Modelling levitated 2-lobed droplets in rotation using Cassinian oval curves

Abstract

A simple model of rotating 2-lobed droplets is proposed by setting the outline shape of the droplet to the Cassinian oval, a mathematical curve that closely resembles in shape. By deriving the governing equation of the proposed model and obtaining its stationary solutions, the relationship between the angular velocity of rotation and the maximum deformation length is explicitly and precisely calculated. The linear stability analysis is performed for the stationary solutions, and it is demonstrated that the stability of the solutions depends only on the ratio of the deformation length to the radius of the central cross-section of the droplet, which is independent of the physical properties of the droplet. Via comparison with an experimental study, it is observed that the calculated result is consistent with the deformation behaviour of actual 2-lobed droplets in the range where the stationary solution of the proposed model is linearly stable. Therefore, the proposed model is a suitable model for reproducing the steady deformation behaviour of 2-lobed droplets in a wide range of viscosities, surface tensions, densities and initial radii of the droplet, and especially if the viscosity of the droplet is low, the entire process of deformation of the 2-lobed droplet, including the unsteady breakup process, can be very well reproduced by the proposed model.