MOTION OF THE GAS SPHERE SIS 



The potential required to describe the flow of fluid around a sphere 

 which has a translational velocity U is, from Eq. (8.21), given by 



cos 6 



r 



and hence this flow can be described by a dipole source of strength 

 M = {}^)(^^U. The equipotentials are given by cos 6 = const X r^, 

 and the streamlines obtained from the Stokes stream function by 

 sin 6 = const X r; these lines are plotted in Fig. 8.12b. It is seen that 

 the dipole source acts like a small tube through which the flow is con- 

 centrated. The drawing also shows that the boundary condition for 

 flow around a sphere is satisfied if it is remembered that the dipole must 

 move with the velocity of the sphere. 



B. The method of images applied to a rigid boundary. The usefulness 

 of the concepts of simple and dipole sources lies in the fact that restric- 

 tions on the flow of fluid imposed by geometrical boundaries can be 

 satisfied by superimposing the flows from suitable combinations of these 

 sources placed at points inaccessible to the fluid. The velocity poten- 

 tials of each such flow, and hence their sum, satisfy Laplace's equation, 

 and from standard theorems of potential theory the resulting potential 

 is the solution of the hydrodynamical problem. Knowing the velocity 

 potential permits calculation of the velocity and pressure distribution, 

 and the equations of motion can therefore be obtained. 



The simplest boundary condition is that of an infinite rigid plane, 

 representing the sea bottom or a confining wall, which requires that the 

 flow adjacent to the plane be parallel to it, the corresponding condition 

 on the potential being that at the plane dcp/dii = 0, where n is a normal 

 to the plane. A pulsating sphere in an infinite medium is equivalent to 

 a point source at its center. If a second like source is placed at a dis- 

 tance 26 from the first it is evident that the combined flow at a plane 

 perpendicular to the line joining the sources and halfway between them 

 will by symmetry be parallel to the plane. 



This is, however, precisely the flow distribution required at a rigid 

 boundary a distance h from the radially pulsating sphere. This restric- 

 tion can thus be analyzed by replacing the boundary plane with a source 

 located at the position of an optical mirror image of the source, i.e., at a 

 distance h behind the plane as shown in Fig. 8.13. The flow from two 

 simple fixed sources cannot, however, be the true one because the second 

 image source produces a flow through the boundary of the sphere. If 

 the sphere is small compared to its distance from the boundary, this flow 

 is nearly uniform at all points on its surface, and the flow distribution 

 would be nearly that around a sphere moving away from or toward the 

 wall as it expands or contracts. It is thus seen that displacements of a 



