On Elastic Complex Fluids with Complex Microstructure

The goal of this project is to study the rheological and hydrodynamic behavior of elastic complex fluids. We are particularly interested in those due to the effects of the fluid microstructures. A unified energetic variational approach, taking into account the competition of the kinetic and the internal elastic energies, is employed to explore the special coupling between the transport of the elastic variables and the induced elastic stresses. We will focus on several problems: 1) free interface motions in the mixtures of different materials; 2) singularity motions and configurations in liquid crystal flows; 3) other elastic complex fluids, including multiscale models of polymeric materials, electro-kinetic fluids and viscoelastic materials. This project will incorporate theories of partial differential equations, calculus of variations, asymptotic analysis, and numerical techniques. It covers both static and time dependent evolution problems arising from different materials. We are interested in molecule configurations; flow patterns; phase transitions; defect dynamics; and the mechanical and rheological properties of these materials. Complex fluids comprise a large class of soft materials, such as polymeric solutions, liquid crystal solutions, pulmonary surfactant solutions, electro-rheological fluids, magneto-rheological fluids and blood suspensions. The physical properties are determined by the interplay of entropic and structural intermolecular elastic forces and interfacial interactions. These materials exhibit many intricate rheological and hydrodynamic features that are very important to biological and industrial processes. Applications include the treatment of airway closure disease by surfactant injection; polymer additives for inkjet printers; fuel injection; fire extinguishers; magneto-rheological damping of structural vibrations; medical micro-fluidic diagnostic devices; and deformation and transport of blood cells. There are many important mathematical problems in the modeling and analysis of complex fluids. The complexity of the systems requires new mathematical methods and in many cases, new concepts. The research of this project is designed to enhance and contribute to this important link between analysis and the understanding of challenging physical problems.