Mathematical Sciences: Instabilities and Bifurcations in Non-Newtonian Shear Flows

在这个项目中，我们将进行非牛顿（或粘弹性）流体流动的不稳定性和分岔分析。将特别强调三维剪切流。将要研究的问题包括：(a)导致粘弹性流动不稳定的机制(b)导致不稳定的流变和流体动力学因素以及(c)一旦不稳定发生时发生的分岔的性质。我们非常感兴趣的是评估流变因素的作用，如第一和第二正应力差，以及剪切变薄在不稳定的发生和发展中的作用。流体动力和几何因素的重要性，如惯性，表面张力和纵横比也将进行研究。为此，我们将考虑不同几何形状的剪切流，并将采用许多不同的本构模型，如Oldroyd- B、Phan- Thien Tanner、Johnson Segalman和Giesekus模型。我们的分析结果将与现有的实验结果进行比较。除了提供不稳定性和分岔的定性和定量结果外，我们的结果还将提供有价值的信息，说明不同的本构模型不仅可以描述简单的剪切流动，还可以描述复杂的流动。非牛顿（或粘弹性）流体是本项目的主题，包括广泛用于工业和科学应用的材料。例如聚合物（用于塑料工业）、油漆、工业油墨、悬浮液、乳剂和生物流体。这些流体的性质和行为可能与我们更熟悉的水（牛顿流体）等普通流体截然不同。在工业加工和科学实验中，这些流体受到剪切运动的影响。例如，为了确定新材料的性质，它们被放置在称为流变仪和剪切的仪器中。从随后的运动中获得的数据然后用于确定相关的材料特性。从应用和实验中可以得知，当粘弹性流体发生剪切时，其流动性质可能会发生巨大变化，从而导致不可预测的结果。这些剧烈的变化被称为不稳定。由于这些不稳定性可能在工业加工过程中产生不良和意外的后果，并对经济产生重大影响，因此了解导致和维持这些不稳定性的因素对我们来说很重要。这样的理解将提供一种预测这种不稳定何时发生的方法，以及当它们发生时对流动的影响。这将使我们能够在实验和工业加工过程中设置参数，从而防止不稳定。在这个项目中，我们将试图通过研究和分析描述粘弹性流体流动的数学方程来提供这些问题和其他相关问题的答案。这项工作将对联邦政府在材料和制造业领域的战略举措作出重要贡献。

PI: David O. Olagunju Proposal DMS-9704622 INSTABILITIES AND BIFURCATIONS IN NON-NEWTONIAN SHEAR FLOWS ABSTRACT In this project we shall undertake the analysis of instabilities and bifurcations in flows of non-Newtonian (or viscoelastic) fluids. Particular emphasis will be paid to three dimensional shear flows. Among the issues that will be investigated are: (a) the mechanism(s) that cause instabilities in viscoelastic flows (b) what rheological and hydrodynamical factors are responsible for the instabilities and (c) the nature of bifurcations that occur once instability has set in. Of great interest will be to asses the role of rheological factors such as first and second normal stress differences, and shear thinning in the onset and development of instabilities. The importance of hydrodynamical and geometrical factors like inertia, surface tension and aspect ratios will also be investigated. To this end we shall consider shear flows in different geometries and will employ a number of different constitutive models such as the Oldroyd--B, Phan--Thien Tanner, Johnson Segalman and the Giesekus models. The results of our analysis will be compared with available experimental results. In addition to providing qualitative as well as quantitative results on instabilities and bifurcations our results will also provide valuable information on how well different constitutive models describe not only simple shear flows but complex flows as well. Non--Newtonian (or viscoelastic) fluids which are the subject of this project include materials used in a wide ranging number of industrial and scientific applications. Examples are polymers (used in the plastic industry), paints, industrial inks, suspensions, emulsions and biological fluids. The nature and behavior of these fluids can be radically different from ordinary fluids such as water (Newtonian fluids) with which we are much more familiar. During industrial processing and scientific experiments thes e fluids are subjected to shearing motions. For example in order to determine the properties of new materials they are placed in instruments called rheometers and sheared. Data obtained from the subsequent motion are then used in determining the relevant material properties. It is known from applications and experiments that when viscoelastic fluids undergo shearing the nature of the flow may change drastically in ways that may lead to unpredictable results. These drastic changes are termed instabilities. Because these instabilities can have undesirable as well as unexpected consequences during industrial processing with great economic implications, it is important for us to understand the factors that cause and sustain them. Such an understanding will provide a means of predicting when such instabilities will occur and what the effect will be on the flow when they occur. This will enable us to set parameters during experiments and industrial processing so that instabilities can be prevented. In this project we will try to provide answers to these and other related issues by studying and analyzing mathematical equations that describe the flow of viscoelastic fluids. This work will be an important contribution to the Federal Government's strategic initiatives in the areas of materials and manufacturing.