In the extended example case considered under Case B, where a vortex is near a 

 cylinder having total circulation F'- F around it, w contains another term and 



dw ^ JA _ JA ^ iV' 1 

 dz z-c z + c 2n z-x^ 



The poles at 2 = - c and z = x^ both lie inside the cylinder and contribute to the integral. The 

 product terms can be treated as before; for example, 



2 2 



{z-c) {z-x.) c-x 



\2-C 3-X.J 



The latter product, arising from two poles that lie inside the path of integration, gives zero 

 in the integration, as is always the case with included poles. The force on the cylinder is 

 thus found to be, using A = F/2n- where F is the circulation around the vortex, 



1 inc 2nh 



A positive value of A'j, means that the force acts toward the vortex. 



Finally, in the case of a vortex moving freely parallel to a wall as described under 



Case F, the motion can be made steady without altering the force by imparting to everything 



a velocity equal and opposite to that with which the vortex is moving. The fluid velocity at 



the wall is then 



r r H 



On the assumiption of zero pressure at infinity, the force on the wall is, from [34h], in which 

 U = r/inH here, 



M[(sF--]-^(i^nlC-^-<^,^)^^ 







To evaluate the second integral, put y = H tan 0. 



(For notation and method; see Section 34; Reference 1, Articles 64, 155; Reference 2, 

 Section 13.30, 13.31, 13.40, 13.41; for line vortex and cylinder, MuUer^^ and Morris. ^^ 



43. LINE SOURCE AND PLANE WALL 



w = - A{\n{z+h) + \n{z-h)'\ ,- T^Sa] 



A and h real constants. 



Some problems are easily solved by superposing known types of flow. 



Consider for example, a uniform line source in fluid that is bounded, at a distance h 

 from the source, by a plane rigid wall parallel to the source, as in Figure 62. If the flow is 



