by taking account of viscosity and of the resistance to slipping over the surface of the lamina. The instability of the surfaces of discontinuity, which remains even when these are modified by viscosity, may be held to account for the turbulent motion of the fluid in the wake; see Arts. 234, 345. The calculations of Rayleigh (Art. 77) give for the mean excess of per unit area on the front of the lamina the expression pressure where U is the velocity of translation, and a the angle which the plane makes with the direction of motion. The proportionality to the square of the velocity is confirmed more or less by experiment, but the numerical factor, and in particular its variation with a, is less exactly verified. It is found indeed that there is not only an excess of pressure on the front face, but a defect of pressure, or 'suction,' on the rear, both circumstances contributing of course to the total resistance. When a does not differ much from, the lamina moving nearly broadside-on, the excess and defect appear to be nearly uniform over the breadth, changing rapidly near the edges*. According to Rayleigh's formulae, on the other hand, the pressure on the front face varies much more gradually from the centre to the edges †. The double trail of vortices with opposite rotations which follow in the wake of a lamina, or a cylinder, has been depicted, sometimes very effectively with the help of photography, by various observers. It is often remarked that vortices appear to detach themselves from the solid on the two sides alternately, the result being similar to the unsymmetrical arrangement discussed by Karmán (Art. 156) except that the vortices are not concentrated, as was there assumed for simplicity. In the paper referred to, Karmán has further calculated the resistance which the continual creation of vortices. implies, on the assumption that the system has the configuration necessary for stability. The relative rate at which the vortex-system recedes from the solid has to be taken from observation; when this is done, the result is stated to be in good agreement with experiment. * Stanton, "On the Resistance of Plane Surfaces in a Uniform Current of Air," Proc. Inst. Civ. Eng. t. clvi. p. 78 (1904). For an account of the whole subject from the experimental side reference may be made to Eiffel, La résistance de l'air; examen des formules et des expériences, Paris, 1910. † If in Art. 76 (5) we put t= cosec x, we find 21 q=tan 1x, from which the value of 1-qa as a function of x can be plotted. See for example Ahlborn, Ueber den Mechanismus des hydrodynamischen Widerstandes, Hamburg, 1902; Mallock, Proc. R. S. t. lxxix. p. 262 (1907), and t. lxxxiv. p. 490 (1910); Karmán (and Rubach), l.c. 371. An explanation of the fact that a body may be supported against gravity, if endowed with a suitable horizontal velocity, has been put forward from a somewhat different standpoint*. The theory is based on the result of Art. 69, where it was shewn that a circular cylinder will describe a trochoidal path, the motion being mainly horizontal, if the surrounding fluid is frictionless, and its motion irrotational, provided there is a circulation (k), in the proper sense, about it. In particular the path may be a horizontal straight line, the lifting force (which is to counteract gravity) being then Y = kρU per unit length, where U is the horizontal velocity. This result is easily extended to any form of section. If (u, v) be the fluid velocity, vanishing at infinity, the formula for the pressure is since the motion relative to the body is steady. Hence if l, m be the direction-cosines of the outward normal to an element ds of the contour of the cross-section, the resultant pressure on the solid parallel to x is We have here omitted two line-integrals taken round an infinite enclosing contour; these vanish since the velocity at infinity is of the order 1/r, where r denotes distance from the origin. In the same way we find The case of an elliptic cylinder, which includes as an extreme form that of a plane lamina, may be examined on the basis of the formula given at the end of Art. 72. With * Lanchester, Aerodynamics, London, 1907, Art. 122: "Peripteroid Motion." † Another proof is given by Kutta, "Ueber eine mit den Grundlagen des Flugproblems in Beziehung stehende zweidimensionale Strömung," Sitzungsber. d. k. bayerischen Akad. d. Wiss. 1910. The theorem was given in an unpublished dissertation of date 1902. A prior publication is attributed to Joukowsky (1906). the notation there adopted, the fluid pressures on an elliptic cylinder of semi-axes a, b reduce (when w= = 0) to a force Kutta (l.c.) has treated the case of a lamina whose section is an arc of a circle. He assumes the circulation to be so adjusted in relation to the velocity of translation that the infinite value of the fluid velocity which would otherwise occur at the following edge is avoided, whilst an infinity remains of course at the leading edge. It is supposed that in this way an approximation to actual conditions is obtained, the 'circulation' representing the effect of the vortices which are produced behind the lamina in real fluids; and a good agreement with experiment is claimed. 372. As regards the total resistance to the translation (through liquids) of similar bodies, of any shape, in corresponding directions, we are led by consideration of dimensions to a formula of the type where a is any length defining the scale of the body (e.g. the radius, in the case of a sphere). The approximate proportionality to U2 which is found in many cases indicates that the function ƒ is nearly constant, and that the resistance is accordingly almost independent of the viscosity. As in former cases this does not mean that viscosity is without influence; it plays its part, along with the resistance to slipping over the surface of the solid, in bringing about the régime which is finally established. The general character of the motion relative to the body does not appear to be very different from that which holds in the case of the plane disk. In the case of a cylinder, for instance, the central stream-line divides where it meets the surface in front, and then follows the surface for some distance on each side, the motion of the fluid on either hand being fairly smooth and regular. At a certain stage, however, the stream-line in question appears to leave the surface, and can no longer be definitely traced, the space between its apparent continuation and the cylinder being filled with eddies. An able attempt to trace this phenomenon mathematically has been made by Prandtl*. The region in front of the solid is regarded as made up of two portions, viz. (i) a thin stratum in contact with the solid, with a rapid variation of relative (tangential) velocity in the direction of the thickness, and (ii) an outer region in which the motion is taken to be irrotational, being practically unaffected by viscosity. Approximate solutions of the equations of motion are sought, appropriate to these two regions, and continuous with one another at the common boundary. The calculations are necessarily * "Ueber Flüssigkeitsbewegung bei sehr kleiner Reibung," Verh. d. III. Internat. Math.Kongresses, Heidelberg, 1904. See also Blasius, Grenzschichten in Flüssigkeiten mit kleiner Reibung, Berlin, 1907. elaborate, but the results obtained, and represented graphically, are interesting. In the case of compressible fluids the formula (1) requires modification. If denote the elasticity, the method of dimensions leads easily to the assumption If U is small compared with the velocity of sound in the gas, viz. √(k/p), this approximates to the form which is equivalent to (1). F =pU2a2f (va› 0), (3) The law of resistance varying as the square of the velocity is found to hold fairly well in the case of a projectile moving through air up to velocities of about 800 ft. per sec. When the velocity approaches or exceeds that of sound, the law changes, as we should expect. We have then a wave-making resistance, analogous to that discussed in Art. 249, in addition to the frictional type. It is hardly necessary to do more than refer to the striking experiments of Mach and Boys* in this connection. Experiments on the resistance to plane surfaces moving lengthways have been made by W. Froude†, Zahm‡, Lanchester §, and others. In Zahm's experiments the fluid was air, and the resistance was found to vary as U2-", where n = 15. The formula for the mean resistance per unit area would accordingly be R = CpU2 (va)". (4) The coefficient C is not however an absolute constant, but varies somewhat with the length of the plane, for a reason already indicated (p. 655). Thus in the case of a board moving at 10 ft. per sec. the average friction in lbs. per sq. ft. fell from 000524 when the length was 2 ft., to 000457 when it was 16 ft. The coefficient is therefore to be regarded as a function of the length, or rather of the ratio of the length to the breadth. A comparison with Froude's results for water verifies the proportionality of the resistance to density. The bearing of these results on the 'skin-resistance' of ships is obvious. * Nature, t. xlvii. p. 440 (1893). †l.c. ante p. 655. $ "On the atmospheric friction on even surfaces," Phil. Mag. (6), t. viii. p. 58 (1904). § Aerodynamics, Art. 247. Stanton, Rep. of Advisory Comm. for Aeronautics, 1909-10, p. 25. CHAPTER XII ROTATING MASSES OF LIQUID 373. THIS subject had its origin in the investigations on the theory of the Earth's Figure which began with Newton and Maclaurin, and were continued by the great French school of mathematicians which flourished near the end of the eighteenth and the beginning of the nineteenth century. It has in recent times undergone great development, at the hands, notably, of Thomson and Tait, Poincaré, and Darwin. The problem is to ascertain the possible forms of relative equilibrium of a homogeneous gravitating mass of liquid, when rotating about a fixed axis with constant angular velocity, and to determine the stability or instability of such forms. We begin with the case where the external boundary is ellipsoidal. We write down, in the first place, some formulae relating to the attraction of ellipsoids. If the density p be expressed in 'astronomical' measure, the gravitationpotential, at internal points, of a uniform mass enclosed by the surface *For references see p. 593. The sign of has been changed from the usual reckoning. |