﻿Motion of a Sphere in a Viscous Fluid. 527 



-obtained by ignoring some of the terms in the general 

 equations of motion, and thus simplifying them to such an 

 extent that a direct mathematical solution is possible. 



These solutions may be divided into two classes, which 

 may be looked upon as limiting cases of very high and very 

 low velocities respectively. The equations of motion of an 

 incompressible fluid as obtained by JSavier and Poisson may 

 be written : — 



~du 1 "dp dw ~du "du 



^t plfrx B^ dy d~ ' ; 



with similar equations for the other coordinates (X, Y, Z are 

 the components of the impressed forces). 



The right-hand sides of these equations contain terms such 



.as u^- , which are of the second power in the velocity _, and 



also terms such as v\/ 2 u, which are of the first power ; if, 

 therefore, in any case the velocity is large and the coefficient 

 of viscosity v is small, the terms v\/ 2 u &c. may be neglected, 

 and we thus arrive at the so-called " perfect fluid " or 

 irrotational equations. These equations can be solved for a 

 large variety of boundary conditions, and it is these solutions 

 -which take up the greater part of the current text-books of 

 hydrodynamics. It is found, however, that the actual motion 

 observed is, in general, very different from that given by the 

 solutions thus obtained, a difference which is perhaps most 

 glaringly shown by the fact that, according to the perfect 

 fluid theory, a body moving with uniform velocity experiences 

 no resistance to its motion. 



This discrepancy is usually ascribed to the occurrence of 

 ''cavitation." At sharp corners and edges the theory makes 

 the velocity of the fluid infinite, the pressure in the fluid has 

 consequently a large negative value, a cavity is formed 

 around the edge, and the instability of the motion around 

 this cavity is supposed to cause a general breakdown of 

 the motion. Without denying the fact that cavitation may 

 occasionally occur, and that its occurrence may alter the 

 whole motion, a careful survey of the experimental facts 

 available will show that the above explanation is totally 

 inadequate. In the first place, the motion of the fluid sur- 

 rounding a moving body is (at ordinary velocities) the same 

 for air and for water, due regard being paid to the differed 

 densities and viscosities, and the irrotational solutions are no 

 more applicable to air than to water. It is difficult, how- 

 ever, to imagine anything resembling cavitation taking place 



