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Full text of "Scientific Papers - Vi"

1916]                     ON THE  DYNAMICS  OF EEVOLVI3STG  FLUIDS
At a constant level the pressure diminishes as we pass inwards.     J^u corresponding rarefaction experienced by a compressible fluid does not ^ca  ^ such fluid to ascend.    The heavier part outside is prevented from coming in below to take its place by the centrifugal force*.
The condition for equilibrium, taken by itself, still leaves v an arbitrary function of r, but it does not follow that the equilibrium is stable. In I-"*'-manner an incompressible liquid of variable density is in equilibrium miner gravity when arranged in horizontal strata of constant density, but stability requires that the density of the strata everywhere increase as we pass (town-wards. This analogy is, indeed, very helpful for our present purpose1. As the fluid moves (u and w finite) in accordance with equations (6), (7), (M), (vr) remains constant (k) for a ring consisting always of the same matter, and v-/r  k-/rs, so that the centrifugal force acting upon a given portion <>t the fluid is inversely as rz, and thus a known function of position. The only difference between this case and that of an incompressible fluid of variable density, moving under extraneous forces derived from a potential, itf that here the inertia concerned in the (u, w) motion is uniform, whereas in a variably dense fluid moving under gravity, or similar forces, the inertia an<l bhe weight are proportional. As regards the question of stability, the dirTon-iicc' J3 immaterial, and we may conclude that the equilibrium of fluid revolving me way round in cylindrical layers and included between coaxial cylindrical walls is stable only under the condition that the circulation (&) alwayn in-jreases with r. In any portion where k is constant, so that the motion IN ihere " irrotational," the equilibrium is neutral.
An important particular case is that of fluid moving between an cylinder (r = a) revolving with angular velocity a> and an outer fixed cylinder ?* = &). In the absence of viscosity the rotation of the cylinder is without jffect. But if the fluid were viscous, equilibrium would require "f
k=vr= a? to (b2 - r*)/(bs - a2),
expressing that the circulation diminishes outwards. Accordingly a fluid vithout viscosity cannot stably move in this manner. On the other hand, if t be the outer cylinder that rotates while the inner is at rest,
Jc = vr = b*a> (V2 - a2)/(&2 - a2), aid the motion of an inviscid fluid according to this law would be stable,
We may also found our argument upon a direct consideration of the kinetic :nergy  (T) of the motion.    For T is proportional to \vzrdr, or   /A^dr'//*.
* When the fluid is viscous, the loss of circulation near the bottom of the containing iodines this conclusion, as explained by James Thomson. t Lamb's Hydrodynamics,  333.
E. VI.                                                                                                                       20higher degree.. Math. Soc. Yol. x. p. 4 (1879); Scientific Papers, Yol. i. p. 361.   Also Theory of Sound, 2nd ed.  357, &c.