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On the Stability of Rotating Drops



Asha K. Nurse, Sam R. Coriell, Geoffrey B. McFadden


We consider the equilibrium and stability of rotating axisymmetric fluid drops by appealing to a variational principle that characterizes the equilibria as stationary states of a functional containing surface energy and rotational energy contributions, augemented by a volume constraint. The linear stability of the drops is determined by solving the eigenvalue problem associated with the second variation of the energy functional. We compute equilibria corresponding to both oblate and prolate shapes, as well as toroidal shapes, and track their evolution with rotation rate. The stability results are obtained for two cases: (i) a prescribed rotational rate of the system (''driven drops''), or (ii) a prescibed angular momentum (''isolated drops''). For families of axisymmetric drops instabilities may occur for either axisymmetric or non-axisymmetric perturbations; the latter correspond to bifurcation points where non-axisymmetric shapes are possible. We employ an angle-arc length formulation of the problem which allows the computation of equilibrium shapes that are not single-valued in spherical coordinates. We are able to illustrate the transition from spheroidal drops with a strong indentation on the rotation axis to toroidal drops that do not extend to the rotation axis. Toroidal drops with a large aspect ratio (major radius to minor radius) are subject to azimuthal instabilities with higher mode numbers that are analogous to the Rayleigh instability of a cylindrical interface. Prolate spheroidal shapes occur if a drop of lower density rotates within a denser medium; these drops appear to be linearly stable. This work is motivated by recent investigations of toroidal tissue clusters that are observed to climb conical obstacles after self-assembly [Nurse et al., Journal of Applied Mechanics 79 (2012) 051013].
Journal of Research (NIST JRES) -


bifurcation, linear stability, rotating drop, toroids, variational principle


Nurse, A. , Coriell, S. and McFadden, G. (2015), On the Stability of Rotating Drops, Journal of Research (NIST JRES), National Institute of Standards and Technology, Gaithersburg, MD, [online],, (Accessed April 21, 2024)
Created April 19, 2015, Updated October 12, 2021