on the solid. The limits of integration for r are from r = 6 at § to 

 integral gives 



Evaluation of the 



F =17, 



[V^-q' 



'^dZ. 







Thus, if the motion is steady, the total force on the solid is the same as if the pressure 

 in the fluid were uniform and equal to its actual value at infinity. 



The form.ulas would also represent the flow inside a shell having the shape of -S, due to 

 a source on its axis. The volume of fluid emitted per second by the source is Itt h^ U. 



Changing the sign of V merely reverses the velocity at all points. To reverse the 

 solid lengthwise, the a:-axis may be drawn in the opposite direction. (See Reference 2, 

 Section 15.23.) 



122. POINT SOURCE AND SINK IN UNIFORM STREAM; RANKINE SOLIDS 



Upon a uniform stream with velocity U in the direction of negative x, superpose the 

 flow due to a point source on the a;-axis a,t x = a and also that due to a sink of equal strength 

 at X = - a. The resulting potential and stream function, fromj Equations [119a, b] and [119d,e], 

 can be written 



d>= U 



2 \r^ rj^ 



[122a] 



lA = — f^[w^ + b^ (cos 0. -cos 6^)] 



[122b] 



where 6 is a positive constant and the significance of r,, r^, 0,, O2 is shown in Figure 203. 

 The figure refers to any plane through the ^--axis, about which the flow is symmetrical. In 

 particular, 



T^ = [(x- a)'^ +7o^] , Tj = [(x+ a)^ +^0^] 



The flow net is symm.etrical also with respect to the plane x = 0. For, the second 

 term in the brackets in [122b] can also be written - 6^[cos (tt-O^) + cos 62], and it is then 

 clear that ip is unaltered whereas <f> is reversed in sign if x, r., and tt-O^ are interchanged 

 with -X, r^, and 62- 



The components of velocity are, from [118h,i,j], 7 = and 



1 + 



b^ I X- a x+ a \ 



"^177 "77) 



[122c] 



308. 



