Numerical simulations of a rising drop with shape oscillations in the presence of surfactants

Abstract

The effect of insoluble surfactants on the dynamics of a drop immersed in another liquid has been numerically investigated. The interface is captured by means of the Level-Set method, and the evolution of the surfactant concentration Γ is solved along the deformable interface. The Marangoni stress resulting from tangential gradients of Γ has been implemented in the Ghost Fluid method, allowing one to compute the viscous-stress jump at the interface. The numerical model is first evaluated in the case of an oscillating droplet at mode 2. Numerical results accurately reproduce the shape-oscillation dynamics and demonstrate that the presence of surfactants has a strong impact on the damping rate. This effect, however, does not evolve monotonically with Γ but rather follows the evolution of the maximal value of the gradient of Γ that develops over the drop surface. The case of rising spherical droplets is next considered, where the distribution of surfactant at steady state results from the balance between tangential advection towards the rear of the droplet and the upward Marangoni flux. It is shown that the droplet velocity matches that of a solid particle, although a significant part of the interface is still surfactant free (around 30%). The gradient of Γ induced by the rising motion does not evolve monotonically with Γ as a result of the competition between Marangoni stress and the shear stress in the external boundary layer. Finally, the study of the case combining shape oscillations and rising motion shows that the rising motion does not influence the oscillation dynamics in the limit of low Weber number. The particular case of a droplet oscillating as a clean droplet (no change of eigenmodes) and rising as a fully contaminated droplet could be exhibited, in agreement with previous experimental observations.