330 HYDRODYNAMICAL PKOBLEMS SUGGESTED BY PITOT'S TUBES [396
although we know that, in consequence of the sharp edges, the electrical law would be widely departed from. In the recesses of the tube there is no motion, and the pressure developed is simply that due to the velocity of the stream.
The problem itself may be treated as a modification of that of Helmholtz*, where flow is imagined to take place within the channel and to come to evanescence outside at a distance from the mouth. If in the usual notationf z = x + iy, and w = 0 + i-ty be the complex potential, the solution of Helm-holtz's problem is expressed by
or
x = 0 + e* cos ^, y = ^ + e* sin ^ ................... (2)
The walls correspond to ^ = ±TT, where y takes the same values, and they extend from « = — oo to os = -l. Also the stream-line -v/r = 0 makes y — 0, which is a line of symmetry. In the recesses of the channel 0 is negative and large, and the motion becomes a uniform stream.
To annul the internal stream we must superpose upon 'this motion, expressed say by 0: + ityi, another of the form 02 + tya where
02 + ftya = - # — iy. In the resultant motion,
0 = 0! + 02 = 0! - #, V = ^i + ^ = fa - V>
so that 0! = 0 + #, tyi — ty + y, and we get
0 = 0 +'«*+* cos (i/r+y), 0 = ^ + e*+*sin(^+#), ...... (3)
whence x = - 0 + log V(02 + ^2), y = - ty + tan~J («^/0) ......... (4)
or, as it may also be written,
£ = _ 10 -f. log w .......... ..................... (5)
It is easy to verify that these expressions, no matter how arrived at, satisfy the necessary conditions. Since x is an. even function of ^, and y an odd function, the line y = 0 is an axis of symmetry. 'When ty = 0, we see from (3) that sin y = 0, so that y = 0 or ± TT, and that cos y and 0 have opposite signs. Thus when 0 is negative, y = 0 ; and when 0 is positive, y = ± TT. Again, when 0 is negative, x ranges from + oo to - oo ; and when 0 is positive x ranges from - oo to - 1, the extreme value at the limit of the wall, as appears from the equation
das/d<j) = -1+1/0 = 0,
making 0 = 1, x = - 1. The central stream-line may thus be considered to pass along y = 0 from x = oo to as = ~ oo . At x = - oo it divides into two '
* Berlin Monatsber. 1868; Phil. Mag. Vol. XXXVI. p. 337 (1868). In this paper a new path was opened. *
t See Lamb's Hydrodynamics, § 66.f similitude the influence of linear scale (I) upon the mean pressure should enter only as a function of v/Vl, where vis the kinematic viscosity of air and V the velocity of travel. In the present case v = 1505, V (4 miles per hour) = 180, and I, identified with the width of the strip, = 1'27, all in C.G.S. measure. Thus