Mathematical Sciences: Problems in Viscoelastic and Multilayer Flows

小行星9306635 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，并包括三个组成部分：i）粘弹性流动中的不稳定性和分叉分析的严格的数学基础的发展。 要研究的问题包括线性稳定性和谱之间的联系，以及中心流形的存在，这使得动力学的常微分方程的减少。 虽然这些问题是很好地理解在牛顿流体动力学，很少有人知道在粘弹性流体;二）研究流体界面的不稳定性和定性动力学。 以前的工作在该地区一直关注的线性稳定性和“简单”的分歧。 拟议的研究将关注更复杂的分叉，如边带不稳定性，和几个不稳定的模式共存的情况下，三）研究开放边界的流动问题。 这种开边界是由于为了数值计算而截断流动区域而产生的。拟议的研究旨在解决有关粘弹性流动中的开放边界的问题，以及开放边界的存在所造成的人为不稳定性的可能性。 一个研究生将工作的存在性稳定的流动模型方程产生的动力学理论的聚合物溶液。 拟议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操作中是重要的。所提出的研究结果将有助于发展一个更好的定性了解这种流动，并在开发方法，其数值模拟。 研究重点特别是对不稳定性的研究，即定性变化的性质的流动时，某些参数发生变化，并在人工边界所产生的问题时，流域被截断的数值目的。 ***

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. ***