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Equilibrium and stability of axisymmetric drops on a conical substrate under gravity



Asha K. Nurse, Sean A. Colbert-Kelly, Sam R. Coriell, Geoffrey B. McFadden


Motivated by recent investigations of toroidal tissue clusters that are observed to climb conical obstacles after self-assembly, we study a related problem of the determination of the equilibrium and stability of axisymmetric drops on a conical substrate in the presence of gravity. A bariational principle is used to characterize equilibrium shapes that minimize surface energy and gravitational potential energy subject to a volume constraint, and the resulting Euler equation is solved numerically using an angle/arclength formulation. The resulting equilbria satisfy a Laplace-Young boundary condition that specifies the contact angle at the three-phase trijunction. The vertical position of the equilibrium drops on the cone is found to vary significantly with the dimensionless Bond number that represents the ratio of gravitational and capillary forces; a global force balance is used to examine the conditions that affect the drop positions. Most of the equilbria correspond to upward-facing cones, and are analogous to sessile drops resting on a planar surface; however we also find equilbria that correspond to downward facing cones, that are instead analogous to pendant drops suspended vertically from a planar surface. The linear stability of the drops is determined by solving the eigenvalue problem associated with the second variation of the energy functional. The drops are generally found to be unstable to non-axisymmetric perturbations that promote a tilting of the drop. Additional points of marginal stability are found that correspond to limit points of the axisymmetric base state. Drops that are far from the tip are subject to azimuthal instabilities with higher mode numbers that are analogous to the Rayleigh instability of a cylindrical interface.
Physics of Fluids A-Fluid Dynamics


Axisymmetric drops, Stability, Conical substrate, Variational principle, Tissue self-assembly


Nurse, A. , Colbert-Kelly, S. , Coriell, S. and McFadden, G. (2015), Equilibrium and stability of axisymmetric drops on a conical substrate under gravity, Physics of Fluids A-Fluid Dynamics, [online],, (Accessed December 8, 2023)
Created October 3, 2015, Updated October 12, 2021