Effect of overall drop deformation on flow-induced coalescence at low capillary numbers

Abstract

Comparison of recent experimental results for flow-induced drop coalescence [H. Yang, C. C. Park, Y. T. Hu et al., “The coalescence of two equal-sized drops in a two-dimensional linear flow,” Phys. Fluids13, 1087 (2001)] with existing theory provides the motivation for an examination of the theory. Specifically, for head-on collisions, the experiments show a plateau in the dependence of drainage time versus capillary number at low capillary number that could not be explained by either the existing scaling analysis or the existing thin-film theory of the film drainage process previously described in the pioneering work of Davis and co-workers [S. G. Yiantsios and R. H. Davis, “Close approach and deformation of two viscous drops due to gravity and van der Waals forces,” J. Colloid Interface Sci. 144, 412 (1991); R. H. Davis, J. A. Schonberg, and J. M. Rallison, “The lubrication force between two viscous drops,” Phys. Fluids A 1, 77 (1989); M. A. Rother, A. Z. Zinchenko, and R. H. Davis, “Buoyancy-driven coalescence of slightly deformable drops,” J. Fluid Mech. 346, 117 (1997); S. G. Yiantsios and R. H. Davis, “On the buoyancy-driven motion of a drop towards a rigid surface or a deformable interface,” J. Fluid Mech. 217, 547 (1990)]. Both of these results indicate that the existing theories, while fundamentally correct in concept, are incomplete in providing a framework for a comprehensive explanation of the experimental results. In the present paper, we reexamine the thin-film theory of Davis et al. in the low capillary number limit. We find that a quasistatic model in which deformation is localized within the thin film is in general not sufficient to describe the leading-order asymptotic approximation of the flow-induced collision and coalescence of two slightly deformable drops at low capillary number. Instead, the overall deformation induced in the drops by the external flow plays a key role in determining the initial film thickness needed for numerical simulation of the thin-film dynamics via the existing theoretical framework. Also, we find that including retardation effects is important to be able to make quantitatively accurate predictions, especially at viscosity ratios below O(1).