Mathematical Sciences: Problems in Viscoelastic and Multilayer Flows

[9306635 . Renardy]该提案解决了粘弹性和多层流动的数学分析中的一些问题，包括三个组成部分：i)为粘弹性流动的不稳定性和分岔分析建立了严格的数学基础。要研究的问题包括线性稳定性和谱之间的联系，以及允许将动力学降低到ode的中心流形的存在。虽然这些问题在牛顿流体动力学中得到了很好的理解，但在粘弹性流体中却知之甚少；Ii)流体界面的不稳定性和定性动力学研究。该领域以前的工作关注的是线性稳定性和“简单”分岔。建议的研究将关注更复杂的分岔，如边带不稳定性，以及几种不稳定模式共存的情况；Iii)开边界流动问题的研究。这种开放边界是为了数值目的而截断流域而产生的。提出的研究旨在解决粘弹性流动中开放边界的问题，以及由于开放边界的存在而导致人为不稳定的可能性。研究生将研究由聚合物溶液动力学理论得出的模型方程的稳定流动的存在性。所提出的研究旨在进一步发展聚合物液体流动的数学理论，以及多层流动。这种流动在许多工业加工作业中是重要的。拟议的研究结果将有助于对这种流动有更好的定性理解，并有助于制定其数值模拟方法。研究的重点是研究不稳定性，即当某些参数改变时流动性质的质变，以及当流域为数值目的被截断时产生的人工边界所引入的问题。＊＊＊

9306635 Renardy The proposal addresses a number of problems in the mathematical analysis of viscoelastic and multilayer flows and encompasses three components: i) the development of a rigorous mathematical foundation for the analysis of instabilities and bifurcations in viscoelastic flows. The issues to investigate include the connection between linear stability and spectrum, and the existence of center manifolds which allow the reduction of the dynamics to ODEs. While these issues are well understood in Newtonian fluid dynamics, very little is known in viscoelastic fluids; ii) the study of instabilities and qualitative dynamics of fluid interfaces. Previous work in the area has been concerned with linear stability and "simple" bifurcations. The proposed research will concern more complicated bifurcations, such as sideband instabilities, and situations where several unstable modes coexist; iii) the study of flow problems with open boundaries. Such open boundaries arise from truncation of flow domains for numerical purposes. The proposed research is aimed at addressing questions relating to open boundaries in viscoelastic flows, and the possibility of artificial instabilities resulting from the presence of open boundaries. A graduate student will work on the existence of steady flows for model equations resulting from kinetic theories of polymer solutions. The proposed research is aimed at further developing the mathematical theory of the flow of polymeric liquids, as well as multilayer flows. Such flows are important in many industrial processing operations. The results of the proposed research will be helpful in developing a better qualitative understanding of such flows and in developing methods for their numerical simulation. The research focusses in particular on the study of instabilities, i.e. qualitative changes in the nature of the flow when certain parameters are changed, and on problems introduced by artificial boundaries which arise w hen the flow domain is truncated for numerical purposes. ***