On a C -integrable equation for second sound propagation in heated dielectrics
aa r X i v : . [ phy s i c s . c l a ss - ph ] A p r Manuscript submitted to doi:10.3934/xx.xx.xx.xxAIMS’ JournalsVolume X , Number , Month pp. X–XX
ON A C -INTEGRABLE EQUATION FOR SECOND SOUNDPROPAGATION IN HEATED DIELECTRICS Ivan C. Christov
School of Mechanical Engineering, Purdue UniversityWest Lafayette, IN 47907, USA
Abstract.
An exactly solvable model in heat conduction is considered. The C -integrable ( i.e. , change-of-variables-integrable) equation for second sound( i.e. , heat wave) propagation in a thin, rigid dielectric heat conductor uni-formly heated on its lateral side by a surrounding medium under the Stefan–Boltzmann law is derived. A simple change-of-variables transformation isshown to exactly map the nonlinear governing partial differential equationto the classical linear telegrapher’s equation. In a one-dimensional context,known integral-transform solutions of the latter are adapted to construct exactsolutions relevant to heat transfer applications: (i) the initial-value problem onan infinite domain (the real line), and (ii) the initial-boundary-value problemon a semi-infinite domain (the half-line). Possible “second law violations” andrestrictions on the C -transformation are noted for some sets of parameters. Introduction.
Micro- and nano-scale heat transfer is a growing area of me-chanical engineering with significant and timely technological applications. How-ever, modeling heat transfer in such situations still remains challenging [25]. Yet,“[i]nterface heat transfer is one of the major concerns in the design of microscaleand nanoscale devices” [34], specifically interface heat transfer between a dielectricand its surrounding medium ( e.g. , a metal as investigated in [34]). Here, following[40] (see also the time-fractional generalization of the latter [28]), we consider heattransfer in a thin, rigid rod of a dielectric material subject to uniform heating of itslateral surface by its surroundings, as shown schematically in Fig. 1. In this work,our goal is to show that such an inherently nonlinear problem can be transformedto a linear one, providing an exactly solvable heat transfer model.In this context, the energy balance equation, in the absence of volumetric sourcesor sinks of heat, takes the form: ∂e∂t + ∇ · q = 0 , x ∈ Ω , (1)where q = q ( x , t ) = ( q x , q y , q z ) and e = e ( ϑ ) denote the heat flux vector and theinternal energy (per unit volume) in the material, respectively. At this stage, weconsider x ≡ ( x, y, z ) ∈ Ω ⊂ R , the boundary of the domain Ω being denotedby ∂ Ω with unit surface normal ˆ n (as shown in Fig. 1), and t ≥
0. We assumethat the heating/cooling of the conductor occurs via the Stefan–Boltzmann law[12, 48] of radiation/absorption of thermal energy across the dielectric’s lateral
Mathematics Subject Classification.
Primary: 35Q79, 80A20; Secondary: 35L20, 74J05.
Key words and phrases.
Second sound, heat waves, C -integrability, dielectrics, radiation heattransfer, integral transforms. surface into/from the surrounding medium. In other words, Eq. (1) is supplementedby the boundary condition q · ˆ n = σǫ (cid:0) ϑ − ϑ ∞ (cid:1) , x ∈ ∂ Ω , (2)where ϑ = ϑ ( x , t ) is the absolute temperature in the dielectric, ϑ ∞ is the sur-rounding medium’s (constant) absolute temperature, the constant ǫ ∈ [0 ,
1] is theemissivity of the surface of the dielectric, and σ ≈ . × − W/(m · K ) is theStefan–Boltzmann constant [48]. Equation (2) is an approximation for the radiativeheat transfer flux valid for convex surfaces placed into large isothermal environments(as in Fig. 1) [48, p. 170]. xyz s J( x , t ) J ¥ J ¥ J( t ) W q ( x , t ) y n A Figure 1.
Schematic of a thin, rigid rod of a dielectric material(contained within the domain Ω with boundary ∂ Ω and unit surfacenormal ˆ n ) subject to uniform heating/cooling of its lateral surfaceby its surroundings. The dielectric is long in the x -direction andthin in the cross-sectional y - and z -directions, so that heat con-duction can be assumed to be unidirectional and radiation to be avolumetric source term in the energy equation. In this particular il-lustration, the temperature ϑ at one end ( x = 0) can be prescribed.The temperature in the surrounding medium ( i.e. , in R \ Ω) is theconstant ϑ ∞ .In Fig. 1, we have taken the dielectric to be (semi-)infinitely long and aligned withthe x -axis, so that ˆ n = (0 , ˆ n y , ˆ n z ) on its lateral surface. Additionally, the dielectrichas a fixed cross-section A of area | A | . In such a configuration, assuming a slendergeometry, i.e. , the rod to be sufficiently long and thin so that lateral conductionin the cross-section establishes itself immediately, we may cross-sectionally averagethe energy Eq. (1): ∂e∂t + ∂q x ∂x + 1 | A | Z Z A ∇ ⊥ · q d y d z = 0 , (3)where we have introduced the notation ∇ ⊥ := ( ∂/∂y, ∂/∂z ) for the gradient in thecross-sectional coordinates, and( · ) := 1 | A | Z Z A ( · ) d y d z. (4)Now, using the divergence theorem, we rewrite Eq. (3) as ∂e∂t + ∂q x ∂x + 1 | A | Z ∂A ( q y , q z ) · (ˆ n y , ˆ n z ) | ∂ Ω d s = 0 , (5) Here, we have assumed, at least for the time being, the requisite smoothness of solutions tointerchange the order of differentiation and integration. C -INTEGRABLE SECOND SOUND EQUATION 3 where ∂A is the boundary of the cross-sectional area A with perimeter | ∂A | , and d s is the length element along it. Next, noting that q · ˆ n = ( q x , q y , q z ) · (0 , ˆ n y , ˆ n z ) =( q y , q z ) · (ˆ n y , ˆ n z ) on ∂ Ω and using the boundary condition from Eq. (2), Eq. (7)becomes ∂e∂t + ∂q x ∂x = − | ∂A || A | | ∂A | Z ∂A σǫ (cid:0) ϑ − ϑ ∞ (cid:1) d s. (6)Note that | ∂A | / | A | ≡ /d h , where d h is, by definition, the hydraulic diameter of thecross-section [6, § § ∂e∂t + ∂q x ∂x = − d h σǫ (cid:0) ϑ − ϑ ∞ (cid:1) . (7)In other words, for the long, slender geometry considered, the heating/cooling ofthe dielectric by its surroundings through radiation/absorption can be treated asa volumetric source term, instead of a boundary condition. Note that Eq. (7)represents a nonlinear partial differential equation (PDE). Henceforth, we dropthe x subscript on q without fear of confusion. For convenience, let us also re-normalize the Boltzmann constant asˆ σ := 4 σ/d h (8)in the present one-dimensional context.We are interested in second sound propagation in this dielectric medium at eitherlow temperatures (on the order of 1 K) or in the presence of “high” heat flux. Forsuch applications, Chester [15] argued that the heat flux of certain dielectric solidsshould obey the Maxwell–Cattaneo (MC) [46, 13] constitutive relation/law (see also[42, 43, 35, 59]): τ ( ϑ ) ∂q∂t + q = − K ( ϑ ) ∂ϑ∂x , (9)where K = K ( ϑ ) > τ = τ ( ϑ ) is its thermal relaxation time. Note that for τ ≡
0, the MC law in Eq. (9)reduces to Fourier’s law of heat conduction, namely q = − K ( ϑ )( ∂ϑ/∂x ) (see also[51, 44]). The significance of the MC law, which has (since the works of Maxwelland Cattaneo) been put on solid ground through the theory of stochastic processes[11] and using thermodynamic free energy and dissipation potentials [52] (see also[51, 44]), is that it allows for the propagation of heat waves , also known as secondsound [42, 43, 35, 59, 17]. For crystalline dielectric solids, Casimir–Debye theorypredicts (see, e.g. , [62, § A related nonlinear model was developed by Antaki [2], in the context of reactive solids athigh-temperature (constant τ and K ), by taking the volumetric heating term to correspond to asingle-step irreversible exothermic Arrhenius reaction. However, Antaki’s model does not appearto be C -integrable. The MC law appears under various names in the literature, including “Cattaneo–Vernotte,”due to Vernotte’s seemingly parallel development of the mathematical model [60]. However, Cat-teneo promptly disputed said novelty claim [14]. In the related context of molecular diffusion andmass transfer, the history of the equivalent of Eq. (9) is even more serpentine, going back at leasttwo decades prior to Cattaneo’s 1948 paper [13] to the Russian-language literature and the worksof Fock and Davydov (see, e.g. , the discussion in [4, 57]).
I. C. CHRISTOV and specific heat at constant volume: K ( ϑ ) ≈ K R (cid:18) ϑϑ R (cid:19) , (10a) c v ≈ βϑ , (10b)both of which are valid for “low” temperatures, e.g. , ϑ ≪ min( ϑ D , ϑ ∗ ), where ϑ D is the Debye temperature and ϑ ∗ is the temperature at which the full conductivitycurve might reach a maximum (see, e.g. , [62, p. 292]). In Eq. (10b), the constant β ( >
0) is a property of the dielectric solid under consideration, which is in some wayssimilar to the Sommerfeld coefficient (per unit density) in metals [22]. Additionally, K R = K ( ϑ R ) is the (reference) thermal conductivity of the solid at the (reference)temperature ϑ R . According to Chester [15, Eq. (12)] (see also [26, p. 186]), thethermal relaxation time for such a dielectric solid should be given by τ = 3 Kς ̺c v , (11)where ̺ ( >
0) denotes the dielectric’s constant (since it is a rigid solid) mass density,and ς represents the average value of the phonon speed in this medium.In the present work, we derive the governing equation for the temperature field ϑ ( x , t ) in Section 2. In Section 3, we show that, in a one-dimensional (1D) context,the latter nonlinear PDE can be exactly linearized though a change-of-variables(“ C ”) transformation. Then, in Section 4, we use exact solutions available for thelatter linear PDE to obtain exact solutions to the original nonlinear boundary-value problem, while also providing graphical illustrations. Finally, in Section 5,conclusions and avenues for future work are discussed.2. The governing equation.
In this section, we seek to obtain the governingequation for second sound propagation in a uniformly heated long, thin and rigiddielectric rod. First, we use the equilibrium thermodynamic relation between spe-cific internal energy u and temperature ϑ via the specific heat at constant volume c v : d u = c v d ϑ . Noting that u = e/̺ , from d e = ̺c v d ϑ and Eq. (10), it follows that e ( ϑ ) := ̺ Z c v ( ϑ ) d ϑ = e + ̺β ϑ , (12)where e is the internal energy (per unit volume) in some reference state, whichcan be taken to be the free vacuum so that e = 0 without loss of generality.Substituting Eq. (12) into Eq. (7), the energy balance equation becomes (cid:18) ̺β (cid:19) ∂ϑ ∂t + ∂q∂x = − ˆ σǫ (cid:0) ϑ − ϑ ∞ (cid:1) . (13)Second, from Eqs. (11) and (10), we obtain τ ( ϑ ) = 3 K R ̺ς ϑ β = const. (14)Next, we substitute Eqs. (10) and (14) for the conductivity and relaxation timeinto the MC law for the heat flux, i.e. , Eq. (9), to obtain:3 K R ̺ς ϑ β ∂q∂t + q = − K R (cid:18) ϑϑ r (cid:19) ∂ϑ∂x . (15) C -INTEGRABLE SECOND SOUND EQUATION 5 To eliminate the heat flux q between Eqs. (13) and (15), we apply ∂/∂t to Eq. (13): (cid:18) ̺β (cid:19) ∂ ϑ ∂t + ∂ q∂x∂t = − ˆ σǫ ∂ϑ ∂t . (16)Then, we substitute for ∂q/∂t from Eq. (9) into Eq. (16): (cid:18) ̺β (cid:19) ∂ ϑ ∂t − ̺ς βϑ K R ∂q∂x − ̺ς β ∂∂x (cid:20) ∂∂x (cid:18) ϑ (cid:19)(cid:21) = − ˆ σǫ ∂ϑ ∂t . (17)Next, we substitute for ∂q/∂x from Eq. (13) into Eq. (17) and simplify to arrive atthe governing nonlinear PDE for the temperature ϑ ( x, t ) field:3 K R ς ϑ ˆ σ ∂ ϑ ∂t + 3 K R ς ϑ ˆ σ (cid:18) ǫ ˆ σ̺β + ̺ς βϑ K R (cid:19) ∂ϑ ∂t = K R ϑ ˆ σ ∂ ϑ ∂x − ǫ (cid:0) ϑ − ϑ ∞ (cid:1) . (18)For convenience, let us now introduce dimensionless variables through the fol-lowing set of transformations: ϑ = ϑ c Θ , x = L c X, t = T c T, (19)where ϑ c and L c are, respectively, characteristic temperature and length scales setby, e.g. , the initial or boundary conditions of a given problem or the geometry of agiven domain. On the other hand, T c is the characteristic time scale emerging fromthe equation itself, namely T c = s K R ς ϑ ˆ σ . (20)Substituting the dimensionless variables from Eq. (19) into Eq. (18) yields the di-mensionless governing PDE: ∂ Θ ∂T + λ ∂ Θ ∂T = c ∂ Θ ∂X − ǫ (Θ − Θ ) , (21)where we have introduced three dimensionless numbers:Θ R := ϑ ∞ ϑ c , (22a) λ := s K R ς ϑ ˆ σ (cid:18) ǫ ˆ σ̺β + ̺ς βϑ K R (cid:19) , (22b) c := s K R ϑ ˆ σL . (22c)Note that λ is a dimensionless effective inverse thermal relaxation time, while c is the dimensionless effective speed of second sound in the dielectric.Obviously, if the length scale L c is not set by the initial and/or boundary condi-tions, it can be taken to be L c = p K R / (4 ϑ R ˆ σ ) thus yielding c = 1. In the presentwork, we consider only problems on the real line X ∈ Ω = ( −∞ , + ∞ ) or the half-space X ∈ Ω = (0 , + ∞ ) for which there is no natural length scale set by the initialor boundary conditions on these domains. Hence, we take L c = p K R / (4 ϑ R ˆ σ ) to bethe “natural” length scale, and the speed of second sound c is, thus, convenientlynormalized to unity. Consequently, the final dimensionless form of the governingPDE (21) is ∂ Θ ∂T + λ ∂ Θ ∂T = ∂ Θ ∂X − ǫ (Θ − Θ ) . (23) I. C. CHRISTOV
For the equivalent of Eq. (23) for the case of constant specific heat c v and constant thermal conductivity K , we refer the reader to [39, p. 1020]. We now set out to exactly solve Eq. (23) given certain initial and boundary conditions on the real lineor on a half-space. Although it is, of course, evident that Eq. (23) is linear in Θ ,the question of whether simply writing down a general solution for Θ ( X, T ) (forgiven initial and boundary conditions) uniquely yields Θ(
X, T ) is far from obvious.3. C -integrability of the governing equation. In this section, we wish to es-tablish whether and when a certain algebraic transformation can exactly linearizeEq. (23). Here, “ C ” stands for “change of variables” [9, p. 2]. Calogero [9, 10]has introduced this terminology to distinguish between nonlinear PDEs that canbe linearized by a change-of-variables transformation from S -integrable nonlinearPDEs that can be linearized via a spectral transform, e.g. , the inverse scatteringtransform [1], which linearizes the celebrated Korteweg–de Vries equation [45, 27].3.1. Qualitative features and non-negativity of solutions.
First, note thatthe equilibrium solution, i.e. , the solution of Eq. (23) that is independent of X and T , is simply Θ ≡ Θ R . Next, suppose a uniform initial condition is chosen such thatΘ | T =0 = Θ = Θ and ∂ Θ ∂T (cid:12)(cid:12)(cid:12)(cid:12) T =0 = ˆΘ ∀ X ∈ Ω . (24)It follows from Eq. (23) that the temperature distribution equilibrates according tothe ordinary differential equation (ODE): d Θ d T + λ dΘ d T = − ǫ (Θ − Θ ) . (25)Treating the latter as a linear ODE in the variable θ = Θ , the characteristicpolynomial of the homogeneous ODE and its roots are easily found to be r + λ r + ǫ = 0 ⇔ r = r ± := 12 (cid:18) − λ ± q λ − ǫ (cid:19) . (26)Clearly, the ODE (25) has oscillatory solutions for λ < ǫ because, then, r ± ∈ C .We readily recognize these characteristic roots, and the ODE (25) as the gov-erning equation for damped, simple harmonic motion. Indeed, the case of λ < ǫ is that of under-damped simple harmonic motion. To determine non-negativity ofsolutions, we must first determine if the absolute minimum of Θ , as a functionof T alone, can be ≤
0. In the simple harmonic oscillator analogy, determiningthis condition can be thought of as a “stopping problem” for if and/or when theoscillator position reaches zero, having started from some non-zero position. Although analyzing a uniform state’s time-evolution is an over-simplification over the originalproblem, there is some value in understanding the solutions to Eq. (25) because, as Jou et al. note“[i]n continuous systems, the exponential decay observed in the discrete model would correspondto a diffusive perturbation, whereas an oscillation in [Θ] would correspond to the propagation ofa heat wave” [44, p. 46]. C -INTEGRABLE SECOND SOUND EQUATION 7 T Θ Figure 2.
Oscillatory behavior of solutions, given in Eq. (27), tothe ODE (25) for λ = ǫ = 1 ⇒ λ < ǫ . Solid curve correspond toΘ i = 1 > Θ R = 0 .
1, dashed curve corresponds to Θ i = 1 < Θ R =2. For this choice of parameters, the solid curve’s first minimum“dips” below Θ = 0; thus, this solution is not strictly non-negativeand Θ can become imaginary.Upon finding the homogeneous and particular solutions to Eq. (25) and applyingthe initial conditions from Eq. (24), we obtainΘ ( X, T ) = e − λ T/ (h + (Θ − Θ ) λ i p ǫ − λ sin (cid:18) T q ǫ − λ (cid:19) + (cid:0) Θ − Θ (cid:1) cos (cid:18) T q ǫ − λ (cid:19)(cid:27) + Θ . (27)Hence,dΘ d T = e − λ T/ (h ǫ (Θ R − Θ i ) − λ ˆΘ i p ǫ − λ sin (cid:18) T q ǫ − λ (cid:19) + ˆΘ cos (cid:18) T q ǫ − λ (cid:19)(cid:27) , (28)Although, in principle, one can now derive conditions, starting from Eq. (28), whichguarantee that the first minimum of Θ remains non-negative, which by the expo-nential decay (in T ) of the solution guarantees that Θ remains non-negative, theseconditions are extremely involved. For the sake of brevity, and to illustrate just themain idea, let us consider the special case ˆΘ = 0.Now, from Eq. (28) with ˆΘ = 0, we find that the first extremum of Θ occursat T = T := 4 π p ǫ − λ , (29)If Θ i > Θ R , this extremum is a minimum (see solid curve in Fig. 2), while forΘ i < Θ R , it is a maximum (see dashed curve in Fig. 2). For Θ i > Θ R , at T = T ,we have Θ ( T , X ) = exp − πλ p ǫ − λ ! (cid:0) Θ − Θ (cid:1) + Θ . (30) I. C. CHRISTOV
Hence, Θ | T = T < − Θ Θ > exp πλ p ǫ − λ ! . (31)Equivalently, assuming that Θ i = Θ R , then strictly negative values of Θ ( i.e. ,strictly imaginary solutions for Θ), occur if λ < ǫ π / ln | − Θ / Θ | (Θ i = Θ R ) . (32)Obviously, for Θ i = Θ R , Θ is simply the uniform steady state of Eq. (25) for all T ≥
0, and there are no extrema in the solution profile. Note that the denominatorin the bound in Eq. (32) becomes singular for Θ / Θ ≈ Hence, this specialcase in which the bound becomes unphysical ( i.e. , λ <
0) is not of significantphysical interest as it is, for all practical purposes, the case of Θ i = Θ R . However,it can easily be shown from Eq. (32) that, for Θ / Θ ∈ (0 , negative . Thus, since this is a bound on λ , then the only choice of λ that guarantees non-negativity of the solution in Eq. (27), for thisrange of Θ / Θ , is the trivial case λ ≡ i.e. , second-sound propagation doesnot occur). Hence, for clarity, we rewrite the bound, which is necessary to ensurenon-negativity of Θ at its first minimum, as λ ≥ ǫ π / ln | − Θ / Θ | , Θ / Θ (0 , \ { } , = 0 , else . (33)For Θ / Θ (0 , \ { } , the bound in Eq. (33) is sharper than λ ≥ ǫ , whichmeans that there are non-negative oscillatory solutions to the ODE (25) for Θ .Clearly, the case λ < ǫ [or λ violating the bound in Eq. (33) for the specialcase of ( ∂ Θ /∂T ) T =0 = 0] is problematic. Of course, we are not the first to pointout such a possibility; see, e.g. , [44, p. 202]. For example, Rubin [56] has argued thatthe possibility of negative temperature could be interpreted as a so-called “secondlaw violation.” Such negative temperatures and second law violations typically ariseif the thermal relaxation time is “too large” (compared to some appropriate timescale) and/or when a finite domain and reflections from boundaries are considered[3]. In our context, λ is a (dimensionless) inverse thermal relaxation time, henceif we wish to avoid completely the issue of oscillatory solutions when λ < ǫ canbe interpreted as a upper bound on the thermal relaxation, beyond which secondlaw violations can occur.It is important to note that, in the present context, the emissivity ǫ of thedielectric plays a key role in setting the bound on λ in Eq. (33). Obviously, in theabsence of heating/cooling of the dielectric ( ǫ ≡
0) in the above analysis, there are no oscillatory solutions to Eq. (25) as long as ( ∂ Θ /∂T ) | T =0 = 0. If ( ∂ Θ /∂T ) | T =0 = 0(but ǫ ≡ λ can be easily derived in terms of the initial Although we use the approximate-equals sign, the critical value Θ / Θ = 1 − e − π is equalto 1 within machine precision, i.e. , to sixteen decimals. Others [44, p. 228] have formulated this condition as a restriction on the maximum magnitudeof the heat flux at the initial instant of time. C -INTEGRABLE SECOND SOUND EQUATION 9 conditions (which are, of course, related to the heat flux): λ ≥ W (cid:20) − ( ∂ Θ /∂T ) | T =0 Θ | T =0 (cid:21) , (34)where W ( · ) is the principal branch of the Lambert W -function [21], a specialfunction with a surprising number of applications in the physical sciences (see, e.g. , [38]). As long as λ satisfies Eq. (34), then solutions to the ODE (25) with ǫ ≡ T ≥
0. Note that, ( ∂ Θ /∂T ) | T =0 > λ >
0. Hence, for clarity, we may write the bound in Eq. (34)as λ > W (cid:20) − ( ∂ Θ /∂T ) | T =0 Θ | T =0 (cid:21) , ( ∂ Θ /∂T ) | T =0 < , ( ∂ Θ /∂T ) | T =0 > . (35)Beyond restricting the value of λ until no negative temperature arise, one canalso consider modifications to the MC law proposed in [20, 56, 3, 44] (amongstothers) that do not exhibit such behaviors. The key modification in some of thesetheories [20, 3] is to introduce a non-equilibrium relation of the form e = e ( ϑ, q ), incontrast to Eq. (12). Another modification could be to replace the MC law with onederived on the basis of Green–Naghdi theory [30, 31, 32], as done in, e.g., [36, 5].While this approach can be employed to study second-sound in rigid conductors,it is also not without criticism (see, e.g., the Mathematical Reviews entries for[30, 31, 32]). Nevertheless, thinking more broadly, perhaps “second law violations”are to be expected on average (in some probabilistic sense) [53, 54] at the physicalscales, temperatures and heat fluxes at which the MC law is believed to apply (and,hence, second sound exists).3.2. The C -transformation. With the aforementioned issues in mind, let us nowrestrict to the case in which negative temperatures are not possible for the spatially-homogeneous temperature initial-value problem for Eq. (23), specifically the bound λ ≥ ǫ . The non-negativity of solutions of the spatially-homogeneous problem nowguarantees that those solutions are bounded between max { Θ i , Θ R } and 0. However,these qualitative estimates do not necessarily apply to the the spatially-extendedproblem. Hence, in order to introduce a valid one-to-one C -transformation, weconsider two cases: Case (I): 0 ≤ Θ( X, ≤ Θ R ∀ X ∈ Ω , (36a)Case (II): 0 ≤ Θ R ≤ Θ( X, ∀ X ∈ Ω . (36b)Then, the appropriate C -transformations areΘ − Θ θ ⇔ Θ q θ + Θ [Case (I)] , (37a)Θ − Θ θ ⇔ Θ q Θ − θ [Case (II)] . (37b)Since the dielectric conductor cannot be heated/cooled above/below the externalmedium’s reference temperature Θ R , we expect that the inequalities in Eq. (36)hold true also for T >
0. Hence, the C -transformations in Eq. (37) remain validone-to-one mappings for T > C -integrable . Now,it is easy to show that in either case (I) or case (II), the mappings introduced above transforms Eq. (23) into the telegrapher’s equation [23, 8, 41]: ∂ θ∂T + λ ∂θ∂T = ∂ θ∂X − ǫθ. (38)In the absence of radiative heat transfer (equivalently, making the emissivity of thelateral surface vanish, i.e. , ǫ ≡ damped wave equation (DWE), which is discussed in the context of heatwaves in [17].4. Exact solutions to the C -linearized governing equation. In this section,we adapt, to the present heat transfer context, known exact solutions to a standardinitial-value problem (IVP) and a standard initial-boundary-value problem (IBVP)for Eq. (38).4.1. d’Alembert problem on the real line.
In this subsection, let us considerthe d’Alembert problem on the real line Ω = ( −∞ , + ∞ ). We subject Eq. (38) tothe following initial and asymptotic conditions: θ | T =0 = F ( X ) , ∂θ∂T (cid:12)(cid:12)(cid:12)(cid:12) T =0 = G ( X ) , lim | X |→∞ θ ( X, T ) = 0 ∀ T ≥ , (39)where F ( · ) and G ( · ) are some arbitrary functions. The initial-value problemthus defined has an exact solution given in the textbook by Webster [61, Ch. IV,Eq. (146)] (see also the equivalent solution in Guenther and Lee’s textbook [33, § k := ǫ − λ /
4, which corresponds to the quantity ( ac − b ) /a in [61, Ch. IV, Eq. (146)].Webster only considers ( ac − b ) /a <
0. Although we have restricted the value of λ based on the considerations in Section 3.1, we allow k ≤
0. Hence, we mustalso consider the special case ( ac − b ) /a = 0 explicitly, which is not difficult to do.Thus, we arrive at the exact solution θ ( X, T ) = 12 e − λ T/ ( F ( X + T ) + F ( X − T ) + Z X + TX − T F ( η ) K ( X, η, T, k ) d η + Z X + TX − T (cid:20) G ( η ) + λ F ( η ) (cid:21) K ( X, η, T, k ) d η ) , (40)where the integration kernels K , are defined as K ( X, η, T, k ) := , k = 0 − I (cid:16)p | k | [ T − ( X − η ) ] (cid:17) | k | T p | k | [ T − ( X − η ) ] , k < , (41a) K ( X, η, T, k ) := , k = 0 I (cid:16)p | k | [ T − ( X − η ) ] (cid:17) , k < , (41b)and I ν ( · ) is the modified Bessel function of the first kind of order ν . Notice thatfor k = 0 (although λ , ǫ = 0) solutions to Eq. (38) on the real line behave similarly C -INTEGRABLE SECOND SOUND EQUATION 11 - X a ) Θ - X ) Θ - X c ) Θ - X ) Θ Figure 3.
Dimensionless temperature Θ profiles versus X at dif-ferent dimensionless times T showing the relaxation of a unit pulsevia the exact solution in Eq. (40). (a) T = 0 .
5, (b) T = 1, (c) T = 2, (d) T = 4. Here, ǫ = 0 . λ = √ ǫ ( k <
0) for solidcurves, while λ = √ ǫ ( k = 0) for dashed curves.to solutions to the “classical” wave equation [ i.e. , Eq. (38) with λ ≡ ǫ ≡ e.g. , [61, Ch. III, Eq. (27)]): θ ( X, T ; λ = ǫ = 0) = 12 " F ( X + T ) + F ( X − T ) + Z X + TX − T G ( η ) d η . (42)The solution in Eq. (40) with k = 0, nevertheless, retains the exponential dampingin time, while also involving both initial conditions, i.e. , F and G into the integralterm. The reader is referred to [17] for further discussion of wave solutions in heattransfer.Of interest is how an initial unit pulse (“box”) of heat, F ( X ) = H ( X + 1) − H ( X −
1) and G ( X ) = 0, relaxes in time. As Θ R has only a minor quantitativeeffect on the solutions depicted, we restrict our attention to Θ R = 1 without lossof generality. This solution is illustrated in Figs. 3 and 4 for ǫ = 0 . . X = ±
1) edges of the initial conditions along the space-time curves X = ± (1 + T ),decaying in time. Additional shocks form and interact in the region in-between, i.e. , − (1 + T ) < X < T . Notice, of course that the dashed curves in both Figs. 3( ǫ = 0 .
1) and 4 ( ǫ = 0 .
5) are qualitatively similar with small quantitive difference inmagnitude. This observation follows from the fact that, for k = 0, the solution inEq. (40) no longer depends explicitly on the surface emissivity ǫ . Nevertheless, wehave chosen λ = √ ǫ , so the exponential damping of the solution changes with ǫ .4.2. Signaling problem on a half-space.
In this subsection, we consider thesignaling problem on the half-space Ω = (0 , + ∞ ). In the absence of radiative heattransfer (equivalently, making the emissivity of the later surface vanish, i.e. , ǫ ≡ § - X ) Θ - X ) Θ - X c ) Θ - X ) Θ Figure 4.
Dimensionless temperature Θ profiles versus X at dif-ferent dimensionless times T showing the relaxation of a unit pulsevia the exact solution in Eq. (40). (a) T = 0 .
5, (b) T = 1, (c) T = 2, (d) T = 4. Here, ǫ = 0 . λ = √ ǫ ( k <
0) for solidcurves, while λ = √ ǫ ( k = 0) for dashed curves.boundary and asymptotic conditions θ | T =0 = ∂θ∂T (cid:12)(cid:12)(cid:12)(cid:12) T =0 = 0 , θ (0 , T ) = H ( T ) F ( T ) , lim X → + ∞ θ ( X, T ) = 0 ∀ T ≥ , (43)where F ( · ) is some arbitrary function, and H ( · ) is the Heaviside unit step function.The initial-boundary-value problem thus defined has an exact solution given byJordan and Puri [41].To adapt the latter exact solution to the present context, we note that Jordanand Puri [41] have considered all three cases k ⋚ k ≤
0. Thus,we arrive at the exact solution θ ( X, T ) = H ( T − X ) " e − λ X/ F ( T − X ) + Z TX F ( T − η ) K ( X, η, | k | ) d η , (44)with k := ǫ − λ / K ( X, η, | k | ) := , k = 0 ,X e − λ η/ I hp | k | ( η − X ) ip | k | ( η − X ) , k < . (45)Of interest is the case of F ( T ) = 1 ∀ T >
0, which corresponds to the so-called heat pulse experiments [24], however, in the present case there is also lateral in-terfacial heating of the dielectric heat conductor. Again, as Θ R has only a minorquantitative effect on the solutions depicted, we restrict our attention to Θ R = 1without loss of generality. This solution is illustrated in Fig. 5. Clearly, a shockpropagates from the heated wall at X = 0 into the medium along the space-time curve X = T , decaying in time. The amplitude of the shock, defined as C -INTEGRABLE SECOND SOUND EQUATION 13 X ) Θ X ) Θ X c ) Θ X d ) Θ Figure 5.
Evolution of a (dimensionless) heat pulse Θ under thesolution from Eq. (44). (a,b) ǫ = 0 .
1, (c,d) ǫ = 0 .
5; (a,c) T = 2,(b,d) T = 4. In all panels λ = √ ǫ ( k <
0) for the solid curves,while λ = √ ǫ ( k = 0) for the dashed curves.[[ θ ]]( T ) := lim X → T − θ ( X, T ) − lim X → T + θ ( X, T ), is easily found from Eq. (44) bytaking X → T ± to obtain [[ θ ]]( T ) = e − λ T/ F (0). Now, using Eqs. (37),[[Θ]]( T ) = q e − λ T/ F (0) + Θ [Case (I)] , (46a)[[Θ]]( T ) = q Θ − e − λ T/ F (0) [Case (II)] . (46b)Here, it is instructive to recall that Jordan and Puri note that “[t]he case of k < k < k <
0) of interest in low-temperature heat transfer in heated dielectrics.However, above we arrived at this condition not by considering the physics of theproblem but by defining conditions on the validity of the C -transformation. Finally,we note that, in the context of the solution in Eq. (44), the most pernicious featureof violating the bound on λ ( i.e. , takin k >
0) is a change in the concavity ofthe profile behind the wave front ( i.e. , for
X < T ), which leads to “sagging” andeventually negative values of θ (and, thus, imaginary values of Θ) at some locationbehind the wavefront.5. Conclusion.
We considered an exactly solvable model in heat conduction, specif-ically the C -integrable ( i.e. , change-of-variables-integrable) equation for secondsound propagation in a thin, rigid dielectric heat conductor uniformly heated onits lateral side by a surrounding medium. The exact linearization of the govern-ing PDE yielded the well known telegrapher’s equation. The region of parameterspace in which we expect the proposed C -transformation to be valid was motivatedthrough a qualitative discussion on the non-negativity of solutions to the spatially-homogeneous problem. Adapting known transform solutions from the literature, we presented some relevant exact solutions to the heat transfer problem on an infinitedomain (initial-value problem), and the heat pulse solution to the heat transferproblem on a semi-infinite domain (initial-boundary-value problem). Of course,due to the presence of jump discontinuities, the solutions discussed in Sections 4.1and 4.2 must be regarded as weak , satisfying the original PDE only in the sense ofdistributions.Although we showed exact solutions are possible to what was, originally, a non-linear problem, it is also possible to “attack” the exactly C -linearized governingEq. (38) by either positivity-preserving nonstandard finite-difference schemes, suchas those constructed in [39, 47], or, if the governing equations are written as afirst-order hyperbolic system, by Godunov-type shock-capturing schemes [18].Note that if the heat conductor being considered is not rigid, then the “ordinary”time derivative in the MC constitutive equation, i.e. , Eq. (9), must be replaced byan appropriate convected time rate as shown by C. I. Christov [16]: the so-called“Cattaneo–Christov” heat flux law [58]. Additional issues arise in fully deformablecontinua (such as dissipative solids [49]) and when mass transfer is also considered[50], leading to the so-called “Christov–Morro” theory [19, 29].Finally, it would be of interest to develop sharper criteria for the non-negativityof solutions to the telegrapher equation, specifically for arbitrary spatially inhomo-geneous initial conditions. Developing such a mathematical theory would extend thevalidity of the proposed C -transformation introduced in Section 3.2, which maps thenonlinear Eq. (23) into the linear Eq. (38), to a wider region of the ( λ , ǫ ) parameterspace of the proposed heat transfer model. Acknowledgements.
The author would like to thank Dr. P. M. Jordan and theguest editors for the invitation to participate in this special issue and for their effortsin organizing it. Helpful discussions with Dr. Jordan regarding Refs. [39, 40] arealso acknowledged. The author is also grateful to the anonymous reviewers for theirhelpful comments, which have improved the presentation and bibliography of thispaper.
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Received January 2018; revised April 2018.
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