Figure 85 — Streamlines past a broader 



Rankine oval with construction curves. 



At P the pressure equals that at 



infinity. See Section 54. 



(Copied from Reference 7.) 



Useful forms resembling the outlines of ships can be obtained in this manner; see 

 McEntee^'*, and Taylor^S. 



The formulas may also represent the flow due to a line source and an equal line sink 

 inside a cylindrical shell having the form of S. 



Changing the sign of V merely interchanges source and sink and reverses all velocities. 



Kinetic Energy 



If the cylinder S is moving through fluid at rest at infinity, the term Uz is missing from 

 w. Then, at large 2, 



--"[(-!)/(-7)]--"(-T--) = 



2aglj 



Hence, in Equation [76c] of Section 76, b^ = 2ag(J, and, from [76d,f], the energy of the fluid, 

 for unit thickness perpendicular to the flow, is 



T^ = %p(inag-S)V^ 



[54k] 



where S is the cfoss-sectional area of the cylinder. 



(For notation and general explanation; see Section 34; Reference 2, Section 8.30.) 



55. VORTEX PAIR IN A UNIFORM STREAM 



The complex potential for a pair of line vortices with equal and opposite circulations, 

 located at {0,-c) and superposed upon a uniform flow at velocity U toward negative x, is 



= iA[\n (z - ic) - \n (z + ic)] + Vz 



[55a] 



where A is & real constant. The circulation about (0,c) is F = 2itA, that about {0,-c), 

 r = -2nA. See Equations [401] and [35a]. Hence • 



= - A(6^-d^)+ Vx, il/ = Aln—+Uy 



'2 



[55b,c] 



127 



