452 ON THE DYNAMICS OF REVOLVING FLUIDS [413
If at any stage the u motion ceases, (6) gives
pv~/r, .............................. (20)
pfp = ? $r* + 2 (JV - #) log r - i W - RJ r-2} + const. . . .(21)
Since, as a function of r, tf continually increases as R diminishes, the same is true for the difference of pressures at two given values of r, say ra and rZf where rz>rl. Hence, if the pressure be supposed constant at r1} it must continually increase at rg.
If the fluid be supposed to be contained between two coaxial cylindrical walls, both walls must move inwards together, and the process comes to an end when the inner wall reaches the axis. But we are not obliged to imagine an inner wall, or, indeed, any wall. The fluid passing inwards at r = n may be supposed to be removed. And it remains true that, if it there pass at a, constant pressure, the pressure at r = r» must continually increase. If this pressure has a limit, the inwards flow must cease.
It would be of interest to calculate some case in which the (u, w) motion is less simple, for instance, when fluid is removed at a point instead of uniformly along an axis, or inner cylindrical boundary. But this seems hardly practicable. The condition by which v is determined requires the expression of the motion of individual particles, as in the so-called Lagrangian method,, and this usually presents great difficulties. We may, however, formulate-certain conclusions of a general character.
When the (u, w) motion is slow relatively to the v motion, a kind of " equilibrium theory " approximately meets the case, much as when the slow motion under gravity of a variably dense liquid retains as far as possible the horizontal stratification. Thus oil standing over water is drawn off by a syphon without much disturbing the water underneath. When the density varies continuously the situation is more delicate, but the tendency is for the-syphon to draw from the horizontal stratum at which it opens. Or if the liquid escapes slowly through an aperture in the bottom of the containing vessel,. only the lower strata are disturbed. ' In like manner when revolving fluid is drawn off in the neighbourhood of a point situated on the axis of rotation,. there is a tendency for the surfaces of constant circulation to remain cylindrical and the tendency is the more decided the greater the rapidity of rotation.. The escaping liquid is drawn always from along the axis and not symmetrically in all directions, as when there is no rotation. The above is, in substance, the. reasoning of Dr Aitken, who has also described a simple experiment in illustration.
P.S.— It may have been observed that according to what has been said above the stability of fluid motion in cylindrical strata requires only that the square of the circulation increase outwards. If the circulation be in botk 20higher degree.. Math. Soc. Yol. x. p. 4 (1879); Scientific Papers, Yol. i. p. 361. Also Theory of Sound, 2nd ed. §§ 357, &c.