Analysis of Viscoelastic and Compressible Flows

Virginia Polytechnic Institute And State University, Blacksburg VA

Investigators

Abstract

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.