The hydrodynamic stability of helium ll between rotating cylinders. I

Abstract

The hydrodynamic instability of helium ll between rotating cylinders is investigated on two assumptions regarding the mutual friction force, F, between the normal and the super­fluid components of the liquid. On both assumptions F is proportional to the constant vorticity which prevails in the stationary state and to the difference in the velocities between the two fluids; however, on one assumption the effect of F is confined entirely to the transverse plane, while on the other it is allowed to be isotropic (with respect to the difference in the velocities). The hydrodynamic problem is solved for the case when the two cylinders (of radii R 1 and R 2 ) are rotated in the same direction and ( R 2 — R 1 ) ≪ ½ ( R 2 + R 1 ). It follows from the theory that when ∂( r 2 Ω)/∂ r < 0 (where Ω denotes the angular velocity and r the distance from the axis) the flow becomes (eventually) unstable along two branches: the first of these is the normal (Taylor) instability of a viscous fluid inhibited by its coupling with an inviscid fluid, and the second is the (Rayleigh) instability of the superfluid inhibited, in turn, by its coupling with a viscous fluid. Further, in all cases the critical Taylor number of instability (suitably defined) becomes asymptotic to a relation which is equivalent to Γ 2 = ½ ( R 2 2 — R 2 1 )R 2 1 , where Γ is the coupling constant. From an experiment of Kolm & Herlin’s (1956), to which the present theory appears applicable, a value of Γ = 0·52 is deduced; this is in very good accord with the value Γ = 0·55 which Hall & Vinen (1956 α ) have deduced from an unrelated experiment.