satisfied are: 



1. The velocity vector v on S , the exterior side of S, satisfies the 

 nonslip condition, v = 0. 



2. The flow exterior to T is irrotational. 

 Then we have the following theorem : 



The disturbance flow field exterior to S can be generated by the vorticity 



in BLW alone . 



Proof : 



First let us suppose that distributions of vorticity and irrotational 

 singularities are present in B, the interior of the region bounded by S. 

 The velocity components normal to S induced by these distributions define 

 a Neumann problem for a source distribution M on S, equivalent to the in- 

 terior distributions. 



We now have a source distribution M on S and vorticity to in BLW. The 



tangential velocity v is continuous across S, and hence, by the nonslip 



condition, v = on the interior side of S. Also the flow induced by the 



distributions M and OJ is irrotational within B. Hence v = within B, and, 



consequently, the normal component of the velocity on the interior side of 



S is zero. But, because of the nonslip condition, v is zero on the ex— 



n 



terior side of S also, hence the strength of the source distribution must 

 be zero. This leaves only the vorticity distribution oa, as we wished to 

 show. 



Verification with Stokes Solution for a Sphere 



Stokes solution for a sphere of radius a_ in a uniform stream in the 

 x-direction, expressed in spherical coordinates (R,0,(j)), is given by 



Figure 7 



16 



