Mathematical Sciences: Modelling, Analysis and Computation in Viscoelasticity

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Intriguing phenomena have been observed experimentally in highly elastic and very viscous fluids, such as polymer solutions and melts. Such non-Newtonian materials can be modeled mathematically as viscoelastic fluids with fading memory, which exhibit behavior intermediate between the nonlinear hyperbolic response of purely elastic materials and the strongly diffusive, parabolic response of viscous Newtonian fluids. In certain flow regimes, these fluids exhibit instabilities that severely disrupt polymer processing. Laboratory observations have found "spurt " instabilities in pressure-gradient driven flows (Vinogradov et al., 1972), persistent oscillations in flow at fixed volumetric flow rate (Lim & Schowalter, 1989), and anomalies in step shear strain experiments (Morrison & Larson, 1991). Many researchers attribute the observations to "slip" or "apparent slip, " i.e., loss of adhesion of the fluid to the wall. This project involves the investigation of an alternative explanation for these phenomena. The hypothesis is that all three have a common origin in bulk material properties, rather than adhesive properties. To test this hypothesis, the corresponding one-dimensional shear flows pressure-driven and piston-driven flow in a slit die, and Couette flow are modeled. The characteristic feature of the fluid models employed is a non-monotone relation between steady shear stress and strain rate. Analysis and numerical simulations show that the polymer system changes state in a thin layer near the wall, giving the appearance of a slip layer. The structure of solutions of initial/boundary-value problems for non-monotone models is remarkably rich, and the development of numerical methods capable of simulating these flow problems has interest in its own right. The same basic system of time-dependent, quasilinear partial differential equations is used to model all three experiments; different forcing terms, as well as boundary and initial conditions, are used in the three cases. The governing system is globally well-posed in time, in two senses: with respect to classical solutions arising from smooth initial data, and with respect to "almost classical" solutions, containing discontinuities in the stress and strain rate. Calculated solutions are in good qualitative agreement with the experiments, showing spurt in pressure-driven flow, persistent oscillations in piston-driven flow, and the development of anomalies in the relaxation modulus after a step strain. In terms of a reduced approximating system, spurt and its related phenomena in pressure-driven flow can be analyzed by using phase-plane techniques. The analysis for piston-driven and step strain experiments is more difficult because the system is infinite-dimensional. For example, numerical simulations strongly suggest that there is a Hopf bifurcation in piston-driven flow, leading to the observed persistent oscillations. In order to compare predictions to experiment, it is crucial to determine how the amplitude and frequency of the limit cycle depends on physical parameters. This requires detailed analysis of the governing equations. This is required in each of the experiments, and that is the central focus of this research. The theme of this project has been and will continue to be the use of numerical simulation to guide analysis of the governing equations and approximations to them, in order to identify the time-scales, amplitudes, and other characteristic features of the predictions of the model.