Elsevier

Wave Motion

Volume 105, September 2021, 102756
Wave Motion

Revisiting Manne et al. (2000): A reformulation and alternative interpretation under the modified internal energy theory of second-sound

https://doi.org/10.1016/j.wavemoti.2021.102756Get rights and content

Highlights

  • The inexact model of Manne et al. (2000) is reformulated under the modified internal energy model.

  • A new link between the modified internal energy model and inertial theory is identified.

  • A nonlinear generalization of the thermasy variable is proposed.

  • New analytical and numerical results for temperature-rate waves are derived.

  • Results obtained are applied to the damped, nonlinear Klein–Gordon class of equations.

Abstract

Returning to Manne et al. (2000), wherein a class of hyperbolic reaction-diffusion-acoustic models is considered, we show that the inexact transport equation derived by these authors becomes the exact equation for the propagation of second-sound (i.e., thermal waves) in a rigid solid when reformulated under the modified internal energy model, one of several formulations of what has come to be known as hyperbolic heat conduction theory. We then compare/contrast this PDE with/to its exact version, which is based on Cattaneo’s equation for the thermal flux, in the context of a signaling problem involving weak discontinuities of second-sound propagating in a solid rod. Our analysis also reveals a connection between the modified internal energy model and what has been termed “inertial theory”, a competing theory of second-sound in solids, and suggests a generalization of the quantity known as the “thermasy”. We conclude with a discussion on the application of the present findings to a well known, widely applicable, class of nonlinear wave equations.

Introduction

Citing the problem of determining stress distributions in granular materials as their motivation, Manne et al. [1], in 2000, investigated the impact of non-zero (and constant) transport correlation times on the propagation of wave fronts described by nonlinear reaction-diffusion-acoustic systems. Restricting their attention to 1D propagation along the x-axis, these authors considered, in effect, the system ut+jx=kf(u),j=D0tϕ(tζ)ux(x,ζ)dζ, where we have introduced the flux vector j=(j(x,t),0,0). Here, as in Ref. [1], the non-negative function u=u(x,t) denotes a per unit length density or concentration; ϕ(t) is known as the memory function; when not identically zero, the source term f(u) is a nonlinear function which contains the upper and lower bounds to which u is subject; k is a constant; and the diffusion coefficient D(>0), which is assumed to be constant, carries (SI) units of m2/s.

After presenting Ref. [1, Eq. (2.1)],1 the transport equation for the case f(u)0 and arbitrary ϕ(t), Manne et al. then sought to derive the transport equation for the particular memory function ϕ(t)=αexp(αt),where the parameter α(>0) carries units of 1/s, and arbitrary f(u). Their efforts yielded the PDE (see Ref. [1, Eq. (2.2)]) utt+αut=v2uxx+αkf(u),where v=αD is the characteristic signal speed.

Because it does not contain the contribution stemming from t acting on kf(u), Eq. (2) is, unfortunately, not the correct transport equation vis-à-vis Sys. (1)(a–c). As such, all results derived from it can, at best, only be expected to apply to the noted phenomena in some approximate sense.

Notwithstanding these issues, our aim is not that of correcting Ref. [1]; indeed, this was carried out, without critique, in 2001 by Abramson et al. [2]. Among other things, these authors derived the correct transport equation corresponding to Sys. (1)(a–c), specifically (see Ref. [2, Eq. (3)]), utt+[αkf(u)]ut=v2uxx+αkf(u),a PDE which has been the topic of a number of studies since (at least) 1991; see, e.g., Sobolev [3], Méndez et al. [4], Needham and Leach [5], Straughan [6], and Coupland [7], and Jordan and Lambers [8].

Instead, the primary aims of the present investigation are as follows: (I) point out an existing theoretical framework, namely, the modified internal energy model, under which Eq. (2) is the correct/exact transport equation; (II) discuss the physical interpretation of said reformulation and derive the corresponding governing system; and (III) compare/contrast, using both analytical and numerical methods, Eqs. (2), (3) in the context of a model problem involving temperature-rate waves (see Section 3.3 below) in a rigid solid. Additionally, and perhaps of deeper significance, our analysis has uncovered a connection – one which appears to have not been previously noted – between the modified internal energy model and the inertial theory of second-sound in solids.

Remark 1

The models considered below represent different versions of what has come to be known as hyperbolic heat conduction theory, the theoretical foundations of which were laid by Maxwell and Nernst; see, e.g., the review articles by Ackerman and Guyer [9] and Joseph and Preziosi [10]. According to this paradigm, the conduction of heat in continuous media occurs via the propagation of second-sound (i.e., thermal waves), a phenomenon which has been observed in a variety of media; see, e.g., Refs. [9], [10], [11], [12], [13], [14] and those cited therein. In this regard it should be noted that none of the models considered below describe (thermal) energy transport via purely ballistic phonons, i.e., phonons which travel without interaction; see Refs. [9], [11], [14], [15].

Section snippets

Governing system and constitutive assumptions

Rubin [16] introduced his modified internal energy (MIE) model in 1992. Under the simplest (i.e., constant coefficient) special case of this version of hyperbolic heat conduction theory, the flow of heat in a stationary, homogeneous and isotropic, 1D rigid solid is described by the system [16] ρ0et+hx=ρ0r,h=K0ϑx,e=c0ϑ+ϰ0ϑt. Here, ϑ=ϑ(x,t) is the absolute temperature; e=e(x,t) is the specific internal energy; h=(h(x,t),0,0) is the heat flux vector; r, the per unit mass rate of internal heat

Second-sound version of Eq. (3)

As a model describing second-sound in a rigid solid with Eq. (9) as source term, Eq. (3) takes the form ϑtt+[α(ϖc0)(12ϑϑs)]ϑt=a2ϑxx+(αϖc0)ϑ(1ϑϑs),which follows, in part, from the assumption that the solid’s specific internal energy is given by e=c0ϑ. Here, we record for later reference that Eq. (14) is readily decomposed into the two-equation system ρ0c0ϑt+hx=ρ0ϖϑ(1ϑϑs),(1+α1t)h=K0ϑx, which like Sys. (12) is a semilinear, strictly hyperbolic one.

System (15) is based on the

Problem formulation

We now turn our attention to what is commonly referred to in the acoustics literature as a “signaling problem”—one, however, in which the inserted signal is thermal, not mechanical. In particular, we consider a thin (stationary) rigid-solid rod, within which the per unit mass rate of heat production is given by Eq. (9), that occupies the interval 0<x<1; here, the rod’s dimensional cross-sectional area (i.e., σ) and length (i.e., l) are assumed to be such that lσ holds, i.e., the rod is thin.

Numerical results

The sequences shown in Fig. 1, Fig. 2 depict particular instances in the evolution of the θ vs. x solution profiles corresponding, respectively, to Case (i) ( MIE only) and Case (iv) ( CHF) of IBVP (24c) with θ+0. Those in Fig. 3, Fig. 4, in contrast, depict T vs. x, where for convenience we have set Tθθ+θ0,and correspond, respectively, to Case (i) (now MIE/IHF) and Case (iv) ( CHF) of IBVP (24c) with θ+=0.25. The parameter values employed, in all four figures, were selected on the

A generalization of the thermasy concept

In the case of an arbitrary (nonlinear) source term, say, r=ϖϑsνR(ϑ), Sys. (12) is easily generalized to ρ0c0ϑt+hx=αρ0c0(ϑϑe)+ρ0αϖΨ,ht=αK0ϑx,Ψt=ϑsνR(ϑ), where the value of ν is such that the product ϑsνR(ϑ) carries the unit kelvin. [In the case of the logistic law, i.e., R(ϑ)= Eq. (9)ϖ, it is clear that ν=0, while in those of the Arrhenius and Zel’dovich laws [30, §2.1] we have ν=±1, respectively.]

Let us now consider the thermasy, which we denote by

. This thermal variable, as Scholle 

CRediT authorship contribution statement

P.M. Jordan: Conceptualization, Writing - original draft, Analytical methodology, Reviewing & editing. J.V. Lambers: Data curation, Software, Numerical methodology, Reviewing & editing.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

The authors thank the five anonymous reviewers for their instructive criticisms and valuable comment. The authors also thank Profs. D. Jou, M.B. Rubin, and B. Straughan for sharing their insight and expertise. P.M.J. and J.V.L. were supported by ONR, USA funding.

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