Darrigol O. Worlds of flow. A history of hydrodynamics from the Bernoullis to Prandtl (794382), страница 8
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In a contemporaryletter to Euler he expressed his exasperation over d'Alembert's treatise:32I have seen with astonishment that apart from a few little things there is nothing to beseen in his hydrodynamics but an impertinent conceit. His criticisms are puerile indeed,and show not only that be is no remarkable man, but also that he never will be.29D'Alembert [1 744] pp. viii-ix.30Ibid. pp. 19-20.31Ibid. p. 139.32D. Bernoulli to Euler, 7 July 1745, quoted in Truesdell [1954] p. XXXVIIn.
Truesdell approves([1954] p. XXXVII): 'D'Alembert's method makes no contribution and has had no permanent influence in fluidmechanics.'WORLDS OF FLOW161 .3.3 The cause of windsIn this judgment, Daniel Bemoulli overlooked the fact that d'Alembert's hydrodynamics,being based on a general dynamics of connected systems, lent itself to generalizationsbeyond parallel-slice flow.
In a prize-winning memoir of1 746 on the cause of winds,d'Alembert offered striking illustrations of the power of this approach. As thermal effectswere beyond the grasp of contemporary mathematical physics, he focused on a cause thatis now known to be negligible: the tidal force exerted by the Moon and the Sun. Forsimplicity, he confined his analysis to the case of a constant-density layer of air covering aspherical globe with uniform thickness. He further assumed that fluid particles originallyon the same vertical line remained so in the course of time (owing to the thinness of the airlayer) and that the vertical acceleration of these particles was a negligible fraction ofgravity, and he neglected second-order quantities with respect to the fluid velocity and tothe elevation of the free surface.
His strategy was to apply his principle of dynamics to themotion induced by the tidal forcef and the force of gravity g (for unit density), bothofwhich depend on the location on the surface of the Earth.33Calling'Ythe absolute acceleration of the fluid particles, the principle requires thatthe fluid layer should be in equilibrium under the forcef + g - 'Y ·From earlier theorieson the shape of the Earth (regarded as a rotating liquid spheroid), d'Alembert borrowedthe equilibrium condition that the net force should be perpendicular to the free surface ofthe fluid.
He also required that the volume of vertical cylinders of fluid should not bealtered by their motion, in conformance with his constant-density model. As the modemreader would expect, from these two conditions d'Alembert derived some sort of momentum equation, and some sort of continuity equation. But he did it in a rather opaquemanner. Some features, such as the lack of specific notation for partial differentials or theabundant recourse to geometrical reasoning, disconcert modern readers only.34 Otherswere problematic to his contemporaries: he often omitted steps and introduced specialassumptions without warning.
Also, he directly treated the utterly difficult problem offluid motion on a spherical surface without preparing the reader with simpler problems.Suppose, with d'Alembert, that the tide-inducing luminary orbits above the equator(with respect to the Earth).35 Using the modem terminology for spherical coordinates,denote bye the colatitude of a given point of the terrestrial sphere with respect to an axispointing toward the orbiting luminary (this is the geographical longitude), </> the longitudemeasured from the meridian above which the luminary is orbiting (this isnot the geographical longitude), TJ the elevation of the free surface of the fluid layer over its equilibriumposition, ve and v.p the 6- and </>-components of the fluid velocity with respect to the Earth,the depth of the fluid in its undisturbed state, and R the radius of the Earth (see Fig.h1 .
7).33D'Alembert [1747]. D'Alembert treated the rotation of the Earth, the Sun's attraction, and the Moon'sattraction as small perturbing causes whose effects on the shape of the fluid surface simply added (ibid. pp. xvii,47). Consequently, he overlooked the Coriolis force in his analysis ofthe tidal effects (ibid. p. 65, he announces thathe will be reasoning as if it were the luminary that rotates around the Earth).34D'Alembert used a purely geometrical method to study the free oscillations of an ellipsoidal disturbance ofthe air layer.35The Sun and the Moon actually do not, but the variable part of their action is proportional to that of such aluminary.THE DYNAMICAL EQUATIONSFig.1.7.17Spherical coordinates for d 'Alembert's atmospheric tides.
The fat line represents the visible part ofthe equator, over which the luminary is orbiting. N is the North pole.D'Alembert first considered the simpler case </>>=:J0, for which he expected the component vq, to be negligible. To first order in 7J and v, the conservation of the volume of avertical column of fluid yields( 1 . 18)which means that an increase in the height of the column is compensated for by anarrowing of its base (the dot denotes the time derivative for a fixed point on the Earth's·surface).
Since the tidal force f is much smaller than the gravity force, the vector sumf + g - 'Y makes an angle (fe - y8)jg with the vertical. To first order in 7J, the inclinationof the fluid surface over the horizontal is 877/R8(). Therefore, the condition that f + g - 'Yshould be perpendicular to the surface of the fluid is approximately equivalent to3 6( 1 . 1 9)As d'Alembert noted, this equation of motion can also be obtained by equating thehorizontal acceleration of a fluid slice to the sum of the tidal component fe and ofthe difference between the pressures on both sides of this slice. Indeed, the neglect of the36D'Aiembert [1747] pp.
88-9 (formulas A and B). The correspondence with d'Alembert's notation is given bye - u, v, -q, dTJ/dB - -v, R/hw - s, w/Rg - b2/2a, R/gK - 3S/4pd3 (with/ = -K sin 28).WORLDS OF FLOW18vertical acceleration implies that, at a given height, the internal pressure of the fluid variesas the product g7). Hence d'Alembert was aware of two routes to the equation of motion,namely, through his dynamic principle, and through an application of the momentum lawto a fluid element subjected to the pressure of contiguous elements.
In some sections he37favored the first route, in others the second.In his expression of the time variations ij and v9, d' Alembert considered only the forcedmotion of the fluid for which the velocity field and the free surface of the fluid rotatetogether with the tide-inducing luminary at the angular velocity -w. Then the values oft71and v9 at the colatitude () and at the time t + d are equal to their values at the colatitude() + wdt and at the time t.
This gives871.&uo , .vo = w) =w() .ae 78(1.20)D' Alembert equated the relative acceleration v9 with the acceleration y9, for he neglectedthe second-order convective terms, and judged the absolute rotation of the Earth irrelevant (he was aware of the centripetal acceleration, but treated the resulting permanentdeformation of the fluid surface separately; and he overlooked the Coriolis acceleration).With these substitutions, his equations(1 .
1 8)andbecome ordinary differential(1.19)equations with respect to the variable ().D'Alembert eliminated 7) from these two equations, and integrated the resulting differential equation for Newton's value -K sin 2() of the tide-inducing forcef9• In particular, heshowed that the phase of the tides (concordance or opposition) depended on whether therotation period 27T/w of the luminary was smaller or larger than the quantity 27TR/ .fili.,which he had earlier shown to be identical to the period of the free oscillations of the fluid38layer.In another section of his memoir, d'Alembert extended his equations to the case whenthe angle </> is no longer negligible.
Again, he had the velocity field and the free surface ofthe fluid rotate together with the luminary at the angular velocity -w. CallingRwdttheoperator for the rotation of angle wdt around the polar axis, and v(P, t) the velocity vectorat pointP and at time t, we havev(P, t + dt) = Rwdtv(RwdtP,t) .(1.21)Expressing this relation in spherical coordinates, d'Alembert obtained. (&ueV9 = WA.[jj} COS '+'-&uo8</>sin </>tan ().- V<f> Sm '+' SillA.•e),(1 .22)37D'Alembert [1747] pp. 88-9.
He represented the internal pressure by the weight of a vertical column of fluid.hls discussion of the condition of equilibrium (ibid. pp. 15-16), he introduced the balance of the horizontalcomponent of the external force acting on a fluid element and the difference in the weight of the two adjacentcolumns as 'another very easy method' for determining the equilibrium. In the case of tidal motion with </> "' 0, hedirectly applied this condition of equilibrium to the 'destroyed motion' f + g - 'Y· In the general case (ibid.
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