Darrigol O. Worlds of flow. A history of hydrodynamics from the Bernoullis to Prandtl (794382), страница 35
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Denoting by w(r)a virtual displacement of the particles of the fluid, and using the notation of the previoussection, the corresponding moment is(3.12)Replacing the sums by integrals, and separating angular variables in the first integrationyieldsJM = - N2 7II\l w d-r,with·(3.13)(3.14)When the fluid is subjected to an internal force density f and to an external pressure P, theequilibrium condition reads:JJ(f w + 7IIN2 \l w) d'J" - w P dS = 0,···(3.15)which has the same form as eqn (1.44) that Lagrange gave for the equilibrium of anincompressible fluid.In conformance with this analogy, Navier took the density N to be nearly constant (hegave it the value one) but made the parameter 'UI vary from one particle of the fluidto another.
This odd assumption (it seems incompatible with the expression for 7II), ofwhich more will be said later, brought him back to the Euler-Lagrange conditions of32Navier [1827] p. 390.116WORLDS OF FLOWf + \J(N2 w) = 0 within the fluid, and ]5 = -N2 w on its surface, so-N2 w plays the role of internal pressure.
In Navier's words, w 'measures theequilibrium, namelythat P =resistance opposed to the pressure that tends to bring the fluid parts closer to each other. mNavier then turned to the case of a fluid moving with a velocity v(r), and he assumedthat 'the repulsive actions of the molecules increased or diminished by a quantity proportional to the velocity with which the distance of the molecules decreased or increased.'Denoting by !f;(R) the proportionality coefficient, this intuition implies a new contributionof the form(3. 1 6)to the moment of the molecular forces. By analogy with the corresponding formula(3.2)for elastic solids, this leads to an additional force �/!;.v in the equation of motion of anincompressible fluid, with(3. 1 7)The new equation of motion readsp[�;](3.18)+ (v \J)v = f - \JP + �/!;.v,·which is now known as the 'Navier-Stokes equation' (for an incompressible fluid).343.2.6Boundary conditionsNavier gave this equation in a memoir read onand published it in summary form in the1 8 March 1 822 at the Academy of SciencesAnnates de chimie et dephysique.
There he assumed,as Girard had, that the velocity v vanished at the wall, in which case the balance of momentsgives no additional boundary condition.3 5 Under this hypothesis, Navier calculated theuniform flow in a pipe of rectangular section and found a discharge proportional to thepressure gradient, as Girard had observed for 'linear motions' (that is, laminar flow).According to the same calculation, the average fluid velocity in a square tube should beproportional to the square of its perimeter (as it is according to Poiseuille's later law forcircular tubes).
Navier (wrongly) believed this result to agree with Girard's observation of adeparture from the expected proportionality to the perimeter (in the case ofcircular tubes). 36At the same time, Navier deplored a contradiction with another of Girard's results,namely, the difference between the discharge in glass and copper tubes. He now faced thefollowing dilemma: either he maintained the boundary condition v=0and thus contradicted Girard's experimental finding, or he gave up this condition and contradicted the most3'Navier [1823c] p.
395. Cf. Saint-Venant [1864b) pp. lxii-lxiv, Dugas [1950) pp. 393-401, Grattan-Guinness[1990) pp. 986-92 (with questionable chronology), Belhoste [1997).34Navier [1823c] p. 414.35However, the tangential stress must vanish at the free surface of the fluid.36Navier [1822) p. 259.1 17VISCOSITYessential assumption of Girard's theory. As he indicated toward the end of his memoir, hepreferred the second alternative. On 16 December 1822, he read a second memoir in which heproposed a new boundary condition based on an evaluation of the moment of the forcesbetween the molecules of the fluid and those of the wall.
The form of this moment isM"where=EIV.wdS,(3.19)E is a molecular constant. This is to be cancelled by the surface termof the momentcondition isM',T J.L(8;vj + Bjv;)w;for any displacementwdSj(3.20)that is parallel to the wall. The resulting(3.21)where81_ is the normal derivative and v11 is the component of the fluid velocity parallel tothe surface.37With this new boundary condition, Navier redid his calculation of uniform square-pipeflow, and also treated the circular pipe by Fourier series. Taking the limit of narrow tubes,he found the average flow velocity to be proportional to the surface coefficientE, to thepressure gradient, and to the diameter of the tube, in rough agreement with Girard's data.Note that he no longer·believed Girard's data to support a quadratic dependence of thevelocity on the diameter.
In fact, Girard's theoretical formula assumed a linear dependence, and his experimental results indicated an even slower increase with diameter. As hehad no reason to distrust Girard's experiments on the differences between glass and coppertubes, Navier built the old idea of fluid-solid slip into the theory of a viscous fluid.383.2.7A useless equationFor large pipes, Navier's theory no longer implies a significant surface-slip effect, but stillmakes the loss of head proportional to the average fluid velocity.
Since Navier knew thatin most practical cases the loss of head was nearly quadratic, he did not bother taking thelarge-section limit of his resistance formulas. He only noted that, in this limit, the flowobviously did not have the (recti)linearity assumed in his calculations. Probably discouraged by this circumstance, he. never returned to his theory of fluid motion. In the hydraulicsection of his course at the Pouts et Chaussees, he only mentioned his formula for capillarytubes, which agreed with 'M. Girard's very curious experiments'.
The theory on which thisformula is based, he immediately noted, 'cannot suit the ordinary cases of application.Since the more complicated motion that the fluid takes in these cases has not beensubmitted to calculation, the results of experience are our only guide.'3937Navier [1823c]. In Cauchy's stress language, the condition means that the tangential stress is parallel andproportional to the sliding velocity.38Navier [1 823c] pp. 432"'40.39/bid.
p. 439; Navier [1838] pp. 88-9.118WORLDS O F FLOWThe two commissioners for Navier's first memoir, Poisson and Joseph Fourier, and thethree for his second memoir, Girard, Fourier, and Charles Dupin, never wrote theirreports, perhaps because Navier was elected to the Academy in1 824,weii before thepublication of his second memoir. However, the mathematician Antoine Cournot wrote areview for Ferussac'sBulletin that may reflect the general impression that Navier's memoirmade at the French Academy. Being Laplace's admirer and Poisson's protege, Coumotwelcomed Navier's theory as a new contribution to the then prosperous molecular physics.Yet he suspected a few inconsistencies in Navier's basic assumptions.40In his derivation of hydrostatic pressure, Cournot noted, Navier assumed incompressibility, which seemed incompatible with the molecular interpretation of pressure as areaction to a closer packing of the molecules.
In fact, according to Navier's formula(3.14)the coefficientdensitywshould be a constant, which excludes a variable pressure if theN is also a constant. Upon closer inspection, Navier's procedure is more coherentthan Coumot believed. Here and elsewhere, Navier's formulas did not quite reflect hisbasic intuition. In his mind, the distance R in the force functionf(R) did not represent thedistance of the molecules in the actual state of the fluid, but their distance beforecompression. For a real substance, which can only be approximately incompressible, thedifference between those two distances is extremely smaii but fmite, so that Navier's/ function could vary with the local state of the fluid.4 1Another worry of Cournot's was that Navier admitted the same equations of equilibrium of a fluid as Euler and Lagrange, and yet obtained different equations of motion,against d'Alembert's principle.
'The matter', Cournot deplored, 'does not seem to be freefrom obscurity.' Today we would solve this apparent paradox by noting that dissipativeforces, such as those expressing fluid viscosity or the viscous friction between two solids,are to be treated, in the application of d'Alembert's principle, as additional, motiondependent forces that are impressed on the system.
At the molecular level, where Navierreasoned, the difficulty is that his calculation seems to rely on velocity-dependent forcesunknown to Laplacian physics. 42Even here Navier's formulas did not directly reflect his intentions. As a close reading ofhis text shows, he meant that the macroscopic motion of the fluid modified the distributionofintermolecular distances: 'If the fluid is moving', he wrote, 'which implies, in general thatthe neighboring molecules come closer to or further from one another, it seems natural toassume that the [intermolecular] repulsions are modified by this circumstance.' This occursin the Laplacian conception of fluids, because the trajectory of an individual moleculeundulates around the path that is imposed overall by the macroscopic motion.
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