Mathematical Analysis of Complex Fluids

Virginia Polytechnic Institute And State University, Blacksburg VA

Investigators

Abstract

Abstract The study of viscoelastic flows leaves many unresolved mathematical issues. The project will resolve a number of open questions. The development of a rigorous foundation for analyzing the stability of viscoelastic flows has long presented an unresolved challenge. Unlike the case of Newtonian flows, there are no applicable theorems of a general nature that link stability to spectral properties or allow deducing nonlinear stability from linearized stability. The project will build on recent developments concerning "advective" systems that promise the possibility of a rigorous study of the stability of creeping flows of viscoelastic fluids. Another part of the project involves formulation and analysis of models for complex yield stress behavior. Phenomena such as yield stress hysteresis, time dependence of yield stress and thixotropy will be explained by a combination of fast and slow dynamics which arises in a singular limit of certain models of viscoelastic flows. The infinite Weissenberg number limit provides another class of problems rife with unresolved mathematical and numerical issues. Work under the project will address the well-posedness of equations which describe the infinite Weissenberg number limit, and singular perturbation problems associated with this limit. The PI will also continue his research on the controllability of viscoelastic flows. Such flows are not fully controllable. The stress tensor in a viscoelastic fluid is subject to certain positive definiteness restrictions which, physically, result from the fact that polymer molecules can be stretched by a flow, but cannot be forced to retract. A precise quantification of this positivity requirement, however, is in general quite difficult. Complex fluids, such as polymers, pastes, and emulsions, arise in numerous applications in the plastics and food industries, as well as biological systems. The equations modeling such fluids are only partly understood. The project will address several important issues arising in the study of these equations, leading to a better understanding of phenomena as well as better methods of their numerical simulation. Flow instabilities, yield behavior of some fluids (which start to flow only when loaded above a certain threshold), problems of the highly elastic limit, and control of flow are among the problems addressed in this work. Several graduate students are already participating in this research.