_J^L COS '-^ cosh 5^^, . = ?^ 3i„^' S,„h ^ 



[47e,f] 



All functions are periodic in the direction of x with the period a, and the a;-axis represents a 



plane of flow symmetry. The lines x = 0, - a/2, -a, represent planes of purely 



geometrical symmetry. 



Near the origin, as in Section 46, w reduces to aB/nz, so that there is a dipole at the 

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



S > 0, toward negative a; if /3 < 0. Similar line dipoles occur at y = and x = ~ a, ^ 2a 



Hence the formulas represent the flow due a row of such dipoles spaced a apart along the 

 aj-axis; see Figure 74. 



2nB I 2?7ar\"i 



On the X-axis, v = Q, u = 1 - cos I L47gJ 



a \ a I 



On the lines x = 0, - a, - 2a, etc., v = 0, 



27tB f 2ny \-l ,_^, 



u = (cosh ^ - l) [47h] 



a \ a f 



In the conjugate flow the dipole axes are directed parallel to the y-axis; this is the 

 flow of Section 46 rotated through 90 degrees. 



Flow Past a Grating 



If a uniform flow at velocity U toward negative x is superposed, a term —V is added 

 in u, Uz in w, and Ux in (fj, and the formula for ijj becomes 



j, = Uy-2. sinh — ? [47i] 



H a 



Assume that B/U is positive, so that the dipole axes are oppositely directed to the stream. 

 Then i/r = on the i-axis and on a dividing surface S, symmetrical with respect to both axes, 

 given by 



B 2ny / 2ny 2ffa!\ . ,., 



y= — sinh /(cosh -cos [47] J 



" V a \ a a j 



113 



