Viscoelastic Flows of White-Metzner Type Fluids

聚合物熔体和溶液是工业中非常重要的材料，表现出粘性流体和弹性固体的特性。因此，这些粘弹性材料可能很难有效地建模，并且需要本构关系才能将任何材料变形与内应力相关联。这在许多聚合物加工应用中尤其如此，在这些应用中，这些材料在诸如绕过折返角流动以及从口模流出的情况下遇到。这些几何形状是由应力奇点的存在联系在一起的，如果不能理解这些奇点的数学原理，最终可能会对任何加工产品的质量造成负面影响。大多数了解这种粘弹性流动的早期工作都是由化学工程师完成的，他们采取了一种主要由数据驱动的方法来提出和测试这些材料的模型；很少有人尝试分析方法。然而，在过去的三十年里，这种情况已经开始通过渐近分析的使用而改变，它可以在保留关键基础物理的情况下产生更简单的流动的主导顺序描述。该项目的主要目的是通过将这些技术应用于由White-Metzner型本构关系控制的粘弹性流体类来建立这些技术的成功。特别是，将研究绕再入角的流动，其中将通过使用边界层理论和利用自然应力公式来描述流体内的应力来构造主导阶控制方程。然后将对这些方程进行数值分析，并通过与OpenFOAM等计算流体力学(CFD)软件的模拟进行比较，评估它们对描述折返角流动的适用性。

Polymer melts and solutions are highly important materials in industry, and display characteristics of both viscous fluids and elastic solids. Due to this, these viscoelastic materials can be quite hard to model effectively, and require a constitutive relation in order to relate any material deformation to the internal stresses. This is particularly the case in many polymer processing applications, where these materials are encountered in situations such as flow round a re-entrant corner, and flow out of a die. These geometries are linked by the existence of stress singularities, and failure to understand the mathematics of these singularities can ultimately lead to negative implications on the quality of any processed products.Most of the early work on understanding this viscoelastic flow was completed by chemical engineers, who took a largely data-driven approach to proposing and testing models for these materials; analytical approaches were rarely attempted. However, in the last thirty years, this has begun to change via the usage of asymptotic analysis, which can produce a simpler 'leading order' description of the flow whilst retaining the key underlying physics.The main aim of this project is to build on the success of these techniques by applying them to classes of viscoelastic fluids governed by White-Metzner type constitutive relations. In particular, flow around a re-entrant corner will be studied, in which leading order governing equations will be constructed via the use of boundary layer theory, and by utilising a natural stress formulation to describe the stresses within the fluid. These equations will then be analysed numerically, and their suitability for describing the re-entrant corner flow will be assessed via comparisons with simulations from computational fluid dynamics (CFD) software such as OpenFOAM.