Oscillations of slightly viscous liquid drops covered with a monolayer of insoluble surfacant

A fundamental understanding of the oscillations of surfactant-laden drops in air is crucial for applications including measurement of interfacial properties and liquid atomization. In this study, the small-amplitude oscillations of a drop of an incompressible Newtonian liquid covered with a monolayer of an insoluble surfactant are analyzed. Here, the Navier-Stokes system governing liquid motion within the drop and the convection-diffusion equation governing surfactant transport along the liquid-gas interface are solved analytically using linear stability analysis and computationally using a finite element-based simulation technique. The presence of surfactant lowers surface tension and thereby reduces the oscillation frequency. On the other hand, its presence generates surface tension gradients and thus enhances damping rate. Special attention is devoted to the variation of the damping rate D with the surfactant strength parameter B. An approximate analytical solution in the limit of small viscosity (M) is examined to elucidate the key role played by a certain dimensionless number involving the ratio B/M. The role of vorticity in the dynamics is also explored, and insights gained from it are used to explain the variation of D as a function of the initial surfactant loading.