Phase field modelling of the detachment of bubbles from a solid substrate

We develop and implement numerically a phase field model for the evolution and detachment of a gas bubble resting on a solid substrate and surrounded by a viscous liquid. The bubble has a static contact angle $\theta$ and will be subject to gravitational forces. We compute, as a function of the static contact angle, the cricital Bond number over which bubbles detach from the substrate. Then, we perform similar studies for bubble resting on inclined substrates and bubbles under the action of an external flow. We provide approximate formulas for the critical Bond number under all these circumstances. Our method is also able to resolve the pinchoff of the bubble and the possible appearence of satellites.