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Full text of "Scientific Papers - Vi"

16                                               HYDRODYNAMICAL NOTES
where n is integral and Rn a function of r only; and in deducin, may perform the differentiations with respect to 0 (as well as wil to r) under the sign of summation, since -^ = 0 at the limits. Thus
j
dr      a?r2
sm
The right-hand member of (3) may also be expressed in a serie
of the form
2&> = SCW/TT . ^n~l sin mrd/a, ..................
where n is an odd integer; and thus for all values of n we have
(? 7?          «27T2 7?          4,M
UsjLUin,       id i'  J-^n      TI" /•,        /     -i \.,i
I     ---7~7----T   '    ---7—   ---            TT
dr~          dr          a2
The general solution of (7) is
Rn = Anrn"'a + Bnr~n"/a +               _
the introduction of which into (4) gives ty.
In (8) An and En are arbitrary constants to be determined by conditions of the problem. For example, we might make En> and •\IT, vanish when r — r^ and when r =r2, so that the fixed boundary the fluid would consist of two radii vectores and two circular arc fluid extend to the origin, we must make Bn — 0; and if the bo completed by the circular arc r = 1, we have An = 0 when n is even, n is odd
A   =                         =0
n
Thus for the fluid enclosed in a circular sector of angle « and radius
rnir/a _ r2
mr ?i      - 4a           a
the summation extending to all odd integral values of n.
The above formula (10) relates to the motion of uniformly rote bounded by stationary radii vectores at d = 0, 6 — a.    We may su containing vessel to have been rotating for a long time and that (under the influence of a very small viscosity) has acquired this r that the whole revolves like a solid 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«
* Camb. Phil. Trans. Vol. vm. p. 533 (1847) ; Math, and Phys. Papers, Vol. i.rd/a,   -s ........ . ....... , ........... (4)