Nonlinear coupling effects of the thermocapillarity and insoluble surfactants to droplet migration under Poiseuille flow

Under a fully developed Poiseuille flow with nonisothermal condition, it has been widely reported that the thermocapillary effects always strengthen the droplet migration velocity as long as the temperature increases along the direction of Poiseuille flow. The insoluble surfactant, on the other hand, always retards the droplet migration. This is due to the fact that, for most of the models, the Langmuir equation of state for the surface tension is usually simplified under the assumption of low surfactant concentration. The coupling term of temperature and surfactant concentration is dropped, and the thermo-induced and surfactant-induced Marangoni forces are therefore decoupled. In the present study, we develop a thermodynamically consistent phase-field model for investigating the coupling effects of temperature and surfactant concentration on droplet migration under a fully de-veloped Poiseuille flow. By choosing the interface free energy sophisticatedly, the surface tension of our model consists of not only the classical linear part for the thermocapillary effects but also a nonlinear coupling term of temperature and surfactant concentration that recovers the Langmuir equation of state. This coupling term allows us to investigate the case of high surfactant concentration. Through 3D numerical simulations, we find that this nonlinear coupling term introduces extra thermo-induced and surfactant-induced Marangoni forces to the droplet migration, leading to a competition between the two, especially for the case of high surfactant concentration. In particular, the initial migration velocity of a surfactant-covered droplet is always faster than that of a droplet with a clean interface. The terminal velocity, on the other hand, does not reach its steady state but instead decreases gradually as the droplet moves toward the hotter region, whereas, for the case without this term, the initial migration velocity of a surfactant-covered droplet is always lower than that of a clean interface and the terminal velocity stays steady.