Effect of contact line dynamics on the thermocapillary motion of a droplet on an inclined plate.

We study the two-dimensional dynamics of a droplet on an inclined, nonisothermal solid substrate. We use lubrication theory to obtain a single evolution equation for the interface, which accounts for gravity, capillarity, and thermo-capillarity, brought about by the dependence of the surface tension on temperature. The contact line motion is modeled using a relation that couples the contact line speed to the difference between the dynamic and equilibrium contact angles. The latter are allowed to vary dynamically during the droplet motion through the dependence of the liquid-gas, liquid-solid, and solid-gas surface tensions on the local contact line temperature, thereby altering the local substrate wettability at the two edges of the drop. This is an important feature of our model, which distinguishes it from previous work wherein the contact angle was kept constant. We use finite-elements for the discretization of all spatial derivatives and the implicit Euler method to advance the solution in time. A full parametric study is carried out in order to investigate the interplay between Marangoni stresses, induced by thermo-capillarity, gravity, and contact line dynamics in the presence of local wettability variations. Our results, which are generated for constant substrate temperature gradients, demonstrate that temperature-induced variations of the equilibrium contact angle give rise to complex dynamics. This includes enhanced spreading rates, nonmonotonic dependence of the contact line speed on the applied substrate temperature gradient, as well as "stick-slip" behavior. The mechanisms underlying this dynamics are elucidated herein.