Mathematical Sciences: Problems in Viscoelastic and Multilayer Flows

数学科学：粘弹性和多层流问题

• 批准号: 9306635
• 负责人: Michael Renardy
• 金额: $ 7.5万
• 依托单位: Virginia Polytechnic Institute and State University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1993
• 资助国家: 美国
• 起止时间: 1993-07-01 至 1996-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9306635&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Problems; Viscoelastic; Multilayer

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. ***

9306635 Renardy 该提案解决了粘弹性和多层流数学分析中的许多问题，包括三个组成部分：i) 为粘弹性流中的不稳定性和分叉分析建立严格的数学基础。 要研究的问题包括线性稳定性和谱之间的联系，以及允许将动力学简化为 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。 该领域之前的工作主要关注线性稳定性和“简单”分岔。 拟议的研究将关注更复杂的分岔，例如边带不稳定性以及多种不稳定模式共存的情况； iii) 研究具有开放边界的流动问题。 这种开放边界是由于出于数值目的而对流域进行截断而产生的。拟议的研究旨在解决与粘弹性流开放边界相关的问题，以及开放边界的存在导致人为不稳定的可能性。 研究生将研究聚合物溶液动力学理论得出的模型方程稳定流的存在性。 拟议的研究旨在进一步发展聚合物液体流动以及多层流动的数学理论。 这种流动在许多工业加工操作中很重要。拟议研究的结果将有助于更好地定性理解此类流动并开发其数值模拟方法。 该研究特别侧重于不稳定性的研究，即当某些参数改变时流动性质的质变，以及当出于数值目的截断流动域时出现的人工边界引入的问题。 ***