Weakly nonlinear shape oscillations of inviscid drops

Abstract

Abstract Nonlinear axisymmetric shape oscillations of an inviscid liquid drop in a vacuum are investigated theoretically for their relevance for transport processes across the drop surface. The weakly nonlinear approach is adopted as the theoretical method. For the two-lobed mode of deformation $m=2$, known nonlinear effects are an asymmetry of the times the drop spends in the oblate and prolate deformed states, and an oscillation frequency smaller than the linear one found by Rayleigh. The present analysis shows that, for $m=2$, the frequency decrease with increasing surface deformation of the droplet is a third-order effect. For higher deformation modes, the frequency decrease shows in the second-order approximation already. The analysis is carried out for modes of initial deformation up to $m=4$, but not limited to that. The nonlinearity is due to two different contributions: the coupling of different modes of deformation; and the forces from capillary pressure acting on different drop cross-sectional areas in different phases of the oscillation. For the two-lobed mode of deformation, at an aspect ratio of 1.5, the two effects reduce the oscillation frequency by 5 %. The present analysis represents the quasi-periodicity of nonlinear drop oscillations found by other authors in numerical simulations. The results show that drops in nonlinear oscillations at strong deformation may never reach the spherical shape, thus exhibiting a resultant increase of their surface area.