The proposed research addresses fundamental mathematical issues in polymer rheology and continuum mechanics. One area of research is the study of high Weissenberg number asymptotics of viscoelastic flows. This concerns the limit when elastic effects are strong. In this case, the mathematical solutions of the equations often show singular features such as singularities at corners and boundary layers at walls and separating streamlines. Failure to resolve these singularities continues to be one of the major impediments towards successful numerical simulation of viscoelastic flows. The proposed research will explore the application of asymptotic methods in this field to clarify the analytical nature of the solution. A second area of the proposed research concerns flow instabilities. Specific problems to be studied are the process of fiber spinning, and the related problems of film casting and film blowing. One aspect of these flows which has received only scant attention in theoretical studies is the liquid to solid transition and the associated dynamics of the freezing point. The proposed research will study this problem with the objective of elucidating stability boundaries and qualitative dynamics as well as basic mathematical issues such as the well-posedness of the free boundary value problem and the relation between stability and spectral properties. Another topic of the proposal is the investigation of the nature of the spectrum which arises in studying the linearized stability of viscoelastic shear flows. Finally, the proposal addresses a question in control of linear elastic solids by imbedded actuators, namely the question which configurations of the boundary can be achieved with certain types of controls. This work is joint with David Russell. The proposed research in fluid dynamics has significant implications for materials science. In the processing of polymeric liquids, e.g. the manufacture of plastics, the flow of the liquid during processing conditions often has a major impact on the quality and properties of the product. The understanding and simulation of these flows is therefore of importance. The proposal addresses mathematical singularities in the high elasticity limit which pose a major challenge to numerical simulations and need to be resolved at a fundamental level if further progress is to be made. Another aspect of the proposal is the study of flow instabilities. Such instabilities can arise in processing operations where they often lead to unacceptable products. Finally, the proposal addresses new mathematical questions which arises in the study of "smart" materials.
