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Full text of "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.                                                                                                                            *