The thermocapillary motion of liquid droplets in fluid media depends on a variety of influential factors, including the not yet fully understood role played by the presence of the walls and other geometrical constraints. In order to address this specific question, in the present work, we rely on a rigorous mathematical and numerical framework (including an adaptive mesh strategy), which is key to perform physically consistent and computationally reliable simulations of such a problem given the different space scales it involves. Our final aim is the proper discernment of the triadic relationship established among viscous phenomena, thermal effects, and other specific behaviour due to the proximity of the droplet to a solid boundary. Different geometric configurations are considered (e.g., straight, converging, and diverging channels and droplets located near a single or adjacent walls), and distinct regimes are examined [including both (Ma, Re) → 0 and finite Ma flows]. The results show that for straight channels the droplet generally undergoes a decrease in the migration velocity due to its proximity to the wall. Such a departure becomes larger as the Marangoni number is increased. In addition, a velocity component directed perpendicularly to the wall emerges. This effect tends to “pull” the droplet away from the solid boundary if adiabatic conditions are considered, whereas for thermally conducting sidewalls and relatively large values of the Marangoni number, the distortion of the temperature field in the region between the droplet and the wall results in a net force with a component directed towards the surface. For non-straight channels, the dynamics depend essentially on the balance between two counteracting factors, namely, the effective distribution of temperature established in the channel (for which we provide analytic solutions in the limit as Re → 0) and the “blockage effect” due to the non-parallel configuration of the walls. The relative importance of these mechanisms is found to change according to the specific regime considered [creeping flow or Re = O(1)].
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