Stability of rigid motions and rollers in bicomponent flows of immiscible liquids

We consider the motion of two rings of liquids with different viscosities and densities lying between concentric cylinders that rotate with the same angular velocity Ω. Gravity is neglected and interfacial tension is included. We show that rigid motions are globally stable and that the shape of the interface which separates the two fluids is determined by a minimizing problem for a potential [Pscr ] defined as the negative of the sum of the kinetic energies of two rigid motions plus the surface energy of the interface. We show that the stable interface between fluids has a constant radius when heavy fluid is outside and J = - d³ [\![ρ]\!]Ω²/T is larger than four, where d is the mean radius, [\![ρ]\!] < 0 the density difference and T the surface tension. When J is negative the heavy fluid is inside and the interface must be corrugated. The potential of flows with heavy fluid outside is smaller, thus relatively more stable, than when light fluid is outside, whenever J is large or for any J when the volume ratio m of heavy to light fluid is greater than one. These results give partial explanation of the stability and shape of rollers of viscous oils rotating in water and the corrugation of the free surface of films coating rotating cylinders.