Counter-current thermocapillary migration of bubbles in self-rewetting liquids.

In this work, we study the counter-current thermocapillary propulsion of a suspended bubble in the fluid flowing inside a channel subject to an axial temperature gradient~when the surface tension~dependence on temperature is non-monotonic. We~use direct numerical simulations to address the two-phase conservation of mass, momentum and energy with a volume-of-fluid method to resolve the deformable interface. Two distinct regimes of counter-current~bubble migration are characterized: i) ``exponential decay'' where the bubble decelerates rapidly until it comes to a halt at the spatial position corresponding to the minimum surface tension and ii) ``sustained oscillations'' where the bubble oscillates about the~point of minimum surface tension. We illustrate how~these sustained oscillations arise at low capillary number O(10$^{\mathrm{-5}})$ and moderate Reynolds~number O(10) and, they are dampened by viscosity at lower Reynolds number. These results are in agreement with~the~experiments~by~Shanahan and Sefiane (Sci. Rep. 4,~2014).