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.
Fluids with complex microstructure, such as molten plastics, foods and biological fluids cannot be described by the classical equations of fluid mechanics. The more complicated equations which arise in the modeling of such fluids pose many new mathematical challenges. In particular, the limit of high elasticity leads to difficult problems. The research under the grant has advanced the understanding of viscoelastic flows in the following areas: 1. The formulation and mathematical analysis of limit equations for the high elasticity limit and the study of boundary layers which arise in this limit. 2. The analysis of flow stability. 3. The modeling and analysis of thixotropic yield stress fluids using methods of asymptotic analysis. Yield stress fluids are fluids which flow only when a critical stress is exceeded. In thixotropic fluids, this critical stress is not a constant, but is dependent on the flow history and evolves on a slow time scale. Typical examples include ketchup and clay pastes. Boundary layers, such as those arising in the high elasticity limit of viscoelastic flow, are an example where the flow domain has very disparate length scales in different directions. This situation is fairly common in applications of fluid mechanics, for instance a pipe is much longer than its diameter, a river is much longer than it is wide and much wider than it is deep, etc. From a mathematical point, it is natural to simplify the governing equations in a manner that takes advantage of this disparity of length scales. This is known as the hydrostatic approximation. The problem is that the approximate equations may not be well-posed due to the occurrence of instabilities leading to growth of perturbations which eventually violate the hydrostatic approximation. For instance, the overall circulation in the atmosphere satisfies the assumption that horizontal motion dominates over vertical motion, but this breaks down at the scale of a thunderstorm. The question when hydrostatic approximations can be justified is therefore not an easy one. The work under the grant has shown how the stabilization of flows by viscoelasticity or magnetic effects leads to well-posed problems in the hydrostatic limit.
