THE EQUILIBRIUM OF REVOLVING LIQUID UNDER CAPILLARY FORCE.
[Philosophical Magazine, Vol. xxvm. pp. 161—170, 1914]
THE problem of a mass of homogeneous incompressible fluid revolving with uniform angular velocity (to) and held together by capillary tension (T} is suggested by well-known experiments of Plateau. If there is no rotation, the mass assumes a spherical form. Under the influence of rotation the sphere flattens at the poles, and the oblateness increases with the angular velocity. At higher rotations Plateau's experiments suggest that an annular form may be one of equilibrium. The earlier forms, where the liquid still meets the axis of rotation, have been considered in some detail by Beer*, but little attention seems to have been given to the equilibrium in the form of a ring. A general treatment of this case involves difficulties, but if we assume that the ring is thin, viz. that the diameter of the section is small compared with the diameter of the circular axis, we may prove that the form of the section is approximately circular and investigate the small departures from that figure. It is assumed that in the cases considered the surface is one of revolution about the axis of rotation.
Fig. 1 represents a section by a plane through the axis Oy, 0 being the point where the axis meets the equatorial plane. One of the principal
* Pogg. Ann. Vol. xovi. p, 210 (1855); compare Poincar^'s GapillariU, 1895. R. VI. 17eres of radii r, ?-', impinging directly with relative velocity v, is worthy of consideration. According to ordinary elastic theory the only remaining data of the problem are the densities p, p, and the elasticities. The latter may be taken to be the Young's moduli q, q', and the Poisson's ratios, a, a-', of which the two last are purely numerical. The same may be said of the ratios q'jq, p'/p, and r'jr. So far as dimensional quantities are concerned, any maximum strain e may be regarded as a function of r, v, q, and p. The two last can occur only in the combination q/p, since strain is of no dimensions. Moreover, q/p = a?, where a is a velocity. Regarding e as a function of r, v, and a, we see that v and a can occur only as the ratio vfa, and that r cannot appear at all. The maximum strain then is independent of the linear scale; and if the rupture depends only on the. maximum strain, it is as likely to occur with small spheres as with large ones. The most interesting case occurs when one sphere is very large relatively to the other, as when a grain of sand impinges upon a glass surface. If the velocity of impact be given, the glass is as likely to be broken by a small grain as by a much larger one. It may be remarked that this conclusion would be upset if rupture depends upon the duration of a strain as well as upon its magnitude.