Darrigol O. Worlds of flow. A history of hydrodynamics from the Bernoullis to Prandtl (794382), страница 49
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1 52-3. For a recent, mathematically advanced study of this problem, cf. Caflisch [1990]. Helmholtz's relevant manuscripts are 'Stabilitiit einer circulierenden Trennungsfliiche auf der Kugel' (HN, in #681),'Wirbelwellen' (HN, in #684), and calculations regarding a vortex sheet in the shape of a logarithmic spiral (HN,in #680).WORLDS OF FLOW162(b)�(d)�(c)(e)Fig. 4.6.The growth of a protuberance on a vortex sheet.component initiates the spiraling motion observed in actual experiments (see Fig.4.6(d,e)). 47Helmholtz described physical circumstances under which the discontinuous motionnecessarily occurred.
According to Bemoulli's theorem, the pressure of a fluid particlediminishes when its velocity increases, by an amount proportional to the variation of itskinetic energy. Therefore, wherever the velocity of the fluid exceeds a certain upper limit,the pressure becomes negative and, according to Helmholtz, the fluid must be 'tom off'.This necessarily happens when the fluid passes a sharp edge, for the velocity of a continuous flow would be infinite at the edge.
For a smoother edge, the discontinuity occurs abovea certain velocity threshold.484.3.2Coriformal mappingIn the final section of his paper, Helmholtz managed to solve exactly a case of twodimensional, discontinuous motion. The complex-variable method he used in this contextwas so influential that a digression on its origins is in order.As we saw in Chapter I, in his memoir of 1749 on the resistance of winds, d'Alembertingeniously noted that for a two-dimensional flow the two conditionsfJu &v- +- = 0ox oyand&vox-ou =0oy(incompressible flow)(4.
12)(irrotational flow)(4. 13)163VORTICESwere equivalent to the condition that (u - iv)(dx + i dy) be an exact differential. In anOpuscule of 176 1 , he introduced the complex potential cp + i!fr such that(u - iv)(dx + i dy) = d(rp + i!fr).(4. 14)The real part cp of this potential is the velocity potential introduced by Euler in 1 752.
Asd'Alembert noted, its imaginary part if! verifiesd!fr = -V dx + U dy,(4.1 5)and is therefore a constant on any line of current. This is why it is now called the 'streamfunction'. 49The first beneficiary of d'Alembert's remarks was Lagrange, in a theory of the construction of geographical maps. In order that infinitesimal figures drawn on a map (x, y)should be similar to their representation on another map (u, v), an infinitesimal displacement on one map must be related to an infinitesimal displacement on the other throughrelations of the formdu = a dx - f3 dy,dv = f3 dx + a dy.(4. 16)In this stipulation, Lagrange recognized d'Alembert's conditions (4.12) and (4.
1 3) for(u - iv)(dx + i dy) to be an exact differential, and thus reduced the problem of conformalmapping to taking the real part of any regular function of the complex variable. This iswhy such functions are now called 'conformal transformations'. 50In general, d'Alembert's remarks indicate that a function (x, y) -> (u, - v) can beexpressed as a regular function of the complex variable if and only if it satisfies theconditions (4. 12) and (4.13).
These conditions are in turn equivalent to the existence of a(holomorphic) complex potential 'P + i.p such that u = ocpjox = oif!Joy andv = ocpjoy = -8.pjox. They are now called the Cauchy-Riemann conditions, becauseAugustin Cauchy and Bernhard Riemann exploited them in their beautiful theories offunctions in the complex plane. Physicists were initially less receptive.
In a mathematicalstudy of incompressible fluids of 1 838, Samuel Earnshaw gave the general integral ofAcp = 0 as cp = f(ax + {3y) + g(ax - {3y) with a2 + {32 = 0, and cp = In r and cp = e asparticular solutions; but he did not introduce the stream function or the complex potential.In 1 842, Stokes introduced the stream function as a way of solving the incompressibilitycondition (4.12), but he did not appeal to complex functionsYHelmholtz was the first hydrodynamicist to take full advantage of d'Alembert's marvelous discovery.
His reading of Riemann's dissertation of 1861 may have alerted him tothe tremendous power of the theory of complex functions. In his lectures on deformable47A more precise argument of the same kind is given in Batchelor [1967] pp.51 1-17.48Helmholtz [1868d] pp.149-50. Negative pressure, or tension, is in fact possible as a metastable condition ofan adequately prepared fluid: cf.
Reynolds [1878] and earlier references therein.49D'Alembert [1752] pp.50Lagrange [17791.51Earnshaw[1838]pp.60--2, [1761] p. 139. See Chapter I, pp. 21-2.207-12; Stokes [1 842] p. 4. Lagrange ([1781] p. 720) also introduced t/J, but without thegeometrical interpretation.WORLDS OF FLOW164media, Amold Sommerfeld reports that, during a vacation in the Swiss Alps, the Berlinmathematician Karl Weierstrass asked Helmholtz to take a look at Riemann's dissertation, for he could not make sense of it. Helmholtz found it very congenial, presumablybecause Riemann's considerations had their roots in the study of physics problems. Also,Hehnholtz's idea of characterizing potential flow through singular surfaces of discontinuity has some similarity with Riemann's program of characterizing analytic functionsthrough their singularities in the complex plane.
52As suggested by d'Alembert, Helmholtz reduced the solution of his problem of twodimensional flow to the search for a complex potential that was a holomorphic function ofthe variable x + iy. This potential (unlike its derivative uiv) satisfies a simple boundarycondition, namely that its imaginary part, the stream function, must be a constant alongthe frontier of an immersed solid and along a line of discontinuity.
It must, of course, besingular on the lines of discontinuity, the form of which is not a priori given. Helmholtzastutely started with a simple analytic form of the inverse function cp + ilj! -+ x + iy, so thatgeometricaily simple boundary conditions could be imposed on the flow. 53Helmholtz first tried the simple form(4. 17)with z = x + iy andw = cp + iifr.This gives x= A(cpe") and y = ±A7T for ifr =that the two parailel, interrupted straight lines defined by x±7T, so:S - A and y = ±A7T can beregarded as wails along which the fluid is constrained to run.
Consequently, eqn(4. 17)expresses the motion of a liquid flowing from an open space into a canal bounded by twothin parallel walls. At the extremities of these walls, for which cp =0, the fluid velocity iseasily seen to diverge. Helmholtz modified the expression (4.17) so that the lines of currentifr =±7T runalong the outer walls of the canal (from the left) and become discontinuitysurfaces after passing the extremity of the walls.
The condition of constant pressure onthese surfaces led him to the not-so-simple expressionJz = A(w + ew) + A )2ew + e2w dw,which gives the flow shown in Fig.(4. 1 8)4.7. As Kirchhoff later remarked, the dead water maybe replaced by air without altering the boundary solutions. With this modification,Helmholtz's flow represents the jet formed by water issuing from a large container througha so-called Borda mouthpiece.
The contraction of the fluid vein is exactly one-half, asCharles de Borda had proved a century earlier by balancing the momentum flux of the jetwith the resultant of the pressures on the wails of the container.54Helmholtz's amazing exploitation of complex numbers quickly attracted the attentionof contemporary physicists. Gustav Kirchhoff and Lord Rayleigh soon derived othercases of discontinuous, two-dimensional motion by Helmholtz's method. Kirchhoffdealt with free fluid jets, such as those emerging from a water nozzle. He also solved the52Sommerfeld [1949] p. 135 (Sommerfeld got the anecdote from Adolf Wiillner).53Helmholtz [1 868d] pp. 1 53-7.54Ibid.; Kirchhoff [1869]; Borda [1766].
On Borda's reasoning, cf. Truesdell [1955] pp. LXXIII-LXXV.VORTICES165--Fig. 4.7.Helmholtz's discontinuity surface (thin line) for the flow (arrows) from an infinite container into apipe (thick lines). From Kirchhoff [1869] p. 423.problem of a plane blade immersed obliquely in a uniform flow. Rayleigh did the same andcalculated the dragging force of the fluid on the blade. Between the lines of discontinuityissuing from the edges of the blade, the fluid remains at rest, so that the pressure on therear side of the plate is smaller than that on the front.
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