The paper deals with the problem of modification of Fourier Law. After a
short summary on the basic problem of the second sound phenomenon the different
opinions are listed and commented. Also the effect of the second sound
phenomenon on the thermoelasticity is analyzed

Szekeres, András

English

Periodica Polytechnica (Budapest University of Technology and Economics)

PERIODICA POLYTECHNICA SER. MECH. ENG. VOL. 48, NO. 1, PP. 83–87 (2004)THE SECOND SOUND PHENOMENON: PRO AND CONTRAAndrás SZEKERESDepartment of Applied MechanicsBudapest University of Technology and EconomicsH–1521 Budapest, Hungarye-mail: szekeres@mm.bme.huReceived: March 1, 2004AbstractThe paper deals with the problem of modification of Fourier Law. After a short summary on the basicproblem of the second sound phenomenon the different opinions are listed and commented. Also theeffect of the second sound phenomenon on the thermoelasticity is analyzed.Keywords: heat conduction, second sound phenomenon, thermoelasticity.1. IntroductionIn 1867 Maxwell turned the attention to the theoretical contradiction hidden inthe Fourier’s law. As a result, also the practical application is limited by thiscontradiction. It has been a surprise because this law (and its relatives, as Fick’s,Darcy’s, etc.) is one of the most effective and widely used tools in physics. Maybethis Janus face is the reason of the discussion on the topic. The problem dividesalmost all the experts having any kind of relation to the topic: thermodynamicians,mechanicians, thermotechnicians, etc.The core of the problem is very simple. The heat propagation task describedby the Fourier’s law leads to the parabolic differential equation and as a consequenceto unlimited speed of propagation. As a resolution of this kind of contradiction,dozens of modified laws [1, 2] and modification methods, e.g. [3], have been workedout. The common feature of these efforts is that all lead to hyperbolic differentialequation and the speed of propagation becomes limited. This is the origin of thename ‘second sound’ that became the emblem of the whole phenomenon (SSP).While the core of the problem is as simple as described above, the practicalaspect is very complicated. First of all, because the modification needs at least onenew material property, the so called ‘relaxation time’. And the reliable experimentaldetermination of this property has not been solved yet [4]. It’s clear that without this,the whole theory is unstable. There are several other problems, too, e.g. the solutionapplicable in heat conduction is not satisfactory in thermoelasticity [5]. It is an otherpractical problem that the question has importance only in the dynamical tasks, thatare very seldom. Similarly, in case of metals, the problem has less importance thanin case of more up to date engineering materials as e.g. composites.84 A. SZEKERESBased on all of these, it is understandable that the opinion of the experts on theSSP is really widely differing. The different opinions have been collected. Besidethis, the paper deals with a basic summary on the SSP and analyzes the relation ofSSP and thermoelasticity.2. Short Summary on the Second Sound PhenomenonThe heat conduction problem by the energy equation, equation of state and Fourier’slaw, neglecting the mechanical interactions, leads to the parabolic differential eq.on the temperature T ,d Txx + Tt = 0, (1)where d = k/ρc is the diffusivity, t and x are the time and space coordinates, thesubscript denotes partial derivation. It results in unlimited speed of temperaturewave propagation, vT → ∞.To resolve this contradiction several attempts were made [1, 2]. One ofthem, maybe the first one was the Cattaneo-Vernotte modified law (C − V ) of heatconduction, asτht + h = −kTx, (2)where h is the heat flux, k is the conductivity and τ is the so called relaxation time,the new material property. Replacing the Fourier’s law by the C − V , the result isthedTxx + Tt + τTtt = 0, (3)hyperbolic equation with the propagation speedvT =√d/τ . (4)An other possible resolution is to change the equation of state toe = cT + c1Tt , (5)where e is the internal energy, c is the heat capacity, c1 = τc, while keeping thelaw of heat conduction (Fourier) unaltered. The result is the same as above [3].The result has been generalized in two directions: several new modified lawsof heat conduction were published [1, 2], and also modification methods wereworked out e.g. [3]. In spite of all these efforts, the problem of the new materialproperties has remained [4, 6].There are further aspects of the SSP. One of them is the other phenomenaof physics described by similar laws: moisture diffusion (Fick’s law), groundwaterflow (Darcy’s law), dislocation diffusion, solution process, etc [7]. Also the problemof material arises. While the relaxation time in case of metals has a very smallvalue, this material property is bigger by order of magnitudes in case of polymers(composites), human tissues, etc.THE SECOND SOUND PHENOMENON 853. Pro and ContraThe variety of the experts’ opinions is very wide. The mechanicians, first of allthose dealing with thermomechanics, support the idea of applying the SSP in me-chanics.The reason seems to be very simple. The theoretical aspects of the SSP arevery convincing and this group has minor problem with the application. Of course,there are exceptions, see later.The thermotechnicians are mainly against it. The reason is clear and practical.The whole theory of heat-exchangers is based on the Fourier’s law. Knowing theimportance of this field of engineering, nobody wants to risk the value of this hugeknowledge accumulated in the last 60-80 years. Of course, there are exceptionsalso in this case.One of my basic experiences concerning the rejection of SSP is coupled toa first class thermotechnician who was the referee of my first paper on the topic.Looking at the τht member of the C − V , he asked me about the physical meaningcompared to the Fourier’s law.Of course, it is impossible to give a satisfactory answer. It is more a philo-sophical problem whether we are able to explain the modification on the basis ofthe law we want to modify.Another example of the rejection is from a mechanician [8]. His argumentwas more covincing, but not satisfactory. Let us start with the C − V , see Eq. (2),but supposing uniform temperature distribution, i.e. Tx = 0 and as a consequenceτht + h = 0. (6)The solution of this differential Eq. ish = A exp (−t/τ) . (7)It means, the heat flux exists even though Tx = 0, i.e. inspite of missing the drivingpower, and that is impossible. The argument is false, because Cattaneo–Vernotte’smain statement is that not only the temperature gradient could be the reason of theheat flux.The argument of the very practical thermotechnician is understandable fromthat point of view, too, that the experience doesn’t help too much. The problemcomes up only in dynamical cases, mostly with nonmetallic materials. By the way,this two parallel features of the problem (process and material) are shown verycharacteristically by the theory of similitude that results the dimensionless numberτπ/t0, where τ is a material while t0 is a process property.The argument of the mechanician is not understandble, because in the con-tinuummechanics the SSP is an everyday problem. Also the modern life of thequestion has been started in the early 1950s in the thermoelasticity. And it is not bychance. The thermoelasticity is the most often used coupled field of the engineeringand within it the dynamical tasks became very frequent, which brought the needof modification, i.e. SSP. DANILOVSKAYA [9], having solved the thermal shocktask of elastic material, experienced disturbances before the wave front and she86 A. SZEKERESfound the reason in the not satisfactory law of heat conduction. Even though, thesymbiosis of the SSP and the thermoelasticity is not bilateral. As we will see inthe next chapter, the solution which is satisfactory in heat conduction doesn’t givea solution in the thermoelasticity.The complicated feature of SSP can be illustrated by personal experience onthe R&D field. A few years ago I was invited to the ANSYS headquarters to discussthe problem of SSP. I tried to convince them to extend their code into this direction.I wasn’t able, and the final argument was: no need for SSP from the customers. Itried to make it clear that it is a ‘vicious circle’: no need to buy it because there isno possibility to buy it. Now, I am in a similar relation to the ESI Group, I hope tohave more success.I have to mention one more aspect of the SSP. The tradition causes very hardproblem. Of course, in some sense it is advantageous, but if it becomes too hard itcreates more problems than advantage. I guess, this is the case with the modificationof one of the most celebrated laws of the physics, i.e. the Fourier’s law.4. Second Sound Phenomenon and ThermoelasticityAs we saw previously, dealing with the pure heat conduction problem the modi-fication in both cases according to the Eq. (2) (modified law of heat conduction),or Eq. (5) (modified state equation) lead to the same hyperbolic equation, and theproblem of unlimited wave propagation is solved.Dealing with the coupled problem of the thermoelasticity, let us derive thegeneralized equations of motion and the heat conduction based on the basic equa-tions of the system:utt − Eρuxx + EαρTx = 0, (8)dTxx + τTtt + Tt − τ Eρc(ux − αT ) uxtt − Eρc(ux − αT ) uxt = 0, (9)where u is the displacement, E is the Young modules and α is the coefficient ofthermal expansion.The following comments apply to the equation system (8) and (9).a. Neglecting the interactions between temperature and displacement, the sys-tem splits into the independent equations ofutt − Eρuxx = 0, (10)dTxx + τTtt + Tt = 0. (11)Eq. (11) shows the different cases of the heat conduction:• by the modified laws for both cases C − V and state equation,THE SECOND SOUND PHENOMENON 87• by the Fourier taking τ = 0.b. For the cross-coupled case the generalized equation of motion, Eq. (8), is notaffected by the modification.c. While the generalized equation of heat conduction, Eq. (9), is different fromFourier (τ = 0), for C−V (all members exist) and for modified state equation(4th member is equal to zero).It verifies the basic statement that the solution of the SSP satisfactory in heatconduction does not give perfect solution in thermoelasticity. Generalized, it showsthe basic role of mechanics in thermodynamics.5. SummaryThe old problem of the SSP hasn’t been solved yet. The very complicated featureof the question has several signs. One of them is the diversity of opinions of thedifferent experts. Another one is connected to the thermoelasticity, i.e. while thecontradiction seems to be solved in the heat conduction, the coupled temperatureand displacement causes unsolved problems.References[1] CHANDRASEKHARAIAH, D. S., Thermoelasticity with Second Sound: A Review, Appl. Mech.Rev., 39 No. 3 (1986), pp. 355–375.[2] CHANDRASEKHARAIAH, D. S., Hyperbolic Thermoelasticity: A review of Recent Literature,Appl. Mech. Rev., Vol 51 No. 12, Part 1 (1998), pp. 705–729.[3] SZEKERES, A., Equation System of Thermoelasticity Using the Modified Law of Thermal Con-ductivity, Periodica Polytechnica, 24 No. 3 (1980), pp. 253–261.[4] SZEKERES, A. – SZALONTAY, M., Experiments on Thermal Shock of Long Rods, PeriodicaPolytechnica, 24 No. 3 (1980), pp. 243–252.[5] SZEKERES, A., Heat Conduction with Second Sound. Thermoelasticity and Other CoupledFields, Seminar in the Institute of Cybernetics, Tallinn, Estonia, Nov 19, 1998.[6] MAURER, M. J., Relaxation Model for Heat Conduction in Metals, J. Appl. Phys., 40 No. 13(1969), pp. 5123–5130.[7] SZEKERES, A., Selected Topics of Thermo-Hygro-Mechanics (THM), Seminar in the Instituteof Cybernetics, Tallinn, Estonia, Aug 31, 2001.[8] VALANIS, K., Personal Discussion, 1999.[9] DANILOVSKAYA, B. I., On the Dynamical Problems of Thermoelasticity (In Russian), Priklad-naya Matematika and Mechanika, 16 (1952), pp. 341–344.

THE SECOND SOUND PHENOMENON: PRO AND CONTRA

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