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

非牛顿剪切流中的不稳定性和分叉在这个项目中，我们将对非牛顿(或粘弹性)流体流动中的不稳定性和分叉进行分析。将特别强调三维剪切流。将研究的问题包括：(A)粘弹性流动中引起不稳定性的机制(S)，(B)什么流变学和流体动力因素导致不稳定性，以及(C)一旦出现不稳定性时发生的分叉的性质。非常有兴趣的是评估流变因素，如第一和第二法向应力差，以及剪切稀化在不稳定性的发生和发展中的作用。此外，还将研究惯性、表面张力和长宽比等流体动力学和几何因素的重要性。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。除了提供关于不稳定性和分叉的定性和定量结果外，我们的结果还将提供关于不同本构模型不仅描述简单剪切流动而且描述复杂流动的有价值的信息。非牛顿(或粘弹性)流体是本项目的主题，包括用于广泛的工业和科学应用的材料。例如聚合物(用于塑料工业)、涂料、工业油墨、悬浮液、乳状液和生物液体。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的仪器中并进行剪切。然后使用从后续运动中获得的数据来确定相关的材料属性。从应用和实验中得知，当粘弹性流体经历剪切时，流动的性质可能会发生巨大的变化，从而可能导致不可预测的结果。这些剧烈的变化被称为不稳定。由于这些不稳定在工业加工过程中可能会产生不良和意想不到的后果，具有重大的经济影响，因此了解造成这些不稳定的因素并使其持续下去对我们来说很重要。这样的理解将提供一种方法来预测这种不稳定将在何时发生，以及当它们发生时将对流动产生什么影响。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为这些和其他相关问题提供答案。这项工作将是对联邦政府在材料和制造领域的战略举措的重要贡献。

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.