is difficult to introduce into the differential equation. Therefore, to account for all components of the motion, it is proposed to use an integral formulation. Next question is the role of the discreteness of the description of the medium in the kinetic theory and the interaction of the discreteness and "continuity" of the media. The question of the relationship between the discreteness of a medium and its description with the help of continuum mechanics arises due to the fact that the distances between molecules in a rarefied gas are finite, the times between collisions are finite, but on definition under calculating derivatives on time and space we deal with infinitely small values. We investigate it

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One-Dimensional Heat Conduction and Entropy Production

Cited by 11 publications

References 13 publications

Steady-state kinetic temperature distribution in a two-dimensional square harmonic scalar lattice lying in a viscous environment and subjected to a point heat source

Steady-state kinetic temperature distribution in a two-dimensional square harmonic scalar lattice lying in a viscous environment and subjected to a point heat source

Recent Advances in Conservation–Dissipation Formalism for Irreversible Processes

The Effect of Angular Momentum and Ostrogradsky-Gauss Theorem in the Equations of Mechanics

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