Numerical Methods and Analysis for Interfacial Flow with Ionic Fluids and Surfactants

This project is an investigation of fundamental problems in the dynamics of fluids, complex fluids, and deformable matter that occur in applications to biology and microtechnology. Its focus is on the development of new mathematical models and efficient numerical methods to study the morphology and control of drops, cells, and vesicles by electric fields in electrolytic fluids and by flow fields with surfactant. Electrokinetic techniques are among the most common methods for manipulating soft particles and ionic fluids in micro-scale and biological applications and can induce, for example, shape changes in cells and vesicles, from which membrane properties can be inferred. They can also induce membrane channel and pore formation for advanced cell treatment and therapy. Surfactants are used to enhance or control a wide range of complex fluid flows that occur in industries ranging from oil extraction to agriculture, food, and pharmaceutical processing. They are also important in microfluidic applications. Impacts of the proposed research include the development of new mathematical models and numerical methods that will be of benefit to scientists and engineers studying electrokinetic and surfactant phenomena in biology and engineering. An additional impact of this project is the education and involvement of graduate students. The interdisciplinary training they receive will be valuable preparation for a range of careers in mathematics and science.

During the deformation of a fluid-fluid interface with surfactant, surfactant exchange between the interface and bulk occur in a thin layer adjacent to the interface, and the layer's dynamics control interfacial surface tension and shape. During the interfacial flow of an ionic fluid that is driven by an electric field, a thin screening cloud of ions develops at the interface to form an electrochemical double layer or 'Debye layer'. The electric field induces motion in the ion cloud, the interface, and the surrounding fluid. The influence of surfactants on interface and flow dynamics and the occurrence of 'induced-charge electrokinetic flow' in an ionic fluid are both important phenomena in a wide range of applications. This project addresses a difficulty in the numerical computation of such flows in the practically important limit of thin surfactant exchange layers and thin electrical double layers, by developing fast and accurate 'hybrid' or multiscale numerical methods that incorporate an asymptotic analysis of the layer dynamics into a boundary integral or similar formulation of the interfacial free boundary problem. Central themes of the current project are the development of a hybrid method for electrokinetic flow about a deformable membrane and the investigation of fast and accurate numerical methods for studying the influence of surfactant and ionic surfactant on interfacial flow. The algorithm for electrokinetic flow will incorporate an analysis of the high-wavenumber or small-scale component of the elastic and electrostatic stresses on a membrane into a nonstiff method that is capable of handling the multiple time and space scales inherent in the problem. The method will be used to study canonical problems in the deformation of drops, vesicles, and cells, and to explore the interaction between membrane and ionic fluid properties. In the context of surfactant-laden flows, the hybrid numerical method will be developed to accurately handle multiply connected domains with close interaction of interfaces, such as occurs with multiple drops, and the presence of surfactant in the interior fluid, both of which are associated with substantial numerical challenges. This project also involves the development of fast or accelerated algorithms, including for drops in 3D flow with soluble surfactant, as well a fundamental analysis of the stability and convergence of the boundary integral methods that are a cornerstone of this work.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.