Mathematical Analysis of Complex Fluids

粘弹性流动的研究留下了许多尚未解决的数学问题。这个项目将解决一些悬而未决的问题。长期以来，粘弹性流动稳定性分析的严格基础的发展一直是一个未解决的挑战。与牛顿流的情况不同，不存在将稳定性与谱特性联系起来或允许从线性化稳定性推导出非线性稳定性的一般性质的适用定理。该项目将以“平流”系统的最新发展为基础，该系统有望对粘弹性流体蠕动流动的稳定性进行严格的研究。该项目的另一部分涉及复杂屈服应力行为模型的制定和分析。屈服应力滞后、屈服应力的时间依赖性和触变性等现象将由某些粘弹性流动模型的奇异极限中出现的快、慢动力学组合来解释。无限Weissenberg数极限提供了另一类充斥着未解决的数学和数值问题的问题。该项目下的工作将解决描述无限Weissenberg数极限的方程的适定性，以及与该极限相关的奇异摄动问题。PI还将继续研究粘弹性流动的可控性。这种流动不是完全可控的。粘弹性流体中的应力张量受到一定的正确定性限制，这在物理上是由于聚合物分子可以被流动拉伸，但不能被强制收缩。然而，对这种积极性要求进行精确的量化通常是相当困难的。复杂的流体，如聚合物、糊状物和乳液，在塑料和食品工业以及生物系统中有着广泛的应用。模拟这种流体的方程只被部分理解。该项目将解决这些方程研究中出现的几个重要问题，从而更好地理解现象以及更好的数值模拟方法。流动不稳定性，某些流体的屈服行为（只有在加载超过一定阈值时才开始流动），高弹性极限问题和流动控制是本工作解决的问题之一。一些研究生已经在参与这项研究。

AbstractThe 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.