Scientific Papers - Vi

334 HYDKODYNAMICAL PKOBLEMS SUGGESTED BY PITOT'S TUBES [396
The second tan"1 thus passes from 0 to TT, thereby completing its course, while | is still small.
When ^ = 0 absolutely, either | or 17, or both, must vanish, but we must still have regard to the relative values of -ty and f. Thus when £ is small enough, ID = 0, and this part of the stream-line coincides with the axis of symmetry. But while f is still small, * changes from 0 to IT, the new value representing the inner face of the wall. The transition occurs when £ = 2i^, •»7 = 1, making in (11)'y= — oo . The point 0' at the edge of the wall (x = TT, y = 0) corresponds to £ = 0, 77 = 0.
For the free part of the stream-line we may put 77 = 0, so that
2£ TT
.(12)
.(13)
.(14)
where tan"1 £ is to be taken between 0 and \ir. Also When £ is very great,
&=£+-|-7r3 2
and the curve approximates to a parabola. When £ is small,
1 J—1 1 £"9 / T CT v
0-7r = ££J, y = tr, ........................(15)
so that the ratio (# - ?r)/?/ starts from zero, as was to be expected.
The upward movement of y is of but short duration. It may be observed that, while dxjdt; is always positive,
which is positive only so long as £ < 1. at -^ = 1-^ = 0-2146,
And when £ = 1, 2/ = - J + log 2 = 0'097.
Some values of a; and y calculated from (12), (13) are given in Table II and the corresponding curve is shown in fig. 3.
TABLE II.— = 0.
f X y * X y
o-o 3-142 0 2-5 4-451 - 0-571
0-5 3-178 + 0-050 3-0 4-892 - 1-098
1-0 3-356 + 0-097 4-0 5-816 - 2-583
1-5 3-659 + 0-027 5-0 6-768 - 4-62
2-0 4-034 -0-195 20-0 21-621 1 -97-00
V