Comments on "Scattering Cancellation-Based Cloaking for the Maxwell--Cattaneo Heat Waves"
aa r X i v : . [ phy s i c s . c l a ss - ph ] A ug Comments on “Scattering Cancellation-Based Cloaking for the Maxwell–CattaneoHeat Waves”
Ivan C. Christov ∗ School of Mechanical Engineering, Purdue University, West Lafayette, Indiana 47907, USA (Dated: August 7, 2019)A number of errors, both mathematical and conceptual, are identified, in a recent article byFarhat et al. [Phys. Rev. Appl. , 044089 (2019)] on cloaking of thermal waves in solids, andcorrected. The differences between the two thermal flux laws considered in the latter article are alsocritically discussed, specifically showing that the chosen model does not, in fact, correspond to theMaxwell–Cattaneo hyperbolic (wave) theory of heat transfer. Introduction.
This Comment presents a critique ofthe recent
Physical Review Applied publication [1], thefocus of which is a proposed cloaking scheme for thermalwaves in rigid solids. To begin, it is instructive to brieflyreview the topic of hyperbolic heat transport, i.e., thetheory of heat waves [2, 3], in rigid solids.Consider a thermally conducting, homogeneous andisotropic, rigid solid at rest. As first suggested by the-ory, and subsequently confirmed by experiment, at suf-ficiently low temperatures the transport of heat in suchbodies occurs not via diffusion, the mechanism under-lying Fourier’s law for the thermal flux, but instead bythe propagation of thermal waves (or second sound ) [2].Many constitutive relations have been proposed to de-scribe this phenomenon [3]. Perhaps the best known isthe Maxwell–Cattaneo (MC) law [4], which in the presentcontext reads (cid:18) τ ∂∂t (cid:19) Φ = − κ ∇ T. (1)Unsurprisingly, the history of this relation is complex:there exists Russian-language literature describing a sim-ilar flux law prior to Cattaneo (but, of course, afterMaxwell) [5]. Here, T = T ( x , t ) and Φ = Φ ( x , t ) denotethe absolute temperature and the thermal flux vector, re-spectively, where x = ( x, y, z ) ∈ R . As in [1], τ ( >
0) isthe thermal relaxation time for phonon processes that donot conserve phonon momentum [2], and κ ( >
0) is thethermal conductivity of the solid under consideration.Equation (1), which reduces to Fourier’s law on setting τ ≡
0, is the latter’s simplest generalization that yieldsa hyperbolic thermal transport equation, unlike theparabolic transport equation that stems from Fourier’slaw. Therefore, the MC law overcomes the so-called“paradox of diffusion”—the philosophically problematicimplication that thermal disturbances in continuous me-dia propagate with infinite speed under Fourier’s law.Delving into the critique of the study carried out in[1] will further illustrate these notions. Unless other-wise stated the same notation used in [1] is employed ∗ [email protected]; http://christov.tmnt-lab.org herein. First, observe that in [1, Eqs. (3)], the energy bal-ance equation is incorrectly stated; specifically, its sourceterm, which is denoted here by S = S ( x , t ), is missingand the ∂T /∂t term it contains should be multiplied bythe product ̺c p to ensure dimensional consistency. Sec-ond, but more troubling, the “flux diffusion” term, whichis introduced into the MC law (1), appears to be an at-tempt to introduce some of the features of the flux rela-tion of Guyer and Krumhansl (GK) [6, Eq. (59)], whichwas derived by GK from the linear Boltzmann equation.Note that under the GK model, heat flow is not neces-sarily down the temperature gradient and certainly “willnot permit the propagation of [heat] waves” [2, p. 46]unless the “flux diffusion” terms are neglected.Now, with the required corrections made to yield theactual GK flux law, [1, Eqs. (3)] can be expressed as ̺c p ∂T∂t = − ∇ · Φ + S , (2a) (cid:20) τ ∂∂t − τ σ (∆ + 2 ∇∇· ) (cid:21)| {z } =: K Φ = − κ ∇ T, (2b)where ∆ := ∇ · ∇ is the Laplacian operator. In Eqs. (2), c p (not c v , see [7, p. 9]) and ̺ are the specific heat at con-stant pressure and the mass density, respectively, of thesolid under consideration. Next, it is easily establishedfrom [2, Sect. IV] that σ = (1 / τ N V , where τ N is therelaxation time for N -processes and V carries SI unitsof m s − . Therefore, the SI units of σ are m s − ; not W m − K − , as reported in [1, p. 3]. The operator K here differs from its counterpart in [1] in that the latteris missing ∇∇· . Remark 1.
Again, while Eq. (1) is the σ ≡ incor-rect to regard the latter as exhibiting a small, “innocent”correction to the former. As shown below, the MC fluxlaw (1) predicts heat waves (hyperbolic thermal trans-port equation), while the GK flux law (2b) predicts heat diffusion (parabolic thermal transport equation). Remark 2.
Observe that, as a result of erroneouslydropping ̺c p in the energy balance, many equations in[1] are dimensionally inconsistent and, therefore, devoidof physical meaning. For example, consider [1, Eq. (4)].The first and second terms on the left-hand side (LHS)have units K s − , while the third and fourth terms onthe LHS have units W m − . The thermal transport equation.
As in [1], regardall coefficients as constant and proceed to eliminate Φ between the equations of Eqs. (2), assuming sufficientsmoothness of the dependent variables. The first step inthis process is employing Eq. (2a) to recast Eq. (2b) as (cid:20) τ ∂∂t − τ σ ∆ (cid:21)| {z } =: H σ Φ = − κ ∇ T + 2 τ σ (cid:20) ∇ S − ̺c p ∂ ( ∇ T ) ∂t (cid:21) . (3)Next, after applying H σ to Eq. (2a), and then usingEq. (3), one obtains the thermal transport equation ∂T∂t + τ ∂ T∂t = τ ˜ σ ∂ (∆ T ) ∂t + κ ∆ T + 1 ̺c p H ˜ σ [ S ] , (4)where κ := κ / ( ̺c p ) is the thermal diffusivity and, forconvenience, ˜ σ := 3 σ has been defined. As the right-hand side (RHS) of [1, Eq. (4)] is not acted upon by theoperator H ˜ σ , nor multiplied by 1 / ( ̺c p ), and its LHScontains σ , not ˜ σ , Eq. (4) above is the corrected versionof [1, Eq. (4)]. Remark 3.
The ˜ σ ≡
0, source-free version of Eq. (4) isthe multidimensional version of the damped wave equa-tion [2, p. 42], which predicts that thermal signals (dis-turbances) propagate at a finite characteristic speed of c := p κ /τ (see also [8]). Meanwhile, the ˜ σ >
0, source-free version of Eq. (4) is a multidimensional
Jeffreys-type equation , which predicts an infinite speedof propagation of signals [2, p. 46].To demonstrate this important difference betweenwave-like and diffusive thermal transport (see also [9]),but in a slightly simpler way, consider a related one-dimensional (1D) initial-boundary value problem (IBVP)posed by Tanner [10] for the Jeffreys-type equation aris-ing in the context of viscoelasticity. (This IBVP is alsothe one considered in [8] for the damped wave equationof hyperbolic heat conduction.) Recasting Tanner’s prob-lem in the present notation: ∂T∂t + τ ∂ T∂t = τ ˜ σ ∂ T∂t∂x + κ ∂ T∂x , ( x, t ) ∈ Ω , (5a) T (0 , t ) = T H ( t ) , lim x →∞ T ( x, t ) = 0 , t > , (5b) T ( x,
0) = ∂T∂t ( x,
0) = 0 , x > , (5c)where H ( · ) denotes the Heaviside unit step function,Ω := (0 , ∞ ) × (0 , ∞ ) is the space-time domain of in-terest, and the constant T ( >
0) is the amplitude of the σ (cid:1) (cid:1) σ (cid:1) ≡ (cid:1) (cid:1) ( (cid:2) ) (cid:3) / (cid:3) (cid:1) ( - ) FIG. 1. Wave-like (dashed curve, ˜ σ ≡
0) versus diffu-sive (solid curve, ˜ σ = 6 . × − m s − ) spread ofheat from the sudden imposition of a temperature jump at( x, t ) = (0 , τ = 0 .
991 s and κ = 2 . × − m s − , as given in [11] for limestone. This plot is at t = 1 s, i.e.,after approximately a single thermal relaxation time. inserted thermal signal. Following Tanner [10], one canapply the Laplace transform to Eq. (5a) and its bound-ary conditions (BCs) (5b). This IBVP correspond to a heat pulse experiment [11, 12]. After making use of theinitial conditions (5c), and then solving the resulting sub-sidiary equation subject to the (transformed) BCs, oneobtains an algebraic expression that can inverted back tothe time domain. This exact inverse is known [10], withseveral other representations summarized in [13].To illustrate this fundamental difference between thethermal transport described by the MC law and the GK-type law used in [1], the respective exact solutions ofIBVP (5) are shown in Fig. 1 for ˜ σ ≡ σ > Mathematica ’s NIntegrate subroutine, to arbi-trary precision, on a finite grid of x values [13]. Fig-ure 1 shows that under the MC law (dashed curve), theheat pulse has only propagated slightly less than 2 mminto the domain. Meanwhile, under the GK law (solidcurve), the normalized temperature T /T is non-zero ev-erywhere in the domain. Since the physical parametervalues given in [1] are incorrect/inconsistent, to generatethe plots in Fig. 1, the values for limestone are takenfrom [11], wherein it was experimentally demonstratedthat material micro-structure can lead to GK-type heatconduction at room temperature (see also [12, Ch. 9]).To summarize: the behavior of the ˜ σ > x = 0 is “felt” instantly , but equally,at every point in the half-space x >
0. In contrast, the˜ σ ≡ T exp[ − t/ (2 τ )], propagating (to the right) with finite speed c ≈ . × − m s − (for the chosen parametervalues); see also [8, p. 545] and [2, p. 45]. Harmonic disturbances.
Returning to Farhat et al. ’sanalysis, set S ( x , t ) = 0 and assume T ( x , t ) =Θ( x ) exp( − i ωt ), where ω ( >
0) is the angular frequencyof some thermal disturbance impacting the solid in ques-tion. Under these assumptions, Eq. (4) is reduced to the(source-free) Helmholtz equation∆Θ + (cid:18) τ ω + iω κ − i τ ˜ σ ω (cid:19) Θ = 0 . (6)It should be noted that, in [1, Eq. (5)], “ T ” is reusedinstead of introducing a new (time-independent) functionsuch as Θ herein. [1, Eq. (5)] also incorrectly features thethermal conductivity, with its subscript (“0”) missing, inplace of the thermal diffusivity κ .Consider plane wave propagation in a direction set bythe unit vector ˆ u . Then, on setting Θ( x ) = Θ exp(i k ˆ u · x ), Eq. (6) yields the dispersion relation k ( ω ) = τ ω ( κ − σ ) κ + τ ˜ σ ω | {z } =: a +i ω ( κ + τ ˜ σ ω ) κ + τ ˜ σ ω | {z } =: b , (7)where, i = √−
1, Θ > k ∈ C . Enforcing Θ < ∞ as | x | → ∞ (and, also, since b >
0) requires Im( k ) ≥ k ( ω ) = s a + √ a + b s − a + √ a + b . (8)The dispersion relation in Eq. (8), as well as the˜ σ ≡ k ) < Im( k ) for the chosen set of (realistic) physi-cal parameters, contrary to what is shown in [1]. How-ever, Re( k ) > Im( k ) does hold true under the MC law(˜ σ ≡ not balanced, because Im( k ) → (4 κ τ ) − / = const. butRe( k ) ∼ τ ω/ √ κ τ , as τ ω → ∞ . Other issues.
In addition to those detailed above, thefollowing other errors/issues were noticed in [1]:(i) In [1, p. 2], it is claimed that the term proportionalto σ is necessary to “make the discretizing processasymptotically stable.” Leaving aside the unclearmeaning of “asymptotically” in this context, thisstatement is false. There is no difficulty whatsoeverin discretizing a hyperbolic heat transport equationby any number of methods, as has been known forover three decades (see, e.g., [14], but note thatmodern schemes [15] should be used nowadays).(ii) Below [1, Eq. (5)], it is stated that “ k is a complexnumber for all frequencies [under the GK-type fluxlaw], which is markedly different from classical heatwaves (Fourier transfer).” Setting aside the factthat heat waves are impossible under Fourier’s law,it is clearly seen, on setting τ ≡ k ( ω ) = p i ω/ κ = (1 + i) p ω/ (2 κ ) ∈ C ; i.e.,there is no marked difference between Fourier andnon-Fourier heat flux laws in this regard. τ (cid:1) ω (-) (cid:1) (cid:1) ( (cid:2) (cid:2) ) (cid:1)(cid:2) (cid:1) (cid:1) ] (cid:1) (cid:1) > (cid:3) (cid:4)(cid:5) (cid:1) (cid:1) ] (cid:1) (cid:1) > (cid:1) (cid:1)(cid:2) (cid:1) (cid:1) ] (cid:1) (cid:1) ≡ (cid:3) (cid:4)(cid:5) (cid:1) (cid:1) ] (cid:1) σ (cid:1) ≡ (cid:3) FIG. 2. Real and imaginary parts of the wavenumber k (darkcolors for the MC law and light colors for the GK law) ascomputed from Eq. (8) for the same parameter values used togenerate Fig. 1. (iii) The unknown coefficients in the expansions in [1,Eqs. (8) and (9)] are found by applying a bound-ary condition involving “the temperature field T ,as well as its flux κ ∇ T .” Under the MC law, theheat flux is not (with misprints corrected) − κ ∇ T ,as it would be under Fourier ’s law; rather, it is theexpression obtained by solving Eq. (1) for Φ . Inthe case of harmonic time-dependence, for which Φ ( x , t ) = F ( x ) exp( − i ωt ), specifying the flux atthe boundary of some spatial domain D ⊂ R , un-der the MC law, would correspond to specifying F = − (cid:18) − i ωτ ω τ (cid:19) κ ∇ Θ on x ∈ ∂ D . (9)The corresponding expression under the GK fluxlaw (2b) is lengthier. This error in imposing theBCs on the series expansion puts into question allsubsequent results in [1, Sec. III and IV].(iv) The conclusion of [1, p. 7] states that “theFourier heat equation is not frame invariant.”However, this statement is false. The thermaltransport equation under Fourier’s law is indeedframe-invariant because the material derivative DT /Dt := ∂T /∂t + v · ∇ T is featured on the LHSof Eq. (2a) in its derivation for heat transfer ina moving (or deforming) medium with velocity v (see, e.g., [16, Sec. 7.1]), whence DT /Dt ≡ ∂T /∂t if v = (stationary conductor). Furthermore,the frame-indifferent formulation of the MC law ismisattributed in [1]; it was, in fact, derived in [1,Ref. 35] not [1, Ref. 43].(v) The word “photon(s)” should be replaced with“phonon(s)” everywhere in [1], given that the con-text is heat conduction, not electromagnetism. Conclusion.
On the basis of the above-identified er-rors and stated criticisms, it must be concluded thatFarhat et al. [1] have failed to provide “the first demon-stration of scattering cancellation cloaking for heat waves [emphasis added] obeying the Maxwell–Cattaneo trans-fer (sic) law.” It would, therefore, be of interest to re-do the study attempted in [1], with the correct physicalmodel [i.e., the σ ≡ ACKNOWLEDGMENTS
Contributions to an earlier draft by an anonymous col-league are acknowledged. [1] M. Farhat, S. Guenneau, P.-Y. Chen, A. Al`u, and K. N.Salama, “Scattering cancellation-based cloaking for theMaxwell-Cattaneo heat waves,” Phys. Rev. Applied ,044089 (2019).[2] D. D. Joseph and L. Preziosi, “Heat waves,” Rev. Mod.Phys , 41–73 (1989); “Addendum to the paper “HeatWaves” [Rev. Mod. Phys. , 41 (1989)],” Rev. Mod.Phys , 375–391 (1990).[3] B. Straughan, Heat Waves , Applied Mathematical Sci-ences, Vol. 117 (Springer, New York, 2011).[4] J. C. Maxwell, “On the dynamical theory of gases,” Phil.Trans. R. Soc. Lond. , 49–88 (1867); C. Cattaneo,“Sulla conduzione del calore,” Atti Sem. Mat. Fis. Univ.Modena , 83–101 (1948).[5] S. L. Sobolev, “On hyperbolic heat-mass transfer equa-tion,” Int. J. Heat Mass Transfer , 629–630 (2018);O. G. Bakunin, “Mysteries of diffusion and labyrinths ofdestiny,” Phys.-Usp. , 309–313 (2003).[6] R. A. Guyer and J. A. Krumhansl, “Solution of the lin-earized phonon Boltzmann equation,” Phys. Rev. ,766–778 (1966).[7] H. S. Carslaw and J. C. Jaeger, Conduction of Heat inSolids , 2nd ed. (Oxford University Press, Oxford, 1959).[8] K. J. Baumeister and T. D. Hamill, “Hyperbolic Heat-Conduction Equation—A Solution for the Semi-InfiniteBody Problem,” ASME J. Heat Transfer , 543–548(1969); “Discussion: “Hyperbolic Heat-ConductionEquation—A Solution for the Semi-Infinite Body Prob-lem (Baumeister, K. J., and Hamill, T. D., 1969, ASMEJ. Heat Transfer, 91, pp. 543–548),” ASME J. HeatTransfer , 126–127 (1971).[9] I. C. Christov, “Wave solutions,” in Encyclopedia ofThermal Stresses , edited by R. B. Hetnarski (Springer,Netherlands, 2014) pp. 6495–6506. [10] R. I. Tanner, “Note on the Rayleigh problem for avisco-elastic fluid,” Z. angew. Math. Phys. (ZAMP) ,573–580 (1962).[11] P. V´an, A. Berezovski, T. F¨ul¨op, Gy. Gr´of, R. Kov´acs,´A. Lovas, and J. Verh´as, “Guyer-Krumhansl–typeheat conduction at room temperature,” EPL (Europhys.Lett.) , 50005 (2017).[12] A. Berezovski and P. V´an, Internal Variables in Ther-moelasticity , Solid Mechanics and Its Applications, Vol.243 (Springer International Publishing, Cham, Switzer-land, 2017).[13] I. C. Christov, “Stokes’ first problem for some non-Newtonian fluids: Results and mistakes,” Mech. Res.Commun. , 717–723 (2010).[14] G. F. Carey and M. Tsai, “Hyperbolic heat transfer withreflection,” Numer. Heat Transfer , 309–327 (1982);D. E. Glass, M. N. ¨Ozi¸sik, D. S. McRae, and B. Vick,“On the numerical solution of hyperbolic heat conduc-tion,” Numer. Heat Transfer , 497–504 (1985).[15] R. J. LeVeque, Finite Volume Methods for HyperbolicProblems (Cambridge University Press, New York, 2002).[16] C. S. Jog,
Continuum Mechanics: Foundations and Ap-plications of Mechanics , 3rd ed., Vol. 1 (Cambridge Uni-versity Press, Delhi, India, 2015).[17] A. Pantokratoras, “Comment on the paper “OnCattaneo–Christov heat flux model for Carreau fluid flowover a slendering sheet, Hashim, Masood Khan, Resultsin Physics 7 (2017) 310–319”,” Res. Phys. , 1504–1505(2017); “Comment on the paper “Three-dimensional flowof Prandtl fluid with Cattaneo-Christov double diffu-sion, Tasawar Hayat, Arsalan Aziz, Taseer Muhammad,Ahmed Alsaedi, results in physics 9 (2018) 290–296”,”Res. Phys. , 1596–1597 (2019); “Four usual errorsmade in investigation of boundary layer flows,” PowderTechnol.353