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

PI：大卫O.奥拉贡朱 提案DMS-9704622 非Newtonian激波流的可解性和分叉 摘要 在本项目中，我们将对非牛顿（或粘弹）流体流动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）在粘弹性流动中引起不稳定性的机制（B）什么流变学和流体动力学因素是造成不稳定性的原因，以及（c）一旦不稳定性开始出现，就会发生分叉的性质。最感兴趣的将是评估流变因素的作用，如第一和第二法向应力差，剪切稀化的发病和发展的不稳定性。流体力学和几何因素，如惯性，表面张力和纵横比的重要性也将进行研究。 为此目的，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，例如Oldroyd-B，Phan-Thien坦纳，约翰逊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。除了提供定性和定量的结果不稳定性和分叉，我们的研究结果也将提供有价值的信息，以及不同的本构模型描述不仅简单的剪切流，但复杂的流动。 非牛顿（或粘弹性）流体是这个项目的主题，包括在工业和科学应用中广泛使用的材料。 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普通流体如水（牛顿流体）完全不同。在工业加工和科学实验中，这些流体受到剪切运动。例如，为了确定新材料的特性，它们被放置在称为流变仪的仪器中并被剪切。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。从应用和实验中已知，当粘弹性流体经历剪切时，流动的性质可能以可能导致不可预测的结果的方式急剧变化。这些剧烈的变化被称为不稳定性。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，并产生巨大的经济影响，因此了解导致和维持这些不稳定性的因素对我们来说非常重要。 这样的理解将提供一种方法来预测这种不稳定性何时发生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，从而防止不稳定性。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回答这些问题和其他相关问题。 这项工作将对联邦政府在材料和制造领域的战略举措做出重要贡献。

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.