Also, since the flow Is axisyrametric , there exists a Stokes stream function 

 ^(x, y) which is related to the velocity potential by the equations 



M> = _ v ag e# = y Q± r ,i 



dx y dy' dy * dx u J 



It is seen that Equation [2] may be interpreted as the necessary and suffi- 

 cient condition insuring the existence of the function i//. As is well known, 

 ip is constant along a streamline and, considering the surface of revolution 

 generated by rotation of a streamline about the axis of symmetry, 2ni// may be 

 considered as the flux bounded by this surface. On the surface of the given- 

 body and along the axis of symmetry outside the body we have \p = 0. ip satis- 

 fies the equation 



afg sfg = ± eg m 



ax 2 a y 2 y d y 



which is obtained by eliminating <f> between Equations [3]. 



The velocity will be taken as the negative gradient of the velocity 

 potential. Let u, v be the velocity components in the x, y directions, Then 

 by [3 ], we have 



u = -^ = - — $& M 



dx y ay L } J 



t.-»-1? [6] 



ay y dx L J 



For a uniform flow of velocity U parallel to the x-axis we have 



<t> = -Ux, 4, = -1 Uy 2 [7] 



The boundary condition for the body to be a stream surface may be 

 written In various ways. If the body is stationary the boundary condition is 



if, (x, VfU) = [8a] 



or, equivalent ly, 



(4S) s -° I"" 



