Stability analysis of viscoelastic flows

Viscoelastic flow instabilities are very common for polymer processing flows and can give rise to severe defects of the final product. In spite of this frequent occurrence, very little is known about these flow phenomena which are found not only in complex flows but also in simple flows (e.g. Taylor-Couette flow). This thesis is motivated by a viscoelastic flow instability during the injection molding process. The occurrence of unstable flows in injection molding can result in specific surface defects that are characterized by alternating shiny and dull bands perpendicular to the flow direction. These defects, which are sometimes referred to as flow marks, tiger stripes or ice lines, have been observed in injection molding of many polymers and polymer blend systems including PP, LDPE, ASA, PC/ABS. Important questions that are addressed are: What are the critical flow conditions for the onset of instability? and: How does this relate to the rheology of the polymer itself? but probably the most intriguing question is: What is the physical mechanism of the driving force for the secondary flow? This work is a numerical investigation to elucidate these questions and to provide the limits of stability of injection molding flows. Important aspects of the modeling of viscoelastic flows are the choice for the numerical algorithm and rheological model. It is well known that the numerical algorithm is essential for the correct prediction of the stability behavior of complex flows since most numerical schemes produce approximate solutions that are not solutions of the original problem. Also, if the numerical analysis is correct and approximate solutions are solutions of the viscoelastic operator, the outcome of the stability analysis will be dependent on the capability of the nonlinear constitutive model to describe real polymer melts. The stability of a generic fountain flow has been investigated . This flow is considered as a prototype flow for the injection molding process. The analysis is performed using a finite element technique which is based on the DEVSS-¯G spatial discretization. Application of this technique is necessary since this is one of the few spatial approximation schemes that is able to capture the correct eigenmodes for several benchmark problems. In order to model real viscoelastic melts, which are described by discrete spectra of relaxation times, an efficient and accurate time integration technique for the viscoelastic operator has been developed. Due to the presence of the free surface during injection molding, it is necessary to include perturbations of the computational domain into the numerical analysis. These domain perturbations are the result of general disturbances of the velocity field and are expressed in the numerical scheme by local deformations of the fountain flow surface of the flow front. Inclusion of these local deformations of the free surface increases the complexity of the numerical analysis substantially. Therefore, the domain perturbation technique that is developed in this work is benchmarked in shear flows of two superposed fluids where local deformations of the fluid/fluid interface are taken into account. A suitable set of constitutive set of equations needs to be selected for the stability analysis of the fountain flow. Apart from providing accurate predictions for the steady base flows, the constitutive equations should also be able to capture the essential dynamics of the polymer melt. The behavior of several nonlinear rheological models that are able to describe the standard viscometric shear data for most polymers has been investigated for simple shear flows (Couette and Poiseuille flows). Since the base flow solutions do not vary along the streamlines for these flows, the stability behavior can be obtained from the solutions of 1-dimensional generalized eigenvalue problems. This analysis shows that the choice of the constitutive model is not a trivial one since not all nonlinear models are able to capture the complex stability behavior of polymer melts. For example, it is shown that critical Weissenberg numbers exist for the Phan-Thien–Tanner and the Giesekus model for simple shear flows. Due to the excellent results obtained for the linear stability analysis in shear flows and the ability to describe full sets of viscometric data of several branched polymer melts, the eXtended Pom-Pom (XPP) model has been selected for the stability analysis of the fountain flows. For a one mode model with a ‘realistic’ set of parameters it is found that a linear instability sets in at We ~ 3. Also, the occurrence of instability could be postponed when the number of arms in the eXtended Pom-Pom model is increased. This corresponds to a higher degree of branching of the polymer melt and although this has some effect on the shear properties of the different XPP fluids, the major influence of varying the number of arms can be found in the extensional behavior. This indicates that the flows are more stable for fluids with increased strain hardening. The structure of the leading eigenmode turns out to be a swirling flow near the fountain flow surface which is consistent with the experimental observations.