Müller I. A history of thermodynamics. The doctrine of energy and entropy (1185104), страница 56
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Thereforethe geological history of the earth had to be shorter than that.That age was of the same order of magnitude as Helmholtz’s result forthe age of the earth, cf. Insert 2.2. So great was Kelvin’s – and, perhaps,Helmholtz’s – prestige that biologists started to revise their time tables forevolution. Geologists were at a loss, however. Fortunately for them it turnedout in the end that both Kelvin and Helmholtz had made wrong assumptions. Indeed, the earth possesses within itself a source of heat byradioactive decay so that, whatever it loses by conduction is replaced by78Well, that statement must be qualified.
Let us say that the book has the appearance of atextbook on analysis written in the mid 20th century. Really modern books on the subjectmake even interested readers give up in frustration and bewilderment on the first half-page.W. Thomson: “On the secular cooling of the earth.” Transactions of the Royal Society ofEdinburgh (1862).Phenomenological Equations237radioactivity. Thus the earth can maintain its present temperature for aslong as needed to guarantee a geological – and biological – history of somebillions of years. Yet Kelvin, who lived until 1907, would never acceptradioactivity, he stuck to his old prediction till the end.
Asimov says:In the 1880’s Thomson settled down to immobility, … and passed his lastdays bewildered by the new developments.9Adolf Fick (1829–1901)Fick was a competent physiologist who did much to increase ourknowledge about the mechanical and physical processes in the human body.Later in life he became an influential professor in Zürich but at the timewhen he published his paper on diffusion10 he was a prosector, i.e.
theperson who cut open dead bodies up to the point where the anatomyprofessor took over for his demonstrations to a class of medical students.Fig. 8.2. Cut from the title page of Fick’s paperFick was interested in diffusion of solutes in solvents and he adopted amolecular interpretation that sounds very peculiar indeed to modern readers,with regard to physics, grammar and style:11When one assumes that two types of atoms are distributed in empty space,of which some (the ponderable ones) obey Newton’s law of attraction,while the others – the ether atoms – repel each other also in the combinedratio of masses, but proportional to a function f(r) of the distance, whichfalls off more rapidly than the reciprocal value of the second power; whenone assumes further that the ponderable atoms and ether atoms attract eachother with a force, which again is proportional to the product of massesbut also to another function ij(r) of the distance which decreases evenmore rapidly than the previous one, when one – this is what I say –assumes all this, then one sees clearly, that each ponderable atom must besurrounded by a dense ether atmosphere, which if the ponderable atommay be thought of as spherical, will consist of concentric spherical shells,which all have the density of the ether, such that the ether density at some91011I.
Asimov: “Biographies ...” loc. cit. p. 380.A. Fick: “Ueber Diffusion.” [On diffusion] Annalen der Physik 94 (1855) pp. 59–86.Since all this was published, we must assume that it represented acceptable scientificreasoning at the time. And indeed, Navier and Poisson argued similarly when theyderived their versions of the Navier-Stokes equations, see below.2388 Thermodynamics of Irreversible Processespoint at the distance r from the centre of an isolated ponderable atom maybe expressed by f1(r), which must certainly for a large argument assume avalue which equals the density of the general sea of ether.Fick continues like that speculating about the form of the functions f(r), ij(r)and f1(r), and effectively weaving a Gordian knot of words and sentencesuntil – on page 7(!) of his paper – he has the good sense of cutting theargument short with the words:Indeed, one will admit that nothing be more probable than this: Thediffusion of a solute in a solvent … follows the same rule which Fourierhas pronounced for the distribution of heat in a conductor…12This is a relief, because now he comes to what has become known asFick’s law for the diffusion flux Ji :--DJi nXiwn.wxin is the number density of solute particles and Xi is their velocity, if oneassumes that the solvent is at rest.
D is the diffusion coefficient.And again, in analogy to heat conduction, Fick assumes that the rate ofchange of n in a corpuscle is proportional to the balance of influx and effluxand thus obtainswnwtDw 2n.wx 2This is known as the diffusion equation; it is formally identical to theequation of heat conduction, so that Fourier’s solutions can be carried overto boundary and initial value problems of diffusion.In particular, for one-dimensional diffusion of a solute in an infinitesolvent, if n(x,t) is initially a constant no in a small interval X– ǻ/2 < x < X+ ǻ/2and zero everywhere else, the solution reads13n ( x, t )n0 ∆È ( x X )2 Ø.exp É 4 Dt ÙÚ4πDtÊIt follows that a maximum of n(x,t) passes through a given point x at thetime12I have taken the liberty to prosect, as it were, Fick’s hemming and hawing from thissentence.
He remarks that Georg Simon Ohm (1787–1854) has seen the same analogy forelectric conduction.13 The solution refers to the limiting case ǻĺ0 and n ĺ, but so that n ǻ is equal to theoototal number of solvent particles.Phenomenological Equationst max( x X )22DxXhence2392 Dt max ,so that, in a manner of speaking, diffusion proceeds in time as t . This isthe hallmark of all random walk processes and we shall encounter it againin connection with Brownian motion, cf. Chap. 9.
The maximum has theuniversal, i.e. D-independent valuen ( x, t max )no '2Se( x X ) 2.George Gabriel Stokes (1819–1903). Baronet Since 1889At the age of thirty Stokes became Lucasian professor of mathematics atCambridge; in 1854, secretary of the Royal Society; and in 1885, presidentof that institution. No one had held all three offices since Isaac Newton.14Stokes’s mathematical and physical papers fill five volumes with a total ofclose to 2000 pages.15 His main topic was fluid mechanics with an emphasison viscous friction in liquids and gases and his name will always beconnected with the Navier-Stokes equations which relate the viscous stresstensor tij + pįij in a fluid to velocity gradients.
In modern form they read16VKL RG KL2KX ix jOX NG KL .xNTo be sure, Stokes missed out on the second term with the bulk viscosityȜ, but the other term is derived. Ș is now called the shear viscosity butStokes does not seem to have named it. He derived the formula from theprinciple:That the difference between the pressure on a plane in a given directionpassing through any point P of a fluid in motion and the pressure whichwould exist in all directions about P if the fluid in its neighbourhood werein a state of relative equilibrium depends only on the relative motion of thefluid immediately about P; and that the relative motion due to any motion14I.
Asimov: “Biographies...” loc. cit. p. 354.G.G. Stokes: “Mathematical and Physical Papers.” Cambridge at the Universities Press(1880 – 1905).16 Angular brackets denote symmetric, trace-free tensors.152408 Thermodynamics of Irreversible Processesof rotation may be eliminated without affecting the differences of thepressure above-mentioned.17Nowadays we would say concisely that the viscous stress is a linear isotropic function of the velocity gradient.
But no matter, Stokes in his ownway reached a result. After 13 pages of cumbersome, yet reproduciblederivation Stokes came up withStokes:È 2 u 2 u 2 u Ø η È u X w Øp. ηÉ 2 2 2 Ù xyz Ú 3 x ÉÊ x y z ÙÚÊ xThis is the stress contribution to the x-component of the momentumbalance.Nobody at that time used vector and tensor notation, and (u,X,w) were thecanonical letters for the velocity components in x, y, z direction.As it was, Stokes had been anticipated by two scientists across theEnglish Channel: Louis Navier18 (1785–1836) and Siméon Denis Poisson19(1781–1840). Both had employed somewhat irrelevant molecular models –much in the manner of Fick whom I have cited at length – but they didcome up with reasonable expressions, viz.Navier:È 2u 2u 2u Øp AÉ 2 2 2 Ùxyz ÚÊ xPoisson:È 2u 2u 2u Øp È u X w Ø.
AÉ 2 2 2 Ù B Éxx Ê x y z ÙÚyz ÚÊ xThus we conclude that the credit should have gone to Poisson who, afterall, had two coefficients which implies that he allowed for shear and bulkviscosity. However, Poisson is nowadays rarely mentioned in this context.It is true though that Stokes did a lot more than set up the equations; hesolved them in fairly complex situations. He was much interested in themotions of the pendulum and how this was affected by friction.
In 1851 hewrote a long article on the question.20 Section II of that article is entitledSolutions of the equations in the case of a sphere oscillating in a mass offluid either unlimited, or confined by a spherical envelope concentric withthe sphere in its position of equilibrium.1718G.G. Stokes: “On the theories of the internal friction of fluids in motion and of theequilibrium and motion of elastic solids.” Transactions of the Cambridge PhilosophicalSociety.
III (1845) p. 287.L. Navier: Mémoires de l´Académie des Sciences VI (1822) p. 389.S.D. Poisson: Journal de l´´Ecole Polytechnique XIII cahier 20 p. 139.20 G.G Stokes: “On the effect of the internal friction of fluids on the motion of pendulums.”Transactions of the Cambridge Philosophical Society IX (1851) p. 8.19Phenomenological Equations241The result could be specialized to the case of uniform motion of a sphereof radius r with the velocity X. The force to maintain the motion is given byF = 6ʌȘrX ,a formula that is universally called the Stokes law of friction. It is nowderived as an exercise in all good books on fluid mechanics.The solution of boundary value problems for the Navier-Stokes equationrequires more than an able mathematician: A decision about the boundaryvalues of the velocity components near the walls of a pipe or the surface ofa sphere must be made. Stokes says:The most interesting questions connected with this subject require for theirsolution a knowledge of the conditions which must be satisfied at thesurface of a solid in contact with the fluid21Fig.
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