MOTION OF THE GAS SPHERE 



317 



the total strength of the combination is zero. There is therefore no 

 net flow introduced by these sources, their effect being merely to give a 

 redistribution of the flow pattern. 



The combination of sources depicted in Fig. 8.14 thus restores the 

 boundary conditions at the sphere, but it is evidently done at the ex- 

 pense of again violating the condition on the plane. It might thus ap- 

 pear that the complications of the method have succeeded only in re- 

 storing the problem to its original status. This, however, is not true 



B 



e 



Fig. 8.14 Distribution of simple sources for a sphere of finite radius and 



infinite rigid wall. 



because the flow pattern obtained at each stage is a better approxima- 

 tion that the preceding one. The method of images in this particular 

 problem is thus one of successive approximations, as is indicated by the 

 fact than the strengths of the sources added inside the sphere in Fig. 

 8.14 are decreased in strength relative to the initial ones by factors of 

 the order (a/26) . The radius a of the sphere is always less than its dis- 

 tance h from the boundary if the method is to apply at all, and these 

 sources are therefore weaker as well as contributing no net flow. Suc- 

 cessive additions of image sources to remedy boundary conditions at the 

 plane and sphere in turn can be carried out to increase the accuracy of 

 the approximation, the process being a convergent one, and it is evident 

 that the number of repetitions needed will increase as the ratio a/25 in- 

 creases. 



the vector OP and the axis of symmetry. Integration for a fine source of strength 

 M' per unit length gives the value ^' = M' (7-2 — I'l) at P, r2 and ri being the dis- 

 tances from the extremities of the line source to P. Adding the functions ^ and ^' 

 for the sources specified gives a constant value of {"if + ^') on the sphere of radius a, 

 which is therefore a streamhne as required. 



