where r = Caj"" + a;^) ; or, if r, ^, and w are employed as polar coordinates, 



V 1 + COS Q 



s (u. [132f] 



r sin y 



(See Reference 7, page 70.) 



133. TRANSVERSE FLOW PAST A SOLID OF REVOLUTION 



Let the uniform line of transverse dipoles on the a;-axis as described in the first part 

 of the last section be immersed in a uniform stream having velocity Y toward negative y. 

 For the stream, ^ = Fy = F oTcos w, hence the resultant potential is, from Equation [132c], 



= [ 



^27+^^ (cos Q - cos (9,) cos oj. [133a] 



Both the xy- and sec-planes are planes of geometrical symmetry, the equipotential sur- 

 face for = - (^ is the mirror reflection in the 3a;-plane of that for = <;6j, so that the stream 

 lines are symmetrically disposed. A third plane of symmetry is the bisector of the segment 

 Cj c^. All of the equipotential surfaces are asymptotic at infinity to planes perpendicular to 

 the y-axis; that for = is the 3a?-plane itself, on which a -- rrfi. 



On any plane through the line of dipoles or the a?-axis, the trace of an equipotential 

 surface is a curve defined by 



V 



y o; + ^;3 (cos - cos Q ) = = constant. [133b] 



m cos CD 



Clearly the same geometrical set of equipotential curves occurs on all of these planes but 

 the value of attached to a given curve is proportional to cos w. 



It suffices, therefore, to study the curves on the a;?/-plane, where |cos &j| = 1. 

 Assume that F > 0, f > 0. Then, since cos 0^ - cos Q^ > 0, it is clear that, on the part of 

 the plane on which y > 0, -> <» both as y = oT-* <=o and as y -♦ with x lying between Cj and 

 c^, so that 0^ -» 0, ^2 "* "■ Hence, in particular, on the line x = (Cj + c^)/^, a relative mini- 

 mum of must occur at some point Q; see Figure 219. From the character of the flow caused 

 by dipoles, it is clear that the fluid will flow away from Q both toward a; > and toward 

 X < 0, and hence that the potential must decrease in both of these directions. The point Q 

 is thus a saddle point for 0, and hence also a stagnation point, since the &j-component of the 

 velocity vanishes by symmetry. On the half-plane where y < 0, symmetrical relations occur. 



332 



