Mathematical Problems in Polymer Rheology

拟议的研究解决了聚合物流变学和连续介质力学中的基本数学问题。其中一个研究领域是粘弹性流动的高Weissenberg数渐近性质的研究。这涉及到当弹性效应很强时的极限。在这种情况下，方程的数学解经常表现出奇异特征，如拐角处的奇异性、壁面的边界层奇异性和分离的流线。未能解决这些奇异性仍然是粘弹性流动数值模拟成功的主要障碍之一。拟议的研究将探索渐近方法在这一领域的应用，以澄清解的分析性质。拟议研究的第二个领域涉及流动不稳定性。要研究的具体问题是纤维纺丝的过程，以及与之相关的薄膜浇铸和吹膜问题。这些流动的一个方面在理论研究中只得到很少的关注，那就是液体到固体的转变以及与之相关的冰点动力学。这项研究将研究这一问题，目的是阐明稳定性、边界和定性动力学，以及基本的数学问题，如自由边值问题的适定性以及稳定性和谱性质之间的关系。该建议的另一个主题是研究粘弹性剪切流的线性化稳定性时产生的谱的性质。最后，该方案解决了嵌入执行器控制线弹性固体的问题，即通过某些类型的控制可以实现哪些边界构型的问题。这项工作是与大卫·罗素共同完成的。提出的流体力学研究对材料科学具有重要意义。在聚合物液体的加工中，例如塑料的制造，在加工条件下液体的流动常常对产品的质量和性能产生重大影响。因此，对这些流动的理解和模拟具有重要意义。该提案涉及高弹性极限中的数学奇点，这些奇点对数值模拟构成了重大挑战，如果要取得进一步进展，就需要从根本上加以解决。该提案的另一个方面是对流动不稳定性的研究。这种不稳定可能出现在加工操作中，在那里它们经常导致不合格的产品。最后，该提案解决了“智能”材料研究中出现的新的数学问题。

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.