Topics in Complex Fluids and Biophysiology: the Energetic Variational Approaches

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Complex fluids are important in both physical and biological sciences. The microscopic structures associated with the materials give rise to many intricate properties and fascinating applications. These include various polymeric materials, viscoelastic fluids and liquid crystalline materials in chemical engineering, magnetohydrodynamics (MHD) in energy and power industry, electro-kinetic and electrorheological (ER) fluids in soft matter physics, as well as ionic solutions and electrolytes in cellular biology and physiology. The study of these materials and their specific properties require state-of-art multiscale and multiphysics theories and techniques. For example, biology and physiology have shown that almost all biological activities involve the transport of charged particles and ions. This project aims at transferring some of those conventional empirical and phenomenal descriptions into modern quantitative models. The general framework of energetic variational approaches (EnVarA), motivated by the seminal work of Rayleigh and Onsager, has been proven to be with great advantages in studying the complex fluids, such as charged ion transport in biological environments. The main theme of the research is to employ the general energetic variational approaches to derive and study the dynamics of various complex fluids with the applications. These real life problems also provide formidable challenges and great motivations for the development of new mathematical theories and techniques. The project will provide many opportunities for students and junior researchers to have multidisciplinary research experiences in mathematics, bioengineering, and physiology.The purpose of this project is to make progress in the study of several physical and biological complex fluids. It focuses on the underlying energetic variational structures of the models, which capture and predict specific physical and biological properties. The investigator will continue the study of the coupling and competing effects from different spatial or temporal scales. One focus is on the diffusion dynamics of various materials in different environments, such as particles with size effects and electric Coulomb interactions, solutions with high concentrations in heterogeneous environments. All these are crucial in the study of dynamics of ion channel proteins in neurophysiology. Energetic variational methods have shown promises for the study of such multiscale multiphysics problems. Often the existing analysis and numerical methods are inadequate for these complicated problems of non-ideal materials, especially with couplings between fields and structures. The investigator and his collaborators will work on both the analysis issues arising from these studies, developing corresponding numerical algorithms and verifying with the experimental results. They will work with biologists on the study of several specific projects, such as the sensoring mechanism in voltage-gated channels. Self-consistent dynamic models will be derived, which include the flow field, dielectrics, hydration, and conformational changes in proteins. The group will also carry out analytical and numerical studies on projects such as nonlocal diffusion, various dynamic boundary effects, and thermal effects.