85. FLOWPAST A PLANE LAMINA 



If 6 -♦ 0, the cylinder of the last two sections becomes a plane lamina of width 2a, with 

 its edges at {- a,0), on which a stream impinges at velocity U and at an angle of inclination 

 a to its faces. Then c = a, ^ = 0. The general formulas need not be repeated, but a few 

 points may be noted. 



On the lam.ina itself, a'= a =c, 6'= b = 0, and from x = a cos tj and Equation [83n], 

 after expanding sin {-q -a), 



u = +q^=ul- cosa - - sina j + -^== , [85a] 



\ Ja^ - x^ I 27T y/a^ - x'^ 



where the upper sign refers to the front face, on which < 17 < 77, and the lower sign to the 

 back face; and q= \u\. 



Thus 5 -» ~ at the edges of the lamina, in general. By assigning the proper value to 

 r, however, q may be made finite at one edge. Thus, \iY = -2naU sina , u approaches 



- V cos a as a!->- a; for, (ar + d)/\j(r - x - [(a + x)/{a - x)]'^ -* as x->-a. 



If r = 0, stagnation points occur on the lamina at tj = a and at 7/ = a + n^, or at 

 x= a cosa on the front face and at a; =- a cos o on the rear face. The hyperbolic dividing 

 streamline meets the lamina at the first of these points and leaves it at the second; the two 

 hyperbolic arcs, with foci at the edges, are asymptotic to a line drawn through the center of 

 the lamina and inclined at an angle a to its plane. On the front face, u - U at x = a cos (a/2) 

 and u=-fatar=-a sin (a/2). 



When r = 0, however, a more direct formulation becomes possible. Then, from 

 Equation [83a] with 6 = 0, w; = aL* cosh (1^- ia) = aU cosh ^cosa - iaU sinh ^sina . Here, 

 since c - a, the term in cosa equals Uz cos a and so represents a uniform flow parallel to 

 the plane of the lamina, which need not be further considered. 



The term in sin a taken by itself represents a stream flowing toward negative y and 

 impinging perpendicularly on the disk. Dropping for a moment the proportionality factor sin 

 a , so that the velocity of the stream at infinity is U, its complex potential is 



-iaU sinh C^-i^ {2^ - a^)'^. _ [85b] 



If M = (/) + iip, 



^^ - 0^ = U^ (a^ + r - « ), 00 = - 6'^ xy. [85c, d] 



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