Instabilities of rotating liquid columns and orientation dynamics of spheroids in shearing flows

Abstract

As suggested by the title, the research reported in this thesis concerns two problems. The first (and main) part of the thesis, which consists of Chapters 1 and 2, concerns the surface-tension-driven instability of rotating liquid columns. This research is motivated by experimental results reported by Prof. Raghuram Govardhan’s group that concerned the interaction of a vortex ring with a bubble. The general sequence of events in the experiment involves the initial capture of the bubble within the low-pressure vortex core, its elongation along the circumferential direction into a near-toroidal shape, and subsequent break-up (which also leads to the disruption of the vortex ring in many cases). The authors suggested a possible instability governing the bubble breakup, and the thesis therefore assesses the possibility of a linear instability of such a configuration. In building towards the stability of the vortex-ring-bubble configuration, the thesis first considers the somewhat classical problem, of the stability of a rotating column of liquid surrounded by air, in Chapter 1. In the absence of rotation, the problem reduces to the classical Rayleigh-Plateau analysis. There has been considerable prior work on this problem in the presence of rotation. Centrifugal forces are expected to destabilize this configuration, and the necessary and sufficient condition for viscous stability has been derived in earlier efforts. In contrast, arguments in the literature, based on the inviscid equations, suggest only a sufficient condition for stability. The clarification of the rather subtle relationship between the inviscid and viscous stability of a rotating column forms the subject matter of Chapter 1. It is shown, based on the inviscid dispersion relation, that there appear inviscidly stable islands within the known viscously unstable region in the Weber-number-axial-wavenumber plane. For each azimuthal wavenumber, there exists a dominant stable island that terminates in a cusp at a critical Weber number. More interestingly, however, it is shown that there very likely exists an infinite number of additional and much smaller satellite islands within the viscously unstable region. The infinite hierarchy of islands that arises is intimately related to the nature of the rotating column spectrum which comprises a pair of capillary modes and an infinity of Coriolis modes.