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Continuum Mechanics and Thermodynamics

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30.01.2023 | Original Article

Unsteady thermal transport in an instantly heated semi-infinite free end Hooke chain

verfasst von: Sergei D. Liazhkov

Erschienen in: Continuum Mechanics and Thermodynamics | Ausgabe 2/2023

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Abstract

We consider unsteady ballistic heat transport in a semi-infinite Hooke chain with free end and arbitrary initial temperature profile. An analytical description of the evolution of the kinetic temperature is proposed in both discrete (exact) and continuum (approximate) formulations. By comparison of the discrete and continuum descriptions of kinetic temperature field, we reveal some restrictions to the latter. Specifically, the far-field kinetic temperature is well described by the continuum solution, which, however, deviates near and at the free end (boundary). We show analytically that, after thermal wave reflects from the boundary, the discrete solution for the kinetic temperature undergoes a jump near the free end. A comparison of the descriptions of heat propagation in the semi-infinite and infinite Hooke chains is presented. Results of the current paper are expected to provide insight into non-stationary heat transport in the semi-infinite lattices.

Vorheriger Artikel Thermodynamic theory of the most energy-efficient natural repose angle

Nächster Artikel Unsteady ballistic heat transport in a 1D harmonic crystal due to a source on an isotopic defect

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i.e., changing in dependence of the particle number.

See also English translation of the Schrödinger article [23].

This is the monoatomic harmonic chain of identical particles, connected by the linear identical springs, see [24].

Ends of chain are connected with the fixed points by linear stiffness springs.

Definition of the Hooke chain is given in Sect. 1 and in [24].

The statement of problem corresponds to experimental heating of the crystal by the ultrashort laser pulse. Since the expression for heat flux in the Hooke chain contains covariances of displacements and velocities (see, e.g., [3, 34]), initial zero field of initial displacements means zero initial heat fluxes.

We determine the kinetic temperature by its statistical definition (see, e.g., chapter 3, Sect. 29 in [35]). Unambiguous definition of the temperature for systems far from equilibrium is still unresolved fundamental problem (see, e.g., [36, 37]). In this paper, we calculate the kinetic temperature as average of kinetic energy over realizations, because it has simple physical meaning. Discussion of the ergodicity remains out of frameworks of this study.

A macroscale can be interpreted as a scale of the order of the length of chain.

Here, evenness property of the Dirac delta function is used.

The type of the initial temperature perturbation contradicts with the assumption, made in Sect. 3.1. However, as it is shown below, the continuum solution has the same physical meaning, which is characteristic for one at arbitrary initial temperature profile.

Simulations are performed for the chain with 500 particles.

Random numbers \(\rho _n\) are uniformly distributed in the segment \([-\sqrt{3}; \sqrt{3}]\), which satisfies condition (4).

As in the case of rectangular perturbation, the continuum solution decays also as 1/t.

i.e., the function \(T(t)\Big \vert _0\) at width of the initial thermal perturbation \(L+\Delta L\) is equal to \(T\left( \frac{L+\Delta L}{L}t\right) \Big \vert _0\) at the width L.

Therefore, before reflection from the boundary, the continuum solution for the kinetic temperature in the semi-infinite chain obeys solution (40).

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Titel: Unsteady thermal transport in an instantly heated semi-infinite free end Hooke chain
verfasst von: Sergei D. Liazhkov
Publikationsdatum: 30.01.2023
Verlag: Springer Berlin Heidelberg
Erschienen in: Continuum Mechanics and Thermodynamics / Ausgabe 2/2023
Print ISSN: 0935-1175
Elektronische ISSN: 1432-0959
DOI: https://doi.org/10.1007/s00161-023-01186-z

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