Müller I. A history of thermodynamics. The doctrine of energy and entropy (1185104), страница 59
Текст из файла (страница 59)
Müller: “A new approach to thermodynamics of simple mixtures.” Zeitschrift fürNaturforschung 28a (1973).43 J. Meixner: Annalen der Physik (1943) loc.cit.44 Also known as the principle of material objectivity.45 H. Giesekus: “Die rheologische Zustandsgleichung.” [The rheological equation of state]Rheologica Acta 1 (1958) pp. 2–20.42Rational Thermodynamics251context of non-Newtonian fluids and was formalized and extrapolated tocontinuum mechanics in general by Walter Noll (1925– ) in 1958.46 Theprinciple refers to Euclidean transformations, i.e. time-dependent rotationsand translations between frames such that, if xi and xi* are the coordinates ofa volume element in the frames S and S*, we havexi* = Oij(t) xj + bi(t) ļ xi = Oji (t) (xj*– bj(t)) .The orthogonal matrix O(t) and the vector b(t) may be arbitrarily timedependent and, if they are, at least one of the two frames is a non-inertialframe; in order to fix the ideas we take S as an inertial frame.
The principleof material frame indifference states that the constitutive functions must notdepend on the frame in which a body, or a volume element of a body, is atrest. This impliesx that only Euclidean vectors and tensors may occur as variables, andx that the constitutive functions are isotropic functions.The validity of hypotheses and postulates in continuum mechanics andthermodynamics – or at least their applicability to gases – can be checkedby the kinetic theory of gases. And when such a check was made,47 it turnedout that the principle of material frame indifference was wrong, cf. Insert8.1. To be sure, it was not very wrong, because the frame dependence is dueto the curvature imparted to the mean free paths of the atoms by the Coriolisforce.
Therefore, in order to see an effect, one would have to use a veryrapidly rotating frame indeed. In this sense the argument even confirmsframe indifference as a practical tool and reconciles it with the idea –prevailing in non-relativistic physics – that the only true invariance ofphysical laws is Galilei invariance.48But this was not the way, the protagonists of rational thermodynamicssaw the matter. There were no approximate principles for them. Somewere prepared to give up the kinetic theory in order to save material frameindifference.
Noll suggested that the whole universe be turned tomaintain the principle; in the meantime he changed the wording of theprinciple, thus excluding the influence of external forces which – in his46W. Noll: “A mathematical theory of the mechanical behaviour of continuous media.”Archive for Rational Mechanics and Analysis 2 (1958).47 I. Müller: “On the frame-dependence of stress and heat flux.” Archive for RationalMechanics and Analysis 45 (1972).48 Galilei transformations form a subgroup of Euclidean ones, where O is time-independentand b is a linear function of time. There are no inertial forces like the Coriolis force inGalilean frames.2528 Thermodynamics of Irreversible Processesunderstanding – include the inertial forces like the Coriolis force.49 This is asomewhat strange idea, because frame indifference can only be violated bythe effect of inertial forces; there is no other way!50 Truesdell,51 referring tothe argument of Insert 8.1, wondered caustically why the physics of ahollow cylinder should be different from the physics of a full cylinder52 andafterwards ignored the objections.
The subject was thus so successfullyobfuscated that the discussion of material frame indifference never endedand is still going on in the years when I write this. However, nothing is saidnow that has not been said before.Frame dependence of the heat fluxWe consider a gas at rest between two concentric cylinders and focus the attentionon a small volume element of the dimension of the mean free path of the atoms.There is a radial temperature gradient, see Fig. 8.4. The atoms at the bottom of theelement have a greater mean kinetic energy than those on top, because thetemperature is bigger. Therefore the atoms moving upwards through the plane H-Hcarry more energy through that plane than the downward moving atoms. Thus thereis a net energy flux, a heat flux, in the upward-direction, opposite to thetemperature gradient, just as predicted by Fourier’s law.
This is true, if the gas is atrest in an inertial frame. But then we take the cylinders and the gas and put them ona carousel with the axis of rotation coinciding with the axes of the cylinders. Thenthe paths of the atoms are curved by the Coriolis force so that there is a heat fluxthrough the plane V-V as well as through the plane H-H, see figure. Therefore inthe non-inertial frame of the carousel the heat flux has a component perpendicularto the temperature gradient and the size of that component is proportional to theangular velocity of the frame. The relation between the heat flux and thetemperature gradient is therefore frame-dependent.49W.
Noll: “A new mathematical theory of simple materials.” Archive for RationalMechanics and Analysis 48 (1972).50 Logically the new principle of material frame indifference is at a par with Henry Ford’swell-publicized advertisement of the customer service of his company: The Model T maybe had in all colours as long as they are black.51 C.
Truesdell: “Correction of two errors in the kinetic theory that have been used to castunfounded doubt upon the principle of material frame indifference.” Meccanica 11 (1976).One of the “errors” in Truesdell’s opinion was supposed to occur in Müller’s argument, cf.Insert 8.1. The other one was suspected by Truesdell to be contained in a paper by D.G.B.Edelen, T.A. McLennan: “Material Indifference: A Principle or a Convenience.”International Journal of Engineering Science 11 (1973).52 The internal cylinder in the argument is needed for setting up a temperature gradient. In afull cylinder a radially symmetric, non-homogeneous temperature field cannot exist.Rational Thermodynamics253Fig.
8.4. On the frame dependence of the heat fluxA similar argument can be made for the relation between the stress and the velocitygradient. The kinetic theory of gases provides concrete equations for the suggestiveargument presented in this insert.Insert 8.1More damage was suffered by rational thermodynamics when it wasfound that the theory could not be applied to non-Newtonian fluids.
Theearly authors in the field were Bernard David Coleman (1930–) andWalter Noll, whose background was continuum mechanics and, inparticular, continuum mechanics of visco-elastic solids and fluids. 53Therefore from the outset rational thermodynamics has put a strongemphasis on constitutive functionals, by which the stress (say) depends onthe history of the velocity gradient. This is fine as far as it goes. But forpractical flow problems it has seemed appropriate to approximate thefunctional of the history by a function of a few time derivatives of thevelocity gradient, say n of them.
In this way one arrives at the theory of nthgrade fluids whose stationary version was widely used to calculate solutionsfor viscometric flows.54 However, then it turned out that rationalthermodynamics predicts a maximum of free energy for a 2nd grade fluid in5354B.D. Coleman: “Thermodynamics of materials with memory.” Archive for RationalMechanics and Analysis 17 (1964).B.D. Coleman, W. Noll: “An approximation theorem for functionals, with applications incontinuum mechanics” Archive for Rational Mechanics and Analysis 6 (1960).E.g.
see C. Truesdell: “The elements of continuum mechanics” Springer, New York(1966).Also: B.D. Coleman, H. Markovitz, W. Noll: “Viscometric flows of non-Newtonianfluids.” Springer Tract in Natural Philosophy 5 (1966).2548 Thermodynamics of Irreversible Processesequilibrium55 instead of the minimum necessary for stability. Later it wasshown that no nth grade fluid with n > 1 has stable solutions.56 After that,there was serious doubt that rational thermodynamics could be used fornon-linear problems, and now it is accepted by most people that it cannot beso used.In some ways this is a pity, because rational thermodynamics did employsome elegant and rational (sic!) arguments for the exploitation of theentropy inequality. These arguments are not lost, however, because they canbe transferred to extended thermodynamics which we proceed to considernow, – after this:Truesdell’s outspoken partisanship of rational thermodynamics and hisflamboyant style fuelled some lively controversies between adherents ofTIP and the protagonists of rational thermodynamics, chiefly Truesdellhimself.
His attacks on Onsagerism were advanced with much satiricalverve, that makes them fun to read for those who were not targeted.However, the defenders of TIP tried their best to pay Truesdell back in hisown coin. Woods pointed out some awkward features of rationalthermodynamics in a paper entitled “The bogus axioms of continuummechanics.”57 And Ronald Samuel Rivlin (1915–2005) delighted aworldwide audience with a frequently repeated humorous lecture under thetitle “On red herrings and other sundry unidentified fish in moderncontinuum mechanics.”Truesdell was a consummate theoretician.He showed nothing but disdain for experiments,be they conducted in the laboratory or on thecomputer.So, when Truesdell visited me in Berlin, on thefirst day he came and said: Ingo, can I ask youfor a favour? ’’ And I, eager to please my visitorand one-time mentor, said: Of course, Clifford,what can I do for you ?’’,Truesdell: Please, don t show me your lab.’’,,,,,,Fig.
8.5. Clifford Ambrose Truesdell III555657J.E. Dunn, R.L. Fosdick: “Thermodynamics, stability, and boundedness of fluids ofcomplexity 2 and fluids of second grade.” Archive for Rational Mechanics and Analysis56 (1974).D.D. Joseph: “Instability of the rest state of fluids of arbitrary grade greater than one.”Archive for Rational Mechanics and Analysis. 75 (1981).L.C.
Woods: Bulletin of Mathematics and its Applications 1 (1981).Extended Thermodynamics255Extended ThermodynamicsFormal StructureThe objective of extended thermodynamics is the determination of n fields,synthetically denoted by uĮ(x,t) (Į = 1,2,…n). Invariably the first five ofthese fields are the densities of mass, momentum and energy. But inextended thermodynamics the number of fields is extended (sic) and it maycontain the stress, the heat flux and more, see below.We need n field equations and these are based upon n equations ofbalanceCwW D w(DwVwZ C3D(D1,2,...P ) .The fields uĮ may therefore be called densities, the FĮa are called thecorresponding fluxes and the ȆĮ are called productions. The first fiveproductions are zero in accord with the conservation of mass, momentum,and energy.
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
