Scientific Papers - Vi

HYDKODYNAMICAL NOTES
17
method of obtaining it. The results are in harmony; but the fact is not immediately apparent, inasmuch as Stokes expresses the motion by means of the velocity-potential, whereas here we have employed the stream-function.
That the subtraction of £«r2 makes (10) an harmonic function shows that the series multiplying r2 can be summed. In fact
sin(n7rO/ot) _ cos (2(9 — a) 1 vnr (nV2 — 4a2) ~ 2 cos a 2 '
o 2
so that
cos (20 -a) ^rn^asinn7T0/a
=-^ - -+8a22 — — -^ — -r1-2 cos a n7r(n~7r2—4<a2)
(11)
^
In considering the character of the motion denned by (11) in the immediate vicinity of the origin we see that if a < |TT, the term in r2 preponderates even when n= 1. When a— \TT exactly, the second term in (11) and the first term under £ corresponding to n = 1 become infinite, and the expression demands transformation. We find in this case
= ir2 + — (0 - ITT) cos 20 + r2 sin 2(9 (-"- log r - ^-\ +
7T \7T 47T/
7T
......... (12)
the summation commencing at n = 3. On the middle line Q = £TT, we have
The following are derived from (13) :
r -^ r -^ ?• -\tf^
o-o •ooooo 0-4 •14112 0-8 •13030
0-1 •02267 0-5 •16507 0-9 : •07641
0-2 •06296 0-6 •17306 i-o •ooooo
0-3 •10521 0-7 •16210
The maximum value occurs when r = '592. At the point r = '592, Q = ITT, the fluid is stationary.
A similar transformation is required when a = 3?r/2.
When a = TT, the boundary becomes a semicircle, and the leading term (n = 1) is
= -!:2/---> ..................(14)
which of itself represents an irrotational motion.
K. VI.id body. The motion expressed that which would ensue if the rotation of the vessel were suddenl; A related problem was solved a long time since by Stokes*, who c the irrotational motion of fluid in a revolving sector. The solution problem is derivable from (10) by mere addition to the latter of i/r0 for then ^ + ^r0 satisfies V2 (^r + ^r0) = 0 ; and this is perhaps th«