Scientific Papers - Vi

450 ON THE DYNAMICS OF REVOLVING FLUIDS [413
Suppose now that two rings of fluid, one with k = h and r = rx and the other with k = ka and r = ra, where ra > »i, and of equal areas di\z or drz", are interchanged. The corresponding increment in T is represented by
and is positive if L->ki-; so that a circulation always increasing outwards makes T a minimum and thus ensures stability.
The conclusion above arrived at may appear to conflict with that of Kelvin*, who finds as the condition of minimum energy that the vorlicity, proportional to r~ldk/dr, must increase outwards. Suppose, for instance, that k = r~, increasing outwards, while r~ldkjdr decreases. But it would seem that the variations contemplated differ. As an example, Kelvin gives for maximum
energy
v = r from r = 0 to r = b,
v = b2/r from r = b to r = a; and for minimum energy
v = 0 from r = 0 to r — 4/(a2 — 62),
v = r — (a2 — b^jr from r = */(a~ — 62) to r — a.
fa In the first case vr~dr = £Z>2 (2a2 — 62),
Jo
ra
and in the second case I vr2dr = 1&4;
Jo
so that the moment of momentum differs in the two cases. In fact Kelvin supposes operations upon the boundary which alter the moment of momentum. On the other hand, he maintains the strictly two-dimensional character of the admissible variations. In the problem' that I have considered, symmetry round the axis is maintained and there can be no alteration in the moment of momentum, since the cylindrical walls are fixed. But the variations by which the passage from one two-dimensional condition to another may be effected are not themselves two-dimensional.
The above reasoning suffices to fix the criterion for stable equilibrium; but, of course, there can be no actual transition from a configuration of unstable equilibrium to that of permanent stable equilibrium without dissipative forces, any more than there could be in the case of a heterogeneous liquid under gravity. The difference is that in the latter case dissipative forces exist in any real fluid, so that the fluid ultimately settles down into stable equilibrium, it may be after many oscillations. In the present problem ordinary viscosity does not meet the requirements, as it would interfere with the constancy of the circulation of given rings of fluid on which our reasoning depends. But
* Nature, Vol. xxni. October, 1880 ; Collected Papers, Vol. iv. p. 175.n inviscid fluid according to this law would be stable,