Viscoelastic Flows of White-Metzner Type Fluids

聚合物熔体和溶液是重要的工业材料，具有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，并且需要本构关系以便将任何材料变形与内部应力相关联。这在许多聚合物加工应用中尤其如此，其中这些材料在诸如围绕凹入拐角流动和流出模具的情况下遇到。这些几何形状通过应力奇点的存在联系在一起，如果不理解这些奇点的数学原理，最终会对任何加工产品的质量产生负面影响。大多数关于理解这种粘弹性流动的早期工作都是由化学工程师完成的，他们在很大程度上采用数据驱动的方法来提出和测试这些材料的模型;很少尝试分析方法。然而，在过去的三十年中，这已经开始改变通过使用渐近分析，它可以产生一个更简单的“领先秩序”的描述的流动，同时保留关键的基础physics.The主要目的是建立在这些技术的成功，通过将它们应用到类的粘弹性流体由怀特-梅茨纳型本构关系。特别是，将研究围绕一个凹入的角落，其中领先的顺序控制方程将通过使用边界层理论，并利用自然应力公式来描述流体内的应力构造。然后将对这些方程进行数值分析，并通过与计算流体动力学（CFD）软件（如OpenFOAM）的模拟进行比较，评估其用于描述再入角流的适用性。

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.