Direct Numerical Simulations of surfactant transport and Marangoni forces at the interface between two fluids

The dissolved molecules of surfactants present in liquid are adsorbed at the interface between two fluids and can modify the dynamics of drops, bubbles or waves. For example, the non-uniform distribution of surfactant while a bubble is rising results in gradients of surface tension at the interface and retards the rise velocity of the bubble (Levich, 1962) and modifies the dynamics (Tagawa et. al., 2013). These effects have been studied experimentally and theoretically. To study the effect of surfactants through Direct Numerical Simulations (DNS), we provide a numerical method for two-phase Navier-Stokes solver coupled with the surface transport equation for insoluble surfactants (Stone, 1990) using the Basilisk solver (Popinet 2023). The Marangoni force due to the surface tension gradient is modeled as a body force using a continuum surface model (Seric et. al 2018, Tripathi and Sahu 2018). We validate the method for mass conservation and surface convective transport of surfactants using analytical solutions. We perform DNS of rising bubble in quiescent liquid and show that the surfactants adsorbed at the bubble interface get transported due to the motion of fluids and get concentrated at the bottom of bubble. The rise velocity of bubble gets reduced compared to the case of clean bubble as well as the terminal shape. Other configurations such as surfactant effect on breaking waves or thin films could be investigated within this numerical framework.