﻿Motion of a Sphere in a Viscous Fluid. 529 



light on the way in which these surfaces arise or enable us to- 

 decide in what cases they will occur. 



If. instead of neglecting the terms of the form vV 2 « in the 

 equations of motion, we retain these and neglect those of the 



type u^- , the equations reduce to a form which can be 



solved for certain cases, and give solutions which are 

 applicable when the velocities are small and the viscosity 

 high. 



The most important of these solutions is that for a sphere 

 moving in a straight line, obtained by Stokes in his memoir 

 on the motion of pendulums. It was shown by Rayleigh 

 that this solution may be expected to hold so long as Ya/v is 

 small compared with unity, and experiments by Ladenburg* 

 and others show that when due account is taken of the 

 boundary conditions the resistance formulae derived from 

 the solution agree with the experimental results to a very 

 high degree of accuracy. The limiting velocity thus defined 

 is, however, very low, and the practical applications of the 

 solutions are confined to motion in very viscous fluids. 



An attempt has been made by Whitehead f to obtain a 

 second approximation to Stokes's solution by expanding the 

 neglected terms in powers of the velocity, and taking- 

 account of second powers only. The method, how T ever, does 

 not lead to any definite results, as the vorticity becomes 

 infinite in certain parts of the field. This is taken by him to- 

 indicate that the motion becomes unstable and breaks down 

 suddenly in the neighbourhood of the critical velocity, and 

 that eddying motion involving high values of vorticity is set 

 up. It will be seen later that this explanation cannot be 

 reconciled with experimental results. 



It will be gathered from the above that nothing is really 

 known as to the nature of the motion for values of the 

 velocity which are neither very high nor very low, and the 

 investigations described below were undertaken with the object 

 of throwing some light on this problem, and in particular 

 to determine the way in which the motion changes as the 

 velocity is increased beyond the critical value. 



The problem has been approached both from the experi- 

 mental and theoretical sides, and in order to simplify matters 

 as far as possible the investigations have been confined to 

 the case of a sphere moving with uniform velocity along a 

 straight line. The actual motion of the fluid surrounding 



* Ladenburg, Diss. Munich, 1900. 



t Whitehead, Quart. Journ. Maths. 1889, p, L43, 



Phil. Mag. S. 6. Vol! 29. No. 172. April 1915. 2 M 



