Analysis of Viscoelastic and Compressible Flows

粘弹性和可压缩流分析

• 批准号: 1514576
• 负责人: Michael Renardy
• 金额: $ 36.31万
• 依托单位: Virginia Polytechnic Institute and State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-09-15 至 2019-02-28
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1514576&HistoricalAwards=false
• 关键词: Analysis; Viscoelastic; Compressible; Flows

Yield stress fluids are liquids that need to be subjected to a critical stress before they will flow. Examples include essentially everything that is squeezed from a tube or spread with a knife, including common household substances such as cosmetics, shampoo, and ketchup, as well as a wide array of industrial compounds. The study of these fluids is important, for instance, in food, cosmetics, and pharmaceutical industries as well as for biological applications. For many such fluids the yield stress is not fixed, but depends on the flow history. A familiar example is the common experience that ketchup will flow more readily for a second helping than it did the first time. This type of complex yield stress behavior is referred to as "thixotropic." There are several approaches to modeling thixotropic fluids. This project follows up on work of the investigator that showed that the essential features of thixotropic behavior can be obtained by considering the limit of certain models of viscoelastic fluids when a relaxation time becomes very long. This kind of model naturally opens up the possibility of applying mathematical methods of asymptotic analysis, which rely on the presence of a small parameter. The investigator explores the systematic application of these methods to study the behavior of thixotropic yield stress fluids. The second part of the project concerns questions of controllability, that is, whether a system can be driven from a given class of input states to a desired state by a control mechanism. This is a natural engineering question that, from a mathematical point of view, poses fundamental and challenging problems in the theory of partial differential equations. The investigator studies this issue for the equations modeling compressible flow and for other equations of a similar mathematical structure that arise in modeling viscoelastic fluids. A question closely related to controllability issues is that of backward uniqueness. Do we change the future state of a system by altering its present state or is the future partly independent of the present? For systems given by partial differential equations, it is not always possible to "invent the future" in this sense. The investigator has developed a technique that can be used to prove backward uniqueness for certain systems of partial differential equations. A graduate student and a postdoctoral student are included in the project. The project addresses the following areas: Thixotropic yield stress behavior arises as a limit of viscoelastic flow when a relaxation time is large. This large relaxation time naturally provides a small parameter for asymptotic analysis. The investigator has recently analyzed the distinct dynamics regimes of fast, slow and yielded dynamics that arise in this limit. In spatially inhomogeneous flows, these regimes coexist in spatial regions separated by sharp boundaries. Here he analyzes the development of these boundaries as functions of time. The second topic concerns questions of controllability. The equations of compressible flow involve the coupling of a transport equation and a parabolic equation. The investigator studies the controllability of this system and other mathematically similar problems. Prior results for linearized problems are extended to the nonlinear situation. The analysis is based on Carleman estimates for the parabolic part and the method of characteristics for the transport equation, along with a suitable splitting method that allows the construction of controls in an iterative fashion. The design of this splitting method is the principal challenge. The problem of backward uniqueness for partial differential equations is closely related to questions of controllability. The investigator has developed a method of establishing backward uniqueness that is based on the Phragmen-Lindeloef theorem. This method can be applied to one-dimensional linearized compressible flow and to the one-dimensional damped wave equation. The investigator aims to extend the technique to problems in higher space dimensions. The essential ingredient in the analysis is the derivation of certain resolvent estimates, which are based on a rigorous application of matched asymptotics.

屈服应力流体是在它们将流动之前需要经受临界应力的液体。例子基本上包括从管中挤出或用刀涂抹的所有东西，包括常见的家用物质，如化妆品，洗发水和番茄酱，以及各种工业化合物。这些流体的研究是重要的，例如，在食品，化妆品和制药工业以及生物应用。对于许多这样的流体，屈服应力不是固定的，而是取决于流动历史。一个熟悉的例子是，番茄酱在第二次帮助时会比第一次更容易流动。这种类型的复杂屈服应力行为被称为"触变性"。"有几种方法来模拟触变流体。该项目是研究人员工作的后续，该工作表明，当松弛时间变得很长时，通过考虑粘弹性流体的某些模型的限制，可以获得触变行为的基本特征。这种模型自然开辟了应用渐近分析的数学方法的可能性，这依赖于一个小参数的存在。研究者探讨了这些方法的系统应用，以研究触变屈服应力流体的行为。该项目的第二部分涉及可控性问题，即系统是否可以通过控制机制从给定的输入状态驱动到期望的状态。这是一个自然的工程问题，从数学的角度来看，在偏微分方程理论中提出了基本的和具有挑战性的问题。调查研究这个问题的方程建模可压缩流和其他方程的一个类似的数学结构中出现的粘弹性流体建模。与可控性问题密切相关的一个问题是向后唯一性问题。我们是通过改变一个系统的当前状态来改变它的未来状态，还是未来部分地独立于当前？对于由偏微分方程给出的系统，在这个意义上并不总是可以"发明未来"。研究人员已经开发出一种技术，可用于证明某些偏微分方程系统的向后唯一性。该项目包括一名研究生和一名博士后。该项目涉及以下领域：触变屈服应力行为作为粘弹性流动的限制时，松弛时间是大的。这个大的弛豫时间自然为渐近分析提供了一个小参数。研究人员最近分析了不同的动力学制度的快速，缓慢和产生的动态，出现在这个限制。在空间不均匀的流动，这些制度共存的空间区域分离的尖锐的边界。在这里，他分析了这些边界作为时间函数的发展。第二个主题涉及可控性问题。可压缩流动方程组包括一个输运方程和一个抛物方程的耦合。研究者研究了这个系统的可控性和其他数学上类似的问题。线性化问题的先前结果扩展到非线性的情况。分析是基于Carleman估计的抛物线部分和传输方程的特征方法，沿着与一个合适的分裂方法，允许建设的控制在迭代的方式。这种分裂方法的设计是主要的挑战。偏微分方程的向后唯一性问题与可控性问题密切相关。研究者开发了一种建立向后唯一性的方法，该方法基于Phragmen-Lindeloef定理。该方法可应用于一维线性化可压缩流动和一维阻尼波动方程。研究人员的目标是将该技术扩展到更高空间维度的问题。在分析的基本成分是某些预解估计，这是基于严格的应用匹配渐近的推导。