Müller I. A history of thermodynamics. The doctrine of energy and entropy (1185104), страница 62
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Let us consider this:Field Equations for MomentsOnce the distribution function is known in terms of the Lagrangemultipliers, see above, it is possible – in principle – to change back from theLagrange multipliers /K1K2 ...KN to the moments W K1K2 ...KN by inverting therelationui1i2 ...ilÔ µci ...ci Y exp k1 ÇlNf10/ i1i2 ...il µ ci1 ci2 ...cil dc .Once this is done, we may determine the last fluxui1i2 ...iN aÔ µ c ...cin terms of W K1K2 ...KN (Ni1iNNcaY exp k1 Ç l 0 / i1i2 ...il µ ci1 ci2 ...cil dc0.1,..0 ) . Also in principle the productions maythus be calculated after we choose an appropriate model for the atomicinteraction, e.g.
the model of Maxwellian molecules, cf. Chap. 4.77T. Ruggeri, A. Strumia: “Main field and convex covariant density for quasi-linearhyperbolic systems. Relativistic fluid dynamics.” Annales Institut Henri Poincaré 34 A(1981).78 T. Ruggeri: “Galilean invariance and entropy principle for systems of balance laws. Thestructure of extended thermodynamics.” Continuum Mechanics and Thermodynamics 1(1989).79 G.
Boillat, T. Ruggeri: “Moment equations …” loc.cit.80 Incidentally, in the relativistic version of extended thermodynamics the maximal pulsespeed for infinitely many moments is c, the speed of light.2668 Thermodynamics of Irreversible ProcessesIn reality the calculations of the flux ui1i2 ...iN a and of the productions3 K K ...KN (N1 26,7...0 ) 81 require somewhat precarious approximations,Xsince integrals of the type occurring in the last equations cannot be solvedanalytically. However, when everything is said and done, one arrives atexplicit field equations, e.g. those of Fig.
8.8, which are valid for N = 3 sothat there are 20 individual equations. The equations written in the figureare linearized and the canonical notation has been introduced like ȡ for u,ȡ i for ui, 3 ȡ k/µT for the trace uii, t<ij> for the deviatoric stress and qi for theheat flux. The moment u<ijk> has no conventional name, – other than traceless third moment – because it does not enter equations of mass, momentumand energy.
But it does have to satisfy an explicit fields equation, see figure.Fig. 8.8. 4 times field equations of extended thermodynamics for N= 3 Top left: Euler. Topright: Navier-Stokes. Bottom left: Cattaneo. Bottom right: 13 moment81Recall that the first five productions are zero which reflects the conservation of mass,momentum and energy.Extended Thermodynamics267Figure. 8.8 shows the same set of 20 equations four times so as to make itpossible to point out special cases within the different frames:x On the upper left side we see the equations for the Euler fluid, which isentirely free of dissipation and thus without shear stresses and heat flux.x The upper right box contains the Navier-Stokes-Fourier equations withthe stress proportional to the velocity gradient and the heat fluxproportional to the temperature gradient.
This set identifies the onlyunspecified coefficient IJ as being related to the shear viscosity Ș. Wehave K 43 WU MP 6 so that Ș grows linearly with T as is expected forXMaxwellian molecules, cf. Chap. 4.x In the fifth equation of the third box I have highlighted the Cattaneoequation which has provided the stimulus for the formulation ofextended thermodynamics, see above. The Cattaneo equation isessentially a Fourier equation, but it includes the rate of change of theheat flux as an additional term even though it ignores other terms.x The fourth box exhibits the 13-moment equations. These are the onesbest known among all equations of extended thermodynamics, becausethey contain no unconventional terms, – only the 13 moments familiarfrom the ordinary thermodynamics, viz. ȡ, i, T, t<ij>, and qi.For interpretation we may focus on the upper right box in Fig.
8.8, theone that emphasizes the Navier-Stokes theory. In this way we see that somespecific terms are left out of that theory, namelywV KLwVandwS KwVandwV KMwZ MandwS K.wZ MFor rapid rates and steep gradients we may suspect that these terms docount and, indeed, they do, and we must go to the full set of 20 equations,or to equations with even more moments. Since rapid rates and steepgradients are measured in terms of mean times of free flight and mean freepaths, we may suspect that extended thermodynamics becomes necessaryfor rarefied gases.Shock WavesProperly speaking shock waves do not exist, at least not as discontinuities indensity, velocity, temperature, etc. What seems like shock waves turns outto be shock structures upon close experimental inspection, i.e. smooth butsteep solutions of the field equations, which assume different equilibriumvalues at the two sides.
Scientists and engineers are interested to calculate2688 Thermodynamics of Irreversible Processesthe exact form of the shock structures; and they have realized that theNavier-Stokes-Fourier theory fails to predict the observed thickness.82 Sincethis is a case of steep gradients or rapid rates, it is appropriate, perhaps, toapply extended thermodynamics.To be sure we cannot use the formulae of Fig. 8.8, because these arelinearized.
Their proper non-linear form is too complicated to be writtenhere. Let it suffice therefore to say that, yes, extended thermodynamics doesprovide improved shock structures. But the work is hard, because even forrather weak shock – which move with a Mach number of 1.8 – the requirednumber of moments goes into the hundreds as Wolf Weiss83 and Jörg Auhave shown.84An interesting feature of that research – first noticed, but apparently notunderstood by Grad85 – is the observation that, when the Mach numberreaches the pulse speed and exceeds it, a sharp shock occurs within theshock structure.
Obviously those Mach numbers are truly supersonic andnot just bigger than 1. That is to say that the upstream region has no way ofbeing warned about the onrushing wave, if that wave comes along fasterthan the pulse speed. For the mathematician this is a clear sign that he hasover-extrapolated the theory: He should take more moments into accountand, if he does, the sharp shocks disappear, or rather they are pushed to ahigher Mach number appropriate to the bigger pulse speed of the moreextended theory.Boundary ConditionsExtended thermodynamics up to 1998 is summarized by Müller andRuggeri.86 Since the publication of that book boundary value problems havebeen at the focus of the research in the field, and some problems of the 13moment theory have been solved:x It has been shown for thermal non-equilibrium between two co-axialcylinders that the temperature measured by a contact thermometer is not8283848586This was decisively shown by D.
Gilbarg, D. Paolucci: “The structure of shock waves inthe continuum theory of fluids.” Journal for Rational Mechanics and Analysis 2 (1953).W. Weiss: “Die Berechnung von kontinuierlichen Stoßstrukturen in der kinetischenGastheorie.” [Calculation of continuous shock structures in the kinetic theory of gases]Habilitation thesis TU Berlin (1997). See also: W. Weiss: Chapter 12 in: I.
Müller, T.Ruggeri: “Rational Extended Thermodynamics” loc.cit.W. Weiss: “Continuous shock structure in extended Thermodynamics.” Physical ReviewE, Part A 52 (1995).Au: “Lösung nichtlinearer Probleme in der Erweiterten Thermodynamik.” [Solution ofnon-linear problems in extended thermodynamics’’]. Dissertation TU Berlin, ShakerVerlag (2001).H. Grad: “The profile of a steady plane shock wave.” Communications of Pure andApplied Mathematics 5 Wiley, New York (1952).I. Müller, T. Ruggeri: “Rational Extended Thermodynamics.” loc.cit.Extended Thermodynamics269equal to the kinetic temperature, a measure of the mean kinetic energyof the atoms,87 cf.
Inserts 8.2, and 8.3 andx It has been shown that a gas cannot rotate rigidly, if it conducts heat.88Both results differ from those that are predicted by the Navier-StokesFourier theory, indeed, they are qualitatively and quantitativelydifferent.Thus some extrapolations away from equilibrium, that we have grownfond of, must be revised in the light of extended thermodynamics. Notablythis is true for the principle of local equilibrium and for the Clausius-Duheminequality. Both lose their validity when non-equilibrium becomes severe.The problem with more than 13 moments is, that there is no possibility toprescribe and control higher moments – like u<ijk>, or uijjk, etc. – initially oron the boundary.
Thus we face the situation that we do have specific fieldequations for those moments, but that we are unable to use them for lack ofinitial and boundary values.On the other hand, it can be shown that an arbitrary choice of boundaryvalues of uijjk (say) may affect the temperature field in a drastic – and totallyunacceptable, since unobserved – manner.
Therefore it seems to beinevitable to conclude that a gas itself adjusts the uncontrollable boundaryvalues and the question is which criterion the gas employs. It has beensuggested89 that the boundary values adjust themselves so as to minimizethe entropy production in some norm. Another suggestion is that theuncontrollable boundary values fluctuate with the thermal motion and thatthe gas reacts to their mean values.90In all honesty, however, the problem of assigning data in extendedthermodynamics must still be considered open so far.
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