Nonlinear dynamics of rotating drops

Rotating drops have been studied for two centuries due to their relevance to astrophysics, nuclear physics, and capillarity and have attracted the interest of the likes of Plateau, Chandrasekhar, Scriven, and Brown. While gyrostatic equilibrium shapes and their linear stability have been extensively analyzed, recent studies using boundary element methods, which have been able to probe the nonlinear dynamics of rotating drops, have remained limited to the creeping flow regime. We investigate the dynamics and stability of viscous rotating drops undergoing arbitrary-amplitude oscillations by solving the full Navier-Stokes equations by direct numerical simulation. We use the Galerkin finite element method (GFEM) and employ degenerate elements to accurately capture the flow near the drop's center. The transient dynamics are governed by two dimensionless groups: rotational Bond number Bo (rotational/surface tension forces) and Ohnesorge number Oh (dimensionless viscosity). Drops lose stability when Bo is sufficiently large, as predicted by the dispersion relation of Hocking (1960), which we use to benchmark our results on small-amplitude oscillations. We conclude with insights into moderate-amplitude oscillations and their associated nonlinear dynamics from the GFEM simulations.