Mathematical Sciences: Problems in Viscoelastic and Multilayer Flows

9306635本建议涉及粘弹性流动和多层流动的数学分析中的一些问题，包括三个部分：i)为分析粘弹性流动中的不稳定性和分叉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，以及允许将动力学约化为常微分方程组的中心流形的存在。虽然这些问题在牛顿流体动力学中都很清楚，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的研究。该领域以前的工作一直关注线性稳定性和“简单”分叉。建议的研究将涉及更复杂的分叉，例如边带不稳定性，以及几个不稳定模式共存的情况；iii)开放边界流动问题的研究。这种开放边界是为了数值目的截断流动区域而产生的。拟议的研究旨在解决与粘弹性流动中的开放边界有关的问题，以及由于开放边界的存在而导致的人为不稳定性的可能性。研究生将研究由聚合物溶液动力学理论产生的模型方程的恒定流的存在性。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。这种流程在许多工业加工操作中都很重要。建议的研究结果将有助于更好地对这种流动进行定性的理解，并有助于发展其数值模拟方法。这项研究特别侧重于不稳定性的研究，即当某些参数改变时，流动性质的质的变化，以及由于为了数值目的而截断流动区域而引起的人工边界所引起的问题。***

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. ***