Energetic Variational Approaches in Complex Fluids

The focus of this research is on the mathematical theory of anisotropic complex fluids, such as viscoelastic materials, liquid crystals, and ionic fluids in proteins and bio-solutions. The research will include two categories of problems: (1) Mathematical analysis of the systems with basic variational structures that are common to complex fluids. (2) The application of such a general framework, in particular to the study of ionic fluids in cells and proteins. Complex fluids, including mixtures and solutions, are abundant in daily life. The complicated phenomena and properties exhibited by these materials reflect the coupling and competition between microscopic interactions and macroscopic dynamics. The project will study the underlying energetic variational structures that are common to all these materials. New mathematical approaches in the modeling, analysis and numerical simulations will be developed in order to understand these multiscale-multiphysics systems. Major efforts will be devoted to the study of ionic solutions in proteins. Molecular biology has shown that understanding behavior of proteins and nucleic acids in biological plasma (ionic solutions) is essential for understanding and controlling many biological systems. Nearly all biological processes involve the transport of electrically charged ions. Ion channels are instrumental for a vast number of biological phenomena, with many diseases (such as cystic fibrosis) and drug treatments dependent on channel functioning. In mathematical analysis of these phenomena, variational methods show a great promise because they deal naturally with systems with many components that flow between definite boundaries. In this research we will apply energy variational methods to study biological plasma, ionic solutions of sodium, potassium, and chloride ions as they flow through specific protein channels.