Oscillations and stability of a rotating drop

The shapes and stability of rotating drops have been of interest for nearly two centuries on account of applications to planetary dynamics and the measurement of interfacial tension by rotating drop tensiometry, and have attracted the interest of scientists including Plateau, Chandrasekhar, and Brown and Scriven. We tackle this problem using two approaches, one relying on analyzing the small-amplitude oscillations of rotating drops by linear stability analysis (LSA) and the other on computing the equilibrium shapes of gyrostatically rotating drops by finite element analysis (FEA). The transient dynamics of rotating drops is governed by several dimensionless groups including the rotational Bond number Bo (rotational/surface tension force) and Ohnesorge number Oh (dimensionless viscosity). Rotating drops can lose stability when Bo is sufficiently large. The dispersion relation from LSA which governs the growth/decay of perturbations is a transcendental equation for which simple solutions can only be obtained for small and large Oh. In addition to such solutions, we solve this equation numerically to gain a thorough understanding of the drop's response over the entire parameter space of interest. The talk will conclude with a discussion of the results obtained from FEA.