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

Thermocapillary migration of bubbles has been studied since Young described a bubble rising in a pure, quiescent liquid subject to a vertical temperature gradient. Pure liquids usually exhibit a linearly-decreasing dependence of surface tension on temperature. Here, we consider so-called `self-rewetting' fluids where surface tension is a parabolic function of temperature with a defined minima. Specifically, we target the counter-current thermocapillary migration of a bubble under temperature gradient. We present DNS using the Basilisk solver to resolve the two-phase continuity, momentum, and energy equations with a VoF method to capture the interface. The simulations agree with the experimental and the theoretical findings of Shanahan and Sefiane (2014). Two distinct regimes are revealed: i) ``steady migration'' where the bubble migrates against flow to an equilibrium position at the surface tension minimum; and ii) ``sustained oscillations'' where the bubble undergoes steady oscillations around the equilibrium position after a transient migration period. We map these in Re and Ca number parameter space and explain sustained oscillations when Ca \textless O($10^{-4})$ , and their damping in the range O($10^{-4})$ \textless Ca \textless O($10^{-2})$.