Müller I. A history of thermodynamics. The doctrine of energy and entropy (1185104), страница 57
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8.3. George Gabriel Stokes. His degrees and honoursHesitantly he proposes the no-slip-condition which is now routinelyapplied for laminar flows:The condition which first occurred to me to assume … was, that the filmof fluid immediately in contact with the solid did not move relatively tothe surface of the solid.22Stokes tends to consider this assumption as valid when the mean velocityof the flow is small. He is aware of the difficulties that turbulence mightraise.
But he is blissfully unaware, of course, of the problems that may arisein rarefied gases; these are problems that haunt the present-day researchersconcerned with re-entering space vehicles.2122G.G. Stokes: “On the theories of the internal friction….” loc.cit. p. 312.Ibidem. p. 309.2428 Thermodynamics of Irreversible ProcessesCarl Eckart (1902–1973)However convoluted the 19th century arguments of Fourier, Fick andNavier, and Stokes may have been, their works provided valid equations forthe fluxes of mass, momentum and energy in terms of the basic fields ofthermodynamics, viz.
mass density, velocity and temperature. Yet, they didnot provide a coherent picture of thermodynamics of processes, or nonequilibrium thermodynamics. The first such picture was created by CarlEckart in 1940 in one stroke, or rather in two strokes, the first one concerning viscous, heat-conducting single fluids,23 and the second one concerning mixtures.24 Both papers form the basis of what came to be calledTIP – short for thermodynamics of irreversible processes. Let us reviewthese papers in the shortest possible form:One may say that the objective of non-equilibrium thermodynamics ofviscous, heat-conducting single fluids is the determination of the five fieldsmass density ȡ(x,t), velocity X i(x,t), temperature T(x,t)in all points of the fluid and at all times.For the purpose we need field equations and these are based upon theequations of balance of mechanics and thermodynamics, viz.
the conservation laws of mass and momentum, and the equation of balance of internalenergy, see Chap. 3ȡ ȡȡX i ȡu wX jwx jwtijwx jwq jwx j00tijwX i.wx jThese equations are also known as the continuity equation, Newton’sequation of motion and the first law of thermodynamics respectively.While these are five equations – the proper number for five fields – theyare not field equations for ȡ,Xi, and T. The temperature does not even occurand, instead, the equations contain new quantities23C.
Eckart: “The thermodynamics of irreversible processes I: The simple fluid.” PhysicalReview 58, (1940)24 C. Eckart: “The thermodynamics of irreversible processes II: Fluid mixtures.” PhysicalReview 58, (1940).Carl Eckart (1902–1973)243x stress tij, x heat flux qi, x specific internal energy u.In order to close the system of equations, one must find relations betweentij, qi, and u and the fields ȡ,Xi , T.In TIP such relations are motivated in a heuristic manner from an entropyinequality that is based upon the Gibbs equation of equilibrium thermodynamics, cf. Chap.
3s1T( u pȡ2ȡ ) .s is the specific entropy. u and p are considered to be functions of ȡ and T asprescribed by the caloric and thermal equations of state, just as if the fluidwere in equilibrium. This assumption is known as the principle of localequilibrium.Elimination of u and U between the Gibbs equation and the equationsof balance of mass and energy and some rearrangement lead to theequation25ȡs È qi ØÉ Ùxi Ê T ÚX i 1qi T 1X ( 13 tkk p ) n ,tij2x j TxnT xi Twhich may be interpreted as an equation of balance of entropy.
Thatinterpretation implies thatqiis the entropy flux andijiTȈX iq T 1X is the dissipative source1 i t ij 1 tkk p n32density of entropy.x j TxnT xi TInspection shows that the entropy source is a sum of products ofthermodynamic fluxes and thermodynamic forces, see Table 8.1The dissipative entropy source must be non-negative. Thus results anentropy inequality – with iji = qi /T as entropy flux – which is often calledthe Clausius-Duhem inequality, because it represents Duhem’sextrapolation of Clausius’s second law to non-homogeneous temperaturefields.
Assuming only linear relations between forces and fluxes, TIPensures the validity of the Clausius-Duhem inequality by constitutiverelations – phenomenological equations in the jargon of TIP – of the type25As before, angular brackets characterize symmetric traceless tensors.2448 Thermodynamics of Irreversible ProcessesTable 8.1. Fluxes and forces for a single fluidThermodynamic FluxesThermodynamic Forcesw6wZ Kheat flux qitemperature gradientdeviatoric stress t ¢ ij²deviatoric velocity gradientdynamic pressure ʌ = – 1/3 tii – pdivergence of velocityqit ijSNwTwxiwX i2Șwx jwX nOwx nN t0wX PwZ PwX ¢KwZ L ².Fourier½Ș t0°O°¾°t0°¿Navier - StokesTogether with the thermal and caloric equations of state p = p(ȡ,T) andu = u(ȡ,T) the phenomenological equations form the set of materialproperties characterizing a fluid.
ț is the thermal conductivity, and Ș and Ȝare the shear- and bulk viscosities respectively; all three may be functionsof ȡ and T that must be found experimentally.In this manner TIP incorporates Fourier’s law and the law of NavierStokes into a consistent thermodynamic scheme. Neither Fourier, norNavier, or Stokes had made use of thermodynamic arguments, or of theGibbs equation, nor did they need them.
They proposed their laws on thebasis of plausible assumptions about the phenomena of heat conduction andinternal friction.The equations of state and the phenomenological equations combinedwith the equations of balance of mass, momentum and energy provide a setof field equations from which – given initial and boundary values – thefields ȡ(x,t), X i(x,t), and T(x,t) may be calculated. And the solutions aresatisfactory for nearly all normal cases. Indeed, it is no exaggeration to saythat 99% of all flow problems in single fluids are solved by use of thesefield equations; and that begins with the calculation of pipe flow of a liquidCarl Eckart (1902–1973)245and ends with the calculation of lift and drag on an airliner.26 To be sure,both problems need numerical methods in general.It is true that all this could have been done before Eckart – except for thenumerical solutions, of course.
After all Jaumann and Lohr did have the fullset of equations.27 Eckart’s achievement is that he formulated a consistentand coherent theory with the phenomenological equations as part of it.And Eckart did not stop with single fluids. He applied his scheme tomixtures of fluids as well. In that case he started with the Gibbs equationfor a mixture, see Chap. 5 and identified thermodynamic fluxes and forcesas shown in Table 8.2.Table. 8.2. Fluxes and forces in a mixture of fluidsThermodynamic Fluxesheat flux qiThermodynamic Forcestemperature gradientw6wZ KChemical potential gradientdiffusion fluxes Ji Įw 61 (I D I Q )wZ Kdeviatoric stress t¢ ij²deviatoric velocity gradientdynamic pressure ʌ = – 1/3 tii – pdivergence of velocityreaction rate densities Oachemical affinitieswX PwZ PQ¦D1wX ¢KwZ L ².I D J D C PDObviously diffusion and chemical reactions are taken into account,and there are different chemical reactions a = 1,2,…n.
Vanishing of thechemical affinities implies the law of mass action, see Chap. 5.Phenomenological relations in the case of mixtures are more rich than for asingle fluid; they read26The exceptional 1%, that cannot be treated with the field equations described here, relateto rarefied gases, non-Newtonian fluids, ultra-low and ultra-high temperatures andexceptional cases like that.27 G. Jaumann: “Geschlossenes System ...” loc.
Cit.E. Lohr: “Entropie und geschlossenes Gleichungssystem,’’ loc. cit.8 Thermodynamics of Irreversible ProcessesλaX246Ç b 1l ab Ç α 1 g α γ α a µα l a x iνnXiÇ b 1lb Ç α 1 gα γ α a µα nʌqiJiDV¢KL ²νλ ixiw 1 ( g gQ )w T1Q 1 ¦E 1 LE T Ewxiwxi11w(g gQ )wQ1~LD T ¦E 1 LDE T EwxiwxiwX2K ¢K .wZ L ²LThe entropy inequality is satisfied, if the matricesË1abÌ bÍÌ 1ËL1a ÛÜ and Ì λ ÝÜÍ LαLβ Ûare positive semi - definite,Lαβ ÜÝand the viscosity Ș must be non-negative.We note that the chemical potentials – functions of p, T, and theconcentrations – play a central role in these laws, as they should.
Clearlyboth Fourier’s and Fick’s laws are now made considerable more generalthan either Fourier or Fick had them. They allow for cross effects such thata temperature gradient may create diffusion and a concentration gradientmay create heat conduction. Moreover, the concentration gradient of oneconstituent may cause the diffusion flux of another one. Analogous crosseffects may occur between the reaction rates and the dynamic pressure,although I believe that they have never been observed.Eckart never received much credit for his work, because shortly after hispublications Josef Meixner (1908–1994) published a very similar theory,28and so did Ilya Prigogine (1917–).29 In contrast to Eckart the latterauthors stayed in the field and monopolized the subject, as it were.
Onsomewhat uncertain grounds they added Onsager reciprocity relations fortransport coefficients, see below. As a result it is not uncommon to hear2829J. Meixner: “Zur Thermodynamik der irreversiblen Prozesse in Gasen mit chemischreagierenden, dissoziierenden and anregbaren Komponenten.” [On thermodynamics ofirreversible processes in gases with reacting, dissociating and excitable components]Annalen der Physik (5) 43 (1943) pp. 244-270.J. Meixner: Zeitschrift der physikalischen Chemie B 53 (1943) p. 235.I. Prigogine: “Étude thermodynamique des phénomènes irréversibles.” Desoer, Liège(1947).Carl Eckart (1902–1973)247Eckart’s theory described as Onsager’s theory.
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