Direct numerical simulations of thermocapillary migration of a droplet attached to a solid wall

This paper is designated to gain further insight into the physical mechanisms of thermal droplet actuation on a wall through direct numerical simulation. Classical theory states that free droplets in a nonuniform temperature field always move towards the hot side. However, when attaching a droplet to a wall with a nonuniform temperature gradient, lubrication theory explains how such a droplet moves towards the colder side. This paper aims at further investigating and clarifying the physical mechanisms and acting forces in the environment of a nonuniform temperature field and offers some explanations. For the numerical simulations of a droplet attached to a wall with a linear temperature gradient and larger contact angles, the full Navier-Stokes equations and energy equation are solved in a Volume of Fluid framework. The solver is extended with a dynamic contact angle treatment and thoroughly validated. The droplet motion is studied both in two and three dimensions, where a movement towards the cold and the warm side can be observed. The forces acting in such a setting are identified and interpreted. A decomposition of the jump conditions shows that the tangential stress due to the temperature dependent surface tension alone would lead to a motion towards the cold side, whereas the normal component alone would move the droplet to the opposite direction. The differences between two- and three-dimensional simulations show that the problem at hand is clearly three-dimensional.