| RESEARCH
ACTIVITIES IN OUR GROUP We have worked in four problem areas which are described below: Film drainage and drop coalescence Recently, we developed new analytical theories and accurate numerical simulations for coalescence of drops with clean and surfactant-covered interfaces. The focus of these studies is on the ratelimiting step of drainage of the thin liquid film between drop interfaces just prior to coalescence. The fundamental understanding of drop coalescence furthered by these studies is important in engineering applications where the evolution of a drop size distribution must be reliably predicted or controlled. We obtained an analytical solution for the long-time asymptotic evolution of the thin liquid film between deformable drops with mobile interfaces. This had been an unsolved puzzle since the 1978 study of Jones & Wilson [J. Fluid Mech. vol. 87, p. 263, 1978] who concluded that there was “no prospect” for solving the nonlinear integro-differential equation which governs the evolution of the thin film. In another recent study, we demonstrated that the previously-ignored influence of internal circulation within drops qualitatively affects coalescence rates. This new hydrodynamic mechanism for controlling drop coalescence has promising practical applications in fields such as microfluidics. A surprising result is the prediction of flow-stabilized non-coalescing drops, which may explain recent experiments, where non-coalescence was observed but attributed to a nonhydrodynamic repulsive force. Drop breakup We developed an adaptive restructuring algorithm for computational meshes on evolving surfaces. Resolution of the relevant local length scale on the evolving surface is everywhere maintained with prescribed accuracy through the minimization of an appropriate mesh energy function by a sequence of local restructuring operations. The resulting discretization depends on the instantaneous configuration of the surface but is insensitive to the deformation history. Our algorithm made feasible well-resolved three-dimensional boundary integral simulations of drop breakup event by our group and is being widely used for a variety of problems by research groups elsewhere. Ultimately, our numerical simulations provide a guide for developing and testing theoretical descriptions of drop breakup. We used our simulations to explore criteria for breakup and the dynamics of breakup events in simple flows, and the statistics of breakup events in stochastic flows. Useful results include the discovery that the volume of daughter drops produced by breakup events scales with the volume of the critical-size drop, not the the mother drop as was previously assumed in population balance models. We developed a theory for the near-critical dynamics of breaking drops, according to which a single slow-mode undergoes a saddle-node bifurcation at the critical point, analogous to the Landau theory for phase transitions. Our theory provides a reliable method for determining critical parameters from experimental data or numerical simulations. In another study, we derived analytical expressions for the critical parameters for drop breakup in strain-dominated flows. Our analysis and numerical simulations demonstrate, for the first time, distinct coexisting stationary states for drops in Stokes flows. The coexisting states are shown to correspond to a balance between distorting viscous stresses, and either the rotation of the drop by the imposed flow or surface-tension-driven relaxation of the drop shape. Coexisting coiled and stretched configurations of polymer molecules in viscous flows may be similarly explained by the balance of viscous stresses, and the rotation or the entropically-driven relaxation of the molecule. Rheology of suspensions with deformable particles We have conducted several studies which provide a rigorous drop-scale interpretation of emulsion rheology. This work involved the development of the first many-drop three-dimensional boundary integral simulations for flows of concentrated suspensions of hydrodynamically-interacting deformable particles, and small deformation analyses for the dynamics of surfactant-covered drops in dilute emulsions, and for emulsions that are concentrated up to and beyond the jamming threshold. These studies explain the rich rheological behavior of emulsions (e.g., shear thinning viscosities, nonzero normal stresses, and nonlinear frequency response) in terms of the interplay between the distinct time scales of the system, including the relaxation times associated with the drop shape and surfactant distribution, and the time scales associated with convective distortion and rotation of the drop shape and surfactant distribution. The shape modes corresponding to the spectrum of relaxation times reveal interesting features, such as fine-scale oscillatory structures at the contact lines in jammed emulsions which contribute significantly to the stress in the system. Qualitative features of the predicted rheology are observed in a variety of other systems, such as shear-thinning resulting from drop rotation in polymer blends and gels and transient shear stress oscillations in micelle solutions which suggests that our work has broad application to complex fluids with deformable particles. Indeed, our results have been used by theorists elsewhere who are attempting to formulate coarse-grained models and generic theories of complex fluids. Hydrodynamics of surfactant-covered interfaces We developed a theory for the hydrodynamics of incompressible surfactant films, which applies to a broad class of problems where capillary stresses dominate viscous stresses. According to the theory, constant surfactant density is maintained by Marangoni stresses (surface tension gradients) on the interface. The situation is closely analogous to the theory of (three-dimensional) incompressible fluid flow, where constant mass density is maintained by pressure in the fluid. We also developed a theory for film drainage between drops with surfactant-covered interfaces which describes a hydrodynamic back-flow mechanism by which compressible surfactant films hinder coalescence more than immobile interfaces (e.g., rigid particles). An expression was derived for the critical concentration of adsorbed surfactant below which coalescence occurs rapidly. Liquid flow and transport in foams is another research area where our group is working. Here, one of the principal questions is the permeability of liquid in foams and the effects of surfactants. We developed an analytical solution for the permeability and surfactant transport in foams. Our analysis is the first to properly incorporate Marangoni stresses and the predicted permeabilities are twenty times larger than expected in the absence of Marangoni stresses. Recent microscopic measurements of the bubble scale flow by research groups elsewhere support the distinct countergravity flow pattern predicted by our theory. SELECTED
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