354.
ON THE MOTION OF SOLID BODIES THROUGH VISCOUS LIQUID.
[Philosophical Magazine, Vol. xxi. pp. 697—711, 1911.]
§ 1. THE problem of the uniform and infinitely slow motion of a sphere, or cylinder, through an unlimited mass of incompressible viscous liquid otherwise at rest was fully treated by Stokes in his celebrated memoir on Pendulums*. The two cases mentioned stand in sharp contrast. In the first a relative steady motion of the fluid is easily determined, satisfying all the conditions both at the surface of the sphere and at infinity; and the force required to propel the sphere is found to be finite, being given by the formula (126)
-F=G7rpaV, ..............................(1)
where /^ is the viscosity, a the radius, and V the velocity of the sphere. On the other hand in the case of the cylinder, moving transversely, no such steady motion is possible. If we suppose the cylinder originally at rest to be started and afterwards maintained in uniform motion, finite effects are propagated to ever greater and greater distances, and the motion of the fluid approaches no limit. Stokes shows that more and more of the fluid tends to accompany the travelling cylinder, which thus experiences a continually decreasing resistance.
| 2. In attempting to go further, one of the first questions to suggest itself is whether similar conclusions are applicable to bodies of other forms. The consideration of this subject is often facilitated by use of the well-known analogy between the motion of a viscous fluid, when the square of the motion is neglected, and the displacements of an elastic solid. Suppose that in the latter case the solid is bounded by two closed surfaces, one of which completely envelopes the other. Whatever displacements (a, ft, <y) be imposed at these two surfaces, there must be a corresponding configuration
. * Canib. Phil. Trans. Vol. ix. 1850; Math, and Phys. Papers, Vol. m. p. 1 presumably the combustion is most intense. It may be protected with celluloid, or other, varnish.