Modelling a surfactant-covered droplet on a solid surface in three-dimensional shear flow

A surfactant-covered droplet on a solid surface subject to a three-dimensional shear flow is studied using a lattice-Boltzmann and finite-difference hybrid method, which allows for the surfactant concentration beyond the critical micelle concentration. We first focus on low values of the effective capillary number (Ca-e) and study the effect of Ca-e, viscosity ratio (lambda) and surfactant coverage on the droplet behaviour. Results show that at low Ca-e the droplet eventually reaches steady deformation and a constant moving velocity u(d). The presence of surfactants not only increases droplet deformation but also promotes droplet motion. For each lambda, a linear relationship is found between contact-line capillary number and Ca-e, but not between wall stress and u(d) due to Marangoni effects. As lambda increases, u(d) decreases monotonically, but the deformation first increases and then decreases for each Ca-e. Moreover, increasing surfactant coverage enhances droplet deformation and motion, although the surfactant distribution becomes less non-uniform. We then increase Ca-e and study droplet breakup for varying lambda, where the role of surfactants on the critical Ca-e (Ca-e,Ca-c) of droplet breakup is identified by comparing with the clean case. As in the clean case, Ca-e,Ca-c first decreases and then increases with increasing lambda, but its minima occurs at lambda = 0.5 instead of lambda = 1 in the clean case. The presence of surfactants always decreases Ca-e,Ca-c, and its effect is more pronounced at low lambda. Moreover, a decreasing viscosity ratio is found to favour ternary breakup in both clean and surfactant-covered cases, and tip streaming is observed at the lowest lambda in the surfactant-covered case.