Müller I. A history of thermodynamics. The doctrine of energy and entropy (1185104), страница 61
Текст из файла (страница 61)
8.6. Pulse speeds in relation to the normal speed of sound. Table and crosses:Calculations by Weiss68. Circles: Lower bound65( 0 12 ) by Boillat and Ruggeri69stops.66 Indeed, Guy Boillat (1937– ) and Tommaso Ruggeri (1947– )have provided a lower bound for Vmax which tends to infinity for N ĺ.67The fact that Vmax is unbounded represents something of an anticlimax forextended thermodynamics, because the theory started out originally as aneffort to find a finite speed of heat conduction. Let us consider this:Carlo Cattaneo (1911–1979)Fourier’s equation of heat conduction is the prototypical parabolic equationand it predicts an infinite speed of propagation of disturbances in temperatures. This phenomenon became known as the paradox of heatconduction.
Neither engineers nor physicists generally were much worriedabout the paradox. It is quantitatively unimportant in solids and liquids andeven in gases under normal pressures and temperatures. And yet, theparadox represented an awkward feature of thermodynamics and in 1948Carlo Cattaneo made an attempt to resolve it.Upon reflection it was clear to Cattaneo that Fourier’s law was to blameand he amended it. We refer to Fig. 8.7 and recall the mechanism of heatconduction in gases as described in the elementary kinetic theory.
If thereis a downward temperature gradient across a small volume element – of thedimensions of the mean free path – an atom moving upwards will, in themean, carry more energy than an atom moving downwards. Therefore there66676869W. Weiss: “Zur Hierarchie der erweiterten Thermodynamik.” [On the hierarchy ofextended thermodynamics] Dissertation TU Berlin.See also: I. Müller, T.
Ruggeri: “Rational Extended Thermodynamics.” loc.cit.G. Boillat, T. Ruggeri: “Moment equations in the kinetic theory of gases and wavevelocities.” Continuum Mechanics and Thermodynamics 9 (1997).W. Weiss: loc.cit.G. Boillat, T. Ruggeri: “Moment equations …” loc.cit.2628 Thermodynamics of Irreversible Processesis a net flux of energy upwards, i.e. opposite to the temperature gradient,associated with the passage of a pair of particles across the middle layer.That flux is obviously proportional to the temperature gradient, just asFourier’s law requires for the heat flux.Fig. 8.7.
Carlo Cattaneo. The Cattaneo equationCattaneo70 changed that argument slightly. He argued that there is a timelag between the start of the particles at their points of departures and thetime of passage through the middle layer. If the temperature changes intime, it is clear that the heat flux at a certain time depends on the temperature gradient at a time IJ earlier, where IJ is of the order of magnitude of themean time of free flight.
Therefore it seems reasonable to write an nonstationary Fourier law in the formqiÈ T T Øț Éτwith τ ! 0 .t xi ÙÚÊ xiNow, this equation is badly flawed, because it predicts that for qi = 0 thetemperature gradient tends exponentially toward infinity. Nor does thismodified Fourier law lead to a finite speed, so that it does not resolve theparadox. Cattaneo must have known this – although he does not say so (!) –because he proceeded by converting his non-stationary Fourier law intosomething else in a sequence of three steps which deserve to be calledmathematically creative.70C.
Cattaneo: “Sulla conduzione del calore.” [On heat conduction] Atti del SeminarioMatematico Fisico della Università di Modena, 3 (1948).Extended ThermodynamicsqiÈ T1 T Øț ÉτÀqiÙt xi ÚÊ xi1 W wwtțØÈÀ É1 τ Ù qiÊt Ú qi Wwqiwt263wTwxițTxițwT.wxiThe end result, now usually called the Cattaneo equation, is acceptable.It provides a stable state of zero heat flux for wwZ6K 0 and, if combined withthe energy equation, it leads to a telegraph equation and predicts a finitespeed of propagation of disturbances of temperature.So, however flawed Cattaneo’s reasoning may have been, he is the authorof the first hyperbolic equation for heat conduction. Let us quote him howhe defends the transition from the non-stationary Fourier law to theCattaneo equation:Nel risultato ottenuto approfitteremo della piccolezza del parametro IJ pertrascurare il termine che contiene a fattore il suo quadrato, conservandoperaltro il termine in cui IJ compare a primo grado.
Naturalmente, perdelimitare la portata delle conseguenze che stiamo per trarre, converràprecisare un po’ meglio le condizioni in cui tale approssimazione è lecita.Allo scopo ammetteremo esplicitamente che il feno-meno di conduzionecalorifica avvenga nell´intorno di uno stato stazionario o, in altri termini,che durante il suo svolgersi si mantengano abbastanza piccole le derivatetemporali delle varie grandezze in giuoco.In the result we take advantage of the smallness of the parameter IJ so thatterms with squares of IJ may be neglected.
First order terms in IJ are kept,however. Of course, in order to appreciate the effect on the consequences,which we are about to derive, it would be proper to investigate theconditions when that approximation is valid. For that purpose we stressthat the heat conduction should remain nearly stationary.
Or, in otherwords, that the time derivatives of the various quantities at play remainsufficiently small, while the stationary state changes slowly.Well, if the truth were known, this is not a valid justification. How couldit be, if it leads from an unstable equation to a stable one and from aparabolic to a hyperbolic equation.Let me say at this point that Cattaneo’s argument leading to the non-stationaryFourier law is the nut-shell-version of the first step in an iterative scheme that isoften used in the kinetic theory of gases.
In that field the objective is animprovement of the treatment of viscous, heat-conducting gases beyond what the2648 Thermodynamics of Irreversible ProcessesNavier-Stokes-Fourier theory can achieve. The iterative scheme is called theChapman-Enskog method and its extensions are known as Burnett approximationand super Burnett. The scheme leads to inherently unstable equations and should bediscarded. The reason why the fact was not recognized for decades is that theauthors have all concentrated on stationary processes.71 And the reason why it isstill used is natural inertia and lack of imagination and initiative.The situation is quite similar mathematically and psychologically to the onementioned in the context of rational thermodynamics of unstable equilibria of nthgrade fluids with n > 1, see above.However, whatever the peculiarities of its derivation may have been, theCattaneo equation on the paradox of heat conduction served as a stimulus.Müller72 generalized Cattaneo’s treatment within the framework of TIP,taking care – at the same time – of a related paradox of shear motion.
Andthen, after rational thermodynamics appeared, Müller and I-Shih Liu(1943–)73 formulated the first theory of rational extended thermodynamics, still restricted to 13 moments, but complete with a constitutiveentropy flux – rather than the Clausius-Duhem expression – and withLagrange multipliers.Thus the subject was prepared for being joined to the mathematicaltheory of hyperbolic systems. Mathematicians had studied quasi-linear firstorder systems for their own purposes, – without being motivated by theparadoxon of infinite wave speeds.
Godunov, 74 Friedrichs and Lax,75 andBoillat76 discovered that such systems may be reduced to a symmetrichyperbolic form, if they are compatible with a convex extension, i.e. anadditional relation of the type of the entropy inequality. Ruggeri and71The instabilities involved in the Chapman-Enskog iterative scheme have recently beenreviewed by Henning Struchtrup (1956–).
H. Struchtrup: “Macroscopic TransportEquations for Rarefied Gases – Approximation Methods in Kinetic Theory” Springer,Heidelberg (2005).72 I. Müller: “Zur Ausbreitungsgeschwindigkeit von Störungen in kontinuierlichen Medien.”[On the speed of propagation in continuous bodies.]. Dissertation TH Aachen (1966).See also: I. Müller: “Zum Paradox der Wärmeleitungstheorie.” [On the paradox of heatconduction]. Zeitschrift für Physik 198 (1967).73 I-Shih Liu, I. Müller: “Extended thermodynamics of classical and degenerate gases.”Archive for Rational Mechanics and Analysis 46 (1983).74 S.K.
Godunov: “An interesting class of quasi-linear systems.”Soviet Mathematics 2 (1961).75 K.O. Friedrichs, P.D. Lax: “Systems of conservation equations with a convex extension.”Proceeding of the National Academy of Science USA 68 (1971).76 Boillat: “Sur l´éxistence et la recherche d´équations de conservations supplémentairespour les systèmes hyperbolique.” [On the existence and investigation of supplementaryconservation laws for hyperbolic systems] Comptes Rendues Académie des SciencesParis. Ser5. A 278 (1974).Extended Thermodynamics265Strumia77 recognized that the Lagrange multipliers – their main field – couldbe chosen as thermodynamic fields and, if they were, the field equations ofextended thermodynamics were symmetric hyperbolic.
The formal structureof the theory was refined by Boillat and Ruggeri,78,79 and eventually theyproved that for infinitely many moments the pulse speed tends to infinity,although it is always finite for finitely many moments, see above.80As mentioned before this phenomenon is a kind of anti-climax for a theory thathad originally set out to calculate finite speeds. However, the infinite limiting casehas its own appeal and anyway: Extended thermodynamics had by this timeoutgrown its original motivation and had become a predictive theory forprocesses with large rates of change and steep gradients, as they mightoccur in shock waves.
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
