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.
