30 ON THE MOTION OF SOLID BODIES THROUGH VISCOUS LIQUID [354
of equilibrium, satisfying certain differential equations. If the solid be incompressible, the otherwise arbitrary boundary displacements must be chosen subject to this condition. The same conclusion applies in two dimensions, where the bounding surfaces reduce to cylinders with parallel generating lines. For our present purpose we may suppose that at the outer surface the displacements are zero.
The contrast between the three-dimensional and two-dimensional cases arises when the outer surface is made to pass off to infinity. In the former case, where the inner surface is supposed to be limited in all directions, the displacements there imposed diminish, on receding from it, in such a manner that when the outer surface is removed to a sufficient distance no further sensible change occurs. In the two-dimensional case the inner surface extends to infinity, and the displacement affects sensibly points however distant, provided the outer surface be still further and sufficiently removed.
The nature of the distinction may be illustrated by a simple example relating to the conduction of heat through a uniform medium. If the temperature v be unity on the surface of the sphere r = a, and vanish when r = b, the steady state is expressed by
When 6 is made infinite, v assumes the limiting form a/r. In the corresponding problem for coaxal cylinders of radii a and b we have
Iog6— logo. ..............................
But here there is no limiting form when b is made infinite. However great r may be, v is small when b exceeds r by only a little ; but when 6 is great enough v may acquire any value up to unity. And since the distinction depends upon what occurs at infinity, it may evidently be extended on the one side to oval surfaces of any shape, and on the ocher to cylinders with any form of cross-section.
In the analogy already referred to there is correspondence between the displacements (a, /3, 7) in the first case and the velocities (u, v, w) which express the motion of the viscous liquid in the second. There is also another analogy which is sometimes useful when the motion of the viscous liquid takes place in two dimensions. The stream-function (ty) for this motion satisfies the same differential equation as does the transverse displacement (w'} of a plane elastic plate. And a surface on which the fiuid remains at rest (i/r = 0, d^/dn = 0) corresponds to a curve along which the elastic plate is clamped.
In the light of these analogies we may conclude that, provided the square of the motion is neglected absolutely, there exists always a unique steady