INSTABILITIES IN ELONGATION FLOWS OF POLYMERS AT HIGH DEBORAH NUMBERS A Dissertation Presented to The Graduate Faculty of The University of Akron In Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy Atanas Gagov December, 2007 INSTABILITIES IN ELONGATION FLOWS OF POLYMERS AT HIGH DEBORAH NUMBERS Atanas Gagov Dissertation Approved: Accepted: ______________________________ ______________________________ Advisor Department Chair Dr. Arkadii I. Leonov Dr. Sadhan C. Jana ______________________________ ______________________________ Committee Member Dean of the College Dr. James L. White Dr. Stephen Cheng ______________________________ ______________________________ Committee Member Dean of the Graduate School Dr. Avraam I. Isayev Dr. George R. Newkome ______________________________ ______________________________ Committee Member Date Dr. Alex Povitsky ______________________________ Committee Member Dr. Gerald W. Young ii ABSTRACT The objective of this work is to study the instabilities of the contraction flows at high Deborah numbers of various polymers. The following topics will be discussed: 1. Linear and nonlinear stability analysis of isothermal fiber spinning 2. Linear stability analysis of nonisothermal fiber spinning 3. Linear and nonlinear stability analysis of contraction flow 4. Propagation of reservoir instabilities in capillary The analysis of these flow problems requires solution of the closed set of PDE’s (or ODE’s), consisting of equations for conservation of mass and momentum, along with an adequate viscoelastic constitutive equation, with appropriate initial/boundary conditions. The goal of this work is to demonstrate a procedure for determining the critical regime beyond which the process becomes unstable and also to determine weather the process is stable when the disturbances grow to a finite size. Linear and non-linear stability theories have been used to describe the fluctuations of fiber spinning and contraction flow. Linear stability analysis determines the onset of the instabilities of the process while nonlinear analysis establishes the complete range of the stable and unstable conditions. The melt fiber spinning is the most common of polymer fiber processing. Finding critical process conditions and the stabilizing effect of the cooling is described in this iii work. The critical draw ratio is established using linear stability analysis and the effect of the finite size imposed disturbances is studied through nonlinear stability analysis. Contraction flow is one of the benchmark problems in computational polymer fluid mechanics and polymer processing. In this modeling, the whole flow region is divided in naturally introduced sub-regions with well-known and highly simplified types of flow. Thus, the model analyzes the entire flow region in a simplified geometric manner with properly matched conditions between adjacent sub-regions. The propagation of the disturbances formed in the reservoir region has been analyzed. Employing the isothermal “Jet approach” followed by linearized perturbation approximation of the governing equations for finding the onset of the instabilities supplies information about the stability of the contraction flow which has been used to describe the mechanism of propagation of the disturbances into capillary up to the die exit, and its numerical implementation. iv TABLE OF CONTENTS Page LIST OF TABLES……………………………………………………………………..….x LIST OF FIGURES………………………………………………………………………xi CHAPTER I. INTRODUCTION ........................................................................................................ 1 II. LITERATURE REVIEW ............................................................................................. 4 2.1 Importance of Stability in Polymer Processing ...................................................... 4 2.2 Isothermal Fiber Spinning....................................................................................... 7 2.3 Fiber Spinning with Cooling Temperature Gradient ............................................ 11 2.4 Non-linear Dynamics of Isothermal Fiber Spinning............................................. 14 2.5 Flows of Polymer Melts in the Capillary Exit Region and the Die Swell ............ 15 2.6 Viscoelastic Constitutive Equations for Polymer Melts and Solutions ................ 17 2.7 Instabilities in Contraction Flow at High Deborah Numbers ............................... 19 2.8 Numerical Studies of Polymer Flows in Channels and Dies ................................ 26 2.9 Significance of Constitutive Equation .................................................................. 28 III. REOLOGICAL BACKGROUND AND CONTRACTION FLOW MODEL........... 32 3.1 Rheological Characterizations of Polymers.......................................................... 32 3.2 The Choice of Constitutive Equation.................................................................... 33 3.3 Development of the Model ................................................................................... 34 v 3.4 Set of Equations for Solving Inherently Difficult Viscoelastic Flow Problems... 38 3.5 The Choice of the Numerical and Iteration Scheme............................................. 39 IV. ISOTHERMAL FIBER SPINNING........................................................................... 42 4.1 Linear Stability Analysis of Isothermal Fiber Spinning ....................................... 42 Process description ............................................................................................... 42 Nondimensional Governing Equations................................................................. 45 Initial and Boundary conditions............................................................................ 46 Steady State Solution............................................................................................ 47 Linear Stability Analysis for Small Disturbances................................................. 49 Numerical Approach............................................................................................. 51 Accuracy of the Numerical Scheme ..................................................................... 54 Results and Discussions........................................................................................ 54 4.2 Nonlinear Stability Analysis of Isothermal Fiber Spinning.................................. 58 Governing Equations ............................................................................................ 59 Initial and Boundary conditions............................................................................ 60 Numerical Approach............................................................................................. 61 Accuracy of the Numerical Scheme ..................................................................... 62 Results and Discussions........................................................................................ 63 V. FIBER SPINNING WITH COOLING TEMPERATURE GRADIENT.................... 67 5.1 Linear Stability Analysis of Nonisothermal Fiber Spinning................................. 67 Process description ............................................................................................... 67 Viscosity Dependence on Temperature…………………………………….........71 Nondimensional Governing Equations................................................................. 72 vi Boundary conditions............................................................................................. 75 Steady State Solution............................................................................................ 75 Linear Stability Analysis for Small Disturbances................................................. 78 Numerical Approach............................................................................................. 80 Accuracy of the Numerical Scheme ..................................................................... 83 Results and Discussions........................................................................................ 83 VI. INSTABILITIES IN HIGH DEBORAH NUMBER CONTRACTION FLOWS...... 91 6.1 Linear Stability Analysis of Contraction Flow ..................................................... 91 Process description ............................................................................................... 92 Flow in the Far Field Entrance Region................................................................. 94 Flow in the Near Field Entrance Region - “Jet Approach” .................................. 96 Matching Condition at the Boundary of two Flow Regions................................. 99 Steady State Solution in Near Entrance Region ................................................. 100 Steady State Solution for Die Developing Flow................................................. 101 Linear Stability Analysis for Small Disturbances............................................... 105 Numerical Approach........................................................................................... 107 Accuracy of the Numerical Scheme ................................................................... 107 Results and Discussions...................................................................................... 109 6.2 Non-linear Stability Analysis of Contraction Flow ............................................ 112 Governing Equations .......................................................................................... 112 Initial and Boundary