In the case of an elliptic section of semi-axes a, b, we assume This bears to the discharge through a circular pipe of the same sectional area the ratio 2ab/(a2 + b2). For small values of the eccentricity (e) this fraction differs from unity by a quantity of the order e1. Hence considerable variations may exist in the shape of the section without seriously affecting the discharge, provided the sectional area be unaltered. Even when a : b = 8: 7, the discharge is diminished by less than one per cent. 333. We consider next some simple cases of steady rotatory motion. The first is that of two-dimensional rotation about the axis of z, the angular velocity being a function of the distance (r) from this axis. Writing u = wy, v = wx, .(1) we find that the rates of extension along and perpendicular to the radius vector are zero, whilst the rate of shear in the plane xy is rdw/dr. Hence the moment, about the axis, of the tangential forces on a cylindrical surface of radius r, is per unit length of the axis, = μrdw/dr. 2πr. r. On account of the steady motion, the fluid included between two coaxal cylinders is neither gaining nor losing angular momentum, so that the above expression must be independent of r. This gives A .(2) If the fluid extend to infinity, while the internal boundary is that of a solid cylinder of radius a, whose angular velocity is wo, we have *This, with corresponding results for some other forms of section, appears to have been obtained by Boussinesq in 1868; see Hicks, Brit. Ass. Rep. 1882, p. 63. 1 The formulae will apply to the case where the outer cylinder is maintained in rotation whilst the inner one is at rest, if we interchange the meanings of a and b. Experiments on this plan have been made by Mallock †, Couette‡, and others, the couple on the inner cylinder being measured by the torsion of a suspending wire, or by some similar contrivance. The results will be referred to later (Art. 366). Other two-dimensional problems of great technical interest, in which the neglect of the inertia terms is fully justified, are presented by the hydrodynamical theory of Lubrication, initiated by Osborne Reynolds §. 334. A similar solution to that of the preceding Art., restricted however to the case of infinitely small motions, can be obtained for the steady motion. of a fluid surrounding a solid sphere which is made to rotate uniformly about a diameter. Taking the centre as origin, and the axis of rotation as axis of where is a function of the radius vector r, only. If we put (1) (2) .(3) and it appears on substitution in Art. 328 (4) that, provided we neglect the terms of the second order in the velocities, the equations are satisfied by If the fluid extend to infinity and is at rest there, whilst wo is the angular velocity of the rotating sphere (r = a), we have * This problem was first treated, not quite accurately, by Newton, Principia, Lib. I. Prop. 51. The above results were given substantially by Stokes, ll.cc. ante pp. 573, 575. † "Determination of the Viscosity of Water," Proc. Roy. Soc. t. xlv. p. 126 (1888); "Experiments on Fluid Viscosity," Phil. Trans. A, t. clxxxvii. p. 41. "Études sur le frottement des liquides," Ann. de chimie et phys. t. xxi. p. 433 (1890). §l.c. ante p. 571. See also Sommerfeld, Zeitschr. f. Math. t. 1. p. 97 (1904); Harrison, Camb. Trans. t. xxii. p. 39 (1913). If the external boundary be a fixed concentric sphere of radius b the solution is The retarding couple on the sphere may be calculated directly by means of the formulae of Art. 326, or, perhaps more simply, by means of the Dissipation Function of Art. 329. We find without difficulty that the rate of dissipation of energy is If N denote the couple which must be applied to the sphere to maintain the rotation, this expression must be equivalent to Nw, whence The neglect of the terms of the second order in this problem involves a more serious limitation of its practical value than might be expected. It is not difficult to ascertain that the assumption virtually made is that the ratio wa2/v is small. If we put v = 018 (water), and a 10, we find that the equatorial velocity w,a must be small compared with 0018 (c.s.)†. = When the terms of the second order are sensible, no steady motion of the above kind is possible. The sphere then acts like a centrifugal fan, the motion at a distance from the sphere consisting of a flow outwards from the equator and inwards towards the poles, superposed on a motion of rotation. In the case to which the formulae (8) and (10) relate the condition for the validity of the approximation is that the expression should be small, it being assumed that a and b are not very different §. *Kirchhoff, Mechanik, c. xxvi. † Cf. Rayleigh, "On the Flow of Viscous Liquids, especially in two Dimensions," Phil. Mag. (4), t. xxxvi. p. 354 (1893) [Papers, t. iv. p. 78]. Stokes, l.c. ante p. 573. § Experiments on the viscosity of air have been made by Zemplén (Ann. der Phys. (4), t. xxix. p. 869 (1909) and t. xxxviii. p. 71 (1912)) on this plan, except that the outer sphere was made to rotate, the couple N being measured by the torsion of a wire from which the inner sphere was suspended. He finds that the formula analogous to (10) gives consistent results for a wide range of wav, and remarks that criteria of this kind are to be taken as indicating an order of magnitude, rather than an absolute standard. This must be admitted; but it should be remarked that the relevant criterion in the present case has rather the form (12). 335. The motion of a viscous incompressible fluid, when the effects of inertia are insensible, can be treated in a very general manner, in terms of spherical harmonic functions. It will be convenient, in the first place, to investigate the general solution of the following system of equations: The functions u', v', w' may be expanded in series of solid harmonics, and it is plain that the terms of algebraical degree n in these expansions, say un', vn', wn', must separately satisfy (2). The equations (1) may therefore be put in the forms where Xn is some function of x, y, z; and it further appears from these relations that V2Xn = 0, so that Xn is a solid harmonic of degree n. with two similar equations. Now it follows from (1) and (2) that where n+1 is a solid harmonic of degree n + 1. Hence (5) may be written The factor n + 1 may be dropped without loss of generality; and we obtain as the solution of the proposed system of equations: 336. If we neglect the inertia-terms, the equations of motion of a viscous liquid reduce, in the absence of extraneous forces, to the forms so that p can be expanded in a series of solid harmonics, thus The terms of the solution which involve harmonics of different algebraical degrees will be independent. To obtain the terms in P, we assume u = Ar2 + Br2n+3 n др дх a Pn dx r2n+1' where r2 Ww= Ar2 + Br2n+3 x2 + y2+ z2. The terms multiplied by B are solid harmonics of degree n+1, by Arts. 81, 83. Now * Cf. Borchardt, "Untersuchungen über die Elasticität fester Körper unter Berücksichtigung der Wärme," Berl. Monatsber. Jan. 9, 1873 [Gesammelte Werke, Berlin, 1888, p. 245]. The investigation in the text is from a paper "On the Oscillations of a Viscous Spheroid," Proc. Lond. Math. Soc. t. xiii. p. 51 (1881). |