Figure 97 — A few streamlines for fluid flowing past a two-dimensional 



pitot tube. The a;-axis is drawn along the median plane of the tube. 



(Copied from 'leference 258.) 



- oo. For, a positive number always exceeds its logarithm, and toward infinity their ratio in- 

 creases without limit. Furthermore, the lower half of the diagram is symmetrical with the up- 

 per. Hence, to sum up, the streamline for i// = follows the ar-axis to a; = - «>, where 0=0, 

 returns along both of the straight lines y ^ n and y = -rr to a; = -1, where = -1, and then 

 retraces these lines to a; = - <», where = - ». 



Streamlines for a value of ijj close to dip a certain distance into the space between 

 the two lines and then emerge again. 



Singularities occur only at (-1, in-), where <^ = -1, ^ = and q ■* x. With the defini- 

 tion of tan~^ that has been adopted, however, the velocity is discontinuous across the lines 

 y = ±77, because of the discontinuity in <^. In the space between these lines, g -» as a; -►-«=, 

 since both ^S and ip then vanish; but elsewhere toward infinity q -* 1, since |0| -► oo or |i/(| -► <» 

 or both, so that dw/dz -> 1. 



In a physical case, therefore, boundaries must be inserted along the straight lines 

 X < -1, y = -n. They form a two-dimensional pitot-tube with parallel plane walls 2 n apart, 

 placed in a stream of fluid approaching at unit velocity in a direction parallel to the walls. 

 A few of the streamlines above the a;-axis or median plane, labeled with values of 0, are 

 shown in Figure 97. Along the walls and also along the a;-axis, a; = (^ + In |0|, q = |w|, 



u = ; on the walls ^ < 0, on the aj-axis, </> > 0. 



1 +(j> 

 The results may be generalized, as in Section 62, by replacing s, x, y, w, <^, ip, in all 



formulas by z./c, x/c, y/c, gw, g<f), gip. The tube is then 277c wide, and the values of all 



velocities are divided by eg. 



(For notation and method; see Section 34; Reference 1, Article 66.) 



145 



