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 times the undeformed drop radius, for a range of capillary numbers, Ca, from and viscosity ratios from 4

The effect of the dispersed to continuous-phase viscosity ratio on the flow-induced coalescence of two equal-sized drops with clean interfaces was experimentally investigated. The experimental systems consisted of polybutadiene drops suspended in polydimethylsiloxane. The bulk-phase rheological properties of the fluids are Newtonian under the very weak flow conditions of the coalescence experiment (strain rate, $G \,{<}\, 0.08\,{\rm s}^{-1})$. Both head-on and glancing collisions were studied in a purely extensional flow (flow-type parameter, $\alpha\,{=}\,1.0$) for the viscosity ratio $(\lambda)$ range from $O(0.1)$ to $O(10)$. For head-on collisions, the dimensionless drainage times increased with the capillary number (Ca) as $\textit{Ca}^{3/2}$ for all the viscosity ratios, which is consistent with theoretical predictions based on a simple film drainage model. The drainage time at a fixed Ca increased with the viscosity ratio and scaled as $\lambda^{0.82}$. In the case of glancing collisions, the critical coalescence conditions were examined by changing the initial offset, which results in different collision trajectories. In an earlier paper (Yang et al. 2001) that studied a system with a viscosity ratio of 0.096, the critical capillary number $(\textit{Ca}_{c})$ for coalescence always decreased with the increasing offset. However, the present study shows that when the viscosity ratio is greater than $O(0.1)$, the critical capillary number decreases with increasing offset only for the smallest offsets, but then increases with increasing offset until a critical offset is reached above which coalescence is not observed. This is because coalescence for the larger offsets occurs in the extensional quadrant $(\phi\,{>}\,45^\circ$) after the external flow has begun to pull the drops apart. At small offsets, drops coalesced in the compression quadrant with an orientation angle, $\phi \,{<}\,45^\circ$. At the larger offsets, drops also coalesced in the compression quadrant for small Ca, but above some critical Ca, the coalescence angle jumped abruptly (i.e. with a very small change in Ca) to coalescence in the extensional quadrant. Coalescence with $\phi\,{>}\,45^\circ$ is more prevalent for the higher viscosity ratio systems. On the other hand, the maximum offset for coalescence decreased with the viscosity ratio as expected.

The present study experimentally investigates the mechanisms involved in the flow-induced coalescence process for two equal-sized drops (polybutadiene drops suspended in a polydimethylsiloxane matrix), by taking advantage of the capability of the computer-controlled “four-roll mill” to carry out head-on collisions. In this work, head-on collision experiments have been carried out for a time-dependent flow that is designed so that the force along the line of centers mimics the force history due to rotation of the two droplets in a glancing collision. One primary goal of these experiments is to assess the importance of global deformation of the drops in the coalescence process. Specifically, we seek to determine whether global deformation plays a role in the observation that coalescence often occurs during the portion of a glancing collision when the drops are actually being pulled apart by the external flow. By comparison of the results for head-on and glancing collisions, we find that coalescence occurs in an apparently identical fashion in spite of the fact that the overall shape of the drops must be different since the velocity gradient is steady during the glancing collision but time dependent in the head-on collision. Specifically, the (near) axisymmetric film drainage process achieved in a head-on collision is apparently a very good approximation to the same process in a nonaxisymmetric glancing collision, suggesting that the coalescence process is dominated by the time history of the force along the line of centers and is at least approximately independent of the degree of asymmetry in the overall collision process.

This paper reports results from an experimental study of the effects of copolymer/compatibilizer on the coalescence of two equal size drops in the flow field produced by a four-roll mill. The data encompass two different fluid systems, both with PDMS as the suspending fluid and PBd as the drops, and an acid-base complex of PDMS–NH3+ −OOC–PBd adsorbed at the interface that we shall refer to as a copolymer. The two systems differ in the ratio of viscosities (λ) of the drop to the suspending fluid, one having λ=0.19 and the other λ=1.3. For the lower viscosity ratio system, as the amount of adsorbed copolymer is increased, the drainage time for coalescence in a head-on collision is increased monotonically and the critical capillary number for coalescence in a glancing collision is also reduced monotonically in a manner that appears qualitatively consistent with a slowing of the film drainage process due to Marangoni stresses. Detailed trajectory measurements for drops with copolymer show agreement with predicted theoretical results for spherical drops without copolymer, but with an increased viscosity ratio. With copolymer present, we also find that coalescence occurs for the largest capillary numbers only after the drops begin to be pulled apart by the external flow. For the higher viscosity ratio system, the effect of increasing the copolymer concentration is nonmonotonic. For very small concentrations, there is a major decrease in the critical capillary number for coalescence and a corresponding increase in the drainage time prior to coalescence, but as the copolymer concentration is further increased, the film drainage time decreases and the critical capillary number increases to a value that is intermediate between the clean interface result, and the result for the smallest copolymer concentration. This is shown to be due to a dependence of the critical coalescence angle on copolymer concentration that was not present in the lower viscosity ratio system. We conclude by speculating about mechanisms, in addition to the Marangoni effect, that might “explain” these observations.

Flow-induced coalescence of a pair of polymeric drops was studied at the level of individual drops, using a four-roll mill, to understand how the process is affected by the presence of a copolymer at the drop interface. The experimental system consisted of polybutadiene (PBd) drops suspended in polydimethylsiloxane (PDMS). Copolymers were produced at the drop interface by a reaction between functionalized homopolymers (PBd-COOH and PDMS-NH2). The experiments were carried out over wider ranges of parameters than our earlier studies of Hu et al. [Phys. Fluids 12, 484 (2000)] and Ha et al. [Phys. Fluids 15, 849 (2003)], in an attempt to understand the puzzling results found in our earlier studies. The experimental results were consistent with a qualitative mechanism of immobilization of the boundaries of the thin film between drops due to a flow-induced Marangoni effect. A critical or minimum copolymer interfacial coverage (Γmin) exists, above which the copolymer effect becomes independent of the coverage or the viscosity ratio. Using self-consistent mean field theory, the Γmin was found to be approximately 0.08chain∕nm2 corresponding to Δσe≅0.45mN∕m, which is around 30% of the saturation concentration, Γ∞≅0.25chain∕nm2. However, a whole new set of phenomena was discovered when the copolymer coverage is smaller (Γe&lt;Γmin). In this case, we found that there was a strong surfactant effect at small Ca values, but that there was a transition capillary number (Cat) above which the Marangoni effect apparently becomes negligible. In this case, two critical capillary numbers for coalescence (Cac,high and Cac,low) exist, and there are two ranges of Ca and offset where coalescence is possible. The first is for Ca&lt;Cac,low and small offsets. The second is for Cat&lt;Ca&lt;Cac,high, where Cac,high has almost the same values as the critical capillary number for a clean interface system. Between Cac,low and Cat, coalescence is not possible. For the copolymer systems, coalescence at Cac occurred at the angle just prior to the apparent separation of the drops in the extensional quadrant. The nonmonotonic change in Cac with copolymer concentration, found in our earlier study, is due to the fact that the separation angle increases with increased concentration, as can be seen by examination of the collision trajectory data. A copolymer with a smaller molecular weight was also used to probe the potential significance of non-hydrodynamic effects related to the molecular weight. We observed the same saturated limit for the copolymer effect (when Γe⩾Γmin) as in the case of the higher molecular weight copolymer system. We conclude that the Marangoni effect is the main mechanism for the suppression of coalescence in the current polymer/copolymer system.