Coalescence of two equal-sized deformable drops in an axisymmetric flow

The coalescence of two equal-sized deformable drops in an axisymmetric flow is studied, using a boundary-integral method. An adaptive mesh refinement method is used to resolve the local small-scale dynamics in the gap and to retain a reasonable speed of computation. The thin film dynamics is successfully simulated, with sufficient stability and accuracy, up to a film thickness of $O(10−4)$ times the undeformed drop radius, for a range of capillary numbers, Ca, from $O(10−4–10−1)$ and viscosity ratios from $O(0.1–10)$⁠. The results are compared with experimental results from our earlier studies as well as the simple scaling theory for film drainage. The collisions for time-independent flow simulating head-on collisions in the experimental studies show two distinctively different regimes. At lower capillary numbers, the interfaces of the thin film between the colliding drops remain almost spherical up to the point of film rupture, and the dimensionless drainage time scales as $tdG∼Ca$⁠. At higher capillary numbers, the film becomes dimpled at an early stage of the collision process, and the rate of the film drainage significantly slows down after the dimple is fully formed. In this case, the drainage time scales approximately as $tdG∼Ca4∕3$⁠. The simulation, using a Hamaker constant with a fixed value calculated via Lifshitz theory, qualitatively agrees with the experimental results for the higher capillary numbers but not for the lower capillary numbers. The critical conditions for head-on collisions are also examined when the internal circulation within the drop, caused by the external flow, arrests the film drainage. Collisions in a time-dependent flow are also examined to simulate glancing collisions. Although the simulations predict many aspects of the experimental results, the results are quantitatively accurate, in comparison to the experimental data, only for the lowest viscosity ratio of 0.19. The interfaces of the thin film locally bulge outward when the drops are being pulled apart due to the suction pressure. This local deformation causes a local minimum in the film thickness. At the larger offsets, the coalescence angle continuously increases with $Ca$ up to the separation angle $(θ=55°–58°)$⁠, for $Ca<Cac$⁠. At smaller offsets, however, the local deformation for $θ>45°$ cannot induce film rupture, even though coalescence is observed experimentally for the higher viscosity ratios.