Topics in Mathematical Theories of Elastic Complex Fluids

Complex fluids are viscoelastic materials that posses both solids and viscous liquids properties. These materials are abundant in our daily life, including wide varieties of mixtures, polymeric solutions, colloidal dispersions, biofluids, electro-rheological fluids, liquid crystals and liquid crystal polymers. Unlike solids and simple liquids, the model equations for complex fluids continue to evolve as new experimental evidence become available. The systems that we propose to study consist of polymers with induced microstructure due to internal elastic morphology and dynamics. The multiscale, multi-physics nature of the mathematical descriptions and self-consistent theories of these materials requires the combination of tools from nonlinear PDE, calculus of variations, asymptotic expansion, kinetic theory and statistical physics. We propose to develop the mathematical theories to understand these fascinating materials. The main focus is on understanding the role of microstructures in the special coupling between the kinetic transport and the induced elastic stresses. Complex fluids 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 additive to jets in inkjet printers, fuel injection, fire extinguishers; magneto-rheological damping of structural vibrations etc. New mathematical descriptions and self-consistent theories are needed to resolve problems such as the intermolecular and distortional elastic interactions, their coupling to hydrodynamics and applied electric or magnetic fields. This proposal is devoted to develop new mathematical theories to use in the modeling, analysis and the designing of numerical algorithms, in order to understand these important materials.