A wall can be inserted along the arc and the outlying parts of the a;-axis on which 

 i// = 0, and the formulas then represent flow past a circular-arc groove in a plane wall; see 

 Figure 142c. The velocity is infinite at the projecting edges of the groove. 



The lower part of Figure 144 represents the flow past such a groove with m = 1.30, 

 y = 233 deg; to match the description, the figure needs to be turned upside down and the 

 a;-axis reversed. The entire figure may also represent flow on both sides, with the same 

 velocity at infinity, past a thin sheet with a circular-arc buckle in it. The sheet cannot be 

 removed, since the pressures are unequal on the two sides; the excess of pressure p - p^ 

 is shown in the figure for both sides, on an arbitrary scale, on the assumption of steady 

 .motion. Streamlines past a deeper groove, with m - 1.75, y = 315 deg, are shown in 

 Figure 145. 



(For notation and method; see Section 34; Reference 2, Section 6.51, where n = 2m; 

 J.L. Taylor,^^ where m = 1/k.) 



Figure 145 — Flow past a sheet or wall with 

 a deep circular-arc groove. 



90. DOUBLE CIRCULAR CYLINDER, OR CYLINDER AGAINST A WALL 



If, in the formulas of the last section, m is made zero while c remains finite, both 

 arcs come into coincidence with the outlying a;-axis. By decreasing c as well, however, the 

 arcs can be kept finite. Their radius is R = c/sin (m n), since each subtends an angle 2(rT - mn) 

 at its center; and R remains equal to a fixed number a if c is kept equal to a sin (m n) as m^O. 

 The arcs thus reduce to two circles of radius a touching both each other and the a;-axis at the 

 origin. 



Then, at a fixed point representing a given value of 2, ^ becomes small in Equation 

 [88k] as c->0, and 2-*2c/C', substituting for 4" in Equation [89b] and noting that in the limit, 

 as m-»0, c/m = a sin {m7T)/m-* an. 



Tta 

 w - nail coth — 



[90a] 



218 



