Capillary breakup of a liquid torus

Capillary instability of a Newtonian liquid torus suspended in an immiscible Newtonian medium is computed using a Cahn-Hilliard diffuse-interface model. The main differences between the torus and a straight thread are the presence of an axial curvature and an external flow field caused by the retraction of the torus. We show that the capillary wave initially grows linearly as on a straight thread. The axial curvature decreases the growth rate of the capillary waves while the external flow enhances it. Breakup depends on the competition of two time scales: one for torus retraction and the other for neck pinch-off. The outcome is determined by the initial amplitude of the disturbance, the thickness of the torus relative to its circumference, and the torus-to-medium viscosity ratio. The linearly dominant mode may not persist till nonlinear growth and breakup. The numerical results are generally consistent with experimental observations.