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Drop shape oscillations are an elementary phenomenon in drop and spray formation processes. They have been a subject of scientific interest for a long time. Rayleigh (1879) and Lamb (1881) developed analytical solutions for the linear interfacial motion and the angular frequency for inviscid and viscous drops, respectively, in an ambient gas. The corresponding linear results for a viscous drop in a dense and viscous immiscible ambient medium were obtained by Miller and Scriven (1968). Solutions for the corresponding nonlinear cases are sparse in the literature. Since nonlinear effects in drop shape oscillations may influence the velocity and pressure fields in the drop, as well as transport processes across the drop surface, there is interest in studying these effects. Our work develops the weakly nonlinear description of shape oscillations of a viscous liquid drop in a gas and compares the results to experiments on shape oscillations of an acoustically levitated drop. We first carry out a normal-mode analysis treating the oscillations as axially symmetric. The initial drop shape, deviating from the sphere, is represented by Legendre polynomials. The oscillation amplitudes are small enough to allow the series expansions of flow field properties in the analysis to converge. Secondly we present experimental results of damped shape oscillations for viscous liquids. The experiments use an acoustic levitator, where the droplet is excited to the resonance frequency of the fundamental, two-lobed oscillation mode. The information from the measurements is the complex angular frequency of the oscillations, as well as the temporal evolution of the drop shape. The aim is an advancement of the description of nonlinear drop shape oscillations beyond the results for inviscid liquid by Tsamopoulos and Brown (1983). It leads to results revealing nonlinear viscous effects on drop oscillations. The work enables us to describe the interplay of the relevant physical parameters related to the drop surface shape, the liquid flow field in the drop, and the time spent in the different states of deformation during the oscillation. A future objective is to extend the analysis to viscoelastic liquids.
