Figure 70 — Flow net due to a row of vortices 



or sources of alternating sign, or due to a 



vortex between walls a apart. 



(Copied from Reference 253.) 



46. ROW OF EQUAL LINE DIPOLES ON A TRANSVERSE AXIS; DIPOLE 

 MIDWAY BETWEEN WALLS, WITH PARALLEL AXIS; FLOW PAST 

 CYLINDER BETWEEN WALLS OR THROUGH A GRATING 



w = B coth — , a and B real; 

 a 



J. B ■ u^"'^ , B ■ ^"^y 



4> =— Sinn , ih = -— sin 



H a H a 



aH^ \ a a I aH^ a a 



rt = 2 I sinh"= — + sin^ — = cosh cos — — 



L ^ fl J o o- 



[46a] 



[46b, c] 



[46d,e] 



[46f] 



All functions are periodic in the y direction wilii a period a. The lines y = 0, - a/2, - a^ . . . 

 represent planes of flow symmetry; the y-axis represents a plane of geometrical symmetry. 



At each of the points (0,0), (0 - a), (0 - 2a) .... on the y-axis there occurs a line 

 dipole with line-dipole moment equal to aB/n and with its axis directed toward positive x 

 (if B > 0). For, near the origin, for example, w = B cosh (ff3/a)/sinh (nz/a) -» aB/m by use 

 of the series [33f,g]; this represents a dipole, as in Section 37; see Figure 71. 



In the conjugate flow the axes of the dipoles are directed along the y-axis; this is the 

 flow of Section 47 rotated through 90 degrees. 



108 



