Stability of rigid motions and coating films in bicomponent flows of immiscible liquids

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Abstract
We consider the problem of global stability of the rigid rotation of two fluids. The realized interfacial configurations minimize a potential. We derive the most general form of the potential in which the working of the contact line may be expressed as a potential. The resulting variational problem for the interfacial potential is solved when the contact-line conditions are prescribed and for coating flows in which the interface makes a tangent contact with the wetted rod. In the former case, good agreement with experiments is obtained except near lines of contact. This shows that a spinning rod interfacial tensiometer is viable. In the latter case of coating flow, we get good agreement with experiments when the effects of gravity are not too large. The problem of bifurcation of coating flow is discussed qualitatively and some experimental results are given. We show how bifurcating sequences fit well into our qualitative description of the solution which must minimize the interfacial potential as the angular velocity is increased. The last bifurcations lead to pendant drops on a rotating ‘ceiling’ under the influence of centripetal forces which replace gravity. The dynamics of rollers of oil in water, or part in water and part in air, are explained in terms of the wavelength dependence of rotating drops.