Critical conditions for the buoyancy-driven detachment of a wall-bound pendant drop

We investigate numerically the critical conditions for detachment of an isolated, wall-bound emulsion droplet acted upon by surface tension and wall-normal buoyancy forces alone. To that end, we present a simple extension of a diffuse-interface model for partially miscible binary mixtures that was previously employed for simulating several two-phase flow phenomena far and near the critical point [A. G. Lamorgese et al. “Phase-field approach to multiphase flow modeling,” Milan J. Math. 79(2), 597–642 (2011)] to allow for static contact angles other than 90◦. We use the same formulation of the Cahn boundary condition as first proposed by Jacqmin [“Contact line dynamics of a diffuse fluid interface,” J. Fluid Mech. 402, 57–88 (2000)], which accommodates a cubic (Hermite) interpolation of surface tensions between the wall and each phase at equilibrium.We show that this model can be successfully employed for simulating three-phase contact line problems in stable emulsions with nearly immiscible components. We also show a numerical determination of critical Bond numbers as a function of static contact angle by phase-field simulation.