Scientific Papers - Vi

1915] HYDRODYNAMIGA.L PROBLEMS SUGGESTED BY PITOT'S TUBES 331
branches along y = ±tr. From as = — oo to « = — 1, the flow is along the inner side of the walls, and from as — —l to # = - oo back again along the outer side. At the turn the velocity is of course infinite.
We see from (4) that when -^ is given the difference in the final values of y, corresponding to infinite positive and negative -values of cf>, amounts to TT, and that the smaller is ty the more rapid is the change in y.
The corresponding values of as and y for various values of $, and for the stream-lines -^ = — 1, — £, - J, are given in Table I, and the more important parts are exhibited in the accompanying plots (fig. 1).
TABLE I.
0 *=-i *=-* >=-!
X y X y X y
-10 12-303 0-2750 12-30 0-550 12-31 '1-100
___ K 6-610 0-3000 6-614 0-600 6-63 1-198
- 3 4-102 0-3333 4-112 0-665 4-15 1-322
— 2 2-701 0-3745 2-723 0-745 2-80 1-464
_ 1 1-030 0-495 1-111 0-964 1-35 1-785
- 0-50 0-081 0-714 0-153 1-285 — —
- 0-25 -0-790 1-035 — — — —
o-oo -1-386 1-821 -0-693 2-071 o-oo 2-571
0-25 -1-290 2-606 — — — —
0-50 -1-081 2-928 -0-847 2-881 - 0-388 3-035
1-0 -0-970 3-147 -0-888 3-178 - 0-653 3-356
2-0 - 1 '299 3-267 -1-277 3-397 - 1-195 3-678
3-0 -1-898 3-308 - 1 -888 3-477 — —
4'0 — — — — - 2-584 3-897
5-0 -3-389.. 3-342 -3-386 3-542 — ' —
10-0 -7-697 3-367 — — - 7-692 4-042
20-0 — — — — -17-00 4-092
In the second form of the problem we suppose, after Helmholtz and Kirchhoff, that the infinite. velocity, at the edge,-encountered when the fluid adheres to the wall, is obviated by the formation of a surface of discontinuity where the condition to be satisfied is that of constant pressure and velocity. It is, in fact, a particular -case of one treated many years ago by Prof. Love, entitled "Liquid flowing against a disc with an elevated rim," when the height of the rim is. made infinite*. .lam indebted to Prof. Love for the form into which the solution then degrades. The origin 0' (fig. 2) of cc + iy or z is taken at one edge. The central stream-line (^ = 0) follows the line of symmetry AB from y = + cctoy = — oo. At i/ = - oo it divides, one half following the inner side of the wall GO' from y = —<x> to y = 0, then becomes a free surface O'D from y = Q to y = — oo. The connexion between
* Comb. Phil. Proc. Vol. vn. p. 185 (1891).pened. *