Analysis of Non-Newtonian and Two-Phase Flows Zur Erlangung des Doktorgrades der Naturwissenschaften von der Fakult¨at fur¨ Elektrotechnik, Informatik und Mathematik der Universit¨at Paderborn genehmigte Dissertation Dipl.-Math. Kerstin Wielage Tag des Kolloquiuums : 21. April 2005 Gutachter : HD Dr. rer. nat. Dieter Bothe, Paderborn Prof. Dr. rer. nat. Jan Pruß,¨ Halle (Saale) I would especially like to express my gratitude to the following persons: HD Dr. rer. nat. Dieter Bothe, for serving as my supervisor, giving me the opportunity to write this thesis. This dissertation would not have been possible without his invaluable advice, numerous discussions and hints, and his continuous support, Prof. Dr. rer. nat. Jan Pruss,¨ for agreeing to serve as my co-supervisor and particularly for the constructive discussions and suggestions, my colleagues Dr. rer. nat. Stephan Blazy and Dr. rer. nat. Nils Lessmann for their endless time, inspiration, and incredible support which made working with them enjoy- able, Prof. Dr. rer. nat. Odej Kao, the managing director of the Paderborn Center of Parallel Computing where I was employed, in particular, for the opportunity to spend so much time on writing my thesis during the previous year, Prof. Dr. rer. nat. Raimund Rautmann for his continuous effort in guiding and sup- porting my studies, and all my co-workers at the Paderborn Center for Parallel Computing for the great working atmosphere. Special thanks to the administrators for the consistent help with all problems concerning the computer cluster. Contents 1 Introduction 1 2 Mathematical Modeling of Non-Newtonian Fluids 5 2.1 Motivation . 5 2.2 Notation . 6 2.3 Balance Equations . 10 2.4 The Total Stress Tensor . 12 2.5 Constitutive Equations . 15 2.5.1 Rate of Deformation Tensor . 16 2.5.2 Rate of Shear . 19 2.5.3 General Principles for Constitutive Equations . 20 2.5.4 Incompressible Pure Viscous Fluids . 21 2.5.5 Other Models . 23 2.6 Conclusion . 26 3 Local Strong Solvability of Non-Newtonian Flow Problems 29 3.1 Notations . 30 3.2 Preliminaries . 30 3.2.1 Sectorial Operators . 34 3.2.2 -Boundedness and Fourier Multipliers . 36 R 3.2.3 -calculus . 40 H1 3.3 Existence Result for Generalized Newtonian Flows . 44 3.3.1 Strong Ellipticity . 46 3.3.2 The Linear Problem with Constant Coefficients . 48 3.3.3 The Linear Problem with Variable Coefficients . 54 3.3.4 The Nonlinear Problem . 63 3.4 Related Work . 68 4 Numerical Analysis of Two-phase Flows 73 4.1 Numerical Treatment of the Navier-Stokes Equations . 74 4.1.1 Boundary Conditions . 74 ii CONTENTS 4.1.2 General Time Splitting Technique . 75 4.1.3 Finite Volume Method . 78 4.2 Numerical Treatment of Two-phase Flows . 80 4.2.1 Notations for Two-phase Flows . 80 4.2.2 Mathematical Modeling of Surface Tension . 80 4.2.3 Interfacial Dirac-Distribution . 82 4.2.4 Interfacial Jump Conditions . 85 4.2.5 Numerical Treatment of the Interface . 89 4.2.6 Interfacial Area . 93 4.2.7 Surface Tension . 96 4.3 Binary Droplet Collision: Experiment and Simulation . 99 4.3.1 Experimental Setup . 100 4.3.2 Admissibility of the Viscosity Function . 101 4.3.3 Numerical Setup and Results . 103 4.4 Existence of Strong Solutions for Two Phase Flows . 108 List of Figures 1.1 Weissenberg Effect . 1 2.1 Rotating Lid Effect . 6 2.2 Extrudate Swell [BAH77]. (left) Newtonian. (right) Non-Newtonian. 6 2.3 Simple Steady Shear Flow . 13 2.4 Stress tensor. 14 2.5 Meaning of the components of D. 18 2.6 Superposition of Deformation and Rotation . 20 2.7 Stress S21 versus shear rate. 21 2.8 Relations among rheological equations of state [BAH77]. 27 4.1 Decomposition of domain Ω in N control volumes Ωi, i = 1; : : : ; N. 78 4.2 Staggered Grid. 79 4.3 Force on a line element ds on the interface Γ (t). 81 4.4 Domain decomposition for the derivation of the jump conditions. 86 4.5 Illustration of the VOF-variable f. 91 4.6 Transition zone introduced by Brackbill . 97 4.7 Experiment of a binary droplet collision [MRB+02] . 100 4.8 Viscosity of the experimental liquids. 101 4.9 Simulation of the binary droplet collision using non-Newtonian fluids . 104 4.10 Collision Complex Diameter versus Time . 105 4.11 Illustration of overwriting the VOF-variable. 106 4.12 Comparison of the viscosity and shear rate distribution. 107 4.13 Shear rate distribution in connection with the droplet shape . 108 Chapter 1 Introduction Fluids can be classified according to various criteria. One opportunity is to distinguish Newtonian and non-Newtonian fluids. The difference between both classes of fluids can be observed in a variety of situations. For example, consider two bowls containing two different types fluids, e.g., water and a polymer solution representing a Newtonian and a non-Newtonian fluid, respectively. Inserting a rotating rod in each bowl (see figure 1.1), we observe that in case of the Newtonian fluid, a characteristic dip arises near the rotating rod due to centrifugal forces pushing the fluid outwards. In contrast, some non-Newtonian fluids climb up along the rod. This effect is known as the Weissenberg effect [BAH77], [Tan88]. (a) Newtonian (b) Non-Newtonian Figure 1.1: Weissenberg Rod Climbing Effect [BAH77]. 2 CHAPTER 1. INTRODUCTION Particularly, in industrial applications non-Newtonian fluids are of great interest, since many used fluids, e.g., lacquers or polymer solutions show non-Newtonian effects. In general, real-world experiments of industrial processes can be complex, cost-inten- sive, and time-consuming. Therefore, numerical simulations are important and can be useful in order to optimize the process with respect to cost or quality aspects. Thus, the investigation of mathematical models governing the flow of non-Newtonian fluids is important regarding both analysis and numerical simulations. In the context of this thesis, the term analysis in the title refers to both the meaning in the mathematical sense and the more informal meaning of "computational analysis", i.e., the numerical investigation of the behavior of non-Newtonian fluids. Non-Newtonian fluids are characterized by different features, such as viscosity, elas- ticity, or memory effects. An important feature of polymeric liquids is the fact that their viscosity changes with the shear rate, so-called generalized Newtonian fluids on which this thesis is focused on. The mathematical model describing the motion of these fluids is given by the system @ u + u u = div S p + f; on [0; T ] Rn @t · r − r × div u = 0; on [0; T ] Rn (1.1) Rn × u t=0 = u0; on : j Here, the stress tensor S is given by S = 2 µ( D 2)D with the rate-of-deformation tensor k k D = 1 u + ( u)T , and the viscosity function µ depending on D 2 respectively on 2 r r k k the shear rate γ_ = p2 D , where D denotes the Hilbert-Schmidt norm. k k k k The main result of this thesis is the proof of existence of problem (1.1) in the maximal Lp-regularity class. By means of maximal Lp-regularity, local (in time) strong well- posedness of this model is obtained under certain restrictions concerning the viscosity function. For example, for the viscosity function 2 2 m−2 µ( D ) = µ (1 + D ) 2 k k 0 k k with µ0 > 0 which is often used in the mathematical literature, we obtain local existence Rn 3n 4 in ; (n > 1) for m > 2(n−1) , i.e., in the 3D case m > 5=4 is sufficient. We emphasize that according to the engineering− literature the range of interest concerning m is m > 1. In the numerical part, the emphasis is on two-phase flows, since many interesting problems appear in this context. Moreover, we compare experiment and simulation of a binary droplet collision using non-Newtonian fluids, the behavior of which is assumed to be that of generalized Newtonian fluids. Furthermore, assuming the situation of system (1.1), we show the admissibility of the viscosity function used in the simulation. The remainder of the thesis is organized as follows. Starting from the physical sci- ence with some general notations and introducing the meaning of non-Newtonian fluids, we obtain an insight into the variety and the modeling of non-Newtonian fluids. The main subject of chapter 2 is the development of constitutive equations, i.e., of relations between the velocity field u and the stress tensor S which depend on the used sub- stance. The simplest constitutive equation for an incompressible viscous Newtonian fluid 3 is given by the linear relation S = 2µD with the constant dynamical viscosity µ and the rate-of-deformation tensor D. Among others, we find out the constitutive equations for describing the Weissenberg effect (see figure 1.1). In particular, we obtain the consti- tutive equations for generalized Newtonian fluids, which result from the linear relation for incompressible viscous Newtonian fluids by replacing the constant viscosity µ by a function depending on the shear rate γ_ . In chapter 3, we concentrate on generalized Newtonian fluids and develop the proof of existence and uniqueness of local (in time) strong solutions of problem (1.1) by means of maximal Lp-regularity. The main result is stated in theorem 3.3.1. We close this chapter with a survey about some results given in mathematical literature and compare our existence result. Chapter 4 completes the expositions about non-Newtonian fluids with a look at the numerical modeling of two-phase flow.