This project investigates fundamental problems that are motivated by applications to surfactant-mediated flow. The theoretical approach has the potential to be adapted to a wider variety of practical situations that are similar from a mathematical point of view. Using this approach, the project investigators have recently begun the development of a fast and accurate hybrid numerical method to study the effects of solubility of surfactant on the two-phase flow of immiscible fluids in the practically important but theoretically challenging limit of large Peclet number or slow diffusion. Surfactants influence the dynamics of fluid mixtures by altering the surface tension at interfaces between immiscible fluids and are energetically favored to remain on an interface. However, in many examples the slow diffusional mobility of a surfactant that is soluble in the bulk flow near to but not on an interface can exert an important influence on the interface dynamics. The large bulk Peclet number limit of this investigation introduces a separation of spatial scales that presents a substantial challenge for traditional numerical methods. The conceptual underpinning of the approach taken here combines analytical, singular perturbation techniques in the small diffusion limit with fast and accurate numerical methods for two-phase interfacial flow. An important benefit of this approach is that highly accurate surface-based methods, such as the boundary integral or boundary element method, can be adapted to the study of surfactant solubility. Without the treatment that is under development by the investigators these methods do not apply.
The project is expected to develop innovative theoretical models and numerical methods for the analysis and simulation of surfactant-mediated drop breakup and tip-streaming with soluble surfactant. It will develop new, fast, efficient and accurate numerical methods that are expected to be useful to scientists and engineers studying emulsion formation and stability as well as emerging microfluidic applications that range from chemical processing techniques to advanced medical applications. An additional and important impact of the project is the education and training of graduate students and postdoctoral fellows. The interdisciplinary training they receive on this project will be valuable preparation for a range of careers in mathematics and science.
This project developed innovative theoretical models and numerical methods for the analysis and simulation of multi-phase fluid flow with soluble surfactant. Surfactants, or surface-active agents, can significantly alter interfacial evolution and flow in a two-phase fluid mixture by changing the surface tension. Surfactants are in widespread use in the formation and stability of emulsions, and are an essential component in microfluidic processes, an example of which is the synthesis of submicrometer-sized droplets by surfactant mediated ‘tipstreaming’ . The ability to synthesize highly uniform droplets, bubbles, and particles in the 10-100 micrometer size range has numerous potential applications in areas such as production of precision emulsions, foams, and suspensions, drug delivery, and medical imaging. Recent experiments on surfactant- mediated tipstreaming in a microfluidic flow focussing device motivate a fundamental understanding of surfactant solubiity effects. The numerical methods and investigations in this project take a first step in this direction. A central theme of the project has been to address a significant difficulty that occurs in the numerical computation of fluid interfaces with soluble surfactant in the practically important limit of small bulk diffusion of surfactant. In the bulk flow, soluble or dissolved surfactant advects (moves with the flow) and diffuses. Because the diffusion is small compared to advection, a thin `transition’ layer develops adjacent to the interface across which the surfactant concentration varies rapidly. Accurately resolving this layer is a significant challenge for traditional numerical methods but is essential to evaluate the exchange of surfactant between the interface, where it influences drop dynamics by changes in surface tension, and the bulk flow. To remedy the difficulties of computing its solution we have combined the results of an analysis of the advection-diffusion equation for the behavior of bulk surfactant concentration adjacent to the interface with a surface-based method for the direct numerical simulation of the interfacial flow with surfactant. This leads to a hybrid numerical scheme that is derived in the limit of small bulk diffusion. The method has been developed to a point that it has led to what we believe are the first fully-resolved simulations of surfactant-mediated tipstreaming of an isolated drop with a surfactant that is soluble. The numerics have been used to provide an improved understanding of surfactant solubility effects in experiments. The investigators have also developed a hybrid numerical method for problems of electrokinetic flow, or electro-osmosis. Here the immiscible fluids contain free ions but are on average electrically neutral, and an imposed DC electric field induces the ions to move, which in turn drives a fluid flow. Ions tend to accumulate near interfaces, and a narrow layer structure develops that is referred to as a Debye layer. This narrow transition layer is similar to that for the soluble surfactant problem considered by the investigators. They have developed a hybrid numerical method for the electro-kinetic flow problem, which incorporates an analytical reduction of the Debye layer structure for arbitrary deformable and moving interfaces into a surface-based numerical solver for the equations of fluid flow. The numerical method can be applied to study the directed motion and manipulation of the shape of biological membranes with applied electric fields, which is important in biological applications. An additional and important broader impact of the project has been the education and training of two graduate students and two postdoctoral fellows. The interdisciplinary training they received on this project will be valuable preparation for a range of careers in mathematics and science.
