Finite Element Simulations of Steady Viscoelastic Free-Surface Flows by Todd Richard Salamon B. S. Chemical Engineering, University of Connecticut, Storrs (1989) B. S. Chemistry, University of Connecticut, Storrs (1989) Submitted to the Department of Chemical Engineering in partial fulfillment of the requirements for the degree of Doctor of Philosophy at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY September 1995 Instituteof TeLhnology 1995. All rights reserved. @ Massachusetts Institute of Technology 1995. All rights reserved. Author ............... ....... ...... ...... ... ........ Department of Chemical Engineering Septerrper 26, 1995 Certified by ........... ................ .. ......... Robert A. Brown Professor of Chemical Engineering Thesis Supervisor Certified by.... .......... ... ... ... .... ... ... ... .... .. y..... .. Robert C. Armstrong Professor of Chemical Engineering Thesis Supervisor Accepted by .... ;,iASSACHUSETTS INSTITUT E Robert E. Cohen OF TECHNOLOGY Graduate Officer, Department of Chemical Engineering MAR 2 2 1996 ARCH'IVaS LIBRARIES Finite Element Simulations of Steady Viscoelastic Free-Surface Flows by Todd Richard Salamon Submitted to the Department of Chemical Engineering on September 26, 1995, in partial fulfillment of the requirements for the degree of Doctor of Philosophy Abstract Many polymer processing flows, such as the spinning of a nylon fiber, have a free sur- face, which is an interface between the polymeric liquid and another fluid, typically a gas. Along with the free surface and the complex rheological behavior of the poly- meric liquid these flows are further complicated by wetting phenomena which occur at the attachment point of this free surface with a solid boundary where a three-phase juncture, or contact line, is formed. This thesis is concerned with developing accurate and convergent finite element simulations of steady, viscoelastic free-surface flows in domains with and without contact lines. The flow of a liquid film down an inclined solid surface is chosen as a model free- surface flow problem without contact lines. Kapitza (1965) was the first to observe experimentally that the base flow of a flat film became unstable to trains of periodic and solitary-like waves which propagated down the film at constant speed and with- out significantly changing their shape. A large number of subsequent experimental, analytical and numerical work has been devoted to understanding the variety of wave profiles which can appear on the surface of the film. A comprehensive study of the validity of various theories for Newtonian film flow which are simplifications of the full Navier-Stokes equations is presented in this thesis. The shape and flow field of a train of periodic waves propagating at a constant speed serves as the model test problem for comparison of these theories with finite element simulation of the Navier-Stokes equations. Long-wave theories are shown to yield quantitative comparisons with the finite element simulations for small amplitude waves, but deviate qualitatively when the wave amplitude exceeds approximately 10 per cent of the average film thickness. Boundary layer theories do not suffer from such amplitude restrictions but are shown to differ qualitatively from the finite element predictions for small values of the surface tension. The effect of viscoelasticity on the film flow is determined by considering the flow of a viscoelastic fluid described by the Oldroyd-B model. Calculations using the EVSS method of Rajagopalan et al. (1990b) confirm earlier predictions of Gupta (1967) and Shaqfeh et al. (1988) that viscoelasticity destabilizes the flow at low values of the Reynolds number, significantly increasing wave amplitudes and wave speeds. In fact, for a non-vertical film, where the Newtonian flat film is always stable, a purely elastic instability is observed, in agreement with the predictions of Shaqfeh et al. (1988). The extrusion of a fluid from a planar die is chosen as a model problem of a free-surface flow which possesses a contact line. At the attachment point of the free surface with a sharp die exit the sudden change in boundary condition gives rise to a singularity in the stress fields. This singularity has been the source of computational difficulties in both Newtonian and viscoelastic extrusion flow calculations. Mesh refinement techniques are used to explore the solution structure near this singularity on length scales three to four orders of magnitude smaller than previous calculations, and to make fairly precise statements about the flow field in this region. For the extrusion of a Newtonian fluid the finite element results indicate that the meniscus attaches to the die exit at an angle determined by coupling with the global flow field and with infinite curvature as the die lip is approached, confirming an earlier hypothesis of Schultz & Gervasio (1990). Allowing the fluid to slip along the die wall in accordance with Navier's slip law (1827) changes the stress singularity from geometric to logarithmic for the case of large (infinite) surface tension. For the extrusion of a Newtonian fluid with finite surface tension and slip the stress field at the die exit is shown to be more singular than the no-slip case. This behavior is explained by considering several model flow problems that use various combinations of no-slip, slip and shear-free boundary conditions, where it is shown that the slip surface behaves asymptotically like a shear-free surface at leading order. The extrusion of a viscoelastic fluid described by the Giesekus model is determined using the EVSS-G method of Brown et al. (1993). The finite element calculations are shown to be convergent with mesh refinement and to yield integrable stresses at the flow singularity at the die exit. Replacing the sharp die exit with a rounded exit and allowing the fluid to wet the die face eliminates the stress singularity. The sharp and rounded exits yield similar predictions for the local flow fields and the total swell for moderate values of the elasticity and surface tension. However, for small values of the surface tension the existence of solutions in the rounded exit case is shown to be sensitive to the details of the die geometry and attachment point of the contact line, in agreement with experimental results of McKinley (1995). Thesis Supervisor: Robert A. Brown Title: Professor of Chemical Engineering Thesis Supervisor: Robert C. Armstrong Title: Professor of Chemical Engineering Acknowledgments There are many people whom I would like to thank for making this thesis pos- sible. First, I would like to thank my advisors Bob Brown and Bob Armstrong for their guidance and support during the course of this research. Their extensive knowl- edge of polymer fluid mechanics and numerical simulations has contributed greatly to this work. I would like to thank Professors Bill Deen, Anthony Patera, and Gareth McKinley and Dr. David Bornside for serving as members of my thesis committee. I am indebted to Dr. David Bornside for his development of a research tool that has allowed me to investigate a variety of interesting and challenging research prob- lems. I would also like to acknowledge the National Science Foundation and the E. I. Dupont de Nemours company for providing financial support for this research and the Pittsburgh Supercomputing Center for providing computational resources. I would like to thank those in 66-246 - Paul, Paul, Gareth, Dilip, Walt, and Aparna early on, and Jeff, Lars, Mike, Radha, Alice, Junji, and Hua more recently, who have made the past six years enjoyable. Thank you to the support staff - Carol, Nancy, Linda, Janet and Elaine, for keeping this place functioning in a seamless fashion. I would particularly like to thank Arline Benford for being the wonderful person that she is. I am going to truly miss her. I have had the good fortune of making several lasting friendships during the past six years at MIT. I would like to thank Ed Browne, Lloyd Johnston, Tom Quinn, Michael Kezirian and Suresh Ramalingam for their friendship and for providing many fond memories. I would like to thank my parents and my brother for their support and encour- agement. I am forever grateful to them for all that they have done for me. Finally, I would like to thank Christine for her love, encouragement and support during the course of this work. Contents 1 Introduction 35 2 Governing Equations 46 2.1 Conservation Equations ......................... 46 2.2 Non-Newtonian Fluids .......................... 49 2.2.1 Shear-thinning viscosity ................... .. 49 2.2.2 Normal Stress Differences .................... 51 2.2.3 Secondary Flows ......................... 52 2.3 Shear and shearfree flows ......................... 53 2.3.1 Shear Flow .. .. .. ... ..... .. .. ... .. 54 2.3.2 Shearfree Flows .......................... 54 2.4 Non-Newtonian Constitutive Equations . ................ 56 2.5 Rheology of Polyisobutylene Solutions . ................ 66 3 Finite-Element Method 69 3.1 A Two-Point Boundary-Value Problem . ................ 70 3.2 Stokes Flow . .. .. .. 81 3.3 Hyperbolic Equations ........................... 83 3.4 Newton's Method and Solution Tracking . ............... 85 3.5 Computational Issues ........................... 90 3.6 Treatment of Free-Surface Problems . .................. 91 3.7 Local Mesh Refinement .......................... 96 3.8 Simulations in Domains with Flow Singularities ...... ...... 99 4 Simulation with Viscoelastic Constitutive Equations 102 4.1 Type Analysis ....