﻿Motion of a Sphere in a Viscous Fluid. 539 



separated from the rest o£ the fluid by a number of eddies 

 for which the stream-lines are closed curves around a 

 point of zero velocity. These eddies are in some respects 

 similar to the vortices whose motion is worked out in the 

 irrotational theory, and the dynamical effects must also be 

 similar, as the very remarkable calculation of the resistance 

 to the motion of plates and cylinders made by Karman 

 shows. A close study of the photograph shows, however, 

 that there is a very great difference between the eddies and 

 true " vortices," for in the eddies the central part of the 

 liquid moves more or less as a solid body, the velocity 

 diminishing toward* the centre of the eddy where there 

 is always a point of zero velocity, whereas in a vortex the 

 velocity is inversely proportional to the distance from the 

 centre, becoming infinite at that point. It is hoped by 

 further experiments to trace out the gradual development of 

 the eddies and the way in which they die away. 



We may now proceed to compare the above results with 

 those obtained by solving the equations of motion, and it will 

 be convenient to begin with the results for very low velocities 

 given in tigs. 2 and 3. It was pointed out on p. 535 that 

 these cannot be directly compared with Stokes's result owing 

 to the influence of the outer boundary. The motion of a 

 sphere inside a cylinder has been solved by Ladenburg *, and 

 the solution for a rectangular vessel might be obtained by 

 the method of images developed by Lorentzf . The solutions 

 obtained are, however, so very complicated that the numerical 

 computation of the scream-lines would be exceedingly 

 laborious, and for the same reason it would be impossible to 

 use the solutions as a base of further approximation for 

 higher velocities. 



A very simple solution may, however, be obtained for the 

 case of a sphere moving in the fluid contained in a con- 

 centric sphere, and the solution will apply with sufficient 

 approximation to the case of a sphere moving at the centre 

 of a cubical vessel which is represented in fig. 3. 



The equations of motion with the inertia terms omitted 

 and in the absence of impressed forces may be written : — 



with two similar equations for v and u\ 



If we now introduce the Stokes's stream I'iiih tion \fr defined 



* Laden burjr, loo. cit. 



t Lorentz. Aokantllunffen, i. p. 30; 



