Mathematical Models in the Manufacturing of Glass: C.I.M.E. Summer School, Montecatini Terme, Italy 2008 (Lecture Notes in Mathematics C.I.M.E. Foundation Subseries)

Lecture Notes in Mathematics Editors: J.-M. Morel, Cachan F. Takens, Groningen B. Teissier, Paris 2010 Fondazione C.I...

Author: Angiolo Farina | Axel Klar | Robert M.M. Mattheij | Andro Mikelic | Norbert Siedow

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Lecture Notes in Mathematics Editors: J.-M. Morel, Cachan F. Takens, Groningen B. Teissier, Paris

2010

Fondazione C.I.M.E., Firenze C.I.M.E. stands for Centro Internazionale Matematico Estivo, that is, International Mathematical Summer Centre. Conceived in the early fifties, it was born in 1954 in Florence, Italy, and welcomed by the world mathematical community: it continues successfully, year for year, to this day. Many mathematicians from all over the world have been involved in a way or another in C.I.M.E.’s activities over the years. The main purpose and mode of functioning of the Centre may be summarised as follows: every year, during the summer, sessions on different themes from pure and applied mathematics are offered by application to mathematicians from all countries. A Session is generally based on three or four main courses given by specialists of international renown, plus a certain number of seminars, and is held in an attractive rural location in Italy. The aim of a C.I.M.E. session is to bring to the attention of younger researchers the origins, development, and perspectives of some very active branch of mathematical research. The topics of the courses are generally of international resonance. The full immersion atmosphere of the courses and the daily exchange among participants are thus an initiation to international collaboration in mathematical research. C.I.M.E. Director Pietro ZECCA Dipartimento di Energetica “S. Stecco” Università di Firenze Via S. Marta, 3 50139 Florence Italy e-mail: [email protected]

C.I.M.E. Secretary Elvira MASCOLO Dipartimento di Matematica “U. Dini” Università di Firenze viale G.B. Morgagni 67/A 50134 Florence Italy e-mail: [email protected]

For more information see CIME’s homepage: http://www.cime.unifi.it CIME activity is carried out with the collaboration and financial support of: – EMS - European Mathematical Society

Angiolo Farina · Axel Klar · Robert M.M. Mattheij Andro Mikelić · Norbert Siedow

Mathematical Models in the Manufacturing of Glass C.I.M.E. Summer School, Montecatini Terme, Italy 2008 Editor: Antonio Fasano

ABC

Editor Antonio Fasano Università degli Studi di Firenze, Dipartimento di Matematica “Ulisse Dini” Viale Morgagni 67/A I-50134 Firenze Italy [email protected] Authors: see List of Contributors

ISBN: 978-3-642-15966-4 e-ISBN: 978-3-642-15967-1 DOI: 10.1007/978-3-642-15967-1 Springer Heidelberg Dordrecht London New York Lecture Notes in Mathematics ISSN print edition: 0075-8434 ISSN electronic edition: 1617-9692 Mathematics Subject Classification (2010): 76D05, 76D07, 76B10, 76B45, 76M10, 80A20, 80A23, 35K05, 35K60, 35R35 c Springer-Verlag Berlin Heidelberg 2011 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: SPi Publisher Services Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

Preface

The EMS-CIME Course on Mathematical Models in the Manufacturing of Glass, Polymers and Textiles was held in Montecatini Terme (Italy) from September 8 to September 19, 2008. The course was co-directed by John Ockendon (OCIAM, Oxford, UK) and myself. The following topics were treated: (1) Nonisothermal flows and fibres drawing (Angiolo Farina and Antonio Fasano, Univ. Firenze, Italy, Andro Mikelic, Univ. Lyon, France) (*) (2) The mathematics of glass sheets and fibres (Peter Howell, OCIAM, Oxford, UK) (3) Radiative heat transfer in glass industry: modelling, simulation and optimisation (Axel Klar and Norbert Siedow, ITWM – Fraunhofer, Kaiserslautern Germany) (*) (4) Modelling and simulation of glass forming processes (Robert Mattheij, TU Eindhoven, The Netherlands) (*) (5) Injection moulding (Hilary Ockendon, OCIAM, Oxford, UK) (6) Fibre assembly modelling (Hilary Ockendon, OCIAM, Oxford, UK) (7) The mathematics of the windscreen sag process (John Ockendon, OCIAM, Oxford, UK) The focus was largely on glass manufacturing processes, with some digression to polymers and textile fibres in a context very close to the area of glass manufacturing. This volume collects the lecture notes of the courses marked with (*), all devoted to problems in glass industry. It is regrettable that the other lecturers could not provide a chapter, because the subjects they illustrated were extremely interesting. John Ockendon presented a fascinating and quite difficult problem: the production of a windscreen by the natural bending under gravity of a still soft glass layer clumped at the boundary. The audience was very excited by his colourful explanation of the underlying mechanics, making use of any deformable object he had at hand. Hilary Ockendon posed stimulating questions about injection moulding and the “flow” of fibres in a fluffy tuft subject to traction. We had exciting afternoon sessions discussing such problems.

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Peter Howell gave a series of lectures on the manufacturing of glass sheets and fibres which provided an excellent complementary view of the subjects treated by Farina, Fasano, Mikelic. Indeed he addressed different problems in the same area (e.g.: how to get a fibre of a desired cross section), each with a different mathematical approach. Fortunately most of the material not included in this volume is retrievable on the CIME web site, either in the form of slides or of excerpts from books. Altogether the Course presented a remarkable review of quite advanced technological problems in the glass industry and of the mathematics involved. It was quite amazing to realize that such a seemingly small research area is on the contrary extremely rich and it calls for an impressively large variety of mathematical methods. Despite the fact that the volume is not collecting all the material presented at the Course, it deals with a number of problems which are very typical in the field of glass manufacturing and it can certainly be useful not only to applied mathematicians, but also to physicists and engineers, who can find in it an overview of the most advanced models and methods. The Chapter by J.A.W.M. Groot, R.M.M. Mattheij, and K.Y. Laevsky illustrates the various processes of glass forming, starting from the basic physical information, developing the mathematical models for each process, and analyzing the procedures of numerical computations. Then we have two Chapters on radiative heat transfer in glass. The first one is by M. Frank and A. Klar, treating in detail the physics of radiation in glass and various approximated methods to model it, with an eye to numerical complexity. This is a quite substantial piece of work, due to the extension and the intrinsic difficulty of the problem. It is followed by the contribution of N. Siedow, who, after continuing the investigation of numerical methods for heat transfer problems including radiation and convection, passes to a question of great importance: the measurement of glass temperature from the observation of the spectrum of emitted optical radiation. From the mathematical point of view this is formulated as an inverse problem, which is typically ill posed. The way of circumventing this difficulty is explained in detail and examples are provided. The last Chapter, by A. Farina, A. Fasano, A. Mikelic, deals with the industrial process of glass fibre drawing, which goes through several stages having different thermal and mechanical characterizations, and analyzes in general non-isothermal motions of viscous fluids which are mechanically incompressible and thermally expansible. I must abstain from commenting the scientific level of the present volume, since I am among the contributors, but at least I wish to express my deep gratitude to the Authors for their valuable work. Finally, also on behalf of John Ockendon, I wish to thank EMS and CIME for having made this Course possible. A particular thank to the Secretary of CIME, Prof. Elvira Mascolo, who took care of so many details. Antonio Fasano

Contents

Mathematical Modelling of Glass Forming Processes . . . . . . . .. . . . . . . . . . . . . . . . . J.A.W.M. Groot, Robert M.M. Mattheij, and K.Y. Laevsky 1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 1.1 Glass Forming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 1.2 Process Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 1.3 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 2 Mathematical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 2.1 Geometry, Problem Domains and Boundaries . .. . . . . . . . . . . . . . . . . 2.2 Balance Laws .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 3 Parison Press Model .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 3.1 Mathematical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 3.2 Slender-Geometry Approximation . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 3.3 Motion of the Plunger . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 3.4 Simulation Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 3.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 4 Blow Model .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 4.1 Mathematical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 4.2 Glass-Air Interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 4.3 Variational Formulation . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 4.4 Simulation Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 4.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 5 Direct Press Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 5.1 Mathematical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 5.2 Simulation Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 5.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . References . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . Radiative Heat Transfer and Applications for Glass Production Processes . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . Martin Frank and Axel Klar 1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 2 Radiative Heat Transfer Equations for Glass . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 2.1 Fundamental Quantities . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .

1 2 2 6 7 8 8 9 16 16 18 23 26 30 31 32 36 39 42 45 46 47 49 51 53

57 57 58 59 vii

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2.2 Blackbody Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 2.3 The Transfer Equation.. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 2.4 Overall Energy Conservation . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 2.5 Boundary Conditions .. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 2.6 Summary .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 3 Direct Numerical Methods .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 3.1 Ordinates and Space Discretizations . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 3.2 Linear System Formulation . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 3.3 Preconditioning Techniques.. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 3.4 A Fast Multilevel Preconditioner . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 3.5 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 4 Higher-Order Diffusion Approximations .. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 4.1 Asymptotic Analysis and Derivation of the SPN Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 4.2 Boundary Conditions for SPN Approximations .. . . . . . . . . . . . . . . . . 5 Moment Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 5.1 Spherical Harmonics . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 5.2 Minimum Entropy Closure.. . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 5.3 Flux-Limited Diffusion and Entropy Minimization . . . . . . . . . . . . . 5.4 Partial Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 5.5 Partial Moment PN Closure . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 5.6 Partial Moment Entropy Closure . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 6 Frequency-Averaged Moment Equations .. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 6.1 Entropy Minimization .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 6.2 Inversion of the System . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 6.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 7 Numerical Comparisons.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 7.1 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 7.2 Grey Transport .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 7.3 Grey Cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 7.4 Multigroup Transport.. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 7.5 Multigroup Cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 7.6 Adaptive methods for the Simulation of 2-d and 3-d Cooling Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . References . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .

61 62 65 66 68 70 71 73 77 82 85 92 94 100 104 105 107 108 110 112 113 115 116 117 118 119 119 119 120 123 125 125 131

Radiative Heat Transfer and Applications for Glass Production Processes II . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .135 Norbert Siedow 1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 135 2 Models for Fast Radiative Heat Transfer Simulation . . . . .. . . . . . . . . . . . . . . . . 137 2.1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 137 3 Indirect Temperature Measurement of Hot Glasses. . . . . . .. . . . . . . . . . . . . . . . . 148 3.1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 148

Contents

The Basic Equation of Spectral Remote Temperature Sensing . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 3.3 Some Basics of Inverse Problems . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 3.4 Spectral Remote Sensing .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 3.5 Reconstruction of Initial Temperature .. . . . . . . . . .. . . . . . . . . . . . . . . . . 3.6 Conclusions .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . References . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .

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149 150 159 161 170 170

Non-Isothermal Flow of Molten Glass: Mathematical Challenges and Industrial Questions . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .173 Angiolo Farina, Antonio Fasano, and Andro Mikelić 1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 173 2 Mathematical Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 176 2.1 Definitions and Basic Equations .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 176 2.2 Fluids Physical Properties and Constitutive Equations . . . . . . . . . . 177 2.3 The General Model .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 181 2.4 Scaling and Dimensionless Formulation . . . . . . . .. . . . . . . . . . . . . . . . . 183 3 Study of the Stationary Non-Isothermal Molten Glass Flow in a Die . . . . 187 3.1 Existence and Uniqueness Result for the Stationary Problem.. . 189 3.2 Oberbeck–Boussinesq Model .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 194 4 Modelling the Viscous Jet at the Exit of the Die . . . . . . . . . .. . . . . . . . . . . . . . . . . 198 4.1 Definition of L and Jet’s Profile at the End of Stage (c) . . . . . . . . . 201 5 Terminal Phase of the Fiber Drawing.. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 207 5.1 Derivation of the Model of Matovich–Pearson for the Thermal Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 209 5.2 Solvability of the Boundary Value Problems for the Stationary Effective Equations.. . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 216 References . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 223 List of Participants .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 225

Contributors

Angiolo Farina Università degli Studi di Firenze, Dipartimento di Matematica “Ulisse Dini”, Viale Morgagni 67/A, I-50134 Firenze, Italy [email protected] Antonio Fasano Università degli Studi di Firenze, Dipartimento di Matematica “Ulisse Dini”, Viale Morgagni 67/A, I-50134 Firenze, Italy [email protected] Martin Frank University of Kaiserslautern, Erwin-Schrödinger-Strasse, 67663 Kaiserslautern, Germany [email protected] J.A.W.M. Groot Department of Mathematics and Computer Science, Eindhoven University of Technology, PO Box 513, 5600 MB Eindhoven, The Netherlands [email protected] Axel Klar University of Kaiserslautern, Erwin-Schrödinger-Strasse, 67663 Kaiserslautern, Germany and Fraunhofer ITWM, Fraunhofer Platz 1, 67663 Kaiserslautern, Germany [email protected]á K.Y. Laevsky [email protected] Robert M.M. Mattheij Department of Mathematics and Computer Science, Eindhoven University of Technology, PO Box 513, 5600 MB Eindhoven, The Netherlands [email protected] Andro Mikelić Université de Lyon, Lyon, F-69003, FRANCE; Université Lyon 1, Institut Camille Jordan, UMR 5208 CNRS, Bât. Braconnier, 43, Bd du onze novembre 1918 69622 Villeurbanne Cedex, FRANCE [email protected] Norbert Siedow Fraunhofer-Institut für Techno- und Wirtschaftsmathematik Kaiserslautern, Germany [email protected]

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Mathematical Modelling of Glass Forming Processes J.A.W.M. Groot, Robert M.M. Mattheij, and K.Y. Laevsky

Abstract An important process in glass manufacture is the forming of the product. The forming process takes place at high rate, involves extreme temperatures and is characterised by large deformations. The process can be modelled as a coupled thermodynamical/mechanical problem including the interaction between glass, air and equipment. In this paper a general mathematical model for glass forming is derived, which is specified for different forming processes, in particular pressing and blowing. The model should be able to correctly represent the flow of the glass and the energy exchange during the process. Various modelling aspects are discussed for each process, while several key issues, such as the motion of the plunger and the evolution of the glass-air interfaces, are examined thoroughly. Finally, some examples of process simulations for existing simulation tools are provided.

Nomenclature Br Fr Nu Pe Re Fe [N]

Brinkman number Froude number Nusselt number Péclet number Reynolds number External force on plunger

J.A.W.M. Groot () Department of Mathematics and Computer Science, Eindhoven University of Technology, PO Box 513, 5600 MB Eindhoven, The Netherlands e-mail: [email protected] R.M.M. Mattheij Department of Mathematics and Computer Science, Eindhoven University of Technology, PO Box 513, 5600 MB Eindhoven, The Netherlands e-mail: [email protected] K.Y. Laevsky e-mail: [email protected]

A. Fasano (ed.), Mathematical Models in the Manufacturing of Glass, Lecture Notes in Mathematics 2010, DOI 10.1007/978-3-642-15967-1 1, c Springer-Verlag Berlin Heidelberg 2011

1

2

Fg [N] L [m] T [K] Tg [K] Tm [K] V [m s−1 ] Vp [m s−1 ] c p [J kg−1 K−1 ] g [m s−2 ] p [Pa] r p [m] t [s] z¯ p [m] α [W m−2 K−1 ] β [N m−3 s] λ [W m−1 K−1 ] μ [kg m−1 s−1 ] ρ [kg m−3 ] n t u [m s−1 ] uw [m s−1 ] x [m] I E˙ [s−1 ] T [Pa]

J.A.W.M. Groot et al.

Force of glass on plunger Characteristic length Temperature Glass temperature Mould temperature Characteristic flow velocity Plunger velocity Specific heat Gravitational acceleration Pressure Radius of plunger Time Vertical plunger position Heat transfer coefficient Friction coefficient Effective conductivity Dynamic viscosity Density Unit normal Unit tangent Flow velocity Wall velocity Position Unit tensor Strain rate tensor Stress tensor

1 Introduction Nowadays glass has a wide range of uses. By nature glass has some special characteristics, including shock-resistance, soundproofing, transparency and reflecting properties, which makes it particularly suitable for a wide range of applications, such as windows, television screens, bottles, drinking glasses, lenses, fibre optic cables, sound barriers and many other applications. It is therefore not surprising that glass manufacture is an extensive branch of industry.

1.1 Glass Forming The manufacture of a glass product can be subdivided into several processes. Below a common series of glass manufacturing processes are described in the order of their application.

Mathematical Modelling of Glass Forming Processes

3

1.1.1 Melting In industry the vast majority of glass products is manufactured by melting raw materials and recycled glass in tank furnaces at an elevated temperature [33, 61]. Examples of raw materials include silica, boric oxide, phosphoric oxide, soda and lead oxide. The temperature of the molten glass in the furnace usually ranges from 1, 200 to 1, 600 ◦C. A slow formation of the liquid is required to avoid bubble forming [61].

1.1.2 Forming The glass melt is cut into uniform gobs, which are gathered in a forming machine. In the forming machine the molten gobs are successively forced into the desired shape. The forming technique used depends on the type of product. Forming techniques include pressing, blowing and combinations of both, and are discussed further on. During the formation the glass slightly cools down to below 1, 200 ◦C. After the formation the glass objects are rapidly cooled down as to take a solid form.

1.1.3 Annealing Development of stresses during the formation of glass may lead to static fatigue of the product, or even to dimensional changes due to relaxation or optical refraction. The process of reduction and removal of stresses due to relaxation is called annealing [61]. In an annealing process the glass objects are positioned in a so-called Annealing Lehr, where they are reheated to a uniform temperature region, and then again gradually cooled down. The rate of cooling is determined by the allowable final permanent stresses and property variations throughout the glass [61].

1.1.4 Surface Treatment An exterior surface treatment is applied to reduce surface defects. Flaws in the glass surface are removed by chemical etching or polishing. Subsequent flaw formation may be prevented by applying a lubricating coating to the glass surface. Crack growth is prevented by chemical tempering (ion exchange strengthening), thermal tempering or formation of a compressive coating. For more information about flaw removal and strengthening of the glass surface the reader is referred to [61]. The process step of interest in this paper is the actual glass formation. Below three widely used forming techniques are discussed. See [22, 61, 73] for further details on glass forming.

4


Fig. 1 Pressing machine

1.1.5 Press Process Commercial glass pressing is a continuous process, where relatively flat products (e.g. lenses, TV screens) are manufactured by pressing a gob that comes directly from the melt [61]. This process is usually referred to as the press process. In this paper also the term direct press is used to distinguish the process from the parison press, which is explained further on. The direct press is depicted in Fig. 1. Initially, the glass gob is positioned in the centre of a mould. Over the mould a plunger is situated. In order to enclose the space between the mould and the plunger, so that the glass cannot flow out during the process, a ring is positioned on top of the mould. During the direct press the plunger moves down through the ring so that the gob is pressed into the desired shape.

1.1.6 Press-Blow Process A hollow glass object is formed by inflating a glass preform with pressurised air. This is called the final blow. The preform is also called parison. In a press-blow process first a preform is constructed by a press stage, to avoid confusion with the direct press here referred to as the parison press. Figure 2 shows a schematic drawing of a press-blow process. In the press stage a glass gob is dropped down into a mould, called the blank mould, and then pressed from below by a plunger (see Fig. 2a). Once the gob is inside the blank mould, the baffle (upper part of the mould) closes and the plunger moves gradually up. When the plunger is at its highest position, the ring closes itself around the plunger, so that the mould-plunger construction is closed from below. Finally, when the plunger is lowered, the ring is decoupled from the blank mould and the glass preform is carried by means of a robotic arm to another mould for the blow stage (final blow). In the blow stage the preform is usually first left to sag due to gravity for a short period. Then pressurised


5

b

a

preform

gob

baffle

mould

plunger

parison press

final blow

Fig. 2 Schematic drawing of a press-blow process

b

a

preform

glass

mould

ring

settle blow

counter blow

Fig. 3 Schematic drawing of the first blow stage of a blow–blow process

air is blown inside to force the glass in a mould shape (see Fig. 2b). It is important to know the right shape of the preform beforehand for an appropriate distribution of glass over the mould wall.

1.1.7 Blow–Blow Process A blow-blow process is based on the same principle as a press-blow process, but here the preform is produced by a blow stage. By means of a blow stage a hollow preform can be formed, which is required for the production of narrow-mouth containers. In practice the glass gob is blown twice to create the preform. Figure 3

6


shows a schematic drawing of the first blow stage of a blow-blow process. First in the settle blow the glass gob is blown from above to form the neck of the container (see Fig. 3a), then in the counter blow from below to form the preform (see Fig. 3b). After the counter blow the preform is carried to the mould for the second blow stage, the final blow. The final blow is basically the same as in the press-blow process, but the preform is typically different. The temperature of the material of the forming machine is typically around 500 ◦ C. Because of the high temperature of the gob, the surface temperature of the material will increase. To keep the temperature of the material within acceptable bounds the mould and plunger are thermally stabilised by means of water-cooled channels.

1.2 Process Simulation In the recent past glass forming techniques were still based on empirical knowledge and hand on experience. It was difficult to gain a clear insight into forming processes. Experiments with glass forming machines were in general rather expensive and time consuming, whereas the majority had to be performed in closed constructions under complicated circumstances, such as high temperatures. Over the last few decades numerical process simulation models have become increasingly important in understanding, controlling and optimising the process [33, 71]. Since measurements are often complicated, simulation models may offer a good alternative or can be used for comparison of results. The growing interest of glass industry for computer simulation models has been a motivation for a fair number of publications on this subject. The earliest papers dealt with computer simulation of glass container blowing [12, 52]. The first publication in which both stages of the press-blow process were modelled was presented in [74], although the model did not include aspects such as the drop of the gob into the blank mould and the transfer of the preform to the mould for the final blow [28]. Shortly afterward also the first PhD theses on the modelling of glass forming processes appeared. A numerical model for blowing was published in [54]. Not much later a simplified mathematical model for pressing was presented in [27]. Both models assumed axial symmetry of the forming process. Subsequent publications reported the development of more advanced models. A fully three-dimensional model for pressing TV panels was addressed in [35]. A complete model for the three-dimensional simulation of TV panel forming and conditioning, including gob forming, pressing cooling and annealing was developed by TNO [7]. A simulation model for the complete press-blow process, from gob forming until the final blow, was presented in [28]. The model also included effects of viscoelasticity and surface tension. Finally, an extensive work on the mathematical modelling of glass manufacturing was published in the book entitled ‘Mathematical Simulation in Glass Technology’ [33].


7

More recent papers focus on optimisation of glass forming processes [25]. For instance a numerical optimisation method to find the optimum tool geometry in a model for glass pressing was introduced in [42]. Optimisation methods have also been developed to estimate heat transfer coefficients [9] or the initial temperature distribution in glass forming simulation models [41]. An engineering approach to find the optimum preform shape for glass blowing was addressed in [39]. Their algorithm attempts to optimise the geometry of the blank mould in the blow–blow process, given the mould shape at the end of the second blow stage. More recently, an optimisation algorithm for predictive control over a class of rheological forming processes was presented in [4]. By far most papers on modelling glass forming processes use FEM (Finite Element Methods) for the numerical simulation. Exceptions are [75], in which a Finite Volume Method is used, and [14], in which Boundary Element Methods are used. FEM in models for forming processes usually go together with re-meshing techniques, sometimes combined with an Eulerian formulation or a Lagrangian method. In [7] remeshing was completely avoided by using an arbitrary Euler– Lagrangian approach to compute the mesh for the changing computational domain due to the motion of the plunger and by using a Pseudo-Concentration Method to track the glass-air interfaces. In [20] a Level Set Method was used to track the glassair interfaces in a FEM based model for glass blowing. In this paper a general mathematical model for the aforementioned forming processes is derived. Subsequently, the model is specified for different forming processes, thereby discussing diverse modelling aspects. The paper focusses on the most relevant aspects of the forming process, rather than supplying a model that is as complete as possible. As discussed previously, a considerable amount of work on modelling glass forming processes has been done in the recent past. For completeness, comparison or reviewing various references to different simulation models are included. Finally, some examples of process simulations for existing simulation tools are provided. Most of the work in this paper has been done by CASA1 (Centre for Analysis, Scientific computing and Applications).

1.3 Outline The paper is structured as follows. First in Sect. 2 the physical aspects of glass forming are described and a general mathematical model for glass forming is derived. Then the mathematical model is specified for the parison press in Sect. 3, for both the counter blow and the final blow in Sect. 4 and for the direct press in Sect. 5. In each of these sections also a computer simulation model for the forming process in question is described and some results are presented.

1 A research group in the Department of Mathematics and Computer Science of Eindhoven University of Technology.

8


2 Mathematical Model This section presents a mathematical model for glass forming in general. The section has the following structure. First Sect. 2.1 defines the physical domains into which a glass forming machine can be subdivided and in which the boundary value problems for glass forming are defined. Then Sect. 2.2 describes the physical aspects of glass forming, sets up the resulting balance laws and derives the corresponding boundary value problems. Sect. 2.2.1 is concerned with the thermodynamics and Sect. 2.2.2 with the mechanics.

2.1 Geometry, Problem Domains and Boundaries In order to formulate a mathematical model for glass forming, the forming machine is subdivided into subdomains. First the space enclosed by the equipment is subdivided into a glass domain and an air domain. Then separate domains for components of the equipment (e.g. mould, plunger) are considered. Subdomains of the equipment are of interest when modelling the heat exchange between glass, air and equipment. On the other hand, for less advanced heat transfer modelling it can be assumed that the equipment has constant temperature, so that the mathematical model can be restricted to the glass and air domain. In this case the equipment domains are disregarded. Figure 4 illustrates the domain decomposition of 2D axi-symmetrical forming machines for the direct press, parison press, counter blow and final blow, respectively. The entire open domain of the forming machine, consisting of equipment, glass melt and air, is denoted by Σ . The ‘flow’ domain Ω consists of the open glass domain Ωg , the open air domain Ωa and the glass-air interface(s) Γi . For blowing Ω := Ωg ∪ Ωa ∪ Γi is fixed, while for pressing Ω changes in time. Furthermore, Ωg and Ωa are variable in time for any forming process. The boundaries of the domains are:

Γb : Baffle boundary Γo : Outer boundary Γs : Symmetry axis

Γi : Glass-air interface Γp : Plunger boundary

Γm : Mould boundary Γr : Ring boundary

Σ Note that not necessarily all boundaries exist for each forming machine. Domain is enclosed by Γo ∪ Γs . Domain Ω is enclosed by ∂ Ω := Ω ∩ Γe ∪ Γo ∪ Γs , with Γe := Γb ∪ Γm ∪ Γp ∪ Γr

(equipment boundary).

(1)

In addition, define ∂ Ωa := Ω a ∩ ∂ Ω and ∂ Ωg := Ω g ∩ ∂ Ω . Finally, the boundaries for the glass domain and the air domain are distinguished:

Γa,e = ∂ Ωa ∩ Γe , Γg,e = ∂ Ωg ∩ Γe , Γa,o = ∂ Ωa ∩ Γo , Γg,o = ∂ Ωg ∩ Γo .


9 Γo

b Γb

baffle

Ωa air Γi

a Γo

Γs

n

Ωg glass Γi

t

plunger

Ωa air

z

ring

Γr

t n

Γp

r

z

plunger

parison press

Γo Γb

baffle

Ωa

d

Γ o,a Γo,g

t

air

n

Γi

Ωa Γo

Ωg Γ m glass

Γs air

mould

Γi

Ωg

z

Γ g,o

r

direct press

Γs

Γo

Γs

Γm Γo

c

Γm mould

Γp

mould

Ωg glass

Γo

Γi Ωa air

Γm

Γo

t n

mould

glass

Γi

Γg,o

Γa,o

r

counter blow

z r

final blow

Fig. 4 2D axi-symmetrical problem domains of glass forming models

Remark 2.1. In pressing sometimes only the forces acting on the glass are of interest. If the density and viscosity of air are negligible compared to the density and the viscosity of glass, the mathematical model may be restricted to the glass domain Ωg . In this case the interface Γi is replaced by Γo .

2.2 Balance Laws In this section the physical aspects of glass forming are described and the resulting balance laws are formulated in the form of boundary value problems. Subsequently a dimensional analysis of the balance laws is applied. Section 2.2.1 is concerned with the thermodynamics and Sect. 2.2.2 with the mechanics. In subsequent sections these balance laws are specified for different forming processes.

10


2.2.1 Thermodynamics Glass forming involves high temperatures within a typical range of 800–1, 400 ◦C. Temperature variations within this range may cause significant changes in the mechanical properties of the glass. • The range of viscosity for varying temperature is relatively large for glass: it

amounts from 10 Pa s at the melting temperature (about 1, 500 ◦ C) to 1020 Pa s at room temperature. The viscosity increases rapidly as a glass melt is cooled, so that the glass will retain its shape after the forming process. Typical values for the viscosity in glass forming processes lie between 102 and 105 Pa s [61, 73]. The temperature dependence for the viscosity of glass within the forming temperature range is given by the VFT-relation, due to Vogel, Fulcher and Tamman [61, 73]: μ (T )\[Pa s] = exp − A + B/(T − TL ) . (2)

Quantities A [−], B [ ◦ C] and TL [ ◦ C] represent the Lakatos coefficients, which depend on the composition of the glass melt. Figure 5 shows how strongly the viscosity depends on temperature for soda-lime-silica glass with Lakatos coefficients A = 3.551, B = 8575 ◦C and TL = 259 ◦C [49]. • The following density-temperature relation can be deduced ρ (T ) = ρ0 1 − αV T − Tref ,

(3)

where – αV [ ◦ C−1 ] is the volumetric thermal expansion coefficient – Tref [ ◦ C] is a reference temperature – ρ0 [kg m−3 ] is the density at the reference temperature The volumetric thermal expansion coefficient is often assumed constant; for molten glass it is typically ranged from 5 10−5 to 8 10−5 ◦ C−1 . The density

·

·

14

log viscosity

12 10 8 6 4 800

Fig. 5 VFT-relation

900

1000 1100 Temperature ( oC)

1200


11

of molten glass is of the order 2,300–2,500 kg m−3 and is 5–8% lower than at room temperature. Thus it is quite reasonable to assume incompressibility for molten glass [73]. Clearly, the mechanics of glass forming is related to the heat transfer in the glass. The heat transfer in glass, air and equipment is described by the heat equation for incompressible continua: ∂T + u ∇T = − ∇ q + 2 μ E˙ ∇ ⊗ u , ρ cp ∂ t

·

advection

·

◦

in Σ ,

:

conduction radiation

(4)

dissipation

◦

where the temperature distribution T [K] is unknown. Here Σ denotes domain Σ minus the interfaces between the continua. On the interfaces between different continua a steady state temperature transition is imposed,

λ n ∇T = 0.

·

(5)

The heat flux q [W m−2 ] is the result of the contribution of both conduction and radiation, (6) q = −λ ∇T,

·

where λ is the effective conductivity [W

m−1 K−1 ],

given by

λ = λc + λr .

(7)

Here λc is the thermal conductivity and λr is the radiative conductivity. The thermal conductivity measures 1.0 W m−1 K−1 at room temperature for soda-lime glass and increases with approximately 0.1 W m−1 K−1 per 100 K. In this paper it is assumed to be constant. The calculation of the radiative conductivity λr is often a complicated process. However, for non-transparent glasses it can be simplified by the Rosseland approximation [16, 51, 73]

λr (T ) =

16 n2 σ T 3 , 3 α

(8)

where • σ is the Stefan Boltzmann radiation constant [W m−2 K−4 ] • n is the average refractive index [−] • α is the absorption coefficient [m−1 ]

The radiative conductivity λr in the sense of (8) is called the Rosseland parameter. Relation (8) cannot be applied for highly transparent glasses, since in this case not all radiation is absorbed by the glass melt [73]. A more simple approach is to omit the radiative term, which is often reasonable for clear glass [34, 37].

12


For more information on heat transfer in glass by radiation the reader is referred to [33, 37, 38, 43, 73]. The specific heat c p is in general slightly temperature dependent. In [3] an increase in the specific heat of less than 10% in a temperature range of 900–1,300 K for soda-lime-silica glass is reported. For various specific heat capacity models the reader is referred to [3, 60]. In order to analyse the energy exchange problem quantitatively the heat equation is written in dimensionless form. Define a typical: velocity V , length scale L, viscosity μ¯ , specific heat c¯ p , effective conductivity λ¯ , glass temperature Tg and mould temperature Tm . Then introduce the dimensionless variables Vt , L cp c∗p := , c¯ p

t ∗ :=

x , L λ λ ∗ := ¯ , λ

x∗ :=

u , V μ μ∗ = , μ¯

u∗ :=

T ∗ :=

T − Tm , Tg − Tm (9)

For convenience all dimensionless variables, spaces and operators with respect to the dimensionless variables are denoted with superscript ∗ . Substitution of the dimensionless variables (9) in the heat equation and splitting up the effective conductivity into (7) lead to the dimensionless form, Pe

◦ ∗ ∗ ∂ T∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ˙ ∇ ⊗ u∗ , in Σ ∗ , (10) + 2 = ∇ + u ∇ T λ ∇ T μ Br E ∂ t∗

·

·

:

where

ρ c¯ pV L , λ¯ μ¯ V 2 Br = ¯ λ Tg − Tm Pe =

(11) (12)

are the Péclet number and the Brinkman number, respectively. The Péclet number represents the ratio of the advection rate to the diffusion rate. On the other hand, the Brinkman number relates the dissipation rate to the conduction rate. The dimensionless numbers are useful for assessing the order of magnitude of the different terms in (10). The energy BCs follow from symmetry and heat exchange with the surroundings: λ ∇T n = 0, on Γs λ ∇T n = α T − T∞ , on Γo ,

· ·

where T∞ is the temperature of the surroundings. The heat transfer coefficient α [W m−2 K −1 ] can differ for separate equipment domains, such as the mould and the plunger. Let α¯ be a typical value for the heat transfer coefficient, then the Nusselt number is defined by α¯ L Nu = ¯ . (13) λ


13

The dimensionless boundary conditions become ∗ ∗ λ ∗ ∇T ∗ n = 0, ∗ ∗ on Γs λ ∇T n = Nu α T − T∗∞ , on Γo ,

· ·

(14)

with α ∗ = α /α¯ . On the other hand, if the heat transfer in the equipment domain is not of interest, the boundary condition on Γo in (14) is imposed on Γe , with T∞ the surface temperature of the mould. 2.2.2 Mechanics A balance law for the mechanics of the glass melt is formulated. In general, glass can be treated as an isotropic viscoelastic Maxwell material [3,53,54,61], that is the strain rate tensor can be split up into an elastic and a viscous part: E˙ = E˙ e + E˙ v,

(15)

where the elastic and viscous strain rate tensors, E˙ e and E˙ v respectively, are given by [54]

˙ + 1 + ν dev(T) ˙ ˙Ee = 1 − 2ν αV ∂ T + 1 tr(T) E ∂t 3 E 1 E˙ v = dev(T). 2μ

(16) (17)

Here αV [ ◦ C−1 ] is the volumetric thermal expansion coefficient, E [Pa] is the Young’s modulus, ν [-] is the Poisson’s ratio and T˙ [Pa s−1 ] denotes the stress rate tensor. However, at relatively low viscosities the relation between shear stress and viscosity becomes approximately linear. For example, for soda lime silica glasses the viscosity as a function of the strain rate and the temperature becomes [6, 63] ˙ T) = μ (E,

μ0 (T ) , 1 + 3.5 · 10−6E˙ μ00.76 (T )

(18)

where μ0 is the Newtonian viscosity. Consequently, the motion of glass is dominated by viscous flow and the influence of elastic effects can be neglected [3, 53, 61]. Moreover, as verified in Sect. 2.2.1, glass is practically incompressible in the forming temperature range, from which it follows that ˙ = 0. tr(E)

(19)

It can be concluded that glass in the forming temperature range behaves as an incompressible Newtonian fluid [3, 53, 73], i.e. ˙ T = −pI + 2 μ E,

(20)

14


with

1 E˙ = ∇ ⊗ u + (∇ ⊗ u)T . (21) 2 For simplicity the pressurised air is considered as an incompressible, viscous fluid with uniform viscosity. Thus the motion of glass melt and pressurised air is described by the Navier–Stokes equations for incompressible fluids. These involve the momentum equations, ∂u + u ∇ ⊗ u = −∇p + ρ g + 2∇ μ E˙ , ρ pressure gravity ∂t

·

·

in Ω \ Γi ,

(22)

viscosity

inertia

and the continuity equation, which follows directly from (19),

·

in Ω \ Γi .

∇ u = 0,

(23)

The unknowns are the flow velocity u [m s−1 ] and the pressure p [Pa]. Flow problem (22)–(23) is coupled to the energy problem (10) in two ways: firstly the viscosity is temperature dependent and secondly the heat transfer is partly described by convection and diffusion. In order to apply a quantitative analysis the Navier–Stokes equations are written in dimensionless form. First a dimensionless pressure is defined by p∗ :=

Lp . μ¯ V

(24)

The gravity force can be written as −ρ gez , where ez is the unit vector in z-direction. Substitution of the dimensionless variables (9) and (24) into the Navier–Stokes equations (22)–(23) and division by the order of magnitude of the diffusion ¯ term, μLV2 , lead to the dimensionless Navier–Stokes equations

∗ ∂ u∗ Re ∗ ∗ Re ∇ ⊗u = −∇∗ p∗ − ez + 2∇∗ μ ∗ E˙ ∗ , in Ω ∗ \ Γi ∗ , +u ∂ t∗ Fr in Ω ∗ \ Γi ∗ , (25) ∇∗ u∗ = 0,

·

·

·

where Re =

ρV L , μ¯

(26)

Fr =

V2 gL

(27)

are the Reynolds number and the Froude number, respectively. The Reynolds number measures the ratio of inertial forces to viscous forces, while the froude number measures the ratio of inertial forces to gravitational forces.


15

The jump conditions between two immiscible viscous fluids are the continuity of the flow velocity, [[u]] = 0, (28) as well as the continuity of its tangential derivative, [[ ∇ ⊗ u t]] = 0,

·

(29)

and a dynamic jump condition stating the balance of stress across the fluid interface [31, 45], [[Tn]] = 0. (30) Remark 2.2. If the influence of surface tension is taken into account, the dynamic jump condition is (31) [[Tn]] = −γκ n, where n points in the air domain. Here γ [N m−1 ] denotes the surface tension and κ [m−1 ] denotes the curvature. The influence of surface tension highly depends on the glass composition [61]. In this paper the influence of surface tension is simply disregarded, although for some glasses a dimensional analysis can point out that the surface tension term is not negligible. Boundary conditions for the flow problem can be determined as follows. On Γs symmetry conditions are imposed. On Γg,e a suitable slip condition for the glass should be adopted. The air can escape through small cavities in (part of) the mould wall. This aspect can be modelled by allowing air to flow freely through the mould wall. Thus free-stress conditions are proposed on this part of the equipment bound(1) ary, which is referred to as Γa,e . A free-slip condition is prescribed on the remaining (2) part, Γa,e . On Γo the normal stress should be equal to the external pressure. A commonly used boundary condition to describe fluid flow at an impenetrable wall [13, 19, 29, 50] is Navier’s slip condition: Tn + β (u − uw ) t = 0,

·

(32)

where β is the friction coefficient [N m−3 s] and uw is the velocity of the wall [m s−1 ]. A similar condition can be obtained by using the Tresca model [16]. Introduce a dimensionless friction coefficient,

β ∗ :=

Lβ , μ¯

(33)

then the dimensionless Navier’s slip condition reads: ∗ T n + β ∗ (u∗ − u∗w ) t = 0.

·

(34)

The order of magnitude of the friction coefficient depends on many parameters, such as the type of glass, temperature, pressure or presence of a lubricant [15, 18, 50].

16


Remark 2.3. For β ∗ → ∞ Navier’s slip condition together with the boundary condition for an impenetrable wall, (u∗ − u∗w ) n = 0, can be reformulated as a no slip condition: u∗ = u∗w . (35)

·

In summary, the boundary conditions for the flow problem can be formulated as:

· · · · ·

u∗ n = n= n= ∗ ∗ (u − uw ) n = T∗ n n =

(u∗ − u∗w ) T∗ n

0, 0, 0, 0, p0 ,

T∗ n ∗ ∗ ∗ T n + β (u − u∗w ) T∗ n T∗ n T∗ n

· t = 0, · t = 0, · t = 0, · t = 0, · t = 0,

on on on on on

Γs , Γg,e , (1) Γa,e , (2) Γa,e , Γo , (36)

where p0 is the external pressure.

3 Parison Press Model This section presents a mathematical model for the parison press. Section 3.1 specifies the mathematical model described in Sect. 2 for the parison press. By restricting the analysis to a narrow channel between the plunger and the mould, an analytical approximation of the flow can be derived. Section 3.2 explains this concept, known as the slender geometry approximation [50]. Section 3.3 describes the motion of the plunger by an ordinary differential equation. It appears that the motion of the plunger is coupled to the flow, which considerably complicates the parison press model. Section 3.4 presents a numerical simulation model for the parison press. The motion of the free boundaries is emphasised. Finally, Sect. 3.5 shows some examples of parison press simulations. The simulation tool used for these results is presented in [34].

3.1 Mathematical Model In Sect. 2.2 the balance laws for glass forming were formulated. In this section these balance laws are further specified for the parison press process. Typical values for the parison press are: Glass density : ρg = 2.5 · 103 kg m−3 Glass viscosity : μ¯ g = 104 kg m−1 s−1 Gravitational acceleration : g = 9.8 m s−2 Flow velocity : V = 10−1 m s−1 Length scale of the parison : L = 10−2 m Glass temperature : Tg = 1, 000 ◦ C Mould temperature : Tm = 500 ◦ C Specific heat of glass : c¯ p = 1.5 · 103 J kg−1 K−1 Effective conductivity of glass : λ¯ = 5 W m−1 K−1 ,


17

As a result the following dimensionless numbers are found: Peglass ≈ 7.6·102, Brglass ≈ 4.0·10−3, Reglass ≈ 2.5·10−4, Fr ≈ 1.0·10−1 . (37) Apparently, the heat transport in glass is dominated by thermal advection. As a result the heat equation (10) simplifies to

∂T∗ dT ∗ = ∗ + u∗ ∇∗ T ∗ = 0, ∗ dt ∂t

·

in Ωg∗ ,

(38)

Thus the temperature remains constant along streamlines. Consequently, if the initial temperature distribution in the glass is (approximately) uniform, the glass viscosity can be considered constant. From the small Reynolds number for glass it can be concluded that the inertia forces can be neglected with respect to the viscous forces. Furthermore, Reglass ≈ 2.5 · 10−3, Fr which means that also the contribution of gravitational forces is rather small. In conclusion, the glass flow can be described by the Stokes flow equations:

·

∇∗ T∗ = 0,

·

∇∗ u∗ = 0,

in Ωg∗ ,

(39)

where T∗ is the dimensionless stress tensor, which satisfies (20) in terms of the dimensionless variables. The air domain is ignored for the following reasons. Firstly, the force of the plunger acts directly on the glass domain. Secondly, the density and viscosity of air are negligible compared to those of glass, so that air hardly forms any obstacle for the glass flow. Thirdly, the simplification of the heat equation to convection equation (38) gives reason to restrict the energy exchange problem to the glass domain, or all together ignore the heat transfer in case of an uniform initial glass temperature. Subsequently, the glass-air interfaces are treated as an outer boundary of the flow domain, Γo , on which free-stress conditions are imposed. Note that although the problem can be considered as a free-boundary problem, the geometry is constrained by the mould and the plunger. Remark 3.1. Close to the equipment wall extreme temperature variations occur over a small length scale. Therefore, the conductive heat flux close to the equipment wall should, strictly speaking, not be disregarded [50,64]. Moreover, the viscosity in this region may increase by several orders, so that the fluid friction may not be negligible and the influence of heat generation by dissipation should be taken into account for optimal accuracy [50]. Although the reader should take notice of these boundary layer effects, in this paper it is simply assumed that they are small enough to be ignored. For more advanced heat modelling during the parison press the reader is referred to [23, 38, 64].

18


The boundary conditions for the flow problem (2.2.2) can be specified for the parison press: u∗ p ez ) u∗ T∗ n

(u∗ − V

· n = 0, · n = 0, · n = 0, · n = 0,

T∗ n ∗ ∗ ∗ T n + β p (u − Vpez ) ∗ T n + βm∗ u∗ T∗ n

· t = 0, · t = 0, · t = 0, · t = 0,

on on on on

Γs , Γg,p , Γg,m , Γo . (40)

In the remainder of this section it is assumed that the glass gob initially has an uniform temperature distribution, so that with (38) it follows that μ = μ¯ is constant.

3.2 Slender-Geometry Approximation In the model for glass pressing the analysis can be restricted to the flow in a narrow channel between plunger and mould. In other words, the analysis focusses on the flow in the slender geometry around the plunger, while the flow between the plunger top and the baffle (Fig. 4b) is considered practically stagnant. For a more complete analysis the reader is referred to [50]. In the slender-geometry approximation of the flow two typical length scales and L are considered, with L, where is the length scale for the width of the channel and L is the length scale for the length of the channel. Thus variations in r-direction are scaled by and variations in z-direction are scaled by L. By means of this scaling the following dimensionless variables can be defined: Vt , L ur , u∗r := εV

t ∗ :=

, L uz u∗z := , V

ε :=

r z , z∗ := , εL L ε 2 Lp p∗ := . μV

r∗ :=

(41)

In the remainder of this section all variables, spaces and operators are dimensionless and the superscript ∗ is ignored. Substitution of the dimensionless variables (41) into the Navier–Stokes equations (22)–(23) leads to the dimensionless 2D axisymmetrical Navier–Stokes equations ∂u

∂ ur ∂ ur ∂p ∂ 1 ∂ 4 ∂ 2 ur +uz , in Ωg , =− +ε 2 rur +ε ∂t ∂r ∂z ∂r ∂r r ∂r ∂ z2 ∂u ∂ uz ∂ uz ∂ p ∂ 1 ∂ 2 ∂ 2 uz Re z +ur +uz =− + ruz +ε , in Ωg , ε Re +ε ∂t ∂r ∂z ∂z ∂r r ∂r ∂ z2 Fr 1 ∂ ∂ uz rur + = 0, in Ωg , (42) r ∂r ∂z

ε 3 Re

r

+ur


19

where Re is the Reynolds number with respect to length scale , i.e. Re =

ρ V = ε ReL . μ¯

(43)

Thus, for small ε , (42) can be simplified to

∂p = O(ε 2 ), in Ωg , ∂r ∂p ∂ 1 ∂ = ruz + O(ε 2 ), in Ωg , ∂z ∂r r ∂r 1 ∂ ∂ uz rur + = 0, in Ωg . r ∂r ∂z

(44)

If the O(ε 2 ) terms are neglected, system of (44) can be recognised as Reynolds’ 2D axial-symmetrical lubrication flow [19, 50]. The equipment boundary conditions in the slender-geometry approximation can be simplified accordingly [50]. Define the plunger surface and the mould surface by: r = r p z − z¯ p (t) ,

r = rm (z),

(45)

respectively, where z = z¯ p (t) is the top of the plunger at time t (see Fig. 6). Consider the plunger position z¯ p ≡ z¯ p (t) at a fixed time t. Then the outward unit normal n p and the counterclockwise unit tangent t p on the plunger wall are given by

ε rp ez − er np = , 1 + ε 2r2 p

−ez − ε rp er tp = . 1 + ε 2r2 p

(46)

o

p m s

rp rm

plunger

Fig. 6 Geometry of the plunger

r zp z

glass

mould

20


Analogously, the outward unit normal nm and the counterclockwise unit tangent tm on the plunger wall on the mould wall are given by −ε r ez + er nm = m , 2 1 + ε 2 rm

e ez + ε rm r tm = . 2 1 + ε 2rm

(47)

By substituting (46)–(47) into Navier’s slip condition (32) and scaling the friction coefficient β by a factor ε the following boundary conditions are obtained β p uz − Vp + ε 2 ur rp =

∂ u z 2 ∂ ur − 2ε 2 r ∂ uz − ∂ ur 1 − ε 2 r2 + ε p p ∂r ∂z ∂z ∂r , 1 + ε 2r2 p on r = r p (z − z¯ p )

−βm uz + ε 2 ur rm

(48)

∂ u ∂ ur z 2 ∂ uz − ∂ ur 1 − ε 2 rm + ε2 − 2ε 2rm ∂r ∂z ∂z ∂r , = 2 1 + ε 2rm on r = rm (z), (49)

where Vp is the velocity of the plunger and β p , βm are the friction coefficients corresponding to the plunger and the mould, respectively. For small ε the Navier’s slip condition on the plunger and mould surfaces can be written as ∂ uz + O(ε 2 ), β p uz − V p = ∂r ∂ uz + O(ε 2 ), −βm uz = ∂r

on r = r p (z − z¯ p ), on r = rm (z).

(50) (51)

The componentwise boundary condition for the impenetrable wall is ur = (uz − Vp)rp , ur =

uz rm ,

on r = r p (z − z¯ p ),

(52)

on r = rm (z).

(53)

Note that for βm,p → ∞ the error in boundary condition (50)–(51) due to the slendergeometry approximation vanishes. Following [40, 50] the analytical solution to system of (44) with set of boundary conditions (50)–(53) on the equipment boundary and free-stress conditions on the other boundaries can be obtained. Neglecting the O(ε 2 ) terms system of equations (44) becomes

∂p = 0, in Ωg , ∂r ∂p ∂ 1 ∂ = ruz , in Ωg , ∂z ∂r r ∂r

(54) (55)


21

1 ∂ ∂ uz = 0, in Ωg . rur + r ∂r ∂z

(56)

Firstly, from (54) it follows that p is a function of z only. Secondly, from (55) and the boundary conditions it follows that uz (r, z) =

ψm (z)χ p (r, z) − ψ p (z)χm (r, z) 1 dp χm (r, z) 2 (z) r + , (57) + Vp 4 dz χm (r, z) − χ p(r, z) χm (r, z) − χ p (r, z) with

1 2 , ψ p (z)=r p (z−¯z p ) r p (z−¯z p )− , β p r p (z−¯z p ) βp

1 2 , ψm (z) = rm (z) rm (z) + χm (r, z) = log rrm (z) + . (58) βm rm (z) βm χ p (r, z) = log rr p (z−¯z p ) −

Thirdly, from (56) and the boundary conditions it follows that

rm 1 d rm ∂ uz 1 (s, z)ds = s s uz (s, z)ds. ur (r, z) = rm rm uz (rm , z) + r ∂z r dz r r

(59)

Finally, to find the pressure gradient, Gauss’ divergence theorem is applied to the continuity equation, 0=

Ωg

·

∇ udΩ =

Γg,o

·

u ndΓ +

Γp

·

u ndΓ +

Ωg ∩Γs

·

u ndΓ +

Γm

·

u ndΓ . (60)

The fluxes through the symmetry axis and the mould wall are zero. Since the continuity equation also holds in the plunger domain Σ p , it holds that Γp

As a result,

·

u ndΓ = −

Γg,o

·

Σ p ∩Γo

u ndΓ = 2π

rm rp

·

u ndΓ = π Vpr2p .

(61)

ruz dr = −π Vpr2p .

(62)

Substitution of (57) into (62) yields r

2

r=rm

χm + 12 1 dp 2 ψm (2 χ p + 1) − ψ p(2 χm + 1) = −Vp r2p . + Vp r − 8 dz χm − χ p χm − χ p r=r p

(63) By solving (63) the following solution for the pressure gradient is found:

22


b

a

11

x 104

0.14 plunger

10 −dp/dz [Pa/m]

z [m]

0.12 0.1 0.08

mould

9 8 7

0.06 0.005

0.01

0.015

0.02

6 0.04

0.06

0.08

r [m]

0.1

0.12

0.14

0.16

z [m]

plunger and mould boundaries

pressure [m]

r-component of flow velocity [m/s]

z-component of flow velocity [m/s]

Fig. 7 Analytical solution of slender-geometry approximation for z¯ p = 0.5

ψ p (z) − ψm (z) dp (z) = 4Vp 2 , dz ψ p (z) − ψm (z) − χ p − χm (z) ω p (z) − ωm (z)

(64)

with

ω p (z) = ψ p (z)2 −

4r p (z − z¯ p )2 , β p2

ωm (z) = ψm (z)2 −

4rm (z)2 . βm2

(65)

Figure 7–8 plot the solution at z¯p = 0.05 m and z¯ p = 0.01 m, respectively, for ε = 0.1 and Vp = −0.1 ms−1 . The (dimensionless) geometries of the plunger and mould have been taken from [50]: √ r p (z) = −0.1 5z,

√ rm (z) = 0.8 5z.

For simplicity no-slip boundary conditions at the mould and plunger wall have been used. In [50] it is reported that the results are in good agreement with the numerical solution obtained by using FEM. For more results the reader is referred to [40, 50].


b

a

6

23 x 105

0.1 plunger

−dp/dz [Pa/m]

z [m]

0.08

5

0.06 0.04

4 3 2

mould

1 0.02 0.005

0.01 r [m]

0.015

0.02

0

0

0.02

0.04

0.06 z [m]

0.08

0.1

plunger and mould boundaries

pressure [m]

r-component of flow velocity [m/s]

z-component of flow velocity [m/s]

0.12

Fig. 8 Analytical solution of slender-geometry approximation for z¯ p = 0.1

3.3 Motion of the Plunger In the previous section it was assumed that the plunger moves with a constant flow velocity. However, in practice the plunger is pushed by a piston. This means that the flow velocity is the result of an external force applied to the plunger. In this section it can be seen that the plunger velocity is coupled to the glass flow, which considerably complicates the parison press model. The press process is initiated by applying an external force Fe to the plunger. This causes the plunger to move with velocity Vp (t). This plunger velocity is the result of the total force F on the plunger, which is the sum of the external force Fe and the force of the glass on the plunger Fg : dVp F(t) Fe + Fg(t) (t) = = , dt mp mp

(66)

where m p is the mass of the plunger [34, 50]. The force Fg is determined by the mechanical forces of the glass acting on the plunger and hence depends on the

24


glass flow. The glass flow at its turn is caused by the plunger motion. Therefore, differential equation (66) is fully coupled to the flow problem. Clearly, if a constant external force is applied to the plunger, the plunger moves until the force of the glass on the plunger is equal in magnitude to the external force. Below the equation for the motion of the plunger (66) is examined more thoroughly. For a complete analysis the reader is referred to [34, 50]. First the force of the glass on the plunger is analysed. At every time t the force can be fully described by the stress tensor (20) integrated over the plunger surface Γp , Fg =

Γp

Tn ez dΓ .

·

(67)

Using the definition of the plunger surface (45) the surface element dΓ can be written as dΓ = 2π 1 + r2 on Γp . (68) p r p dz, By means of the definition of the stress tensor (20) and the expressions for the unit normal and tangent (46) and the surface element (68), the expression for the force of the glass on the plunger becomes Fg = 2π

z0 z¯ p

p − 2μ

∂u ∂ uz ∂ uz r + rp + μ r p dz. ∂z ∂z ∂r

(69)

where z = z0 (t) is the bottom glass level at time t. Substitution of the dimensionless variables (41) into (69) yields Fg = 2π μ V L

z∗ 0 z¯∗p

p∗ − 2ε 2

∂ u∗z ∗ 2 ∂ ur ∂ u∗z ∗ ∗ r + ε + r p dz . ∂ z∗ p ∂ z∗ ∂ r∗

(70)

Considering expression (70) it makes sense to define the dimensionless force of the glass on the plunger as Fg . (71) Fg∗ := 2π μ VL For small ε the dimensionless force can be simplified to Fg∗ =:

z∗ 0 z¯∗p

p∗ r∗p +

∂ u∗z ∗ ∗ r dz + O(ε 2 ). ∂ r∗ p

(72)

The flow velocity and pressure in (72) are implicit functionals of the plunger velocity Vp . However, the slender-geometry approximation in Sect. 3.2 can be used to find approximate solutions for u∗z and p∗ . By substituting (57) and (65) into (70)–(71) the dimensionless force can be written as Fg∗ (t ∗ ) = Vp∗ (t ∗ )I ∗ (t ∗ ),

(73)


25

with an assumed error of O(ε 2 ), where the dimensionless function I ∗ (t ∗ ) only depends on the geometry and the friction coefficient β ∗ [34, 50]. Then the equation for the motion of the plunger (66) in dimensionless form becomes dVp∗ ∗ 2π μ L2 ∗ ∗ ∗ ∗ Vp (t )I (t ) + Fe∗ + O(ε 2 ), (t ) = dt ∗ m pV

(74)

where the dimensionless external force Fe∗ is defined in the same way as Fg∗ in (71). Typical values in the slender-geometry approximation are: Viscosity : μ¯ g = 104 kg m−1 s−1 Flow velocity : V = 10−1 m s−1 Length scale of the parison : L = 10−1 m Mass of the plunger : m p = 1 kg As a result the dimensionless coefficient in (74) is typically 2π μ L2 ≈ 104 . m pV This value is rather large, which indicates that (74) is a stiffness equation. According to [34] this phenomenon can also be observed if (74) is solved numerically using the Euler forward scheme for time integration. This means that one would have to resort to an implicit time integration scheme in order to solve (74). Unfortunately, a fully implicit scheme is practically impossible, since the plunger velocity is not known explicitly. See [34] for further details on this stiffness phenomenon. The stiffness phenomenon previously described indicates that the coupling of the equation for the plunger motion (66) to the boundary conditions on the plunger for the Stokes flow problem is undesirable. Therefore, in the following the plunger velocity Vp(t) is decoupled from the parameter Vp in the boundary conditions for the flow problem. For a more detailed analysis the reader is referred to [34]. The following lemma is used: Lemma 3.2. Let (uυ , pυ ) be the family of solutions of Stokes flow problem (39)–(40) with plunger velocity Vp = υ . Define (u, p)υ := (uυ , pυ ). Then u, p k

1 υ1 +k2 υ2

= k1 uυ1 + k2 uυ2 , p0 + k1 (pυ1 − p0 ) + k2(pυ2 − p0 )

for arbitrary constants k1 , k2 .

The lemma can be proven bydirect substitution of the solution k1 uυ1 + k2 uυ2 , p0 + k1 (pυ1 − p0 ) + k2 (pυ2 − p0 ) into the Stokes flow problem with plunger velocity Vp = k1 υ1 + k2 υ2 [34]. In the remainder of this section all variables, spaces and operators are dimensionless and the superscript ∗ is ignored. From Lemma 3.2 it immediately follows that u, p υ = υ u1 , p0 + υ (p1 − p0) ,

(75)

26


and as a result,

Fg,υ = Fg,0 + υ Fg,1 − Fg,0 .

(76)

Substitution of (76) with υ = Vp (t) into the dimensionless form of (66) yields

dVp 2π μ L2 (t) = Vp(t) Fg,1 (t) + Fg,0(t) + Fg,0(t) + Fe . dt m pV

(77)

The force Fg,1 (t) can be calculated by solving the Stokes flow problem in Ωg (t) with plunger velocity Vp = 1. Thus the equation for the motion of the plunger is decoupled from the boundary conditions for the Stokes flow problem. Note that the Stokes flow problem is still coupled to differential equation (77) as the geometry of the glass domain is determined by the plunger velocity. The time dependency of the plunger velocity seems a bit awkward as the Stokes flow problem is not explicitly time dependent; the flow merely changes in time through the changing geometry. It seems more convenient to define the force of the glass on the plunger and the plunger velocity as functions of the plunger position z := z¯ p : Vp := Vp (z). (78) Fg := Fg (z), As a result also the motion of the plunger can be described by the plunger position. By the chain rule of differentiation, dVp dVp (t) = (z)Vp (z). dt dz

(79)

As for the initial condition, let z = 0 and Vp = V0 at t = 0. Then the motion of the plunger involves the following initial value problem: ⎧

2 2π μ L2 ⎨ 1 dVp (z) = Vp (z) Fg,1 (z) + Fg,0 (z) + Fg,0(z) + Fe 2 dz m pV ⎩ Vp (0) = V0

(80)

Since the plunger velocity does not need to be known to determine the glass domain at given plunger position z, an implicit time integration scheme can be used to solve (80), thus overcoming the stiffness problem [34].

3.4 Simulation Model A simulation tool for the parison press process was designed [34]. The tool is able to compute and visualise the velocity field, the pressure and also the temperature in the parison during the press stage. The input parameters for the simulations include the 2D axi-symmetrical parison geometry, i.e. the description of the mould and plunger surfaces, the initial positions of the plunger and the glass domain, as well as the


27

Fig. 9 Mesh for press simulation of a jar parison [34]

physical properties of the glass. The geometry of the initial computational domain is defined by that of the glass gob (see Sect. 2.1). At the beginning of each time step the computational domain is discretised by means of FEM. A typical mesh for the initial glass domain in a press simulation of the parison for a jar is depicted in Fig. 9. The mesh distribution depends on the geometries of the mould and the plunger. For example, the mesh in the ring domain requires a relatively small scale in comparison with the mould domain and the plunger domain. The implicit Euler method is used to solve the initial value problem for the motion of the plunger (80): ⎧

2 2 ⎪ V k+1 − Vpk 2π μ L2 ⎪ k+1 k+1 k+1 k+1 ⎨1 p Fg,1 (z ) + Fg,0 (z ) + Fg,0 (z ) + Fe = Vp 2 zk+1 − zk m pV ⎪ ⎪ ⎩ Vp0 = V0 (81) The plunger position and hence the geometry of the glass domain can be updated using the approximation zk+1 = zk + Δ t kVp (zk ),

t k+1 = t k + Δ t k ,

(82)

where Δ t k denotes the kth time step. The motion of the free boundaries Γo is described by the ordinary differential equation dx = u(x), fort ∈ [0,tend ). (83) dt Let xi , i = 1, . . . , N be the nodes on a free boundary Γf ⊂ Γo and let xki be an approximation of x(tk ) (see Fig. 10). Then the new position of the ith node can be obtained by the explicit scheme xk+1 = xki + Δ t uki , (84) i

28


Ωg (tk ) Γm

Fig. 10 Time integration on the free boundary [34]

Γp

t t @ xk Γf " i @ @t " t i hhh t((( @ A k !At R @ vi ! t XX ! AUt Y !H XX H! Xt ! ti H ! PP ! HH ! P ! PP xk+1 H Ωg (t˜k+1 ) Pt! i

x˜ k+1 i

Fig. 11 Clip algorithm

Γm

Γp

Ω tk

t t @ xk " i @th i hh t((( t" Γf A A i t x˜ k+1 i Y !AUt H QQ HH !! Q !! HH Ω Qt! tk+1

with uki := u(xki ). A particular question is how to deal with this moving boundary. Depending on the velocity a situation may be encountered where the obtained posilies outside Ωg (see Fig. 10). tion xk+1 i In this situation one of the strategies described below may be used. For simplicity only explicit integration is considered. One approach to deal with the moving boundary is to decrease the time step: = xki + αik Δ t uki , x˜ k+1 i

αik ∈ (0, 1],

(85)

is the new position of the ith node obtained with the decreased time where x˜ k+1 i step. The kth time step can be defined by Δ t˜ik := mini αik Δ t, such that the nodes are situated inside the glass domain at time t˜k+1 := t˜k + Δ t˜ik , as depicted in Fig. 10. Unfortunately, this algorithm introduces a variable time step that turns out to be too irregular in practice and can be excessively small. In order to have consistency in the topology of the computational domain the time step should be constant [34]. An alternative is illustrated in Fig. 11; the so-called clip algorithm leads the nodes along a discrete ‘solution curve’ until they reach the boundary, thereby clipping the trajectories on the boundary. Thus node i, which would originally leave the physical domain at time t k+1 , is clipped on the boundary by (85), but with αik = 1 for a


29

Fig. 12 Modified clip algorithm

t J

J J J

J xki Ji t Q @Q vki @Q Q @ R Q @ v˜ ki · t Q @ @ v˜ ki @ k @ v˜ i · n @ @ @ti @ non-clipped node; the time step only differs for clipped nodes. Regrettably, also this algorithm has an evident drawback: the clipping influences the mass conservation property of the glass domain [34]. The clip algorithm can be modified to enforce better mass conservation [34]. To this end alter the velocities at the nodes that would otherwise end up outside the glass domain, such that their normal component stays the same, i.e.

·

u˜ ki n = uki · n,

(86)

·

while the tangential component uki t is obtained by rotating the velocity vector, such that xki ends up on the equipment boundary (see Fig. 12). This can be formulated as u˜ ki := αik Rki uki ,

(87)

where αik is the scaling parameter and Rki is the 2 × 2 rotation matrix, ⎛ Rki := ⎝

cos γik − sin γik sin γik

cos γik

⎞ ⎠.

(88)

The net outflow for the modified velocity field remains zero and hence the algorithm should give better mass conservation, i.e. Γf

·

u n dΓ =

Γf

·

u˜ n dΓ = 0.

(89)

30


An approximate position of a node i that would be outside the glass domain at time t k+1 is obtained using the modified velocity field, x˜ k+1 = xki + Δ t u˜ ki . i

(90)

Instead of (84) also an implicit time integration scheme can be used: = xki + Δ t uk+1 . xk+1 i i

(91)

In order to use (91) it is necessary to compute the flow velocity in Ωg (tk+1 ). However, even though the plunger position is known in the sense of (82), the free boundary of Ωg (tk+1 ) is still unknown. This difficulty can be overcome by employing an algorithm that iterates on xk+1 . Unfortunately, this straightforward approach i requires each iteration the solution of a Stokes flow problem, which is computationally too costly. In [34] a numerical tool is introduced that overcomes the essential difficulty of the implicitness of the scheme by using the fact that the flow velocity is autonomous. In [69] this matter is examined into more detail.

3.5 Results In this section simulations of the pressing of a jar and a bottle parison are shown. This includes the visualisation of the velocity and pressure fields, the tracking of the free boundaries of the glass flow domain and the computation of the motion of the plunger. For more results of the simulation tool used the reader is referred to [34]. Furthermore, see [28] for results of a different parison press model. First consider the simulation of the pressing of a jar parison. The simulation starts at the moment the gob of glass has entered the mould and the baffle has closed. Figure 13 visualises the flow velocity field in the glass domain during pressing. Here full slip of glass at the equipment boundary is assumed, that is β = 0. The plunger moves upward forcing the glass to fill the space between the mould and the plunger. When the glass hits the mould the pressure increases. Figure 14 depicts the flow velocity of the glass in the neck part ring of the jar in the final stage of the parison press. Figure 15 shows the pressure field. It is important to know the pressure of the glass onto the mould during the process, so that a similar pressure can be applied from the outside in order to keep the separate parts of the mould together [34]. Next consider the simulation of the pressing of a bottle parison. The initial position of the top of the plunger in the simulation is close to the attachment of the ring to the mould. When the glass gob is dropped into the mould it almost reaches the ring before the plunger starts moving (see Fig. 16a). Figure 16 visualises the flow velocity of the glass during pressing. Figure 17 depicts the flow velocity in the bottle neck in the final stage of the parison press. Figure 18 shows the pressure field. The results are similar to the jar parison simulation. Again full slip of glass at the equipment boundary is assumed.


31

Fig. 13 Pressing of a jar parison: flow velocity [m/s] [34]

Fig. 14 Pressing of a jar parison (neckring part): flow velocity [34]

4 Blow Model This section presents a mathematical model for blowing. Section 4.1 specifies the mathematical model described in Sect. 2 for the counter blow and the final blow. In addition to the flow of glass, also the flow of air is modelled. A particular question is

32


Fig. 15 Pressing of a jar parison: pressure [bar] [34]

how to deal with the glass-air interfaces. Section 4.2 discusses several techniques. Subsequently, Sect. 4.3 explains how a variational formulation is used to combine the physical problems for glass and air as well as the jump conditions on the interfaces into one statement. Section 4.4 presents a numerical simulation model for the blow–blow process. Finally, Sect. 4.5 shows some examples of process simulations. The simulation tool used for the results is presented in [21].

4.1 Mathematical Model In Sect. 2.2 the balance laws for glass forming were formulated. In this section these balance laws are further specified for glass blowing. Since the final blow stage starts with the preform obtained in either the parison press stage of the counter blow stage, the orders of magnitude of most physical parameters for these forming processes are typically the same. The temperature of the glass melt in the final blow is usually slightly lower than in the preceding stage, but this does not lead to a significant difference in the order of magnitude of the physical parameters. Therefore, the same typical values for both the counter blow and the final blow are considered: Glass density : Glass viscosity : Gravitational acceleration : Flow velocity :

ρg μ¯ g g V

= 2.5 · 103 kg m−3 = 104 kg m−1 s−1 = 9.8 m s−2 = 10−2 m s−1

Mathematical Modelling of Glass Forming Processes Fig. 16 Pressing of a bottle parison: flow velocity [m/s] [34]

Fig. 17 Pressing of a bottle parison (neckring part): flow velocity [34]

Length scale of the parison : L = 10−2 m Glass temperature : Tg = 1, 000 ◦ C Mould temperature : Tm = 500 ◦ C Specific heat of glass : c¯ p = 1.5 · 103 J kg−1 K−1 Effective conductivity of glass : λ¯ = 5 W m−1 K−1 ,

33

34


Fig. 18 Pressing of a bottle parison: pressure [bar] [34]

The main difference from the physical parameters in the parison press is that the time duration of a blow stage is typically much larger, hence the flow velocity is much smaller. The dimensionless numbers corresponding to either blow stage are: Peglass ≈ 76, Brglass ≈ 4.0 · 10−5, Reglass ≈ 2.5 · 10−5, Fr ≈ 1.0 · 10−3. (92) The Péclet number for glass is moderately large, while the Brinkman number is negligibly small. Thus the heat transfer is dominated by advection, convection and radiation: ∂T∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ , (93) + u ∇ T λ ∇ T = ∇ ∂ t∗

·

·

with

λ ∗ :=

λ , ρ c¯ pV L

(94)

rather than its definition (9) in Sect. 2.2.1. The Reynolds number for glass is sufficiently small to neglect the inertia terms in the Navier–Stokes equations (25). The order of magnitude of the gravity term is given by Reglass ≈ 2.5 · 10−2, Fr


35

which is not extremely small. Therefore, the glass flow is described by the Stokes flow equations: ∇∗ u∗ = 0, (95) ∇∗ T∗ = g∗ ,

·

·

where T∗ is the dimensionless stress tensor, which satisfies (20) in terms of the dimensionless variables, and g∗ is the dimensionless gravity force, given by g∗ =

Re ez . Fr

(96)

Subsequently, the air domain is considered. Note that the arguments to ignore the air domain in the parison press model do not apply to the blow model. Firstly, the inflow pressure is applied at the mould entrance and not directly on the glass, so that in this case the transport phenomena in air are of higher interest for the blow model. Secondly, because the influence of the heat flux cannot be ignored, the energy exchange between the glass and its surroundings should be taken into account. For hot pressurised air the following typical values are considered2: Initial air temperature Specific heat of air Thermal conductivity of air Air density Air viscosity

: T0 : cp : λ : ρ : μ

= 750 ◦ C, = 103 J kg−1 K−1 , = 10−1 W m−1 K−1 . = 1 kg m−3 , = 10−4 kg m−1 s−1 .

The resulting dimensionless numbers are: Peair ≈ 102 ,

Brair ≈ 4 · 10−8,

Reair ≈ 1.

(97)

It can be concluded that the heat transfer in air is described by (93), while the flow of air is described by the full Navier–Stokes equations (25). This means that the model for the air flow is more complicated model than for the glass flow, while the motion of glass is most interesting. Therefore, air is replaced by a fictitious fluid with the same physical properties as air, but with a much higher viscosity, e.g. μa = 1. Then the Reynolds number of the fictitious fluid Re ≈ 10−4 is small enough to reasonably neglect the influence of the inertia forces. On the other hand, the viscosity of the fictitious fluid is still much smaller than the viscosity of glass, so that the pressure drop in the air domain is negligible compared to the pressure drop in the glass domain [2]. Thus the flow of the ficitious fluid can be described by the Stokes flow equations (95). Remark 4.1. If the glass temperature and the pressure in air can be reasonably assumed to be uniform, the calculations can be restricted to the glass domain and the glass-air interfaces can be treated as free boundaries with a prescribed pressure. In this case it can be recommended to use Boundary Element Methods to solve the flow problem [13, 14]. 2

The true orders of magnitude may be slightly different from their rough estimates in the table, but this will not affect the final results.

36


The boundary conditions for the flow problem are given in Sect. 2.2, but can be further specified for glass blowing. Free-stress conditions are imposed for air at the mould wall. If no lubricate is used, usually a no-slip condition is imposed for glass at the mould wall. For modelling of the slip condition in the presence of a lubricate the reader is referred to [15,16,36]. As a result the boundary conditions for the flow problem become: u∗ u∗ T∗ n T∗ n

· n = 0, · n = 0, · n = 0, · n = p0 ,

with

T∗ n u∗ T∗ n T∗ n

p0 =

· t = 0, · t = 0, · t = 0, · t = 0,

on on on on

Γs , Γg,e , Γa,e , Γo , (98)

pin on Γa,o 0 on Γg,o ,

(99)

where pin is the pressure at which air is blown into the mould. Alternatively, one may prefer to introduce the boundary condition

·

u∗ n = 0,

·

T∗ n t = 0,

on Γg,o .

(100)

This boundary condition avoids outflow of glass through the mould entrance during blowing. However, this involves the definition of separate boundaries Γa,o and Γg,o , rather than the single boundary Γo . Moreover, these boundaries can change in time. In the next section it can be seen that this is not always convenient, particularly if a single fixed mesh is used for the discretisation of the flow domain. Instead Γg,o can be conceived as the boundary between the glass and the ring, thereby imposing a no-slip condition, on Γg,o . (101) u∗ = 0, Note that in this case Γa,o and Γg,o remain fixed. The boundary conditions for the energy exchange problem are defined on the boundary of the flow domain: ∗ ∗ λ ∗ ∇T ∗ n = 0, ∗ ∗ on Γs ∪ Γe , λ ∇T n = Nu α T − T∗∞ , on Γo , (102)

· ·

4.2 Glass-Air Interfaces A two-phase fluid flow problem is considered, involving the flow of both glass and air. The flow domain Ω is described by the geometry of the mould and hence fixed. On the other hand, the air domain Ωa and the glass domain Ωg are separated by moving interfaces Γi , as depicted in Fig. 4c–d, and therefore change in time. In order to model the two-phase fluid flow problem, the glass-air interfaces have to captured. There are different numerical techniques to deal with the moving interfaces in


37

two-phase fluid flow problems. They can be classified in two main categories [20]: interface-tracking techniques (ITT) and interface-capturing techniques (ICT). Interface-tracking techniques (ITT) attempt to find the moving interfaces explicitly. ITT involve separate discretisations of domains Ωa and Ωg ; the meshes of both domains are updated as the flow evolves by following the flow velocity on the interfaces (see e.g. Sect. 3.4). The major challenge of ITT is the mesh update. The procedure of updating the mesh can become increasingly computationally expensive as the mesh size decreases or the mesh has to be updated more frequently. See [12,28,52] for examples of ITT in mathematical modelling of glass blow processes. Interface-capturing techniques (ICT) are based on an implicit formulation of the interfaces by means of interface functions, which allow ICT to function on a fixed mesh for domain Ω . An interface function marks the location of the corresponding interface by a given level set. Two widely used ICT are Volume-Of-Fluid (VOF) Methods [26, 56] and Level Set Methods [1, 8, 58, 68]. In VOF methods the interface function denotes the fraction of volume within each element of either fluid. VOF methods are conservative and can deal with topological changes of the interface. However, they are often rather inaccurate; high order of accuracy is hard to achieve because of the discontinuity of the interface function [46]. Furthermore, they can suffer from small remnants of mixed-fluid zones (‘flotsam and jetsam’) [44, 48]. Still VOF methods are attractive because of their rigorous conservation properties. In Level Set Methods the interface is generally represented by the zero contour of the interface function. Level Set Methods automatically deal with topological changes and it is in general easy to obtain high order of accuracy [46]. In addition, properties of the interfaces, such as the normal and the curvature, are straightforward to calculate. Also Level Set Methods generalise easily to three dimensions [65]. A drawback of Level Set Methods is that they are not conservative. Poor mass conservation of Level Set Methods for incompressible two-phase fluid flow problems is addressed in [17, 46, 72]. A major concern in Level Set Methods is the re-initialisation of the interface function in order to avoid numerical problems. For examples of Level Set Methods in mathematical modelling of glass blow processes the reader is referred to [2, 20]. Of the aforementioned methods Level Set Methods seem to be most attractive for this application. The interfaces are accurately captured, topological changes are naturally dealt with, a generalisation to three dimensions is relatively easy and complicated re-meshing algorithms are avoided. In order to compensate for the mass loss or gain coupled Level Set and VOF Methods have been developed [48,62,67]. However, [20] reports a change in mass of less than 1% during the glass blow process simulations, which can be further improved by using higher order time integration schemes or by taking smaller time steps. This indicates that in this case Level Set Methods are suited to be used as ICT. The basic idea of Level Set Methods is to embed the moving interfaces as the zero level set of the interface function φ , the so-called level set function (see Fig. 19): φ x,t = 0,

x ∈ Γi (t).

(103)

38


Fig. 19 Level set function for a glass preform

q=0

air

air

q ν1 : Ω · ∇ψ = κ (B − ψ ) + σ (φ − ψ ),

cm ρ m

(44a) (44b)

Radiative Heat Transfer and Applications for Glass Production Processes

where

69

1 s(Ω , Ω )ψ (x, Ω , ν )d Ω 4 π S2 is the scattered intensity. Here, we have explicitly included the opaque band for frequencies in the interval [0, ν1 ], where absorption is infinitely high and thus ψ ≡ B. On the boundary, for x ∈ ∂ V , the ingoing radiation is prescribed by semitransparent boundary conditions

φ (x, Ω , ν ) =

I(Ω ) = ρ (n · Ω )I(Ω ) + 1 − ρ (n · Ω ) Ib (Ω ),

∀n · Ω < 0,

(44c)

B(ν , Tb ) − B(ν , T ) d ν .

(44d)

while the temperature is assumed to obey k n · ∇T = h(Tb − T ) + απ

n 2

ν1

2

n1

0

The additional term comes from the fact that in the opaque band 0 < ν < ν1 the medium behaves like a perfect black body. At initial time t = 0, the temperature shall be given by T (x, 0) = T0 (x).

(44e)

In these equations, ψ (t, x, Ω , ν ) denotes the specific radiation intensity at point x ∈ V traveling in direction Ω ∈ S2 with frequency ν > 0 at time t ≥ 0. The outside radiation Ib is assumed to be known for the ingoing directions (i.e. n · Ω < 0) on the boundary. We denote the outward normal on ∂ V by n. Furthermore, T (t, x) denotes the material temperature and Tb is the exterior temperature on the boundary. The equations contain as parameters the opacity κ , the scattering coefficient σ , the heat conductivity k and the convective heat transfer coefficient h. Moreover, B denotes Planck’s function B(ν , T ) = n21

−1 2hP ν 3 khPTν B −1 e c2

for black body radiation in glass which involves Planck’s constant hP , Boltzmann’s constant kB and the speed of light in vacuum c. The integration in the second term of the temperature boundary condition (44d) is done on the opaque interval of the spectrum [0, ν1 ], where radiation is completely absorbed. At the interface between glass and surrounding air with refractive indices n1 > n2 , respectively, light rays are reflected and refracted. This is modeled by the so-called semi-transparent boundary conditions (44c). The reflectivity ρ ∈ [0, 1] is the proportion of radiation that is reflected. It is equal to 1 if total reflection occurs i.e. if θ1 > θc where θc is the critical angle given by sin θc = n2 /n1 . Otherwise ρ is calculated according to Fresnel’s equation 1 tan2 (θ1 − θ2 ) sin2 (θ1 − θ2 ) , + ρ (μ ) = 2 tan2 (θ1 + θ2 ) sin2 (θ1 + θ2 )

70

M. Frank and A. Klar

where the refraction angles θ1 and θ2 are given by cos θ1 = |n · Ω | = μ and Snell’s law of refraction n1 sin θ1 = n2 sin θ2 . The solid angles of the reflected ray in (44c) is

Ω = Ω − 2(n · Ω )n. Finally, the hemispheric emissivity α of the boundary surface in (44d) is related to the reflectivity ρ by

α = 2n1

1 0

1 − ρ (μ ) d μ .

For these equations and other applications in glass manufacturing problems we refer, for example, to [39, 40, 46, 87], and the monographs [31] and [60]. Analytical results concerning the existence and uniqueness of solutions to the transfer equation itself and to the radiative heat transfer equations, where also energy conservation and additionally heat conduction are considered, have been obtained by many authors. A rather recent review on methods for transport equations can be found in [6], cf. also [5]. The transfer equation together with energy conservation is considered in [28, 58]. The issue of heat conduction is addressed in [37, 43, 44]. Convection, conduction and radiation is treated in [53, 69].

3 Direct Numerical Methods The main difficulties in solving numerically the integro-differential equation (44) are the large set of unknowns and the coupling between the transport and the integral operators. For instance, ψ is a function of time variable t, space variable x, frequency variable ν , and direction variable Ω . Solving the large linear system of algebraic equations induced by discretizing these variables is computationally very demanding. Here, we focus on the solution of steady-state, mono-energetic, frequency decoupled, isotropic radiative transfer problems in three space dimensions. However, all the methods presented in this paper can be straightforwardly extended to the more general problem (44). Hence the radiative transfer equation we consider reads

Ω · ∇ψ + (σ + κ )ψ = σ φ + κ B

(45)

with boundary values ψ = g. The (45) models the changes of an intensity ψ (x, Ω ) as particles are passing through the domain V at position point x = (x, y, z)T in the direction Ω = (μ , η , ξ )T and are subject to loses due to absorption κ and scattering σ , while their number grows due to the source B inside the domain V . We assume that σ and κ are nonnegative functions and we introduce the mean intensity φ as

φ (x) =

1 4π

S2

ψ (x, Ω )d Ω .

(46)


We also define the scattering ratio γ and the opacity coefficient ϑ as

σ (x) γ = max and ϑ = min σ (x) + κ (x) diam(V ). x∈V x∈V σ (x) + κ (x)

71

(47)

Here diam(V ) is the diameter of the space domain V . In applications, γ and ϑ are used to characterize the convergence rates of the iterative methods and the diffusion limits in the optically thick medium. Many numerical methods have been used to solve the (45). For a review on some of these methods see [52, 75]. It is well known [1] that the standard Source Iteration (SI) becomes extremely costly when the scattering ratio γ ≈ 1. The standard Diffusion Synthetic Acceleration (DSA) has been widely used to accelerate the source iteration [4, 10]. The SI and DSA methods can be seen respectively, as Richardson iteration and preconditioned Richardson iteration with the diffusion approach as preconditioner. We implement SI, DSA and a Krylov subspace method to solve the (45). We also propose a fast multilevel algorithm [36] which uses the approximate inverse operator as a preconditioner and solves the linear system only in the coarse meshes. Numerical results show this algorithm to be faster than DSA in many regimes. The robustness, efficiency and convergence rates of these methods are illustrated by several numerical test examples in both one and two space dimensions. Comparison of the results obtained by different methods is also included in this section. The material in this section is taken from [76].

3.1 Ordinates and Space Discretizations We start with a discrete ordinates discretization in angle. This corresponds to expanding the integrals on the unit sphere S2 in terms of N weighted quadrature rules, S2

ψ (x, Ω )d Ω

N

∑ wl ψ (x, Ωl ),

(48)

l=1

where Ωl = (μl , ηl , ξl )T , for all l = 1, . . . , N, with N = n(n + 2), and n is the number of direction cosines. Since Ωl ∈ S2 , we have

μl2 + ηl2 + ξl2 = 1,

for all l = 1, 2, . . . , N.

We assume n an even number of quadrature points so that the points (μl , ηl , ξl ) are nonzero, symmetric with respect to the x-, y- and z-axis and they are invariant under 90◦ rotations. Furthermore they satisfy the relation

ξi2 = ξ12 + 2 for i = 1, 2, . . . , n/2 and 0 < ξ1 < 1/3.

i−1 (1 − 3ξ12), n−2

72


In (48) wl are the corresponding weights chosen to be positive and satisfy N

N

N

∑ Ωl = 4π , ∑ Ωl μl = 0, ∑ Ωl ηl = 0

l=1

l=1

N

∑ Ωl ξl = 0.

and

l=1

l=1

In practice we choose wl = 4π /N and (μl , ηl , ξl ) are set in such a way the above conditions are guaranteed. Let SN be a given set of N discrete directions in S2 , then a semi-discrete formulation of (45) is

μl

∂ ψl ∂ ψl ∂ ψl + ηl + ξl + (σ + κ )ψl = σ φ (x) + κ q(x), ∂x ∂y ∂z

(49)

ψl (x) denotes approximation to ψ (x, μl , ηl , ξl ) and φ is given by φ (x) =

1 4π

N

∑ wl ψl (x).

l=1

To discretize the (49) in space we suppose for simplicity, that the spatial domain is a box, V = [ax , bx ] × [ay , by ] × [az , bz ]. Then we cover the domain V with a uniform numerical mesh defined by Vh = xi jk = (xi , y j , zk )T ,

xi = iΔ x,

i = 0, 1 . . . , I,

y j = jΔ y,

zk = kΔ z, k = 0, 1 . . . , K ,

j = 0, 1 . . . , J,

where x0 = ax , xI = bx ; y0 = ay , yJ = by ; z0 = az , zK = bz ; and h denotes the maximum cell size. We define the averaged grid points

Δ x = xi − xi−1 ,

Δ y = y j − y j−1,

Δ z = zk − zk−1 ,

y j−1 + y j xi−1 + xi zk−1 + zk , y j− 1 = , zk− 1 = , 2 2 2 2 2 2 for i = 1 . . . , I, j = 1 . . . , J and k = 1 . . . , K. By using the notation fi jk to denote the approximation value of the function f at the grid point (xi , y j , zk ), the fully discrete approximation for the (45) can be written as xi− 1 =

μl

ψl,i jk −ψl,i−1 jk Δx

+ ηl

ψl,i jk −ψl,i j−1k Δy

+ σi− 1 j− 1 k− 1 2

2

2

ψ

−ψ

+ ξl l,i jk Δ zl,i jk−1 + κi− 1 j− 1 k− 1 ψl,i− 1 j− 1 k− 1 2

2

2

2

2

2

= σi− 1 j− 1 k− 1 φi− 1 j− 1 k− 1 κi− 1 j− 1 k− 1 qi− 1 j− 1 k− 1 , 2

2

2

2

2

2

2

where the cell averages values of ψ are given by

2

2

2

2

2

(50)


ψl,i−1 jk = ψl,i j−1k ψl,i jk−1 ψl,i jk

73

1 y j zk

ψl (xi , y, z)dydz, Δ x y j−1 zk−1 1 xi zk = ψl (x, y j , z)dxdz, Δ y xi−1 zk−1 1 xi y j = ψl (x, y, zk )dxdy, Δ z xi−1 y j−1 xi y j zk 1 = ψl (x, y, z)dxdydz, Δ xΔ yΔ z xi−1 y j−1 zk−1

(51)

In this paper we use the Diamond difference method to approximate the fluxes in (51). The method consists on centred differences and approximating the function values at the cell centres fl,i− 1 j− 1 k− 1 by the average of their values at the eight 2 2 2 neighbouring nodes as fl,i− 1 j− 1 k− 1 = 2

2

2

1 fl,i−1 j−1k−1 + fl,i−1 jk−1 + fl,i−1 j−1k + fl,i−1 jk 8

+ fl,i j−1k−1 + fl,i jk−1 + fl,i j−1k + fl,i jk .

(52)

Hence the discrete mean intensity φi− 1 j− 1 k− 1 in (50) is given by 2

2

2

N

φi− 1 j− 1 k− 1 = ∑ wl ψl,i− 1 j− 1 k− 1 . 2

2

2

l=1

2

2

2

Other discretizations using Legendre polynomial collocation in ordinates and finite element or Petrov–Galerkin methods in space can be used in the same manner, we refer to [10, 52, 81, 82] for details. For the discretization of the boundary conditions in (49) we can proceed as follows: when xˆ = ax , the normal n(xˆ0 jk ) = (−1, 0, 0)T , then n(xˆ0 jk ) · Ωl = −μl , and for μl > 0 we have ψl,0 jk = g0 jk when yˆ = ay , the normal n(xî0k ) = (0, −1, 0)T , then n(xî0k ) · Ωl = −ηl , and for ηl > 0 we have ψl,i0k = gi0k when zˆ = az , the normal n(xî j0 ) = (0, 0, −1)T , then n(xî j0 ) · Ωl = −ξl , and for ξl > 0 we have ψl,i j0 = gi j0 The other three cases can be discretized in a similar way. Needless to say that for a given l = 1, 2, . . . , N no component of Ωl is ever zero and only three of the above six cases can hold. Furthermore, in the discretization (50) there are (I +1)(J +1)(K +1) unknowns ψl,i jk and IK + JI + JK + I + J + K + 1 boundary equations.

3.2 Linear System Formulation In order to simplify the notations and to get closer to a compact linear algebra formulation of (50), we first define the matrix entries

74


|μl | |ηl | |ξl | σi− 12 j− 21 k− 12 + κi− 21 j− 12 k− 12 + + + , Δx Δy Δz 8 −|μl | −|ηl | −|ξl | σi− 12 j− 12 k− 12 + κi− 21 j− 21 k− 12 + + + . = Δx Δy Δz 8

dl,i− 1 j− 1 k− 1 = 2

2

2

el,i− 1 j− 1 k− 1 2

2

2

and ul,i− 1 j− 1 k− 1 = 2

2

2

u¯l,i− 1 j− 1 k− 1 = 2

2

2

ul,i− 1 j− 1 k− 1 = 2

2

2

vl,i− 1 j− 1 k− 1 = 2

2

2

v¯l,i− 1 j− 1 k− 1 = 2

2

2

vl,i− 1 j− 1 k− 1 = 2

2

2

|μl | |ηl | −|ξl | σi− 12 j− 21 k− 12 + κi− 21 j− 21 k− 12 + + + , Δx Δy Δz 8 |μl | −|ηl | |ξl | σi− 12 j− 21 k− 12 + κi− 21 j− 21 k− 12 + + + , Δx Δy Δz 8 |μl | −|ηl | −|ξl | σi− 12 j− 12 k− 12 + κi− 21 j− 12 k− 12 + + + , Δx Δy Δz 8 −|μl | |ηl | |ξl | σi− 12 j− 21 k− 12 + κi− 21 j− 21 k− 12 + + + , Δx Δy Δz 8 −|μl | |ηl | −|ξl | σi− 12 j− 12 k− 12 + κi− 21 j− 12 k− 12 + + + , Δx Δy Δz 8 −|μl | −|ηl | |ξl | σi− 12 j− 12 k− 12 + κi− 21 j− 12 k− 12 + + + , Δx Δy Δz 8

Next, we define the vectors ⎞ Ψl,0 ⎟ ⎜ Ψl ≡ ⎝ ... ⎠ ∈ R(I+1)(J+1)(K+1) , ⎛

with

Ψl,K ⎞ Ψl,0k ⎟ ⎜ Ψl,k ≡ ⎝ ... ⎠ ∈ R(I+1)(J+1) ,

⎞ ψl,0 jk ⎟ ⎜ Ψl, jk ≡ ⎝ ... ⎠ ∈ R(I+1) ;

⎛

Ψl,Jk

⎛ ⎜ Φ ≡⎜ ⎝

⎛

Φk− 1

2

⎞

2

ψl,I jk

⎞

Φ1− 1 2 .. ⎟ IJK ⎟ . ⎠∈R , ΦK− 1

Φ1− 1 k− 1 2 2 ⎜ ⎟ .. ⎟ ∈ RIJ , ≡⎜ . ⎝ ⎠ ΦJ− 1 k− 1 2

⎛

2

Φ j− 1 k− 1 2

2

with ⎛

⎞ φ1− 1 j− 1 k− 1 2 2 2 ⎜ ⎟ .. ⎟ ∈ RI ; ≡⎜ . ⎝ ⎠ φI− 1 j− 1 k− 1 2

2

2


⎛ ⎜ Q≡⎜ ⎝ ⎞

⎛

Qk− 1 2

Q1− 1 2 .. ⎟ IJK ⎟ . ⎠∈R , QK− 1

Q1− 1 k− 1 2 2 ⎟ ⎜ .. ⎟ ∈ RIJ , ≡⎜ . ⎠ ⎝ QJ− 1 k− 1 2

2

75

⎞

2

Q j− 1 k− 1 2

2

with ⎞ q1− 1 j− 1 k− 1 2 2 2 ⎟ ⎜ .. ⎟ ∈ RI ; ≡⎜ . ⎠ ⎝ qI− 1 j− 1 k− 1 ⎛

2

2

2

In what follows we define the matrix Hl (known as sweep matrix) for the first sweep case μl < 0, ηl < 0, ξl < 0 and the other seven sweep cases can be derived similarly. In order to simplify the notation, we drop hereafter the space grids subscripts from the matrix entries unless otherwise stated. Thus, ⎛

⎞ Dl E l ⎜ ⎟ .. .. ⎜ ⎟ . . ⎜ ⎟ Hl ≡ ⎜ ⎟ ∈ R(I+1)(J+1)(K+1)×(I+1)(J+1)(K+1), D E l l ⎟ ⎜ ⎝ Dl S ⎠ S

with

⎛

⎞ Dl Ul ⎜ ⎟ .. .. ⎜ . . ⎟ Dl ≡ ⎜ with ⎟ ∈ R(I+1)(J+1)×(I+1)(J+1) , ⎝ Dl Ul ⎠ S ⎛ ⎛ ⎞ ⎞ d u u u¯ ⎜ .. .. ⎟ ⎜ .. .. ⎟ ⎜ ⎜ . . ⎟ . . ⎟ Dl ≡ ⎜ ⎟ ∈ R(I+1)×(I+1) , Ul ≡ ⎜ ⎟ ∈ R(I+1)×(I+1) ; ⎝ ⎠ ⎝ d u u u¯ ⎠ 1

1

⎛

⎞ Vl Wl ⎜ .. .. ⎟ ⎜ ⎟ . . El ≡ ⎜ with ⎟ ∈ R(I+1)(J+1)×(I+1)(J+1) , ⎝ Vl Wl ⎠ S ⎛ ⎛ ⎞ ⎞ v v¯ v e¯ ⎜ .. .. ⎟ ⎜ .. .. ⎟ ⎜ ⎜ . . ⎟ . . ⎟ Vl ≡ ⎜ ⎟ ∈ R(I+1)×(I+1) , Wl ≡ ⎜ ⎟ ∈ R(I+1)×(I+1) . ⎝ ⎝ v v¯ ⎠ v e¯ ⎠ 1

1

76


⎛ ⎜ ⎜ S≡⎜ ⎝

⎛ ⎜ ⎜ Σl ≡ ⎜ ⎜ ⎝

S S .. .. . .

⎞

⎟ ⎟ ⎟ ∈ R(I+1)(J+1)×(I+1)(J+1) , S S⎠ S ⎛ ⎞ 1 1 ⎜ .. .. ⎟ ⎜ . . ⎟ S≡⎜ ⎟ ∈ R(I+1)×(J+1) . ⎝ 1 1⎠ 1 ⎞

Σl,1− 1 2

with

..

⎟ ⎟ ⎟ ∈ R(I+1)(J+1)(K+1)×IJK , ⎟ ⎠

.

Σl,K− 1 2

with

0 ⎛

Σl,k− 1

2

⎜ ⎜ ≡⎜ ⎜ ⎝

⎞

Σl,1− 1 k− 1 2

2

..

⎟ ⎟ ⎟ ∈ R(I+1)(J+1)×IJ , ⎟ ⎠

.

Σl,J− 1 k− 1 2

2

and

0

Σl, j− 1 k− 1 ≡ 2

2

⎞

⎛σ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

+κ 1 1 1 1− 21 j− 21 k− 12 i− 2 j− 2 k− 2 8

..

.

⎟ ⎟ ⎟ ⎟ ∈ R(I+1)×I . ⎟ ⎠

σ

+κ 1 1 1 i− 12 j− 21 k− 12 i− 2 j− 2 k− 2 8

0 Using these definitions with Ψ and Φ being the unknowns, the fully discrete equation (50) can be written in matrix form as ⎛ ⎜ ⎜ ⎜ ⎝

H1 ..

.

HN w1 − 4π S . . . − w4πN S

⎞ ⎞⎛ −Σ 1 Ψ1 .. ⎟⎜ .. ⎟ ⎟ ⎜ . ⎟ ⎟⎜ . ⎟ = −ΣN ⎠⎝ ΨN ⎠ I

Φ

⎞ Q1 ⎜ .. ⎟ ⎜ . ⎟ ⎟, ⎜ ⎝ QN ⎠ ⎛

(53)

0

where I is the IJK × IJK identity matrix and 0 is the IJK null vector. The linear system (53) can be rewritten in common linear algebra notation as Ax = b

(54)


77

with ⎛ ⎜ ⎜ A≡⎜ ⎝

H1 ..

.

HN − w4π1 S . . . − w4πN S

⎛

⎛

⎞ Ψ1 ⎜ .. ⎟ ⎜ ⎟ x ≡ ⎜ . ⎟, ⎝ ΨN ⎠

⎞ −Σ 1 .. ⎟ . ⎟ ⎟, −Σ N ⎠

⎞ Q1 ⎜ .. ⎟ ⎜ ⎟ and b ≡ ⎜ . ⎟ . ⎝ QN ⎠

Φ

I

0

3.3 Preconditioning Techniques In computational radiative transfer the desired quantity is usually the mean intensity Φ which is a function only of position x. Therefore we use the Gaussian elimination to eliminate the intensity Ψ1 , . . . , ΨN from (53) and the reduced equation 1 I− 4π

1 −1 w SH Σ ∑ l l l Φ = 4π l=1 N

N

∑ wl SHl−1Ql ,

(55)

l=1

is solved for Φ . We rewrite (55) in compact form as

I −A Φ = f,

(56)

where the Schur matrix A and the right hand side f are given by A =

1 4π

N

∑ wl SHl−1Σl

and

l=1

f=

1 4π

N

∑ wl SHl−1Ql .

l=1

In this section we briefly discuss some numerical methods used in the literature to solve the linear system (56).

3.3.1 Source Iteration The most popular iterative method to solve (55) is the Richardson iteration known in the radiative transfer community as Source Iteration (SI) method. Given an initial guess Φ (0) , the (m + 1)-iterate solution is obtained by

Φ (m+1) =

1 4π

N

∑ wl SHl−1

Ql + Σl Φ (m) ,

m = 0, 1, . . . .

(57)

l=1

It is easy to see that iteration (57) is equivalent to preconditioned block Gauss– Seidel method applied to (54), where the preconditioner is the block lower triangle of the matrix A. Thus, if M is the block lower triangle of A,

78


⎛ ⎜ ⎜ M≡⎜ ⎝

⎞

H1 ..

.

HN − 4wπ1 S . . . − w4πN S I

⎟ ⎟ ⎟, ⎠

x and b are as given in (54), then Mx(m+1) = (M − A)x(m) + b, and x(m+1) = (I − M −1 A)x(m) + M −1 b.

(58)

Therefore the (m + 1)-iterate mean intensity satisfy

Φ (m+1) =

1 4π

N

∑ wl SHl−1Ψl

(m+1)

l=1

=

1 4π

N

∑ wl SHl−1

Ql + Σl Φ (m) ,

l=1

which is identical to (57). Formal results from linear algebra [29, 35] demonstrate that the preconditioned Richardson iteration (58) converges rapidly as long as the norm of the matrix (I − M −1 A) is small. This condition is ensured by taking the scattering ratio γ small, compare [1] for analysis. For γ 1 the SI method converges rapidly, but for γ ≈ 1 (large optical opacity) convergence becomes slow and may restrict the efficiency of the SI algorithm. The SI algorithm can be implemented as follows Algorithm 1: SI algorithm

a. b. c. d. e.

given the initial guess Φ (0) do m = 0, 1, . . . , itmax do l = 1, 2, . . . , N set wl ←− Ql + Σl Φ (m) solve for yl : Hl yl = wl set wl ←− Syl end do 1 N compute Φ (m+1) = ∑ wl wl 4π l=1 compute r(m) = Φ (m+1) − Φ (m) (m) r L2 if ≤ τ stop r(0) L2 end do

Here itmax is the maximum number of the iterations m, τ is a given tolerance and .L2 is the discrete L2 -norm. The step (b) can be solved directly using Gaussian elimination known in computational radiative transfer as sweeping procedure. Additionally, for each direction Ωl in SN only one of the eight possible sweeps is needed.


79

3.3.2 Diffusion Synthetic Acceleration Among the methods used to accelerate the SI algorithm are the synthetic acceleration procedures [1, 4]. The procedures consist on splitting the SI in to two1 step iterations. Thus, we denote ψ (m+ 2 ) as first iteration for the SI in the continuous form of the problem (45), 1

1

Ω · ∇ψ (m+ 2 ) + (σ + κ )ψ (m+ 2 ) =

σ 4π

S2

ψ (m) (x, Ω )d Ω + κ q(x),

1

ψ (m+ 2 ) (x, ˆ Ω ) = g(x), ˆ

(59)

and an equation for ψ (m+1) is required in such a way to be more accurate approxi1 mation to ψ than ψ (m+ 2 ) . To perform this step with synthetic acceleration method, we subtract (59) from (45),

1 1 σ Ω · ∇ ψ − ψ (m+ 2 ) + (σ + κ ) ψ − ψ (m+ 2 ) = ψ − ψ (m) (x, Ω )d Ω , 4 π S2

1 ψ − ψ (m+ 2 ) (x, (60) ˆ Ω ) = 0,

then (60) are replaced by an approximate problem. The Diffusion Synthetic Acceleration (DSA) method [4] approximates the (60) by the diffusion problem −∇ ·

σ 1 ∇ϕ + κϕ = 3(σ + κ ) 4π

S2

ψ − ψ (m) (x, Ω )d Ω ,

x ∈ V,

Here ϕ (x) is an approximation to the mean intensity

ϕ (x) ≈

1 4π

1 ψ (m+1) − ψ (m+ 2 ) (x, Ω )d Ω . S2

Thus the (m + 1)-iterate mean intensity is given by 1

φ (m+1) = φ (m+ 2 ) + ϕ . Note that (61) does not depend on the angle variable Ω , is linear elliptic equation and simple to solve numerically with less computational cost and memory requirement. In order to build a discretization for the diffusion problem (61) which is consistent to the one used for the radiative transfer equation (45), we consider the same grid structure and the same notations as those used in Sect. 2. Hence a space discretization for the (61) reads as −Dh2

1 ϕ + κi− 1 j− 1 k− 1 ϕi− 1 j− 1 k− 1 = pi− 1 j− 1 k− 1 , 2 2 2 2 2 2 2 2 2 3(σ + κ ) i jk

(61)

80


(m+ 1 ) where pi− 1 j− 1 k− 1 = σi− 1 j− 1 k− 1 φ 1 2 1 2

operator Dh2

2

2

(m) −φ 1 1 1 i− 2 j− 2 k− 2 i− 2 j− 2 k− 12 Dh2 = Dx2 + Dy2 + Dz2 , with 2

is given by

2

2

and the difference

βi jk + βi+1 jk ϕi+1 jk − ϕi jk βi−1 jk + βi jk ϕi jk − ϕi−1 jk − , 2 (Δ x)2 2 (Δ x)2 βi jk + βi j+1k ϕi j+1k − ϕi jk βi j−1k + βi jk ϕi jk − ϕi j−1k = − , 2 (Δ y)2 2 (Δ y)2 βi jk + βi jk+1 ϕi jk+1 − ϕi jk βi jk−1 + βi jk ϕi jk − ϕi jk−1 = − . 2 (Δ z)2 2 (Δ z)2

Dx2 (β ϕ )i jk = Dy2 (β ϕ )i jk Dz2 (β ϕ )i jk

The functions κi− 1 j− 1 k− 1 , ϕi− 1 j− 1 k− 1 and pi− 1 j− 1 k− 1 appeared in (61) are given 2 2 2 2 2 2 2 2 2 as in formula (52). The gradient in the boundary conditions is approximated by upwinding without using ghost points. For example, on the boundary surface x = ax of the domain V , the boundary discretization is

ϕ 1 j+ 1 k+ 1 − 2

2

2

ϕ 3 j− 1 k− 1 − ϕ 1 j− 1 k− 1 2 2 2 2 2 2 2 = 0, 3(σ 1 j− 1 k− 1 + κ 1 j− 1 k− 1 ) Δx 2

2

2

2

2

2

and similar work has to be done for the other boundaries. All together, the above discretization leads to a linear system of form Dϕ = p,

(62)

where D is IJK × IJK nonsymmetric positive definite matrix obtained from the difference diffusion operator (61) with boundary conditions included, and p is IJK vector containing the right hand side term. Once again the DSA method can be viewed as preconditioned Richardson iteration for the linear system (54) with the diffusion matrix D like preconditioner,

x(m+1) = I − D−1 A x(m) + D−1b, and D−1 is obtained by solving the diffusion linear system (62). In terms of Φ this is equivalent to

Φ (k+1) = I − (I − D−1 )A Φ (k) + (I − D−1 )b. The implementation of DSA method to approximate the solution of the radiative transfer equation (45) is carried out in the following algorithm Algorithm 2: DSA algorithm given the initial guess Φ (0) do m = 0, 1, . . . , itmax 1 (a) – (d) are similar to Algorithm 1 for the intermediate solution Φ (m+ 2 )


(e) (f) (g) (h)

81

1 compute p = σ Φ (m+ 2 ) − Φ (m) solve for ϕ : Dϕ = p 1 set Φ (m+1) = Φ (m+ 2 ) + ϕ (m) (m+1) − Φ (m) compute (m)r = Φ r L2 if ≤ τ stop r(0) L2 end do

Note that the first lines in Algorithm 2 are similar to the Algorithm 1. How1 ever, the source iteration algorithm gives only the intermediate solution Φ (m+ 2 ) which has to be corrected by adding the solution ϕ obtained by the diffusion approach. Furthermore, if iterative methods are used for the diffusion approach, then an inner iteration loop has to be added to the iteration used by the SI algorithm and an outer SI iteration may require less accuracy from the inner iterations.

3.3.3 Krylov Subspace Methods In general the matrices A and A in (54) and (56) respectively are nonsymmetric and not diagonally dominant. Furthermore, since σ and κ are nonnegative functions and SN has nonzero directions, the matrix A has positive diagonal elements and nonpositive off-diagonal elements. In addition, if el,i− 1 j− 1 k− 1 ≤ 0, for all l, i, j, k, 2 2 2 then the matrix A is weakly diagonally dominant. This condition is equivalent to

h = max Δ x, Δ y, Δ z ≤ max i jk

8|μl | , σi− 1 j− 1 k− 1 + κi− 1 j− 1 k− 1 2

2

2

2

2

2

8|ηl | 8|ξl | , σi− 1 j− 1 k− 1 + κi− 1 j− 1 k− 1 σi− 1 j− 1 k− 1 + κi− 1 j− 1 k− 1 2

2

2

2

2

2

2

2

2

2

2

,

∀ l,

(63)

2

which means physically that the cell size is no more than eight mean free paths of the particles being simulated. Needless to say that the condition (63) gives the bound of the coarser mesh should be used in the computations. In this paper we propose two Krylov subspace based methods, namely the BI-Conjugate Gradient Stabilized (Bicgstab) [86] and the Generalized Minimal Residual (Gmres(m)) [71], where m stands for the number of restarts for Krylov subspace used in the orthogonalization. Bicgstab method has been applied early in [82] to solve (56). The main idea behind these approaches is that the Krylov subspace methods can be interpreted as weighted Richardson iteration

x(m) = I − α P−1 A x(m−1) + α P−1b,

0 < α < 2,

m = 1, . . . ,

(64)

applied to the linear system (54), where the relaxation parameters α and the preconditioner P are variables within each iteration step. Note that when α = 1 and P = M the iteration (64) is reduced to the SI method.

82


The Bicgstab and Gmres(m) algorithms to solve the linear system (56) can be implemented in the conventional way as in [36,71,86], with the only difference that the dense matrix A can not be explicitly stored. All what is needed, however, is a subroutine that performs a matrix-vector multiplication as shown in the following algorithm Algorithm 3: Matrix-vector multiplication

a. b. c. d.

given a vector u, to apply the matrix A to u we proceed as do l = 1, . . . , N set vl ←− Σl u solve for wl : Hl wl = vl set vl ←− Swl end do 1 N set u ←− u − ∑ wl vl 4π l=1

Note that only three vectors (u, vl and wl ) are needed to perform the multiplication of the matrix A to the vector u. Moreover, only three calls for the algorithm 3 are required from the Bicgstab or Gmres(m) subroutines. Preconditioned Bicgstab or Gmres(m) methods can be also used. For instance, in the case when the matrix A is diagonally dominant, the Bicgstab or Gmres(m) methods can be accelerated by using the diagonal as a preconditioner. This approach which requires additional computational work can be easily implemented. It is worth to say that incomplete Cholesky or ILU type preconditioners can not be used to solve (56) because the matrix A is never formed explicitly.

3.4 A Fast Multilevel Preconditioner We describe in this section multilevel solvers for the linear system (56) using an approximate inverse operator as preconditioner on each level of the multigrid hierarchy. Multilevel methods were first applied to radiative transfer problems in [36]. The author proposes two different type of smoothings to approximate solutions for the one dimensional version of (45) in slab geometry. To formulate multilevel solvers we first modify our notation slightly. Using the discretizations introduced in Sect. 2 we assume for simplicity, a given sequence of uniform, equidistant nested grids V1 ⊂ V2 ⊂ · · · ⊂ VL−1 ⊂ VL = Vh , on V with respective mesh sizes Δ x = Δ y = Δ z = 2−l , l = 1, . . . , L. We use the subscripts l and L to refer to the coarse and fine level respectively. Therefore the problem statement (56) becomes

I − AL ΦL = fL .

(65)


83

With M being the iteration matrix, multilevel can be written as follows (m+1)

M ΦL

(m) + I − AL − M ΦL = fL .

This formulation is equivalent to, (m+1)

ΦL

(m) (m) (m) = ΦL + M −1 r(m) , = ΦL + M −1 fL − (I − AL )ΦL

where r denotes the residual associated to (65) and is defined by r = fL − (I − AL )ΦL . The preconditioner we consider in this section is the Atkinson–Brakhage approximate inverse [8] given as M −1 = BLl = I + (I − Al )−1 AL .

(66)

Then the (m + 1)-iterate solution for (65) is simply (m+1)

ΦL

(m)

= ΦL + BLl r(m) .

Analysis of convergence for this kind of multilevel methods has been done in [36]. The central ideas in this analysis are the strong convergence and collective compactness of the operators generated by Al . Let us first define the two-level (Grid2) algorithm. Applying the multilevel preconditioner (66) to the problem (65) we need the fine-to-coarse grid transfer operator RLl defined by RLl ΦL = Φl , and the coarse-to-fine grid transfer operator PlL defined by PlL Φl = ΦL . A natural way to choose these operators is, bilinear interpolation for PlL and simple injection for RLl as in the standard multigrid literature [30]. However, for radiative transfer equation with discontinuous variables these operators have to be changed to those given in [3] which are specially designed for problems with jumping coefficients. Note that to use these operators we require that any discontinuity of κ , σ or q in (45) is a spatial mesh point. The Grid2 algorithm for solving (65) is detailed in the following steps Algorithm 4: Grid2 algorithm given the fine level {L, AL , fL }, the coarse level {l, Al , fl }, the initial guess (0) ΦL and the tolerance τ do m = 0, 1, . . . , itmax

84


a. b. c. d. e. f. g. h. i. j.

(m)

compute the residual r(m) = fL − (I − AL )ΦL set uL ←− AL r(m) restriction ul ←− RLl uL solve on the coarse level for vl : (I − Al )vl = ul Nyström interpolation compute wl ←− Al vl prolongation wL ←− PlL wl set zL ←− uL + wL compute the preconditioner p ←− r + zL (m+1) (m) update the solution ΦL = ΦL + p (m+1) compute the residual r(m+1) = fL − (I − AL )ΦL (m+1) r L2 if ≤ τ stop r(0) L2 end do

The step (d) usually solves the coarse problem exaclty using direct methods. However, since Al is dense matrix which is never explicitly computed nor stored, iterative solvers are required to perfom the step (d). In our numerical examples we used Gmres(m) method which has been discussed in Sect. 4. Note that in our context the notation Grid2 for two-level algorithm does not necessarily mean that we consider two levels of mesh refinements. Thus, Algorithm 4 is also applicable in cases where we have two different space discretizations on the same mesh (L = l + 1). The fully multilevel algorithm (Gridnest) or nested iteration as refered to in [30] can be implemented recursively as follows Algorithm 5: Gridnest algorithm

a. b. c. d. e. f. g.

given the finest level {lmax , Almax , flmax }, the coarsest level {lmin , Almin , flmin } and the tolerances {τl }, l = lmin , . . . , lmax Solve on the coarsest level for Φlmin : (I − Almin )Φlmin = flmin do k = lmin + 1, . . ., lmax set l ←− k − 1 set L ←− k compute the right hand side fL (0) set ΦL ←− PlL Φl set τ ←− τl call Grid2 to solve for ΦL : (I − AL )ΦL = fL end do

Some comments are in order. The steps (b)–(f) are needed only to set the inputs, (0) fine level {L, AL , fL }, coarse level {l, Al , fl }, initial guess ΦL and tolerance τ to the algorithm Grid2. Recall that Gridnest uses coarse levels to obtain improved initial guesses for fine level problems. The tolerance parameters {τl }, which determine how many iterations of the multilevel algorithm to do on each level, can be considered either fixed or adaptively chosen during the course of computation.


85

3.5 Numerical Results The methods discussed in the above sections are tested on a PC with AMD-K6 200 processors using Fortran compiler, see [76] for details. In all these methods the iterations are terminated when r(m) L2 ≤ 10−5. r(0) L2

(67)

To solve the diffusion problem in DSA we used a preconditioned Bicgstab whereas, Gmres(10) is used in Grid2 and Gridnest to solve the coarse problems. These inner iterations are stopped as in (67) but with 10−2 instead of 10−5. In all our computations for the two space dimension cases we used a discrete SN -direction set with 60 directions from [22].

3.5.1 Radiative Transfer Equation in 1D Slab Geometry Our first example is the (45) in 1D slab geometry

μ

σ 1 ∂ψ + (σ + κ )ψ = ψ (x, μ )d μ + κ q(x) ∂x 2 −1 ψ (0, μ ) = 1, μ > 0, and ψ (1, μ ) = 0,

(68)

μ < 0.

We set q = 0, we used 64 Gauss quadrature nodes in the discrete ordinates and a fine mesh of 512 gridpoints in the space dicrestization. The convergence results for two different values of σ and κ are shown in Fig. 3. The fast convergence of Grid2 is well demonstrated in both cases. Although the scattering ratio for the two cases is 0.99, Grid2 method shows strong reduction of number of iterations comparing to the other methods. Same observation is shown when the regime is optically thick (σ = 99, κ = 1). Next we want to compare the efficiency of these methods in terms of CPU time and number of iterations when the scattering ratio γ runs in the range (0, 1). To this end we set σ = 10 and we vary κ keeping the fine gridpoints fixed to 512. In Fig. 4 we plot the scattering ratio versus the number of iterations in the left and versus the CPU time in the right. Grid2 and Gmres(40) preserve roughly the same amount of computational work (referring to number of iterations and CPU time) in the whole interval, while the SI and DSA become costly for values of γ near 1.

3.5.2 Radiative Transfer Equation with Thermal Source The second example is the (45) in the unit square with a thermal source q = B(T ), with B is the frequency-integrated Planckian B = aR T 4 with aR is the radiation constant (aR = 1.8 × 10−8). We fix the temperature to have the linear profile

86

M. Frank and A. Klar σ = 0.99,

100

κ = 0.01 SI DSA Gmres(40) Grid2

L2−norm of residual

10−1 10−2 10−3 10−4 10−5 10−6 10−7

0

5

10 15 Number of iterations

σ = 99,

100

25

κ=1 SI DSA Gmres(40) Grid2

−1

10 L2−norm of residual

20

10−2 10−3 10−4 10−5 10−6 10−7

0

10

20 30 40 Number of iterations

50

60

Fig. 3 Convergence plots for the radiative transfer equation in 1D slab geometry

T (x, y) = 800x + 1, 000 and the boundary function g(x, ˆ y) ˆ = B(T (x, ˆ y)). ˆ In Fig. 5 we show the convergence plots for different values of σ and κ using a fine mesh of 257 × 257 gridpoints. In all cases Grid2 algorithm presents faster convergence behaviour than DSA even in the diffusion regime (σ = 100 and κ = 1). We summarize in Table 3 the CPU time consumed for each method to perform the computations with the different values of σ and κ . In Table 4 we report, the number of gridpoints I × J at each level, the iteration counter m for that level, the number of iterations in Gmres(10) iGmres , the L∞ -norm of the residual, r(m) L∞ , and the factor r(m) L∞ /r(m−1) L∞ . The results presented in Table 4 do not include those obtained for σ = 1, κ = 1 or σ = 100, κ = 1 because the number of iterations m in Grid2 is very large. For instance, for σ = 100, κ = 1 this number surpasses 15 on the coarsest mesh.


a

400 SI DSA Gmres(40) Grid2 30 25 20 15 10 5 0 0 0.1

350 300 250 CPU time

87

200 150

Clipping

0.2

0.3

0.4

0.5

0.6

0.7

100 50 0

b

0

0.1

0.2

0.3

0.4 0.5 0.6 Scattering ratio

0.7

0.8

0.9

1

0.9

1

5 SI DSA Gmres(40) Grid2 0.4

4.5 4

Clipping

3.5 CPU time

0.3 3 0.2 2.5 0.1 2 0 0

1.5

0.1

0.2

0.3

0.4

0.5

0.6

0.7

1 0.5 0

0

0.1

0.2

0.3

0.4 0.5 0.6 Scattering ratio

0.7

0.8

Fig. 4 Plots of scattering ratio γ versus the number of iterations in (a) and the CPU time (seconds) in (b). In both figures σ is fixed to 10

Nevertheless, in all these test cases we have observed that there is very little variation in the number iGmres and the reduction factor r(m) L∞ /r(m−1) L∞ as the meshes are refined. The results of these tables and Fig. 5 show various interesting features about the behaviours of the preconditioner used by each method. First, where the scattering ratio γ = 0.99 (σ = 100 and κ = 1), it is clear that the SI method is unacceptably slow to converge. The convergence rate is improved significantly by Gmres(40), and it is improved even more by DSA method but at the cost of extra work and storage.

88

M. Frank and A. Klar σ = 1, κ = 10

100

SI DSA Gmres(40) Grid2


10−1 10−2 10−3 10−4 10−5 10−6 10−7

0

1

100

2 3 4 Number of iterations σ = 100, κ = 1

5

6


10−1 SI DSA Gmres(40) Grid2

10−2 10−3 10−4 10−5 10−6 10−7

0

10


50

60

Fig. 5 Convergence plots for the radiative transfer equation with thermal source Table 3 CPU time (in seconds)

σ = 1, κ = 10 σ = 1, κ = 1 σ = 100, κ = 1

SI

DSA

Gmres(40)

Grid2

17.8 33.7 2389.8

29.1 77.4 82.3

15.5 20.8 174.4

17.1 23.5 121.7

The most effective method for solving this example, however, is the multilevel Grid2 and Gridnest methods. Second, the number of iterations iGmres in Gridnset remain nearly the same in all levels and is bounded by the number in the coarsest level.


89

Table 4 Results obtained by Gridnest for σ = 1 and κ = 10 at different levels I ×J

m

iGmres

r(m) L∞

r(m) L∞ /r(m−1) L∞

33 × 33

0 1 2 3

2 2 3

0.73E+00 0.32E-01 0.12E-02 0.56E-05

0.43E-01 0.37E-01 0.46E-04

0 1 2

2 2

0.11E-01 0.33E-03 0.12E-05

0.30E-01 0.36E-04

0 1 2

2 2

0.83E-02 0.67E-03 0.53E-05

0.81E-01 0.79E-04

0 1 2

2 2

0.22E-02 0.73E-04 0.15E-06

0.33E-01 0.20E-04

65 × 65

129 × 129

257 × 257

1 σ=0 σ=1 κ=0

σ = 100

κ = 10

κ=0

s=0

s=0

s=1 0.23

σ=0

κ = 0.001 s=0

0

0.44

0.88

σ=0

κ = 100 s=0

1.22

2

Fig. 6 Geometry and values of σ , κ , s = κ q for the discontinuous equation

3.5.3 Radiative Transfer Equation with Discontinuous Variables The aim of this example is to test the multilevel algorithm for radiative transfer problem with jumping coefficients. Thus, the problem statement is the (45) augmented by discontinuous scattering, absorption and source term [2]. The space domain geometry and the values of σ , κ and s = κ B for each subdomain are given in Fig. 6. We take in the first run of this example vacuum boundary conditions whereas, in the second run we use the nonhomogeneous boundary condition g, ⎧ 1 ⎪ ⎪ ⎨ 1 g(x, ˆ y) ˆ = ⎪1 ⎪ ⎩ 0

when xˆ = 0 and 0 ≤ yˆ ≤ 1, when yˆ = 0 and 0 < yˆ ≤ 0.44, when yˆ = 1 and 0 < yˆ ≤ 0.44, otherwise.

90

M. Frank and A. Klar 100 10−1

Clipping


100 10−2

10−2

10−3

10−4 10−6

10−4


0

5

10

15

20

25

30

10−5 10−6

0

50

100 150 Number of iterations

200

100 10−1

Clipping


100 10−2

10−2

10−3

10−4 10−6

10−4


0

5

10

15

20

25

30

10−5 10−6

0

50


250

300

Fig. 7 Convergence rates for the discontinuous equation subject to vacuum boundary conditions (top) and nonhomogeneous boundary conditions (bottom) Table 5 CPU time (in seconds)

Vacuum boundary condition Nonhomogeneous boundary condition

SI

DSA

Gmres(40)

Grid2

794.04 1100.18

304.42 397.53

192.25 110.07

205.22 201.91

The space domain is discretized uniformly into 400 × 200 gridpoints at the finest level. We display in Fig. 7 the convergence rates for the two runs. Table 5 provides the running time used by each method for these computations. A simple inspection


91

Table 6 Gridnest results for the discontinuous problem with vacuum boundary I ×J

m

iGmres

r(m) L∞

r(m) L∞ /r(m−1) L∞

100 × 50

0 1 2 3 4 5

9 9 8 7 7

0.13E+01 0.50E+00 0.13E+00 0.24E-01 0.14E-02 0.45E-05

0.39E+00 0.27E+00 0.19E+00 0.57E-01 0.32E-02

0 1 2 3

8 7 7

0.51E-01 0.10E-02 0.78E-05 0.32E-07

0.21E-01 0.73E-02 0.41E-02

0 1 2

7 7

0.79E-02 0.40E-04 0.13E-06

0.51E-02 0.33E-02

0 1 2

7 7

0.38E-02 0.24E-04 0.94E-07

0.64E-02 0.39E-02

200 × 100

400 × 200

800 × 400

of Fig. 7 shows that Grid2 algorithm solves this problem more effectively than SI or Gmres(40) methods and with less iterations than DSA method. Furthermore, we note that the Gmres(40) method performs poorly after the seventh iteration in both runs. This may be partly due to the fact that the discontinuous σ and κ coefficients change the matrix structure very badly. While, it is not surprising that the SI algorithm performs very poorly in this case. An examination of the CPU time in Table 5 reveals that Gmres(40) consumes less computational work than the other methods. We have observed that the main part of the CPU time needed in DSA or Grid2 is used by Bicgstab or Gmres(10) to solve the diffusion problem in DSA or the coarse linear system in Grid2, respectively. However, by limiting the number of iterations in Bicgstab and Gmres(10) to 1, or increasing the tolerance from 10−2 to 10−1, the results change favourably with significant advantage for Grid2. Table 6 tabulates the results obtained by Gridnest using four levels. It can be clearly seen that there is very little variation in the number iGmres and the reduction factor r(m) L∞ /r(m−1) L∞ as the meshes are refined.

3.5.4 Radiative Transfer Equation with Frequency Dependence Our final example is the frequency-dependent problem

σ Ω · ∇ψ (x, Ω , ν ) + σ + κ ψ (x, Ω , ν ) = ψ (x, Ω , ν )d Ω + κ B(T, ν ), (69) 4 π S2

92


with boundary condition ψ = B(T (x), ˆ ν ). Here ψ = ψ (x, Ω , ν ), T = T (x), σ = σ (x, ν ) and κ = κ (x, ν ) denote respectively, the radiation intensity, the temperature, the scattering and the opacity within the frequency ν > 0. In order to discretize the (69) with respect to the frequency variable ν , we assume N frequency bands [νι , νι +1 ], ι = 1, . . . , N with piecewise constant absorption,

κ = κι ,

∀ ν ∈ [νι , νι +1 ] ι = 1, . . . , N.

We define the frequency-averaged intensity in the band [νι , νι +1 ] by

ψι =

νι +1 νι

ψ (x, Ω , ν )d ν .

Then, the (69) are transformed to a system of N radiative transfer equations of the form νι +1

σι Ω · ∇ψι (x, Ω ) + σι + κι ψι (x, Ω ) = ψι (x, Ω )d Ω + κι B(T, ν )d ν . 4 π S2 νι (70) To approximate the frequency integrals we used trapezoidal formula. In our numerical simulations we use eight frequency bands [νι , νι +1 ], ι = 1, . . . , 8 from glass manufacturing [47]. These frequencies are given in Table 2 in the introduction. We compute the solution of the (69) in a unit square, on the refined grid with 100 × 100 gridpoints, and 64 discrete ordinates. Hence the number of unknowns for each frequency band is 64 × 104, and these computations are done for the 8 frequencies such that the overall number of equations amounts 5.12 × 106. The scattering parameter σ is varying in the set {1, 10, 100}. For every frequency band we calculate it corresponding scattering ratio and, we store the number of iterations and the running time obtained by each method. The results are given in Table 7. All the algorithms iterate the solution in a large iteration numbers for the first frequency bands (with large scattering ratio), then these numbers go decreasing as the frequency bands grow until their reach the minimum for the last frequency band. In this case when the size of scattering ratio is changing dramatically over the frequency bands, only the DSA and Grid2 algorithms lead to satisfying results for all frequencies and also in diffusive limit (σ = 100). The superiority of Grid2 is clearly demonstrated in Table 7.

4 Higher-Order Diffusion Approximations Approximations that are widely used are the PN approximations, cf. Sect. 5.1. A major drawback in higher dimensions and for complicated problems is the large number of equations which have to be solved. We propose the SPN approximations as alternatives to the full glass equations. This class of approximations uses diffusion equations instead of the radiative transfer equations. The number of equations is


93

Table 7 Number of iterations and CPU time for the eight frequency-bands problem Band

σ =1

1 2 3 4 5 6 7 8

Scattering ratio SI

Gmres(40)

DSA

Grid2

0.71428 0.66666 0.11494 0.06077 0.03450 0.00371 0.00175 0.00014

13 13 7 6 5 4 3 3

6 6 5 4 4 3 3 2

6 6 4 4 3 3 3 2

3 3 3 3 3 2 2 2

30.45

26.36

39.38

29.29

117 114 21 14 10 5 4 3

17 17 8 7 5 3 3 2

8 8 5 5 5 3 3 2

5 5 4 3 3 3 3 2

108.30

36.61

35.13

38.40

1577 1637 151 80 48 10 8 4

79 79 22 16 12 5 4 3

16 17 15 14 15 5 4 3

15 14 10 9 8 4 4 2

1190.73

107.80

57.27

122.05

CPU(sec) ———

σ = 10

1 2 3 4 5 6 7 8

0.96153 0.95238 0.56497 0.39292 0.26329 0.03597 0.01732 0.00173

CPU(sec) ———

σ = 100

1 2 3 4 5 6 7 8

0.99601 0.99502 0.92850 0.86617 0.78137 0.27175 0.14985 0.01381

CPU(sec) ———

considerably reduced compared to the PN equations. The method originates from neutron transport theory in nuclear physics where it was successfully introduced. Nevertheless, it suffered from a lack of theoretical foundation with the result that it was not completely accepted in the field. This has been remedied, however, during the last decade, such that the method has now been substantiated. We want to study the optically thick regime where the opacity κ is large and the radiation is conveyed in a diffusion-like manner. Therefore, we rewrite the above equations in non-dimensional form introducing reference values which correspond to typical values of the physical quantities. In order to obtain a diffusion scaling we impose the relations tre f = cm ρm κre f x2re f

Tre f , Ire f

and kre f =

Ire f , κre f Tre f

94


on these reference values and define the non-dimensional parameter

ε=

1 κre f xre f

(71)

which is small in the optically thick, diffusive regime. Then the rescaled equations read (without marking the scaled quantities):

ε2

∂T = ε 2 ∇ · k∇T − ∂t

∞ ν1 S2

κ (B − ψ ) d Ω d ν ,

(72a)

∀ν > 0, Ω ∈ S2 : εΩ · ∇ψ = κ (B − ψ ).

(72b)

Here, we have neglected scattering. This can be incorporated in a straightforward way, however. The boundary condition for the temperature changes into

ε k n · ∇T = h(Tb − T ) + απ

n 2

ν1

2

n1

0

B(ν , Tb ) − B(ν , T ) d ν .

(72c)

It is well known that an outer asymptotic expansion of (72a) and (72b) leads to equilibrium diffusion theory, which is, in the cases considered here, expected to be valid in the interior of V , see e.g. [46,67,68]. The diffusion or Rosseland approximation is

∂T = ∇ · k + kr (T ) ∇T, ∂t

with

kr (T ) =

4π 3

∞ 1 ∂B ν1

κ ∂T

dν .

However, this diffusion theory is not capable of describing boundary layers and the question arises whether more sophisticated approximations can suitably model the boundary layer effects. In the realm of neutron transport, such higher-order asymptotic corrections to diffusion theory exist, and are reasonably well understood; they are the so-called simplified PN (SPN ) theories, see [9, 80]. These SPN theories are, in fact, diffusion in nature i.e. diffusion equations or coupled systems of diffusion equations are employed. They contain boundary layer effects and can be remarkably accurate, much more accurate than the standard Rosseland approximation. In practice, these equations are viewed as an extended form of the classical diffusion theory. No separate boundary layer treatment is necessary because the boundary layers are included in the SPN equations. For other approximate theories for the above equations and applications, see for example [17, 49]. The material of this section is taken from [47].

4.1 Asymptotic Analysis and Derivation of the SPN Approximations To solve (72a) in the domain V , we write this equation as

ε 1 + Ω · ∇ ψ (x,t, Ω , ν ) = B(ν , T ). κ


95

and apply Neumann’s series to formally invert the operator

−1 ε ψ = 1+ Ω ·∇ B κ ε ε2 ε3 ε4 ∼ = 1 − Ω · ∇ + 2 (Ω · ∇)2 − 3 (Ω · ∇)3 + 4 (Ω · ∇)4 · · · B. κ κ κ κ

(73)

Integrating with respect to Ω and using the result S2

(Ω · ∇)n d Ω = [1 + (−1)n] :

2π n ∇ , n+1

where ∇2 = ∇ · ∇ = Δ is the Laplacian, we get

ϕ=

ε2 ε4 ε6 ψ d Ω = 4π 1 + 2 ∇2 + 4 ∇4 + 6 ∇6 · · · B + O(ε 8). 3κ 5κ 7κ S2

Hence, −1 ε2 ε4 ε6 4π B = 1 + 2 ∇2 + 4 ∇4 + 6 ∇6 ϕ + O(ε 8 ) 3κ 5κ 7κ ε2 ε4 ε6 ∇2 + 4 ∇4 + 6 ∇6 = 1− 2 3κ 5κ 7κ 2 ε2 ε4 ε6 ∇2 + 4 ∇4 + 6 ∇6 2 3κ 5κ 7κ ε2 4 ε ε 6 6 3 2 4 − ∇ + 4 ∇ + 6 ∇ · · · ϕ + O(ε 8 ), 3κ 2 5κ 7κ +

so we have the asymptotic expansion ∀ν > 0 :

ε2 4ε 4 4 44ε 6 6 4π B = 1 − 2 ∇2 − ∇ − ∇ ϕ + O(ε 8 ). 3κ 45κ 4 945κ 6

(74)

If we discard terms of O(ε 4 ), O(ε 6 ) or O(ε 8 ) we obtain the SP1 , SP2 and SP3 approximations, respectively. All these equations contain the frequency ν as a parameter.

4.1.1

SP1 and Diffusion Approximations

From (74), we obtain 4π B = ϕ −

ε2 2 ∇ ϕ + O(ε 4 ) 3κ 2

96


and up to O(ε 4 ) we may write the equation in the form ∀ν > 0 :

−ε 2 ∇ ·

1 ∇ϕ + κϕ = κ (4π B). 3κ

(75a)

In this equation, ν is simply a parameter. Thus, in practice, these equations would be solved independently for each frequency or frequency group and subsequently coupled via (72a). By (75a), ∞ ν1

S2

κ (B − I) d Ω d ν =

∞ ν1

κ (4π B − ϕ ) d ν = −ε 2

∞ ν1

∇·

1 ∇ϕ d ν + O(ε 4 ). 3κ

Thus, the energy equation (72a) becomes up to O(ε 2 ):

∂T = ∇ · k∇T + ∂t

∞ ν1

∇·

1 ∇ϕ d ν . 3κ

(75b)

Equations (75b) and (75a) are the SP1 approximation to (72a) and (72b). Since (75b) is only of order O(ε 2 ) the approximation is O(ε 2 ). Using ϕ = 4π B+ O(ε 2) in (75b) one obtains up to O(ε 2 )

∂T = ∇ · k∇T + ∂t

∞

1 ∇(4π B) d ν 3 κ ν1

4π ∞ 1 ∂ B d ν ∇T, = ∇ · k∇T + ∇ · 3 ν1 κ ∂ T ∇·

(76)

i.e. we have obtained the conventional equilibrium diffusion or Rosseland approximation (73). However, (75a) permits a boundary layer behaviour near the boundary ∂ V that is not present in (76).

4.1.2 SP2 Approximation From (74), we get for ε going to 0 4π B = ϕ − This implies

ε2 2 4ε 2 2 ε 2 2 ∇ ϕ− ∇ ∇ ϕ + O(ε 6 ). 2 3κ 15κ 2 3κ 2 ε2 2 ∇ B = ϕ − 4π B + O(ε 4). 3κ 2

Hence, with O(ε 6 ) error, the expansion above gives 4π B = ϕ −

ε2 2 ε 2 2 4ε 2 2 4 ( ∇ ϕ − ∇ [ ϕ − 4 π B] = ϕ − ∇ ϕ + ϕ − 4 π B) , 3κ 2 15κ 2 3κ 2 5


97

or equivalently, −ε 2 ∇ ·

1 4 ∇ ϕ + (ϕ − 4π B) + κϕ = κ (4π B). 3κ 5

(77)

Equation (77) implies ∞ ν1 S2

κ (B − I) d Ω d ν = −ε 2

∞ ν1

∇·

1 4 ∇ ϕ + (ϕ − 4π B) d ν + O(ε 4 ). 3κ 5

Thus, the energy equation (72b) becomes up to O(ε 4 )

∂T = ∇ · k∇T + ∂t

∞ ν1

∇·

1 4 ∇ ϕ + (ϕ − 4π B) d ν . 3κ 5

(78)

These two equations can be written in a more advantageous way if we define 4 ξ = ϕ + (ϕ − 4π B). 5

(79)

Introducing the new variable ξ into (77) and (78) we obtain the SP2 equations:

∞ ∂T 1 = ∇ · k∇T + ∇· ∇ξ d ν , and ∂t 3 κ ν1 3 −ε 2 ∇ · ∇ξ + κξ = κ (4π B). 5κ

(80a) (80b)

There is a remarkable similarity between the SP2 (80) and the SP1 (75). This is because the SP1 equations contain some, but not all, of the O(ε 4 ) correction terms. In the realm of neutron transport, the SP2 approximation has not found favour because, in the presence of material inhomogenities, it yields discontinuous solutions. However, it is obvious that (80b) and (80a) can not produce a discontinuous solution. Also, in the realm of neutron transport, the SP1 and SP2 solutions are not capable of exhibiting boundary layer behaviour, while the more complicated SP3 solution described below does incorporate this in a remarkably accurate way. However, the radiative transfer SP1 and SP2 equations stated here can contain boundary layer behaviour. The SP3 approximation derived in the following should capture significant radiative transfer boundary effects that are not captured by the SP1 and SP2 approximations.

4.1.3 SP3 Approximation Ignoring terms of O(ε 8 ) in (74), we get

98


ε 2 2 4ε 2 2 44ε 4 4 ∇ ϕ+ ∇ ϕ+ ∇ ϕ + O(ε 8 ) 2 2 3κ 15κ 315κ 4 ε2 11ε 2 2 4ε 2 = ϕ − 2 ∇2 ϕ + 1 + ∇ ϕ + O(ε 8 ) 3κ 21κ 2 15κ 2 ε2 11ε 2 2 −1 4ε 2 ∇ ϕ + O(ε 8 ) = ϕ − 2 ∇2 ϕ + 1 − 3κ 21κ 2 15κ 2

4π B = ϕ −

(81)

Hence, if we define 11 ε 2 2 −1 2ε 2 ϕ2 ≡ 1 − ∇ ϕ , 21 κ 2 15κ 2

(82)

then (81) becomes up to O(ε 8 ): 4π B = ϕ − or ∀ν > 0 :

−ε 2 ∇ ·

ε2 2 ∇ (ϕ + 2ϕ2 ) 3κ 2

1 ∇(ϕ + 2ϕ2 ) + κϕ = κ (4π B), 3κ

(83a)

while (82) becomes −

2 ε2 2 2 ε2 2 11ε 2 2 ∇ ϕ + ϕ = ∇ ϕ = ∇ ϕ 2 2 21κ 2 15 κ 2 5 3κ 2 2 2ε 2 −4π B + ϕ − 2 ∇2 ϕ2 = 5 3κ

or

4 11 ε 2 2 2 − ∇ ϕ2 + ϕ2 = (ϕ − 4π B) 15 21 κ 2 5

or eventually ∀ν > 0 :

−ε 2 ∇ ·

9 2 2 ∇ϕ2 + κϕ2 − κϕ = − κ (4π B). 35κ 5 5

(83b)

By (83a) we get up to O(ε 6 ) ∞ ν1

S2

κ (B − I) d Ω d ν = −ε 2

∞ ν1

∇·

1 ∇(ϕ + 2ϕ2 ) d ν . 3κ

Thus, the energy equation (72b) becomes

∂T = ∇ · k∇T + ∂t

∞ ν1

∇·

1 ∇(ϕ + 2ϕ2) d ν . 3κ

(83c)


99

Equation (83c) together with the two approximate (83a, 83b) form the SP3 approximation to the system (72a) and (72b). These equations can be rewritten in a computationally more advantageous way. Let us calculate θ {(83a)} + {(83b)} : −ε 2 ∇ ·

1 9 2 2 θ ∇ (ϕ + 2ϕ2) + ϕ2 + κ θ ϕ + ϕ2 − ϕ = κ θ − (4π B). κ 3 35 5 5

We seek linear combinations of both equations such that the two functions in the brackets on the left are scalar multiples. More explicitly, we look for θ that fulfills the condition θ 9 2 (84) (ϕ + 2ϕ2) + ϕ2 = μ 2 θ ϕ + ϕ2 − ϕ , 3 35 5 where μ 2 > 0 is a constant to be determined later. Equation (84) holds for arbitrary ϕ and ϕ2 iff 9 θ 2 2θ = μ2 θ − + = μ 2. and 3 5 3 35 The second of these equations may be solved for θ and we get a quadratic equation in μ 2 : 2 1 2 9 11 = μ2 μ2 − . μ − 2 70 3 14 Its discriminant is positive and thus we obtain two positive real solutions

μ12

3 2 = − 7 7

6 , 5

and

μ22

3 2 = + 7 7

6 , 5

and the corresponding values of the scalar θ are 3 9 − θ1 = 35 7

6 , 5

3 9 and θ2 = + 35 7

6 . 5

Now relation (84) implies, for n = 1, 2,

1 2 2 2 −ε ∇ · ∇μn + κ θn ϕ + ϕ2 − ϕ = θn − κ (4π B). κ 5 5 2

(85)

This suggests that we define two new independent variables for n = 1, 2 In =

1 θn ϕ + ϕ2 − 2/5ϕ =ϕ+ ϕ2 = ϕ + γn ϕ2 θn − 2/5 θn − 2/5

where 5 1 = 1 + (−1)n3 γn = θn − 2/5 7

! 6 . 5

(86)

100


The two equations in (85) are now 1 ∇I1 + κ I1 = κ (4π B), κ 1 −ε 2 μ22 ∇ · ∇I2 + κ I2 = κ (4π B). κ

−ε 2 μ12 ∇ ·

(87a) (87b)

The advantage of this form of the SP3 equations is that the diffusion equations are uncoupled. It will be seen below that there remains, however, a weak coupling in the boundary conditions. The linear transformation of variables above is inverted according to the formulae

ϕ=

γ2 I1 − γ1 I2 , γ2 − γ1

and ϕ2 =

I2 − I1 . γ2 − γ1

(88)

Defining three constants 1 7 w0 = = γ2 − γ1 30 1 γ2 w1 = = γ2 − γ1 6

5 3+ , 6

7 5 = 6 36

6 , 5

−γ1 1 w2 = = γ2 − γ1 6

and

5 3− 6

(89a)

(89b)

we can write ϕ = w1 I1 + w2 I2 and ϕ2 = w0 (I2 − I1) and we have furthermore 1 1 1 (ϕ + 2ϕ2 ) = (w1 − 2w0 )I1 + (w2 + 2w0 )I2 = a1 I1 + a2I2 . 3 3 3 Here again we introduced constants 1 1 5 5 w1 − 2w0 w2 + 2w0 = 5−3 , and a2 = = 5+3 . a1 = 3 30 6 3 30 6 In this way, the SP3 energy equation (83c) above becomes:

∂T = ∇ · k∇T + ∂t

∞ ν1

∇·

1 ∇(a1 I1 + a2 I2 ) d ν . κ

(90)

4.2 Boundary Conditions for SPN Approximations The boundary conditions for the SPN equations in neutron transport come from a variational principle, see [9, 80]. Here, we use the boundary conditions developed for the transport case to state (and rewrite in a more suitable form) the boundary conditions for SP1 , SP2 and SP3 approximations to the transport problem (72b) with the boundary condition (44c).


101

We consider the transport equation (72b) ∀ν > 0 :

εΩ · ∇ψ (x, Ω ) + κ I(x, Ω ) = κ B,

x ∈ V,

with semi-transparent boundary conditions on ∂ V

ψ (x, Ω ) = ρ (n · Ω )ψ (x, Ω ) + 1 − ρ (n · Ω ) ψb (x, Ω ),

n · Ω < 0.

Let us define the scalar flux as before

ϕ (x) =

S2

ψ (x, Ω ) d Ω ,

and define two integrals of the influx into the domain for m = 1, 3

ψm (x) =

1 − ρ (n · Ω ) Pm (|Ω · n|)Ib (x, Ω ) d Ω , x ∈ ∂ V.

(91)

n·Ω 1, Tw − TR TR

(29)

where Tw , Tw > TR , is another characteristic temperature, as we shall see in next section. Note that (28) reads now ρ (ϑ ) = ρR − βRρR TR T!w − 1 ϑ .

(30)

Next, we introduce the hydraulic head P = p + ρRgx3 .

(31)

Hence −ρ ge3 − ∇p in (26) becomes

" −ρ ge3 − ∇p = (ρR − ρ ) ge3 − ∇P = ρR βR TR T!w − 1 ϑ ge3 − ∇P.

We note that, after “eliminating” the hydrostatic pressure, the effect of the temperature field on the flow is more easy to observe. Concerning the shear viscosity μ we assume the well known Vogel–Fulcher– Tamman’s (VFT) formula log μ (ϑ ) = −Cμ +

Aμ , TR − Bμ + TR T!w − 1 ϑ

Aμ , Bμ ,Cμ > 0.

(32)

In particular, μ is monotonically decreasing with ϑ . For more details we refer e.g. to [26], Chap. 6. We note that the temperature in the problems we are considering is such that the denominator in (32) is always positive. Consequently, μ is a given strictly positive and C∞ function of the temperature. In the polymer physics, formula (32) is known as Williams–Landau–Ferry (WLF) relation. Finally, we suppose c p1 and λ to be smooth functions of ϑ , bounded from above and from below by positive constants.

2.4 Scaling and Dimensionless Formulation The scaling of model (25)–(27) will be operated paying particular attention to the specific problem we are interested in. We recall that we are considering a gravity driven flow of molten glass through a nozzle in the early stage of a fiber manufacturing process. At this stage we consider the inlet and outlet temperatures of the fluid as prescribed quantities. In particular, referring to Fig. 2, the fluid temperature on Γin is higher than the one on Γout ,

184

A. Farina et al.

T (Γin ) = Tw , T (Γout ) = TR < Tw . We introduce the following dimensionless quantities ⎧ x v t H ⎪ ! x= , ! v= , t˜ = with tR = , ⎪ ⎪ ⎪ H VR tR VR ⎪ ⎨ c p1 μ ˜ λ μ˜ = , λ= , c!p1 = , ⎪ μR λR c pR ⎪ ⎪ ⎪ ρ ! P ψ ⎪ ⎩ ρ! = , P= , ψ˜ = , ρR PR TR c pR H playing the role of a length scale (see again Fig. 2). The choice of the reference velocity VR has to be made according to the particular flow conditions we deal with3 . We will return to this point later on. As the reference pressure PR we take the point of view that flows of glass or polymer melts are essentially dominated by viscous effects. Accordingly we set4 μR VR . PR = (33) H Concerning the reference quantities μR , λR and c pR we identify them with the values taken by the respective quantities μ , λ and c p for P = PR and T = TR . Suppressing tildes to keep notation simple, model (25)–(27) rewrites5 # # #Kρ # Dϑ (Tw − 1) , (34) div v = ρ (ϑ ) Dt Dv = ρ (ϑ ) Dt

$# # % #Kρ # (Tw − 1) Fr2 +

Dϑ = ρ (ϑ )c p1 (ϑ ) Dt

3

$

1 Div Re

# # #Kρ # PR

ϑ e3 −

PR ρRVR2

∇P

2 μ (ϑ ) 2 μ (ϑ )D(v) − div vI , 3 %

(35)

DP ρR gH − v3 ρR c pRTR (Tw − 1) Dt PR 1 Ec 1 + div (λ ∇ϑ ) + 2 μ (ϑ ) |D(v)|22 − ( div v)2 , Pe Re(Tw − 1) 3 (36) 1 + (Tw − 1)ϑ ρ (ϑ )

This makes our approach different from the ones presented in [22] and in [11] where there is no velocity scale defined by exterior conditions. 4 Notice that P → 0 as V tends to 0 and, as a consequence p tends to the hydrostatic pressure. This R R is consistent with the fact that P represents the deviation of the pressure from the hydrostatic–one due to the fluid motion. 5 In the phenomena we are considering T − 1 is small but not negligible. Typically T − 1 is of w w order 10−1 .

Non-Isothermal Molten Glass Drawing

185

with c p 1 (ϑ ) = −

1 + (Tw − 1)ϑ d 2 ψ . (Tw − 1)2 d ϑ 2

(37)

We list the non-dimensional characteristic numbers appearing in (34)–(36): Kρ

= −βR TR , thermal expansivity number (negative, according to the usual notation) and we may write (30) as # # ρ (ϑ ) = 1 − #Kρ # (Tw − 1)ϑ .

(38)

VR = √ , Froude’s number. gH μR νR = , kinematic viscosity. ρR VR H , Reynolds’ number, which, because of (33), can be also written as Re = νR ρRVR2 Re = . PR VR H ρR c pR Pe = Re·Pr = , Peclet’s number , with c pR = ψR /TR . λR μR c pR Pr = , Prandtl’s number. λR V2 Ec = R , Eckert’s number. c pR TR Fr

As mentioned, we are interested in studying vertical slow flows of very viscous heated fluids (molten glass, polymers, etc.) which are thermally dilatable. So, introducing the so–called expansivity coefficient (or thermal expansion coefficient) # # α = #Kρ # (Tw − 1).

(39)

we will consider the system (34)–(36) in the realistic situation in which the parameter α is small. Typically (e.g. for molten glass) 10−3 α 10−2 . Equation (38) can be rewritten as

ρ (ϑ ) = 1 − αϑ . Next, we define the Archimedes’ number # # #Kρ # (Tw − 1) α Ar = 2 = gH, ⇒ α = Ar Fr2 . VR2 Fr So, system (34)–(36) rewrites as6

6

We remark that

186

A. Farina et al.

div v =

α Dϑ , ρ (ϑ ) Dt

(40)

Dv 1 2 μ (ϑ ) = Ar ϑ e3 + div vI , (41) ρ (ϑ ) −∇P + Div 2 μ (ϑ )D(v) − Dt Re 3 Fr2 DP − v3 Re Dt 1 Ec 1 + div (λ ∇ϑ ) + 2 μ (ϑ ) |D(v)|2 − ( div v)2 . Pe Re(Tw − 1) 3 (42)

1 + (Tw − 1)ϑ Ec Ar Dϑ = ρ (ϑ )c p1 (ϑ ) Dt ρ (ϑ ) (Tw − 1)2

Fr2 Ec and which appear naturally in the equation above can Notice that the ratios Re Re be interpreted as ratios of characteristic times Ec = Re

νR c pR TR

Note also that Ec Ar = α

1 Fr2 = , tR Re

νR gH

1 . tR

gH Ec =α 2. c p R TR Fr

We consider a flow regime such that Ar = O (1), Re = O (1), Pe = O (1) and Ec 1 (e.g. Ec 10−9). The terms in energy equation (42) containing the Eckert are dropped. So (40) and (41) remain unchanged while (42) reads as follow

ρ (ϑ )c p1 (ϑ )

1 Dϑ = div λ (ϑ )∇ϑ . Dt Pe

(43)

We remark that in the limit α small, while Ar and Re order 1, the buoyancy and the viscous forces cannot be negligible. We will show that in such a case the system (40), (41) and (43) can be approximated by a system similar to the Oberbeck–Boussinesq system. Indeed our choice of the parameters takes us close to the conditions of the formal derivation of the Oberbeck–Boussinesq system for bounded domain. Remark 2.3. Let us note that the model (40)–(42) is criticized in [2] and [3], on the basis of the fact that giving the density as a function of the temperature contradicts the Gibbs convexity inequalities thus generating a rest state which is not stable. In order to overcome such a difficulty, they propose to replace the constraint ρ = ρ (ϑ )

α α = . ρ (ϑ ) 1 − αϑ Hence, if order α 2 terms are neglected, we may approximate

α ρ (ϑ )

by α .


187

with the constraint ρ = ρ (s). However, we emphasize that the system (40), (41), (43) (i.e. with Ec = 0) has a rest state (in absence of any body force) linearly stable. We have also to remark that, when Eckert’s number is very small, the rest state instability of the system (40)–(42) would show up after such a long time to be of no interest in technical problems. Clearly, for some other problems (e.g. atmospheric flows) the presence of instabilities, pointed out in [2] and [3], is a real difficulty and the meaning of the stationary problems is not clear. Furthermore, we neglected some other terms from the full compressible Navier–Stokes system, whose influence, in the authors’ knowledge, on linear stability of the rest state has not been investigated. In Appendix 1 we show a direct analysis of the linear stability of the rest state according to model (25), (26), (27), (28) and model (25), (26), (43), (28).

3 Study of the Stationary Non-Isothermal Molten Glass Flow in a Die In Sect. 2 a mathematical model which describes the motion of a mechanically incompressible, but thermally expansible viscous fluid was derived. Such model can be widely used in industrial simulations of flows of hot melted glass, polymers etc. The model for a mechanically incompressible, but thermally expansible viscous fluid could be thought as a particular case of the compressible heat-conducting Navier–Stokes system. Nevertheless, in the mechanically incompressible but thermally expansible case the pressure is not linked any more to the density and to the temperature, very much the same as in the incompressible case. Consequently, we had to be careful with the thermodynamical modelling. An important quantity that we have introduced is the thermal expansion coefd log ρ , where ρ is the molten glass density and T is the absolute ficient β = − dT temperature, and its typical value βR . In this section we study the stationary flow within the die, i.e. the stationary flow in stage (b). The flow domain (see Fig. 2) is the set Ω = {{r < R(x3 , φ )} × [0, 2π ] × (0, H)}, where R : [0, H] × [0, 2π ] → [Rmin , Rmax ] is a C∞ -map. For the reference problem we have in mind R and H are lengths of comparable size. The boundary of Ω contains 3 distinct parts (see again Fig. 2): • Lateral boundary Γlat = {r = R (x3 , φ ), x3 ∈ (0, H) , φ ∈ [0, 2π ]}. • Inlet boundary or upper boundary Γin = {x3 = H and r ≤ R(H, φ ), φ ∈ [0, 2π ]}. • Outlet boundary Γout = {x3 = 0 and r ≤ R(0, φ ), φ ∈ [0, 2π ]}.

The stationary problem in non–dimensional form, for the velocity v, the dimensionless hydraulic head P = (p + ρRgHx3 )/PR and the rescaled temperature ϑ reads as follows: Problem A. Find (v, P, ϑ ) such that

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Fig. 2 The domain Ω

Gin

x3 H

Glat

R(x3,f)

O

Gout

div (ρ (ϑ ) v) = 0, ρ (ϑ ) = 1 − αϑ , (44) 1 2 μ (ϑ ) div v − div (2 μ (ϑ )D (v)) , (45) ρ (ϑ ) (v∇) v = Ar ϑ e3 − ∇ P+ Re 3 c p1 (ϑ ) ρ (ϑ ) v · ∇ϑ = v = v1 e3 ,

ϑ = 1,

v = 0,

−

on Γin ,

1 div (λ (ϑ )∇ϑ ) , Pe v = v2 eg ,

ϑ = 0,

1 λ (ϑ )∇ϑ · n = q0 ϑ + S , Pe

(46) on Γout ,

on Γlat ,

(47) (48)

where the fluid’s thermal expansivity α ≥ 0 is defined by formula (39) and c p1 is a function of temperature which can be expressed by means of Helmholtz free energy, as shown in (37). We refer the reader to Sect. 2 for the notations which, on the other hand, are quite standard. In (48) S is a non–negative prescribed function. The plan of this section is the following: • In Sect. 3.1 we will explain the basic steps of an existence proof for the boundary

value problem (44)–(48), which does not require small data. Furthermore, we will give a hint on uniqueness for small data and prove that solution is infinitely differentiable, if the data are so. • In Sect. 3.2 we exploite the uniqueness and regularity results concerning system (44)–(48) to obtain a rigorous mathematical justification of the Oberbeck– Boussinesq approximation (in the limit of small expansivity parameter), a long debated question, and also to compute the next order approximation (the correction). We note that this is another singular limit, not linked at all to the zero Mach number limit (see [19] or [28]).


189

3.1 Existence and Uniqueness Result for the Stationary Problem In order to study the existence of a solution to the problem (44)–(48), we list the assumptions on the data: We suppose that ⎧ 1 ∞ ⎪ ⎪ v1 ∈ C0 (Γin ) ∩C (Γ¯in ), ⎪ ⎪ ⎪ ⎨ (49) v2 ∈ C01 (Γout ) ∩C∞ (Γ¯out ), ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ S ∈ C∞ (Γ¯ ), S ≥ 0, lat and that Γin

ρ (1)v1 rdrd φ =

Γin

(1 − α )v1 rdrd φ =

Γout

v2 rdrd φ ,

(50)

since ρ (0) = 1. Concerning the coefficients, it is natural that μ corresponds to Vogel–Fulcher– Tamman’s VFT formula (32), that we rewrite log μ (ϑ ) = −Cμ +

Aμ , Bμ − ϑ

Cμ > 0.

(51)

Next, as mentioned we suppose that c p1 , μ , ρ , λ ∈ C∞ (R),

(52)

and take values between two positive constants. We note that the coefficients are defined only on an interval and we may need to extend them on R in some obvious way, not affecting the final estimates.

3.1.1 Existence Result The existence is proved by constructing an iterative procedure, where we first calculate the temperature for given velocity; then for given temperature we calculate the velocity and the pressure. Procedure is repeated until getting a fixed point. Our idea is to prove that such iterations give a uniformly bounded sequence in appropriate Sobolev spaces. Then elementary Sobolev compactness leads to the result. We explain here the main steps of the proof. For details we refer to the article [9]. Step 1. We start by studying the energy equation for a given w = ρ v ∈ H (Ω ), where H (Ω ) = {z ∈ L3 (Ω )3 | div z = 0 in Ω , z · n = 0 on Γlat , z · n|Γin = ρ (1)v1 , z · n|Γout = ρ (0)v2 }.

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Our nonlinearities c p1 and λ are defined only for ϑ such that the density is not negative. We extend it on R by setting & c p1 (ϑ ) =

c p1 (1)/ϑ 2 for ϑ > 1 (

S L∞ (Γ ) 2 S L∞ (Γ ) lat lat ) c p1 (− ) q0 ϑ q0

&

λ (ϑ ) =

λ (1) for ϑ > 1 λ (−

We will prove that ϑ ∈ [− tension. Now for a given

for ϑ < −

S L∞ (Γ ) lat ) q0

S L∞ (Γ ) lat , 1] q0

for ϑ < −

S L∞ (Γ ) lat , q0

S L∞ (Γ ) lat . q0

(53)

(54)

and the result is independent of the ex-

w ∈ H (Ω ), S ∈ L∞ (Γlat ), S ≥ 0 λ , c p1 ∈ W 1,∞ (R), λ ≥ λ0 and constants q0 , Pe > 0,

(55)

we consider the problem Problem Θ : Find ϑ ∈ H 1 (Ω ) ∩ L∞ (Ω ) such that c p1 (ϑ )w∇ϑ =

1 div λ (ϑ )∇ϑ , Pe and ϑ = 0 on Γout

ϑ = 1 on Γin 1 − λ (ϑ )∇ϑ · n = q0 ϑ + S Pe

on Γlat

(56) (57) (58)

We list some results for Problem Θ , whose prove is given in [9]. Proposition 3.1. Under the stated assumptions, Problem Θ has at least one variational solution in H 1 (Ω ), satisfying the estimate

λ0 q0 ∇ϑ 2L2 (Ω )3 + ϑ 2L2 (Γ ) ≤ A0 + B0wL1 (Ω )3 , lat 2Pe 4

(59)

where B0 and A0 are positive constants. Corollary 3.2. Under the assumptions of Proposition 3.1, we have the following estimate ' Pe 1 ' A0 + B0 wL1 (Ω )3 . (60) , '∂x3 R'∞ ϑ 2L2 (Ω ) ≤ 8 max λo qo Lemma 3.3. Any variational solution ϑ ∈ H 1 (Ω ) to Problem Θ , satisfying the a priori estimate (59), satisfies also −

S L∞ (Γlat ) ≤ ϑ ≤ 1 a.e. on Ω . q0

(61)


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Remark 3.4. Let us now recall a very general result for a class of elliptic problems, including Θ (see [1]). Under a suitable condition on the interior angle between the lateral boundary r = R(x3 , φ ) and the upper and lower surfaces, 3 (Ω ) is the elliptic regularity constant, the solution u for the mixed problem div (∇u − f) = 0 in Ω ∇u · n = g on Γlat and u = 0 on Γout ∪ Γin

(62) (63)

≤ 3 (Ω ) gL3 (Γlat ) + fL3 (Ω )3 .

(64)

is estimated as ∇uL3 (Ω )3

Hence, applying such a result, we have ∇ϑ L3 (Ω )3 ≤ E00 + E01wL3 (Ω ) ,

(65)

where E00 =

S L∞ (Γlat ) |Γlat |1/3 3 (Ω ) Pe S L∞ (Γlat ) + qo max 1, λ0 q0 λ ∞ |Ω |1/3 + 3 (Ω )|Γlat |1/3 , + λ0 S L∞ (Γlat ) 3 (Ω ) E01 = Pe c p1 ∞ max 1, . λ0 q0

and

(66)

(67)

Step 2. Now we turn to the continuity and momentum equations, determining the impulse w = ρ v and the pressure p. Our system reads as follows: For given ϑ determine {w, p} satisfying div w = 0 in Ω w 2μ w div ) (w∇) = −Arϑ e3 − ∇(p + ρ 3Re ρ 1 w + Div 2 μ (ϑ )D( ) in Ω Re ρ w = 0 on Γlat , w = (1 − α )v1e3 on Γin and w = v2 e3 on Γout ,

(68)

(69) (70)

Next, there exists

ζ ∈ H 2 (Ω )3 , ∇ζ ∈ L3 (Ω )9 ,

with curl ζ satisfying (70)

(71)

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Now we adapt the well-known Hopf construction to our non-standard nonlinearities. we have Proposition 3.5. (see [9]) Let us suppose that 1/ρ ∈ L∞ (Ω ), 1/ρ ≤ 1/ρmin. Then, for every γ > 0 there is a ξ , depending on γ and ρmin , such that

ξ ∈ H 1 (Ω )3 , div ξ = 0 in Ω , ξ = ζ on ∂ Ω and 1 (φ ∇)φ · ξ dx| ≤ γ φ 2H 1 (Ω )3 , ∀φ ∈ H01 (Ω )3 | Ω ρ Let us introduce now some useful constants: √ ⎧ μ 6 μmin 2 ⎪ min ⎪γ = , A00 = 4 2 , B00 = , C01 = 2 · 61/6|Ω |5/6 ⎪ ⎪ ⎪ ρ Re ρ ρ max max ⎪ min ⎨ 4μmax 61/6 1/3 1 B = 2 · 6 ξ B = 6 (Ω )3 , 01 02 L 2 3 ⎪ ρmin ⎪ ρmin ⎪ ⎪ ⎪ μ 1 2 max ⎪ ⎩ C00 = D(ξ )L2 (Ω )9 . ξ 2L6 (Ω )3 + ρmin Re

(72) (73)

(74)

Our basic result is the following: Theorem 3.6. (see [9]) Let us suppose that 1 B00 − α B02∇ϑ L3 (Ω )3 − α B01 ∇ϑ L2 (Ω )3 > 0 Re = ΔB2 − 4A00α ∇ϑ L2 (Ω )3 (C00 + ArC01 ∇ϑ L2 (Ω )3 ) > 0,

ΔB = and Δdet

(75) (76)

Then there is a solution w ∈ H 1 (Ω )3 for the problem (68)–(70), satisfying the estimate D(w)L2 (Ω )9 ≤ D(ξ )L2 (Ω )9 +

C00 + ArC01 ∇ϑ L2 (Ω )3 √ Δdet

(77)

Step 3. Next we define our iterative procedure: μmin and ξ be the corresponding vector valued function from Hopf’s Let γ = ρmax Re construction. For a given wm = Wm + ξ , such that Wm ∈ BR = {z ∈ H01 (Ω )3 : div z = 0 in Ω and D(Wm )L2 (Ω )9 ≤ R}, we calculate ϑ m , a solution to (56)–(58). Next, with this ϑ m , we determine a solution wm+1 = Wm+1 + ξ for the problem (68)–(70), satisfying the estimate (77). The natural question arising in the iterative process is does Wm+1 remains in BR ? Of course R has to be selected in a suitable way.


193

We have the following result Proposition 3.7. (see [9]) Let the constants A0 and B0 be given as in Proposition 3.1 and let E00 and E01 be given by (66)–(67). Let the constants B00 , B01 , B02 , A00 ,C00 and C01 be given by formula (74). Let ξ be generalized Hopf’s lift, given by Proposition 3.5 and corresponding to γ . Let R be given by R=

√ ρmax 2Re μmin

S L∞ (Γlat ) 2Ar|Ω |5/6 61/6 max 1, q0 μ 1 2 max 2 D(ξ )L2 (Ω )9 + ξ L6 (Ω )3 + . ρmin Re

(78)

Then for all α > 0 such that

Δ1 =

1 B00 − α B02 E00 + E01 (ξ L3 (Ω )3 + 481/12|Ω |1/6 R) Re ( ) √ Pe ) 1/2 −α B01 2A0 + 2B0 ξ L1 (Ω )3 + |Ω |1/4 HR1/2 > 0 and (79) λ0 (

) √ " 2 Pe ) 1/2 A0 + B0 ξ L1 (Ω )3 + |Ω |1/4 R · λ0 S L∞ (Γlat ) " B2 C00 + ArC01 max{1, } > 002 , q0 2Re

Δ2 = Δ12 − 4A00α

(80)

Wm ∈ BR implies Wm+1 ∈ BR . After this invariance result, a simple compactness argument gives the existence of at least one solution. The result could be summarize in the following theorem: Theorem 3.8. (see [9]) There is a weak solution {ϑ , v} ∈ W 1,3 (Ω ) × H 1 (Ω )3 for Problem A, such that ⎧ * * ) ⎪ 1/2 ⎨ ∇ϑ 2 3 ≤ 2Pe A0 + B0ξ L1 (Ω ) + B0 |Ω | R , L (Ω ) λ0 (81) S ∞ ⎪ L (Γlat ) ⎩− ≤ ϑ ≤ 1, and D(v − ξ ) ≤ R, 2 9 L (Ω ) q0 where ξ is given by Proposition 3.5 with γ = 2 μmin /(ρmax Re).

3.1.2 Regularity and Uniqueness Regularity of solutions and the uniqueness are also studied in [9]. Getting uniqueness is quite technical. After proving the regularity, one obtains that for small data

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there is a unique weak solution {v, ϑ } ∈ H 1 (Ω )3 × (W 1,3 (Ω ) ∩ C(Ω¯ )) for the problem (44)–(48) , satisfying the bounds (81). Detailed calculations are in [9]. Let us say a bit more concerning the regularity of solutions: Lemma 3.9. Let c p0 and λ ∈ C∞ (R) . Furthermore let Γlat ∈ C∞ and S ∈ C∞ (Γ¯lat ). Then ϑ ∈ W 2,6 (Ω ) ⊆ C1,1/2 (Ω¯ ). Lemma 3.10. Let ϑ ∈ W 2,6 (Ω ), let Γlat ∈ C∞ and let v j ∈ C∞ , j = 1, 2 satisfy (49). Then {w, p} ∈ W 2,q (Ω ) × W 1,q (Ω ), ∀q < +∞. Theorem 3.11. Let the assumptions on the data from two previous Lemmata hold true. Then every weak solution {v, p, ϑ } ∈ H 1 (Ω )3 × L2 (Ω ) × (H 1 (Ω ) ∩ L∞ (Ω )) for the Problem A is an element of W 2,q (Ω )3 ×W 1,q (Ω ) ×W 2,q (Ω ) , ∀q < ∞. Furthermore {v, p, ϑ } ∈ C∞ (Ω )5 .

3.2 Oberbeck–Boussinesq Model Now we are in position to pass to the limit when the expansivity parameter α tends to zero. First we remark that the a priori estimates from the previous section are independent of α , |α | ≤ α0 , where α0 is the maximal positive α satisfying (79)–(80). Consequently, by a simple weak compactness argument, we have Theorem 3.12. (see [9]). Let {ϑ (α ), v(α )}, α ∈ (0, α0 ), be a sequence of weak solutions to Problem A, satisfying the bounds (81). Then there exists {ϑ OB , vOB } ∈ W 1,3 (Ω ) × H 1 (Ω )3 and a subsequence {ϑ (αk ), v(αk )} such that ⎧ ⎨ ϑ (αk ) → ϑ OB , uniformly on Ω¯ ϑ (αk ) ϑ OB , weakly in W 1,3 (Ω ) ⎩ v(α ) vOB , weakly in H 1 (Ω )3 . Furthermore, {ϑ OB , vOB } is a weak solution for the equations div v OB = 0, + , 1 div 2μ (ϑ OB )D(vOB ) , (v OB ∇)v OB = Arϑ OB e3 − ∇POB + Re 1 OB OB OB c p1 (ϑ )v ∇ϑ = div λ (ϑ OB )∇ϑ OB , Pe

(82) (83) (84)

satisfying the boundary conditions (47)–(50) and the bounds (81). For the mathematical theory of the system (82)–(84), subject to various boundary conditions, but with constant viscosity, we refer to [27], pages 129–137. The existence and uniqueness for the complete non–stationary problem (82)–(84) is in the article [8].


195

3.2.1 Discussion of the Formal Limit For the glass molt in the die typical values of the thermal expansivity number Kρ are of order 0.5 · 10−2, typical temperature oscillation is 10−1 . The buoyancy forces 1 (mixed forced and natural convection) are proportional to 2 = gH/VR2 and their Fr typical order of magnitude is 104 . Therefore it is important to consider a flow regime such that Ar = O (1), Re = O (1) and Pe = O (1). So we have α = Kρ (1 − Tw ) → 0 with the Archimedes number remaining constant. We have ρ (ϑ ) → 1, as α → 0 and the continuity equation (44) becomes the incompressibility condition (82). In (45), the density ρ (ϑ ) becomes 1 and we get the (83). Note that extracting the static pressure and working with the hydraulic head was crucial for correct passing to the limit. Also, in the energy equation (46) the density ρ (ϑ ) becomes equal to 1 and we get the (84). This limit was justified rigorously in Theorem 3.12. Therefore, in practical situations of industrial importance the system (44), (45) and (46) can be approximated by a system similar to the Oberbeck–Boussinesq system. Indeed our choice of the parameters takes us close to the conditions of the formal derivation of the Oberbeck–Boussinesq system for bounded domain. Such conditions (see [25], pages 86–91) are VR ≈ 0,

Ma ≈ 0,

Re = O(1) and Ar = O(1),

(85)

with Ma being Mach’s number, which in our situation is of order 10−6 . Under conditions (85), the non–isothermal compressible Navier–Stokes system can be approximated (formally) by the Oberbeck–Boussinesq system (82)–(84). We note that the same reasoning applies to the situations in which other boundary conditions are given, just modifying the definition of ϑ and of Archimedes’ number Ar. For details we refer again to [25], page 89. In the case of natural convection flows, differently from gravity driven flows, there is no characteristic velocity and one takes VR = μR /(H ρR ). In this case the relevant characteristic number is the Grashof number, Gr= Ar Re2 . Instead of imposing Ar = O (1), one requires Gr = O (1) and the condition VR ≈ 0 is replaced by μR /ρR ≈ 0. Another approximation for the natural convection in the case of a layer of fluid of thickness H, with the top and bottom surfaces held at constant temperature, is de) rived in [22]. The authors take the Chandrasekhar’s velocity VR = gLβ (Tw − TR) (linked to buoyancy) as characteristic. Then they introduced the small parameter 3 1/3 VR ρR ε= , obtaining formally the Oberbeck–Boussinesq equations at order g μR O(ε 3 ). This approach was developed further in [16] in order to get a modified Oberbeck–Boussinesq system, with important viscous heating. A similar derivation of the Oberbeck–Boussinesq approximation for the natural convection in the case of a layer of fluid, can be found in [11], pages 43–53. There, for the sake of definiteness, a perfect gas is considered but the scalings and the orders of magnitude involved are similar to [22].

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Next we mention a number of papers by R. Kh. Zeytounian on the thermocapillary problems with or without a free boundary. They are reviewed in [29] and [30]. He considered the natural convection in an infinite horizontal layer of viscous, thermally conducting and weakly expansible fluid, heated from below. The small parameter is α and for

α ≈ 0,

Fr ≈ 0,

Ma ≈ 0,

and Gr =

α Re2 = O(1), Fr2

(86)

he got the Oberbeck–Boussinesq approximation. In his derivation, the viscous dissipation term is absent if 1 mm ≈

μR2 gρR2

1/3 0 + Pe

√ 3 2 μmin > 4( )1/4 HD(v)L2 (Ω )9 + HLΘ Re ρmax 2

2α μmax vL∞ (Ω )3 Re ρmin H2 2 2 μ L∞ (R) D(v)L3 (Ω )9 C6 (Ω ) + α HvL∞(Ω )3 , + Ar √ + 2 Re

where LΘ =

(89)

Hc p1L∞ (R) ϑ L∞ (Ω ) √ 2N

(90)

(91)

Then derivatives of the solution, with respect to α , exist at all orders as continuous functions of α . With this result, we are ready to state the error estimate for Boussinesq’s limit. First, we write the 1st order correction, i.e. the system defining the first derivad tives {w0 , π˜ 0 , θ 0 } = {v, p, ˜ ϑ }|α =0 : dα , + div w0 − ϑ OB vOB = 0 in Ω (92) , + −ϑ OB (vOB ∇)vOB + (w0 ∇)vOB + (vOB∇)w0 = −Ar θ 0 e3 − ∇π˜ 0 + , 2 Div μ (ϑ OB )D(w0 ) + μ (ϑ OB )θ 0 D(vOB ) in Ω Re λ (ϑ OB ) 0 λ (ϑ OB ) div − ∇θ + (vOBc p (ϑ OB ) − ∇ϑ OB )θ 0 Pe Pe OB OB OB 0 OB −ϑ v C p (ϑ ) + w C p (ϑ ) = 0 in Ω +

θ 0 = 0, w0 = 0 on Γout ;

θ 0 = 0, w0 = V2 e3 on Γin

(93)

(94) (95)

1 (λ (ϑ OB )∇θ 0 + λ (ϑ OB )θ 0 ∇ϑ OB ) · n = q0 θ 0 on Γlat , (96) Pe Under the conditions of the preceding Proposition, with α = 0, the system (92)– (96) has a unique smooth solution. Hence we have established rigorously the O(α 2 ) approximation for Problem A. Clearly, one could continue to any order. The result is given by the following theorem, which is a straightforward corollary of Proposition 3.13. w0 = 0 and −

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Theorem 3.14. (see [9]). Let us suppose the assumptions of Proposition 3.13. Then we have v − vOB − α w0 W 1,∞ (Ω )3 + ϑ − ϑ OB − αθ 0 W 1,∞ (Ω ) ≤ Cα 2 inf p − p

C ∈R

OB

− α p + C L∞ (Ω )3 ≤ Cα 0

2

(97) (98)

where p0 = π˜ 0 − 2μ (ϑ OB ) div w0 /(3 Re).

4 Modelling the Viscous Jet at the Exit of the Die In this section we discuss briefly the stage (c) (recall Sect. 1), that is the free boundary flow in the air. Our goal is to derive a mathematical model, making some simplifying assumptions, in order to discuss the influence of some physical parameters involved. In this stage one meets several difficulties. The first is that we deal with the non-isothermal flow of a thermally expansible fluid. Next, when the fluid leaves the die it experiences strong cooling. It is observed that, in a suitable range of temperature and of pulling velocity, defining the so called “cold breakdown”, the jet detaches from the solid wall inside the die. Of course, phenomena of wetting of the outer surface of the die are also observed if temperature is large enough, making the molten glass less viscous, and/or if the pulling velocity is sufficiently small. Our approach could describe the flow in stage (c) for a general geometry of the die. Nevertheless, since the main difficulties are not really linked with the form of the die and in order to simplify our exposition, we consider cylindrical geometry with azimuthal symmetry. In particular, x will denote the axial coordinate, r the radial–one and the symmetry axis coincides with the x axis. The dependence of the fluid viscosity μ , on the temperature T plays a major role in the process. Therefore, in order to simplify modelling at this stage we neglect variations of the fluid density ρ and of the surface tension σ . Hence, within the framework of incompressibility, the fluid constitutive model becomes T = −pI + 2μ D, and the continuity equation, in the stationary case, writes

∂ (v1 ) 1 ∂ (r v2 ) + = 0, ∂x r ∂r

(99)

where v1 is the fluid axial velocity and v2 the radial–one, that is v = v1 (x, r,t) ex + v2 (x, r,t) er . We consider a die which is a cylinder of radius R and high L1 . After entering the die the molten glass wets only a part of the die walls and then produces a viscous jet. We confine our analysis to the case in which the jet starts at some point xT ∈ (0, L1 ) and then we have a free boundary S , r = h(x,t), for xT < x < L, separating the fluid


199

r L1

Γwall

h(x,t)

R Γin

Γout

RL

xT 0

L

x

Fig. 3 A schematic of stage (c). The x-axis coincides with the cylinder symmetry line. ex , er denote the unit vectors, x = xex +rer . The velocity field has the form v(x,t) = v1 (x,t)ex + v2 (x,t)er . The gravity acceleration is parallel to the x axis, g = gex

from the surrounding air (see Fig. 3). The fiber is cooled down by a stream of air, whose interaction with the fibers is also important, but we will not deal with this aspect here. Referring to Fig. 3, the flow domain is the set

Ω = Ω die ∪ Ω jet , with Ω die = {0 < x < xT , 0 < r < R} , Ω jet = {xT < x < L, 0 < r < h (x,t)} , where, as mentioned, r = h (x,t) is the unknown boundary of the jet. The boundary of Ω contains four distinct parts • • • •

Inlet boundary Γin = {x = 0, 0 ≤ r ≤ R}. Wall boundary Γwall = {0 < x ≤ xT , r = R}. Free boundary S = {r = h (x,t) , x ∈ (xT , L)}. Outlet boundary Γout = {x = L, 0 ≤ r ≤ RL < R}, where RL = h (L), fiber radius at x = L.

We observe that the whole process is governed by the following easily accessible quantities (besides the temperature of the cooling stream of air): • Spool radius, Rsp . • Spool angular velocity, ω . • Power dissipated by the spinning device, W. Indeed, for estimating W it is enough

to measure the intensity of the electric current I entering the electric engine and

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the electric potential difference Δ V at the poles of the engine, so that W = I Δ V − Wo , where Wo is the power consumption of the engine. • Discharge, Qout , i.e. rate of increase of the volume of collected fiber. From these directly measured quantities, one can deduce: • Spinning speed, Vsp = ω Rsp , i.e. fibers velocity entering the spinner. • • • •

Fiber radius, R f = (Qout /π Vsp)1/2 . Torque applied to the spinner, M = W /ω . Traction force applied to the fiber, Fsp = M/Rsp = W /Vsp. Mean stress applied to the fiber surface by the spinning device, Φsp = Fsp /π R2f = W /Qout .

We denote by VL the mean fluid velocity on Γout VL =

2 R2L

RL 0

v (L, r) · ex r dr.

(100)

Now, since the discharge Qout is prescribed, we have (

π VL R2L

= Qout , ⇒ RL =

Qout . π VL

(101)

which links together VL and RL . From the analysis of the simple Matovich–Pearson model for stage (d ), that will be performed in Appendix 2, it will be clear that, once L has been fixed, quantities such VL (the mean fluid velocity on Γout ) and the mean traction stress on Γout

ΦL =

2 R2L

RL 0

T (L, r) ex · ex rdr =

2 R2L

RL 0

(−p (L, r) + 2 μ (L)D ex · ex ) rdr, (102)

as well as the fiber radius RL can be explicitly related with the terminal quantities Vsp , Rsp and Qout . We note that the outlet boundary x = L is an artificial mathematical boundary, because the jet continues beyond that point. In this section, we confine our attention to the portion of jet in the interval xT < x < L, defining L as a location at which the fiber is cooled down to a point such that the fluid viscosity is large enough to prevent any further important fiber radius variations. On the other hand, the temperature should still be larger than the glass transition temperature. The value of L is an unknown of the problem and in fact after the point x = L one is likely to use the averaged equations presented in the next section. Actually, giving a definition of L is a delicate point of the model and depends on how μ varies with the temperature. Although there are authors [13] who, on the basis of empirical observations, suggest to take L = 12R, here we proceed in a rather different way, analyzing carefully the jet profile at the end of stage (c). Such an issue will be discussed in Sect. 4.1.


201

Concerning the boundary conditions to be imposed on the jet boundary S (see Fig. 3), we first assume a purely kinematic condition: the fluid velocity is tangent to the free surface, namely (103) v · n = wS · n = 0. where wS is the free boundary velocity. Next, we impose a dynamic condition: equilibrium of the forces acting on S , that is

−p + 2μ (D n) · n = −σ κ , 2μ (D n) · τ = (τ · ∇)σ ,

(104)

where: 1 h 1/2 − 3/2 . 1 + (h )2 h 1 + (h )2 • The air pressure, p∞ (assumed to be constant), has been rescaled to 0. In other words, in place of p we consider p − p∞. • 2κ =

Concerning the boundary conditions on Γin and Γwall , we have: • On Γin , we prescribe pressure, temperature and no radial velocity

p (0, r) = Pin , 0 ≤ r ≤ R, v2 (0, r) = 0, 0 ≤ r ≤ R.

• On Γwall , we assume no–slip v (x, R) = 0, 0 < x < xT . Of course, some slipping

occurs near to xT , but it is generally admitted that the slipping length s is extremely small.

The abscissa of the triple point, xT , is not a priori given and has to be determined as a part of problem (as well as RL ). In most of the references, one imposes the contact angle (e.g. equal to 0) and uses this additional condition for having a totally determined problem. The boundary conditions that have to be specified on Γout required coupling with the stage (d). This will be the subject of next section. In this section we will suppose that the temperature is a given function of x and r. Actually, the main difficulties will remain, but our calculations will be less lengthy.

4.1 Definition of L and Jet’s Profile at the End of Stage (c) In this section, still considering stationary conditions, we will focus on these issues: 1. To define local boundary conditions on Γout to be used when studying stage (c). 2. To formulate a rational procedure allowing to determine L, replacing the heuristic rule L = 12R.

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A. Farina et al. Table 1 Typical order of magnitude of the main physical quantities at x = L. Notice that the traction force Fsp corresponds to a weight of 3 gr ! VL RL ΦL μ σ 10 m/s 10−5 m 108 Pa 107 Pa s 10−1 N/m

We point out that, for our purposes, L can be chosen in some range of values. Therefore we look for some criterion which has a physical motivation, but is based just on the analysis of magnitude orders. So for the purposes we have in mind, i.e. locating L within a reasonable approximation, we may confound VL and ΦL with their terminal values Vsp and Φsp , neglecting the small variations that will occur from x = L to the collecting spool. Therefore, in place of expressions (188), we set VL = Vsp

and

ΦL = Φsp .

(105)

A full justification of this approach will be provided in Appendix 2. Indeed we are mainly interested to the orders of magnitude, listed in Table 1, below: Remark 4.1. After many studies (see e.g. [23] and references therein) and experimental observations (see e.g. [13] or [21]) we know that even for x > L fiber could exhibit a complicated behavior, oscillations and instabilities are possible. Here we suppose a stabilized behavior when x > L and the asymptotic analysis, which follows in this subsection applies to such situation. In general, one should couple together the stages (c) and (d), through an iterative procedure. Here we do not enter this issue. For the determination of a physically acceptable value of L we proceed considering a characterization of L in terms of the dimensionless quantity h . Hence we define the endpoint of Ω , i.e. of stage (c), in the following way: Definition 4.2. Given a sufficiently small number ς , typically ς = O 10−6 , L is chosen so that # # #h (L)# = ς . (106) We will also make use of the following relationships, that will be checked a–posteriori # # ς 2 # # #h (L)# ⇔ #h (L)# RL ς 2 . (107) RL Our strategy is the following: (i) To define local boundary conditions such that (100) and (102) are fulfilled within a tolerance of order ς . (ii) To determine L according to Definition 4.2. Next, we define pL = p(L, RL ). Of course, pL is unknown at this stage. In order to determine pL and L let us write explicitly (103) and (104) on x = L, r = RL


203

⎧ ⎪ v1 (L, RL ) h (L) − v2 (L, RL ) = 0, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 2μ (L) ∂ v2 ∂ v2 ∂ v1 2 σ ⎪ ∂ v1 ⎪ h + − − (h ) = pL + ⎪ 1/2 ⎪ 2 ⎪ ∂ r ∂ x ∂ r ∂ x 1 + (h ) ⎪ )2 ⎪ 1 + (h 2R L ⎨ σ h ⎪ − 3/2 , ⎪ ⎪ 2 ⎪ ⎪ 2 1 + (h ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ∂ v1 ∂ v2 ∂ v1 ∂ v2 (τ · ∇) σ ⎪ 2 ⎪ + − 1 − (h ) − 2 1 + (h )2 . h = ⎩ ∂r ∂x ∂x ∂r μ (L) (108) 2 Focusing on equations (108)2 and (108)3 we neglect O ς terms (recall Definition 4.2) and7 (τ · ∇) σ /μ . Hence, (108)2 and (108)3 can be rewritten as ⎧ ∂ v1 ∂ v2 ∂ v2 σ ⎪ ⎪ pL + 2μ (L) h + , − = ⎪ ⎪ ∂r ∂x ∂r 2RL ⎨ ⎪ ⎪ ∂ v1 ∂ v2 ∂ v1 ∂ v2 ⎪ ⎪ ⎩ + −2 − h = 0. ∂r ∂x ∂x ∂r

(109)

Next, (109) can be further simplified. Indeed, neglecting once more O ς 2 terms, we have ∂ v2 (L, RL ) σ = . (110) pL − 2μ (L) ∂r 2RL We now assume: A.1 The horizontal fluid velocity on x = L, r = R is VL , namely v1 (L, RL ) = VL ,

(111)

Hence, going back to (108)1, we obtain # # v2 (L, RL ) = − #h (L)# VL .

(112)

We also recall that by symmetry v2 (L, 0) = 0. Thus, following the results of Sect. 5, we also assume: A.2 The profile of the radial velocity on Γout is v2 (L, r) = −

|h (L)|VL ς VL r=− r, RL RL

(113)

dσ ∇T and x = L is located in a region where the dT temperature variations are very small, i.e. σ (T (L)) 1.

7

If σ is just a function of temperature, ∇σ =

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yielding

|h (L)|VL ∂ v2 =− . ∂r RL

Hence, from (110)

|h (L)|VL σ + , RL 2RL

(114)

ς VL |h (L)|VL = −2 μ (L) , RL RL

(115)

pL = −2 μ (L) or pL = −2 μ (L)

since the second - term on the r.h.s. of (114) is negligible with respect to the first–one. ς VL σ ≈ 10−3, according to Table 1. Indeed 2RL μ (L) RL Of course, we still have to determine L. At this point, consistently with the approximations already introduced, we take p(L, r) = pL , 0 ≤ r ≤ RL . Then we exploit (102) and (99), obtaining

ΦL = −pL +

4μ (L) R2L

RL 0

∂ v1 ∂x − 1r

rdr = −pL −

4μ (L) v2 (L, RL ) , RL

∂ (rv2 ) ∂r

which, recalling (113) and (115), gives8

ΦL = 6μ (L)

ς VL ς VL , or ΦL = 6 μ (L) Qout . RL π R3L

(116)

This formula allows us to identify L according to the profile of viscosity, which in turns is known (with a reasonable approximation) in terms of the profile of temperature. Thus L is defined via ΦL RL μ (L) = . (117) 6ς VL According to Table 1, the r.h.s. of (117) is ∼107 Pa s, consistently with the expected value of μ (L). Hence we conclude that L is the distance at which the glass has cooled down to the temperature corresponding to μ ≈ 107 Pa s. We can proceed further obtaining the information on pL and the expected profile of the fiber in the vicinity of x = L. Concerning pL , from (116) and (115) we have 1 pL = − ΦL , 3 It is easy to verify that ς ∼ 10−6 is compatible with the order of magnitude of the quantities entering in formula (116).

8


205

so that the viscous stress on Γout is 4 μ (L) R2L

RL ∂ v1

∂x

0

2 rdr = ΦL , 3

Next, we write also v2 (L, r) in terms of ΦL using (113) and (116), namely v2 (L, r) = −

ΦL r, 6μ (L)

(118)

Summarizing, the boundary conditions on Γout read as follows ⎧ 1 ⎪ ⎨ p = − ΦL , 3 on Γout , ΦL ⎪ ⎩ v2 (r) = − r, 6μ (L)

(119)

with ΦL expressed in terms of Φsp . Going back to (113), we just suppose that it is valid for x from a neighborhood of x = L: |h (x)| |L − x| VL r, 0 ≤ r ≤ h (x) , 1. (120) v2 (x, r) ≈ − h (x) L Consistently with this assumption we have that, in the vicinity of x = L, v2 (x, r) can be also expressed exploiting (118), namely v2 (x, r) ≈ −

|L − x| ΦL r, 0 ≤ r ≤ h (x) , 1. 6μ (x) L

(121)

In this way, comparing (120) and (121), the position of the free boundary r = h(x) is given by h (x) ΦL =− VL , 6μ (x) h (x) so that the following Cauchy problem is obtained ⎧ ⎪ ⎪ ⎨ h (x) = − ⎪ ⎪ ⎩

1 ΦL |L − x| 1, h (x) , 6μ (x) VL L

(122)

h (L) = RL ,

which provides a sufficiently accurate information on the local profile of the fiber. Problem (122) can be integrated, getting the following profile h (x) = RL exp

ΦL 6VL

L x

1 ds , μ (s)

referred to as “transition profile”, in the sequel denoted by hT P (x).

(123)

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We can now check that hT P fulfills condition (107), i.e. if our result is consistent with assumption (107). Evaluating h we have h = h Since, close to x = L we have h = h

2 6VL μ . 1+ ΦL

6VL |μ | 1, that yields ΦL

2

ΦL 6μ VL

ΦL 6μ VL

=

2 2 h , ⇒ h h = h = O ς 2 . h

(124)

Hypothesis (107) is thus fulfilled. As final check of our procedure we show that, considering the fiber profile (123), the assumption (111), the approximation (121) for v2 and boundary conditions (119), we can evaluate v1 (L, r), 0 ≤ r ≤ RL , so that the estimated volumetric discharge differs from the actual one, i.e. Qout , within a tolerance O ζ 2 . We thus use the above mentioned approximations in ⎧ ⎪ ⎪ v1 (x, h (x)) h (x) − v2 (x, h (x)) = 0, ⎨ ∂ v1 ∂ v2 ∂ v1 ∂ v2 (τ · ∇) σ ⎪ ⎪ ⎩ + − , −2 h = ∂r ∂x ∂x ∂r μ (x)

(125)

and look for an estimate of ∂ v1 /∂ r, since we approximate v1 (L, r) in the following way # ∂ v1 ## v1 (L, r) = VL − (RL − r) + O((RL − r)2 ). ∂ r #x=L, r=RL Evaluating (125)2 at x = L, r = RL , we have

∂ v1 ∂ v2 + ∂r ∂x

=− x=L, r=RL

h (L)2 VL ΦL ## ## h (L) = −6 , μ (L) RL

where we have used (99) and (118) for expressing ∂ v1 /∂ x, namely |h |VL ∂ v1 ∂ v2 v2 1 ΦL =− − = = −2 , ∂x ∂r r 3 μ (L) RL and where ( τ · ∇) σ /μ has been neglected. Differentiating (125)1 w.r.t. x we get

∂ v1 ∂ v1 2 ∂ v2 ∂ v2 h+ h + v1 h − − h = 0. ∂x ∂r ∂x ∂r

(126)


207

We now use again (99) and then exploit (118) and (111), obtaining

∂ v1 2 ∂ v2 h (L) − ∂r ∂x

=3 x=L, r=RL

h (L)2 VL − VL h (L) . RL

Next, using (124) for evaluating h (L), i.e. h (L) = h (L)2 /RL and (126) for ∂ v2 /∂ x, we have9

1 + h (L)2

∂ v ## h (L)2 VL 1# = −4 . ∂ r #x=L, r=RL RL

# # # ∂ v1 # # (L, RL )## RL , We are now in position to estimate Δ v1 = max |v1 (r, L) − VL | ≈ # ∂r namely 2 Δ v1 = O ς VL , thus estimating the “relative error” on the outlet discharge # # RL # Δ Qout 1 ## 2 = π RLVL − 2π v1 (L, r) rdr## = O ς 2 , # 2 Qout π RLVL 0 which is thus shown to respect the desired tolerance.

5 Terminal Phase of the Fiber Drawing It is considered that the stage (d) starts when the fiber is sufficiently “cooled down.” This means that the viscosity is sufficiently large (larger than 105 Pa s) and the fiber radius is already rather small (smaller than hundred micrometers). Contrary to the stage (c), where one should treat the 3D Navier–Stokes system, with the free boundary, coupled with the nonlinear conduction of heat, here we have a long (several meters) and tiny filament (radii of hundred micrometers) of molten glass. In the stage (c) one needs an industrial code to solve the corresponding partial differential equations. Here it is possible to employ the asymptotic analysis and come out with a simplified effective model (Fig. 4). Fundamental equations, describing the stage (c), are the temperature dependent incompressible Navier–Stokes equations with free boundary coupled with the energy equations. Their derivation from the first principles is in the article [9], where also the Oberbeck–Boussinesq approximation is justified rigorously, by passing to the singular limit when expansivity parameter goes to zero. For more details see Sects. 2 and 3.2. 9

Notice that ∂ v1 /∂ r is negative, as one would have expected.

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Fig. 4 A simulated glass fiber being drawn down

In the engineering literature on the fiber drawing, this system is frequently approximated by a quasi 1D approximation for viscous flows, in which the radius of the free boundary r = R(z, α ,t), axial speed w and the temperature ϑ are independent of the radial variable and depend only on the axial coordinate z and of the time t. This approximation, summarized in the so–called the “equations of Matovich–Pearson” was introduced in the papers by Kase and Matsuo [17, 18] and Matovich and Pearson [20]. The model was obtained heuristically and reads

∂ A ∂ (wA) + = 0; ∂t ∂z

√ ∂ ∂ w ∂ (σ (T ) A) 3μ (T )A + = 0, ∂z ∂z ∂z

(127)

where A = A(z,t) is the area of fibre section, w = w(z, t) is effective axial velocity, 3 μ is Trouton’s viscosity and σ is the surface tension. μ and σ depend on the temperature, and it is necessary to add the equation for the temperature T = T (z,t). The equations of Matovich and Pearson were obtained under the assumptions (H), listed below: (a) The viscous forces dominate inertia (b) Effects of the surface tension are in balance with the normal stress at the free boundary (c) The heat conduction is small compared with the heat convection in the fiber (d) All the phenomena are axially symmetrical and the fiber is nearly straight


209

5.1 Derivation of the Model of Matovich–Pearson for the Thermal Case The derivation presented in [13] and in [14] surprisingly neglects the fact that the viscosity changes over several magnitude orders performing asymptotic expansions in ε as it was of order one, ε being the ratio of the characteristic thickness RE in the radial direction with the characteristic axial length of fibre L. The correct formal derivation is in [5] and it confirms the model announced in [14]. We will present here the ideas and results. Since we deal with an axially symmetric and long fiber, we slightly change the notation and denote with subscript z the axial component and with the subscript r the radial one. Instead of x we use z to denote the axial variable. Our unknowns are the following: • • • •

Velocity v = vz ez + vr er Hydrodynamic pressure p Temperature T Fiber radius (being the distance from the symmetry axis ) R = R(t, z)

Moreover the following physical quantities are given function of T : • • • • •

Specific heat c p = c p (T ) Density ρ = ρ (T ) Surface tension σ = σ (T ) Viscosity μ = μ (T ) Thermal conductivity λ = λ (T )

We will suppose that the fibre is long and thin, so that the principal flow is directed along the axis z and the velocity is basically one dimensional. In particular, the property of “small thickness” leads to the “lubrication approximation” of Reynolds. The strategy of the “lubrication approximation” is to expand the velocity field with respect to ε . The zero order terms should be sufficient to describe the motion. Effective equations are then derived starting from the compatibility conditions for the solvability of the problems. This idea is traditionally applied to flows through thin domains. Treating the flows with a free boundary is much more complicated. Initially, the radius r describes the position of the free boundary r = R(x,t). The smallness of the expansion parameter ε = maxx,t R(x,t)/{ fiber length } will depend on the solution itself. In the second place, it is not obvious which forces in the equations must be taken into account (for example torsion can be important or negligible). In order to correctly model the heat exchange at the free boundary, we adopt for the heat transfer coefficient the empirical formula of Kase–Matsuo 2ρ∞ vz (z,t)R(z,t) m λ∞ C , h= R(z,t) μ∞

(128)

where the subscript ∞ denotes the parameters of the surrounding air and C > 0 and m, 0 < m < 0.4, are constants determined from the experimental data.

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ρE RE c pE vE 2hE (1 − Tg /TE ), where Tg is the glass transition temperature and TE is temperature at the orifice. Other possibility is to take the distance between the spooler and extrusion die. This will be our choice. We will restrict our considerations to the stationary case, even if the generalization to the non-stationary case is straightforward. Our first difficulty is that viscosity changes over several orders of magnitude. This motivates us to write the Oberbeck– Boussinesq equations in Ω = {z ∈ (0, L); 0 ≤ r < R(z)}, in the following form: Concerning characteristic length, in [13] the authors take L =

2 2 p p + div v − ∇2 v + + div v I − 2D(v) μ (T ) 3 μ (T ) 3 ρ (T ) ρ (T ) ∂v ×∇ log μ (T ) − ∇ div v = gez − (v∇)v + in μ (T ) μ (T ) ∂t

∇

∂ ρ (T ) + div (ρ (T )v) = 0 in Ω ∂t ∂T + v∇T = div (λ (T )∇T ) in ρ (T )c p (T ) ∂t

Ω (129)

(130)

Ω.

(131)

The stress tensor is now

Σ = −μ (T )

p 2 + div v I − 2D(v) . μ (T ) 3

(132)

RE . We introduce the following dimensionless The natural small parameter is ε = L quantities: L T − T∞ = T˜ , t˜, vz = vE v!z , vr = ε vE v!r , vE ΔT 2 vE p˜ p ˜ + div v = , λ = λE λ˜ . μ = μE μ˜ , σ = σE σ˜ , h = hE h, μ (T ) 3 L r = RE ! r, t =

Equations in non-dimensional form numbers: ρ E vE L Re = is Reynolds number; μE c pE ρ vE L is Peclet’s number; Pe = λE vE Fr = √ is Froude’s number; gL

will contain the following dimensionless

μE vE is capillary number; σE h E RE Bi = is Biot’s number; λE Re Bo = is Bond’s number Fr2

Ca =

Vf , where V f is the outlet fiber velocity. α = log E is usually vE known as Hencky strain. and draw ratio is E =


211

In the text, which follows, we will omit the wiggles. The axially symmetric generalized Oberbeck–Boussinesq system takes the following form:

∂ vr vr ∂ vz + + = −vz ∂z log ρ (T ) − vr ∂r log ρ (T ) − ∂t log ρ (T ), in Ω ; (133) ∂r r ∂z 2 ρ (T ) ∂ vr ∂ vr ∂p ∂ vr 1 ∂ vr + vz + ε 2 Re ∂t vr + vr + =− μ (T ) ∂r ∂z ∂r ∂ r2 r ∂r 2 vr ∂ vr ∂ log μ ∂ vr ∂ log μ ∂ vr ∂ vz − 2 + ε2 2 − + ε2 p−2 + r ∂z ∂r ∂r ∂z ∂z ∂r ∂ (vz ∂z log ρ (T ) + vr ∂r log ρ (T ) + ∂t log ρ (T )), in Ω ; (134) − ∂r 2 2 ρ (T ) ∂ vz ∂ vz ∂ vz 1 ∂ vz 2 2∂p 2 ∂ vz + vz + +ε ε Re ∂t vz + vr + = −ε μ (T ) ∂r ∂z ∂z ∂ r2 r ∂r ∂ z2 ∂ log μ ∂ vz ∂ log μ ∂ vr ∂ vz Re ρ (T ) − ε2 + ε2 p−2 + + ε2 2 ∂z ∂z ∂r ∂z ∂r Fr μ (T ) ∂ (135) − ε 2 (vz ∂z log ρ (T ) + vr ∂r log ρ (T )∂t log ρ (T )), in Ω ; ∂z ∂T ∂T ∂T 1 ∂ + vz ε 2 Peρ (T )c p (T ) ∂t T + vr = rλ (T ) ∂r ∂z r ∂r ∂r ∂ ∂ T λ (T ) (136) + ε2 , in Ω . ∂z ∂z Next we have

∂ R(z,t) ∂ R(z) + vz = vr ∂t ∂z

on r = R(z) (the kinematic condition)

(137)

and the dynamic conditions at the free boundary read: $

μ (T )Ca 2ε

2 ∂ R(z)

∂z

∂ vr ∂ vz − ∂r ∂z

$

+ 1−ε .

2

∂R ∂z

2 %

% ∂ vz + ) (ε ∂z ∂r 2 ∂ vr

∂ R(z) ∂ σ (T ) ∂ σ (T ) ∂ R(z) 2 + 1 + ε2 on r = R(z); (138) ∂z ∂r ∂z ∂z $$ $ 2 % ∂ vr ∂ R(z) 2 ∂ vz 2 ∂ R(z) + ε2 ε Caμ (T ) 1+ε p−2 ∂z ∂r ∂z ∂z . ∂ R(z) ∂ vr ∂ R(z) ∂ vz σ (T ) ∂ R(z) 2 − ε2 − 1 + ε2 = ∂z ∂z ∂z ∂r R(z) ∂z . ∂ 2 R(z) ∂ R(z) 2 2 − ε 2 σ (T ) / 1 + ε on r = R(z). (139) ∂ z2 ∂z = ε

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Finally, the heat transfer to the environment is given by Newton’s cooling law h ∂T ∂R ∂T − ε2 = −Bi T ∂r ∂z ∂z λ (T )

.

1 + ε2

∂ R(z) ∂z

2 on r = R(z);

(140)

Before starting the two-scale expansion it is essential to relate with ε the magnitude of all non–dimensional coefficients entering the system. The experimental data from [13] imply that Re ∼ ε ,

Bo ∼ 1,

1 Ca ∼ , ε

1 Pe ∼ , ε

Bi ∼ ε

as

ε → 0.

(141)

In order to perform the asymptotic analysis of our equations, we expand all unknown functions with respect to ε , i.e. for an arbitrary function f = f (r, z) we set f = f (0) + ε 2 Pe f (1) + ε 2 f (2) + . . . . Let Ω0 = { 0 ≤ r < R(0) (z,t), 0 < z < 1 }. After inserting the expansions for unknowns into the system (133)–(140), we find out that the zero order terms in the system (133)–(136) are: (0)

(0)

(0)

vr ∂ vr ∂ vz (0) (0) + + = −vz ∂z log ρ (T (0) ) − vr ∂r log ρ (T (0) ) ∂r r ∂z − ∂t ρ (T (0) ) in Ω0 ;

(142)

(0) (0) ∂ (0) ∂ p(0) ∂ 2 vr 1 ∂ vr (0) (0) + ∂t vz + vz ∂z log ρ (T (0) ) + vr ∂r log ρ (T (0) ) = − + ∂r ∂r ∂ r2 r ∂r $ % (0) (0) (0) (0) (0) vr ∂ log μ (T ) ∂ vr ∂ log μ (T ) ∂ vz − 2 + −p + in Ω0 ; 2 (143) r ∂r ∂r ∂z ∂r

0=

(0)

(0)

(0)

∂ 2 vz ∂ log μ (T (0) ) ∂ vz 1 ∂ vz + + 2 ∂r r ∂r ∂r ∂r 1 ∂ T (0) ∂ 2 T (0) 0= + in Ω0 , r ∂r ∂ r2

in Ω0 ;

(144) (145)

with the boundary conditions (0) ∂ R

(0)

vr = ∂t R(0) + vz

(0)

∂z

on r = R(0) (z);

(146)

(0)

∂ vz = 0 on r = R(0) (z,t); ∂r$ % (0) (0) σ (T (0) ) ∂ vr 1 (0) (0) ∂ vz − ∂z R on r = R(0) (z,t); p = +2 ε Ca μ (T (0) )R(0) ∂r ∂r ∂ T (0) = 0 on r = R(0) (z,t). ∂r

(147) (148) (149)


213

We note that it is advantageous to have log μ , in the differential equations, since, contrary to μ , its variation is not dramatic. Using (145) we obtain T (0) = T (0) (z,t). (0) (0) Next, (144) yields vz = vz (z,t). Radial component of the velocity is calculated using the (142). We integrate it and get (0)

r ∂ vz ∂ T (0) (0) =0= , vr (r, z) = − ∂r ∂r 2

$

% (0) ∂ vz (0) + ∂z log ρ (T (0) )vz + ∂t log ρ (T (0) ) ∂z (150)

∂ p(0) Consequently, (143) reads = 0 and, after using the boundary value (148), we ∂r obtain p(0) =

(0)

1 σ (T (0) ) ∂ vr σ (T (0) ) 1 = + 2 ε Ca μ (T (0) )R(0) ∂r ε Ca μ (T (0) )R(0) −

(0)

∂ vz (0) − ∂z log ρ (T (0) )vz − ∂t log ρ (T (0) )). ∂z

(151)

The kinematic condition (146) now transforms to 0=

∂ ∂ (0) ρ (T (0) )(R(0) )2 + ρ (T (0) )vz (R(0) )2 . ∂t ∂z

(152)

At the order O(ε 2 Pe ), we have the following boundary value problem for the temperature correction T (1) : $

ρ (T

(0)

)c p (T

(0)

)

(0) ∂ T (0) (0) ∂ T + vz ∂t ∂z

ε 2 Peλ (T (0) )

% =

1 ∂ ∂ T (1) (rλ (T (0) ) ) in Ω0 , r ∂r ∂r

(153)

(0)

(vz R(0) )m T (0) ∂ T (1) + Bi = 0, on r = R(0) (z), ∂r ε 2 R(0)

(154)

where m is the exponent from Kase–Matsuo’s formula. The Neumann problem (153)–(154) for T (1) with respect to r has solution if and only if the usual compatibility condition from Fredholm’s alternative is satisfied. It reads % (0) (0) 2Bi (0) (0) m (0) ∂ T ∂ T (0) + vz + 2 vz R ρ (T (0) )(R(0) )2 c p (T (0) ) T = 0 on (0, 1). ∂t ∂z ε Pe (155) $

Next, at the order O(ε 2 ), (135) and condition (138) give ∂ p(0) ∂ log μ (T (0) ) ρ (T (0) ) (0) (0) (0) + + ∂ v + v ∂ v Re z z z t z ∂z ∂z μ (T (0) )

$ p

(0)

(0)

∂ vz −2 ∂z

%

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=

(0)

(2)

(2)

∂ 2 vz ∂ 2 vz ρ (T (0) ) 1 ∂ vz + Bo + + ∂ z2 ∂ r2 r ∂r μ (T (0) ) ∂ (0) − vz ∂z log ρ (T (0) ) + ∂t log ρ (T (0) ) in Ω0 . ∂z ∂ R(0) (z) 2 ∂z =

$

(0)

(0)

∂ vr ∂ vz − ∂r ∂z

% +

(156)

(0)

(2)

∂ vr ∂ vz + ∂z ∂r

1 ∂ σ (T (0) ) on r = R(0) (z,t). ∂z ε Caμ (T (0) )

(157)

Now (150) yields ∂z μ (T 0 )(p0 − 2∂zv0z ) = ∂z

(0)

1 σ (T (0) ) ∂ vz − 3μ (T 0 ) (0) ε Ca R ∂z

(0) −μ (T 0 )∂z log ρ (T (0) )vz − μ (T 0 )∂t log ρ (T (0) ) (158) and integration of the (156) yields (2) ∂r vz |r=R(0)

$ R(0) ρ (T (0) ) (0) ρ (T (0) ) (0) (0) Re = ∂ v + v ∂ v − Bo z t z z z (0) 2 μ (T ) μ (T (0) ) $ $ (0) ∂ ∂ vz 1 (0) (0) +∂z log ρ (T (0) )vz − μ (T ) 3 (0) ∂ z ∂ z μ (T ) %% $ %% 1 1 σ (T (0) ) (0) + ∂t log ρ (T ) + ∂z . (159) ε Ca R(0) μ (T (0) )

On the other hand, after inserting (150) into the boundary condition (157) we get (2) ∂r vz |r=R(0) =

1 ∂z σ (T (0) ) R(0) 0 0 (0) (0) ∂zz vz + ∂z vz ∂z log ρ (T ) + ∂t log ρ (T ) + ε Ca μ (T (0) ) 2 % $ (0) ∂ vz (0) 0 (0) (0) +∂z R + vz ∂z log ρ (T ) + ∂t log ρ (T ) . (160) 3 ∂z

After comparing (159)–(160), we obtain the effective momentum equation: (0) ∂ ∂ vz (0) + μ (T (0) )(R(0) )2 vz ∂z log ρ (T (0) ) + ∂t log ρ (T (0) ) 3 μ (T (0) )(R(0) )2 ∂z ∂z 1 (0) (0) (0) σ (T (0) )R(0) = ρ (T (0) )(R(0) )2 Re ∂t vz + vz ∂z vz − Bo . (161) + ε Ca


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Although Re ∼ ε , we retain the inertia term, just for sake of completeness. One can imagine situations corresponding to large Reynolds numbers. As conclusion we summarize our results in dimensional form: (0)

Proposition 5.1. Let veff = vE vz be the effective axial velocity, Reff = RE R(0) the effective fiber radius and Teff = TE T (0) the effective temperature. Let us suppose that the quantities Q0 (the mass flow), R f (the final fiber radius), V f (the pulling velocity), FL (the traction force) and TE (the extrusion temperature) are given positive constants. Then all other relevant physical quantities are determined by {veff , Reff , Teff } and given by: effective radial velocity: r ∂z veff (z) + veff ∂z log ρ (Teff (z)) + ∂t log ρ (Teff (z)) (162) veff r (r, z) = − 2 σ (Teff (z)) − μ (Teff (z))∂z veff (z) effective pressure: peff (z) = Reff (z) μ (Teff (z)) (veff (z)∂z log ρ (Teff (z)) + ∂t log ρ (Teff (z))); (163) − 3 effective axial stress: Σeff (r, z)ez = μ (Teff (z)) 3∂z veff (z) + ∂t log ρ (Teff (z)) σ (Teff (z)) ez + μ (Teff (z)) 3∂z veff (z) +veff (z)∂z log ρ (Teff (z)) ez − Reff (z) ∂z Reff (z) ∂z σ (Teff (z)) er + r er . (164) +∂t log ρ (Teff (z)) + veff (z)∂z log ρ (Teff (z)) r Reff (z) Reff (z) π R2eff effective traction : Feff = μ (Teff (z)) 3∂z veff (z) + ∂t log ρ (Teff (z)) g σ (Teff (z)) . (165) +veff (z)∂z log ρ (Teff (z)) − Reff (z) Functions {veff , Reff , Teff } are given by the Cauchy problem ∂ ∂ (166) ρ (Teff )(Reff )2 + ρ (Teff )(Reff )2 veff = 0, 0 < z < L; ∂t ∂z ∂ veff D ∂ + log ρ (Teff (z)) + σ (Teff (z))Reff (z) μ (Teff (z))(Reff (z))2 3 ∂z ∂z Dt D veff − g ρ (Teff (z))(Reff (z))2 , 0 < z < L; (167) = Dt DTeff c p (Teff )ρ (Teff )R2eff Dt 2ρ∞ veff Reff m = −Cλ∞ (Teff − T∞ ), 0 < z < L; (168) μ∞ (169) Reff (L) = R f , veff (L) = V f , Teff (L) = TE ,

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D ∂ ∂ = + veff denotes the total derivative. Finally, we have Dt ∂t ∂z vz (r, z) = veff (z) + O(ε 2 Pe ); R(r, z) = Reff + O(ε 2 Pe );

2 vr (r, z) = veff r (r, z) + O(ε Pe );

p(r, z) = peff (z) + O(ε 2 Pe );

Σ (r, z) = Σeff (r, z) + O(ε 2 Pe ). D veff is negligible if Remark 5.2. It is important to recall that in (167) the term Dt compared to g. Such a term, namely the fluid acceleration, is present in (167) because we decided to retain inertia although O(ε ).

5.2 Solvability of the Boundary Value Problems for the Stationary Effective Equations Clearly, the values at the extrusion boundary could be replaced by the values at the interface SE between the stages (c) and (d) of the fiber drawing process. In the industrial simulations it makes sense to solve the full 3D Navier–Stokes system in the stage (c), to solve the (166)–(168) corresponding to the stage (d) and to couple them at the interface SE . Coupling at the interface requires construction of the boundary layer. Following ideas from [24], a typical iterative procedure for the Navier–Stokes equations with free boundary is the following: for a given free boundary we solve the Navier–Stokes equations with normal stress given at the lateral boundary. Then we update position of the free boundary using the kinematic free boundary condition. Iterations are repeated until the stabilization. Such procedure requires solving the (166)–(168) with the boundary conditions (169) replaced by veff (L) = V f , veff (0) = vE , Teff (0) = TE .

(170)

In this section we present the study of the stationary version of the boundary value problem (166)–(168), (170), which was undertaken in [5]. We simply drop the index eff and set Q = Q0 /π . In the absence of the gravity and inertia effects, with constant density and with the heat transfer coefficient depending only on the temperature, the problem was solved in the reference [14]. Using the temperature as variable, it was possible to write an explicit solution for the radius and prove existence and uniqueness. In the general situation, the approach from [14] is not possible any more. Nevertheless, their change of the unknown function will be useful in our existence proof. We prove an existence result under the following physical properties on the coefficients:


217

σ (T ) are defined on R, bounded from above and ρ 1/3 from below by positive constants and decreasing. We suppose them infinitely differentiable. ρ f = minT∞ ≤T ≤TE ρ (T ). (H2) 0 < vE = v|z=0 < V f = v|z=L and TE > T∞ . (H3) c p is an infinitely differentiable strictly positive function.

(H1) Functions μ , ρ and

We introduce the new unknown w by 1/3

w = log

Vf ρ f

vρ 1/3 (T )

.

(171)

$ √ %m C λ∞ 2ρ∞ G and C1 = , m being the Kase–Matsuo exponent, Let Q μ∞ see(128). Then the boundary value problem (166)–(168), (170) transforms to 1/3 G = V f ρg

1 ∂ μ (T ) ∂ w σ (T ) w/2 +√ e −3 ∂z ρ (T ) ∂ z QG ρ 1/3 (T ) g 5G −4/3 −w = − ρ 1/3 (T )ew − Gρ −1/3e−w ∂z w − ρ e ∂z ρ , 0 < z < L; G 6 ∂T c p (T ) + C1 ρ −2m/3 (T )e−mw/2 (T − T∞ ) = 0, 0 < z < L; ∂z

(172) (173)

1/3

w(0) = w0 = log

Vf ρ f

vE ρ 1/3

> 0, w(L) = 0, T (0) = TE .

(174)

We will obtain existence of C∞ -solutions to problem (172)–(174), such that w ≤ w0 . 1/3 ρf Then the velocity v is calculated using the formula v = V f e−w . It is a ρ (T ) C∞ -function and satisfies v(z) ≥ vE on [0, L]. We note as well that ρ f = ρ (T (L)) is not given. For simplicity, we suppose w0 known. Otherwise, we should do one more fixed point calculation for ρ f , which does not pose problems. Definition 5.3. The corresponding variational formulation for problem (172)–(174) is: Find functions w ∈ H 1 (0, L) and T ∈ H 1 (0, L), ∂z T ≤ 0, such that the boundary conditions (174) are satisfied and L μ (T ) ∂ w ∂ ϕ

L

1 σ (T ) min{w,w0 }/2 ∂ ϕ √ dz e ρ (T ) ∂ z ∂ z ∂z QG ρ 1/3 (T ) 0 L L g 1/3 ∂w + ρ (T )emin{w,w0 } ϕ dz + Gρ −1/3(T )e−w ϕ dz ∂z 0 G 0 L 5G −4/3 ρ (T )e−w ∂z ρϕ dz = 0, ∀ϕ ∈ H01 (0, L). + 0 6 C1 −2m/3 ∂T =− ρ (T )e−m min{w,w0 }/2 (T − T∞ ), 0 < z < L. ∂z c p (T ) 3

0

dz −

(175) (176)

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Proposition 5.4. (see [5]). Let μ , σ and ρ satisfy (H1)–(H2) and let {w, T } be a variational solution to (174), (175) and (176). Then we have w ≤ w0 and TE ≥ T ≥ T∞ . Corollary 5.5. (see [5]). Under hypothesis (H1)–(H2), any variational solution {w, T } to (174), (175) and (176) solves equations (172)–(173). Theorem 5.6. (see [5]). Let us suppose hypotheses (H1)–(H3) hold true. Let T˜ be the solution for C1 −2m/3 ˜ −mw0 /2 ˜ ∂ T˜ =− ρ (T )e (T − T∞ ), 0 < z < L; T˜ (0) = TE . ∂z c p (T˜ ) Let κ = maxT∞ ≤T ≤TE |∂T ρ (T )| and let A =

L

dz 0 μ (T˜ (z)) .

(177)

Then there is δ0 > 0 such

that for κ A < δ0 , problem (172)–(174) admits a solution {w, T } ∈ C∞ [0, L]2 , such that ∂z T ≤ 0 and w(z) ≤ w0 . Remark 5.7. We see that in the case of constant density, the necessary condition from the Theorem is always fulfilled. Furthermore, since viscosity takes large values as temperature decreases, A is a very small quantity. Consequently, the Theorem covers all situations of practical interest. Proof uses Brouwer’s fixed point theorem.

Appendix 1 Let us consider the state S = {v = 0, ϑ = 1, ρ = 1, P = 0} , corresponding to the fluid at rest subject only to a uniform pressure and to isothermal conditions. We now study the linear stability of S , in absence of any body forces10 and external heat sources, in two cases: 1. The system dynamics is governed by ⎧ Dϑ ⎪ ⎪ ρ (ϑ ) div v = α , ⎪ ⎪ Dt ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ 1 Dv 1 = ρ (ϑ ) −∇P + Δ v + ∇ ( div v) , Dt Re 3 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 1 2 Ec Dϑ 1 + ϑ (Tw − 1) α Ec DP ⎪ ⎪ + |D(v)|2 , ⎩ ρ (ϑ ) Dt = Pe Δ ϑ + Re Dt (Tw − 1) Re (Tw − 1)2 (178) 10

We consider g = 0, and, as a consequence, p = P from (31).


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where, for the sake of simplicity, we have considered β , c p 1 , μ and λ constant, i.e. β = βR , c p 1 = c p R, μ = μR and λ = λR . 2. The system dynamics is governed by ⎧ Dϑ ⎪ ⎪ ρ (ϑ ) div v = α , ⎪ ⎪ Dt ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ 1 Dv 1 = ρ (ϑ ) (179) −∇P + Δ v + ∇ ( div v) , ⎪ Dt Re 3 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ρ (ϑ ) Dϑ = 1 Δ ϑ , Dt Pe corresponding to Ec = 0, i.e. to the fact that the kinetic energy of the fluid is really negligible when compared to its thermal energy. Let us first linearize the equations considering small (dimensionless) perturbations of the state S , namely v = v ∗ (x,t) , ϑ = 1 + ϑ ∗ (x,t) , ρ = 1 + ρ ∗ (x,t) , P = P∗ (x,t) . In particular, we have also

ρ = 1 − αϑ ∗, ⇒ ρ ∗ = −αϑ ∗ . Case 1. Dynamics governed by (178). We will see, as put in evidence in [2], that, in such a case, the rest state is linearly unstable. The linearized form of system (178) is ⎧ ∂ϑ∗ ⎪ ⎪ div v ∗ = α , ⎪ ⎪ ∂t ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ∂ v∗ 1 1 = −∇P∗ + Δ v ∗ + ∇ ( div v ∗ ) , ⎪ ∂t Re 3 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ∗ ∗ ⎪ ⎪ ⎩ ∂ϑ = 1 Δϑ∗ + β ∂P , ∂t Pe ∂t where

β=

(180)

α Ec . (Tw − 1) Re Tw

2

We now take the Fourier transform11 with respect to the spatial variable x of the dependent variable v ∗ , P∗ and ϑ ∗ , namely

11

The Fourier transform, with respect to the spatial variable x, of a function f (x,t) is defined by fˆ (k,t) =

1 (2π )3/2

R3

f (x,t) exp (−ik · x) dx.

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F.T. v (k,t) , v ∗ (x,t) −→ ∗ ϑ (x,t) −→ ϑ (k,t) , P ∗ (x,t) −→ P (k,t) , getting

⎧ ∂ ϑ ⎪ ⎪ ⎪ ik · v=α , ⎪ ⎪ ∂t ⎪ ⎪ ⎪ ⎨ v ik k2 1 ∂ = − v− k (k · v) , P− ⎪ ∂ t Re Re 3Re ⎪ ⎪ ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎩ ∂ ϑ = − k ϑ + β ∂ P , ∂t Pe ∂t where k = |k|. Decomposing v as follows

(181)

v= v + v⊥ , ⎧ k k k ⎪ ⎨ v· = v , v = k k k ⎪ ⎩ v⊥ = v− v ,

where

system (181) rewrites ⎧ v⊥ ⎪ k2 ∂ ⎪ ⎪ =− v⊥ , ⎪ ⎪ ∂t Re ⎪ ⎪ ⎪ ⎪ ⎪ ∂ v ⎪ ik 4 2 ⎪ ⎪ =− k v , P− ⎪ ⎨ ∂t Re 3Re ⎪ ⎪ k ∂ ϑ ⎪ ⎪ = i v , ⎪ ⎪ ∂t Kρ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ k2 ∂ P k ⎪ ⎪ = ϑ +i v . ⎩ ∂t β Pe αβ

(182)

So, we immediately realize that v⊥ vanishes as t → ∞. Therefore we are left with ⎛

⎞ k 0 ⎟ i ⎜ ⎟ ⎜ ⎟ ⎜ 0 α ⎟ ⎜ ⎟ ⎜ 4 ik ⎟ ∂ ⎜ ⎜ ⎟ ⎟ ⎜ ⎟ ⎜ k2 − ⎜ v ⎟ = A ⎜ v ⎟ , A= ⎜ 0 − ⎟. 3Re Re ⎟ ⎜ ⎟ ⎜ 2 ∂t ⎜ ⎟ ⎝ ⎠ ⎠ ⎝ ⎠ ⎝ k k i 0 P P β Pe αβ

ϑ

⎞

⎛

ϑ

⎞

⎛


221

We have to evaluate the eigenvalues of the matrix A. Denoting them by λi , i = 1, 2, 3 we have k4 ≥ 0. detA = λ1 λ2 λ2 = β Pe Re Now, since the secular equation is a third degree equation in λ whose coefficient are real, at least one eigenvalue is real. The above relations therefore implies that there exists at least a positive eigenvalue. We thus conclude that the state S is instable. Case 2. Dynamics governed by (179). Considering, as before, the Fourier transform of the linearized form of system (179) we obtain ⎧ v k2 ∂ ⎪ ⎪ ⊥ =− v⊥ , ⎪ ⎪ ∂t Re ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ∂ v ik 4 2 ⎪ ⎪ ⎪ ⎨ ∂ t = − Re P − 3Re k v , (183) ⎪ ⎪ k ∂ ϑ ⎪ ⎪ = i v , ⎪ ⎪ ∂t α ⎪ ⎪ ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎩ ∂ ϑ = − k ϑ. ∂t Pe Form (183)4and (183) v⊥ vanishes as t → ∞. In particular, 1 we deduce that ϑ and 2 k ϑ = ϑo exp − t and Pe 2 k ∂ ϑ ϑo k2 ∂ ϑ =− exp − t , ⇒ → 0, as t → ∞. ∂t Pe Pe ∂t So, by (183)3, we have that also v vanishes as t → ∞. Finally, (183)2 implies that P → 0, as t → ∞. We therefore conclude that the state S is linearly stable.

Appendix 2 Referring to Fig. 5, let us consider the simplest version of the Matovich–Pearson model for the stage (d). We have ⎧ ∂ ⎪ ⎪ ⎪ ⎨ ∂ x (v1 A) = 0, ⎪ ∂ v1 ⎪ ∂ ⎪ A = 0, ⎩ ∂x ∂x

(184)

where, as in formula (127), A = A (x) is the fiber cross section area, A = π R2 .

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x 0

L

xT

Lf

Stage (c)

Stage (d)

Fig. 5 A schematic of stages (c) and (d). The spinneret is located at x = L f . x = 0 is the die inflow surface. Picture not in scale

As boundary conditions, following [23], we consider Fsp , v1 x = L f = Vsp , and Φ x = L f = Φsp = π R2sp

(185)

where Φ denotes the longitudinal stress

Φ = 3μ f

∂ v1 , ∂x

with μ f fluid viscosity (considered constant along stage (d)). Solving (184), (185) we obtain Φsp v1 (x) = Vsp exp − Lf − x , (186) 3μ f Vsp Φsp Lf − x . R (x) = Rsp exp (187) 6μ f Vsp Hence the explicit expressions of VL and ΦL , in terms of the terminal quantities are the following Φsp Φsp L f − L , ΦL = −Φsp exp − Lf − L . VL = Vsp exp − 3μ f Vsp 3μ f Vsp (188) As we have seen in Sect. 5.1, the Matovich–Pearson model can be generalized considering the residual spatial dependence of viscosity, still neglecting inertia, surface tension and density variation. In that framework, according to (127) and (167) (see also [7]), (184)2 modifies to ∂ ∂ v1 = 0. μ (x) A (x) ∂x ∂x


223

The expressions replacing (186) and (187) are Φsp L f dx v1 (x) = Vsp exp − , 3Vsp x μ (x ) Φsp L f dx . R (x) = Rsp exp 6Vsp x μ (x )

(189) (190)

Thus, coming back to Sect. 4.1, namely to the definition of L, on the basis of (189), (190) we may confirm the consistency of our approach, taking x = L and observing L f dx L f −L Φ L that 6Vspsp L f μdx ) = O (1) if L μ (x ) = μ . Since L f −L ≈ 10 m, this requires μ = (x 7 O 10 Pa s , consistently with the thermal field experimentally known in stage (d).

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17. Kase, S., Matsuo, T.: Studies of melt spinning I. Fundamental Equations on the Dynamics of Melt Spinning, J. Polym. Sci. A 3, 2541–2554 (1965) 18. Kase, S., Matsuo, T.: Studies of melt spinning. II, Steady state and transient solutions of fundamental equations compared with experimental results. J. Polym. Sci. 11, 251–287 (1967) 19. Lions, P.L., Masmoudi, N.: Incompressible limit for a viscous compressible fluid. J. Math. Pures Appl. 77, 585–627 (1998) 20. Matovich, M.A., Pearson, J.R.A.: Spinning a molten threadline-steady state isothermal viscous flows. Ind. Engrg. Chem. Fundam. 8, 512–520 (1969) 21. von der Ohe, R.: Simulation of glass fiber forming processes. Thesis PhD (2005) – AUC Imprint: ISBN: 8791200253. – Aalborg : Department of Production, Aalborg University (2005) 22. Rajagopal, K.R., R˙uzˇ iˇcka, M., Srinivasa, A.R.: On the Oberbeck-Boussinesq approximation. Math. Models Methods Appl. Sci. 6, 1157–1167 (1996) 23. Renardy, M.: Draw resonance revisited, SIAM J. Appl. Math. 66, 1261–1269 (2006) 24. Renardy, M.: An existence theorem for a free surface flow problem with open boundaries. Commun. Partial Differ. Equ. 17, 1387–1405 (1992) 25. Schlichting, H., Gersten, K.: Boundary-Layer Theory, 8th edn. Springer, Heidelberg (2000) 26. Shelby, J.E.: Introduction to Glass Science and Technology, 2nd edn. RSCP Publishing, London (2005) 27. Temam, R.: Infinite-Dimensional Dynamical systems in Mechanics and Physics. Springer, New York (1988) 28. Zeytounian, R.Kh.: Modélisation asymptotique en mécanique des fluides newtoniens. Springer, New York (1994) 29. Zeytounian, R.Kh.: The Bénard-Marangoni thermocapillary instability problem. Phys. Uspekhis 41, 241–267 (1998) 30. Zeytounian, R.Kh.: Joseph Boussinesq and his approximation: A contemporary view. C.R. Mécanique 331, 575–586 (2003)

List of Participants

1. Aionicesei Elena Univ. of Maribor, Slovenia [email protected] 2. Antonietti Paola Francesca Politecnico of Milano, Italy [email protected] 3. Benedetti Irene Univ. of Firenze, Italy [email protected] 4. Borsi Iacopo Univ. of Firenze, Italy [email protected] 5. Butt Azhar Iqbal Kashif TU Kaiserslautern, Germany [email protected] 6. Ceseri Maurizio Univ. of Firenze, Italy [email protected] 7. Curkovic Andrijana Univ. of Split, Croatia [email protected] 8. Dierich Frank Freiberg Univ. of Mining and Technology, Germany [email protected] 9. Doschoris Michael Univ. of Patras, Greece [email protected] 10. Ebert Svend Freiberg Univ. of Mining and Technology, Germany [email protected] 11. Faienza Loredana Univ. of Firenze, Italy [email protected] A. Fasano (ed.), Mathematical Models in the Manufacturing of Glass, Lecture Notes in Mathematics 2010, DOI 10.1007/978-3-642-15967-1, c Springer-Verlag Berlin Heidelberg 2011

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12. Fusi Lorenzo Univ. of Firenze, Italy [email protected] 13. Geyer Anna Univ. of Vienna, Austria [email protected] 14. Hadjloizi Demetra Univ. Patras, Greece [email protected] 15. Lebedyanskaya Elena Voronezh State agricolture Univ., Russia [email protected] 16. Nagwanshi Rekha Vikram Univ. Ujjain MP, India [email protected] 17. Niedziela Maciej Univ. of Zielona Gora, Poland [email protected] 18. Nouri Fatma Zohra Univ. Badji Mokhtar, France fz [email protected] 19. Olech Michal The Univ. of Wroclaw, Poland [email protected] 20. Pazanin Igor Univ. of Zagreb, Croatia [email protected] 21. Peter Cristian Univ. of Vienna, Austria [email protected] 22. Ricci Riccardo Univ. of Firenze, Italy [email protected] 23. Rosso Fabio Univ. of Firenze, Italy [email protected] 24. Senger Benno Technische Univ. Bergakandemie Freiberg, German [email protected] 25. Serpa Cristina University of Lisbon, Portugal [email protected] 26. Speranza Alessandro Univ. of Firenze, Italy [email protected]



27. Starcevic Maja Univ. of Zagreb, Croatia [email protected] 28. Sulkowski Tomasz Univ. of Zielona Gora, Poland [email protected] 29. Togobytska Nataliya Weierstrass Inst. for Appl Anals and Stoch, Germany [email protected] 30. Turbin Mikhail Voronezh State Univ., Russia [email protected] 31. Verani Marco Politecnico of Milano, Italy [email protected] 32. Vorotnikov Dmitry Voronezh State Univ., Russia [email protected] 33. Zubkov Vladimir Univ. of Limerick, Ireland [email protected] 34. Zvyagin Andrey Voronezh State Univ., Russia [email protected]

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Mathematical Models in the Manufacturing of Glass: C.I.M.E. Summer School, Montecatini Terme, Italy 2008 (Lecture Notes in Mathematics C.I.M.E. Foundation Subseries)

Mathematical Models in the Manufacturing of Glass: C.I.M.E. Summer School, Montecatini Terme, Italy 2008 (Lecture Notes in Mathematics C.I.M.E. Foundation Subseries)

Mathematical Epidemiology (Lecture Notes in Mathematics Mathematical Biosciences Subseries)

Teoria dei numeri: CIME; Varenna , Italy, 1955

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Dynamical systems and small divisors: Lectures CIME school

Dynamical Systems. Lectures 2nd Session C.I.M.E., Montecatini Terme, Italy, 1994

Models of ZF-set theory (Lecture notes in mathematics 223)

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Axiom of Choice (Lecture Notes in Mathematics)

The Dirac Spectrum (Lecture Notes in Mathematics)

Geometric topology: recent developments. Lectures CIME

Stochastic geometry CIME lecs Martina Franca

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Geometric Mechanics (Lecture Notes in Mathematics 1798)

Geometric Mechanics (Lecture Notes in Mathematics 1798)

Spin Glasses (Lecture Notes in Mathematics, 1900)

Invariant theory (Lecture notes in mathematics ; 585)

Noncommutative Stationary Processes (Lecture Notes in Mathematics)

Markov Set-Chains (Lecture Notes in Mathematics)

p-Adic Analysis (Lecture Notes in Mathematics)

Minimal Surfaces (Courant Lecture Notes in Mathematics)

Halbexakte Homotopiefunktoren (Lecture Notes in Mathematics, 12)

Coding Theory (Lecture Notes in Mathematics 201)

*- Autonomous Categories (Lecture Notes in Mathematics)

Commutative group schemes (Lecture Notes in Mathematics)

Complementarity Problems (Lecture Notes in Mathematics)

Mathematical Models in the Manufacturing of Glass: C.I.M.E. Summer School, Montecatini Terme, Italy 2008 (Lecture Notes in Mathematics C.I.M.E. Foundation Subseries)