174 



Mr. A. B. Basset. On the 



[Nov. 24, 



IV. " On the Motion of a Sphere in a Viscous Liquid." By A. B. 

 Basset, M.A. Communicated by Lord Rayleigh, D.C.L., 

 Sec. R.3. Received November 10, 1887. 



(Abstract.) 



The determination of the small oscillations and steady motion of a 

 sphere which is immersed in a viscons liquid, and which is moving in 

 a straight line, was first effected by Professor Stokes in his well- 

 known memoir " On the Effect of the Internal Friction of Fluids on 

 the Motion of Pendulums;"* and in the appendix he also determines 

 the steady motion of a sphere which is rotating about a fixed diameter. 

 The same subject has also been subsequently considered by Helmholtz 

 and other German writers ; but, so far as I have been able to discover, 

 very little appears to have been effected with respect to the solution of 

 problems in which a solid body is set in motion in a viscous liquid in 

 any given manner, and then left to itself. 



In the present paper I have endeavoured to determine the motion 

 of a sphere which is projected vertically upwards or downwards with 

 given velocity, and allowed to ascend or descend under the action of 

 gravity (or any constant force), and which is surrounded by a viscous 

 liquid of unlimited extent, w 7 hich is initially at rest excepting so far as 

 it is disturbed by the initial motion of the sphere. 



In solving this problem, mathematical difficulties have compelled 

 me to neglect the squares and products of velocities, and quantities 

 depending thereon, which involves the assumption that the velocity of 

 the sphere is always small throughout the motion ; and I have also 

 assumed that no slipping takes place at the surface of the sphere. 

 The problem is thus reduced to obtaining a suitable solution of the 

 differential equation 



where 



yjr is Stokes's current function, and fx is the kinematic coefficient of 

 viscosity. The required solution is obtained in the form of a definite 

 integral by a method similar to that employed by Fourier in solving 

 analogous problems in the conduction of heat ; the resistance expe- 

 rienced by the sphere is then calculated, and the equation of motion 

 written down and integrated by successive approximation on the sup- 

 position that u is a small quantity. The values of the acceleration 

 and velocity of the sphere to a third approximation are found to be 



* ' Cambridge Phil. Soc. Trans.,' vol. 9. 



