COMSOL Multiphysics ® M ODEL L IBRARY V ERSION 3.5a How to contact COMSOL: Germany United Kingdom COMSOL Multiphysics GmbH COMSOL Ltd. Benelux Berliner Str. 4 UH Innovation Centre COMSOL BV D-37073 Göttingen College Lane Röntgenlaan 19 Phone: +49-551-99721-0 Hatfield 2719 DX Zoetermeer Fax: +49-551-99721-29 Hertfordshire AL10 9AB The Netherlands [email protected] Phone:+44-(0)-1707 636020 Phone: +31 (0) 79 363 4230 www.comsol.de Fax: +44-(0)-1707 284746 Fax: +31 (0) 79 361 4212 [email protected] [email protected] Italy www.uk.comsol.com www.comsol.nl COMSOL S.r.l. Via Vittorio Emanuele II, 22 United States Denmark 25122 Brescia COMSOL, Inc. COMSOL A/S Phone: +39-030-3793800 1 New England Executive Park Diplomvej 376 Fax: +39-030-3793899 Suite 350 2800 Kgs. Lyngby [email protected] Burlington, MA 01803 Phone: +45 88 70 82 00 www.it.comsol.com Phone: +1-781-273-3322 Fax: +45 88 70 80 90 Fax: +1-781-273-6603 [email protected] Norway www.comsol.dk COMSOL AS COMSOL, Inc. 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Robert Schuman Fax: +46 8 412 95 10 [email protected] F-38000 Grenoble [email protected] www.comsol.com Phone: +33 (0)4 76 46 49 01 www.comsol.se Fax: +33 (0)4 76 46 07 42 For a complete list of international [email protected] Switzerland representatives, visit www.comsol.fr FEMLAB GmbH www.comsol.com/contact Technoparkstrasse 1 CH-8005 Zürich Company home page Phone: +41 (0)44 445 2140 www.comsol.com Fax: +41 (0)44 445 2141 [email protected] COMSOL user forums www.femlab.ch www.comsol.com/support/forums COMSOL Multiphysics Model Library © COPYRIGHT 1998–2008 by COMSOL AB. All rights reserved Patent pending The software described in this document is furnished under a license agreement. The software may be used or copied only under the terms of the license agreement. No part of this manual may be photocopied or reproduced in any form without prior written consent from COMSOL AB. 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Version: November 2008 COMSOL 3.5a Part number: CM020004 CONTENTS Chapter 1: Introduction Typographical Conventions . 3 Model Library Guide 5 Chapter 2: Acoustics Models Acoustics of a Muffler 12 Introduction . 12 Model Definition . 12 Results and Discussion. 13 Modeling in COMSOL Multiphysics . 14 Modeling Using the Graphical User Interface . 15 Eigenmodes of a Room 21 Model Definition . 21 Results and Discussion. 23 Modeling Using the Graphical User Interface . 24 Chapter 3: Chemical Engineering Models Transport and Adsorption 32 Model Definition . 33 Modeling in COMSOL Multiphysics . 35 Results. 36 Modeling Using the Graphical User Interface . 38 Tubular Reactor 44 Introduction . 44 Model Definition . 44 Results. 47 CONTENTS | i Modeling Using the Graphical User Interface . 49 Chapter 4: Electromagnetics Models Electrostatic Potential Between Cylinders 57 Model Definition . 57 Results. 57 Modeling Using the Graphical User Interface . 58 Pacemaker Electrode 68 Introduction . 68 Model Definition . 69 Results and Discussion. 70 Modeling Using the Graphical User Interface . 71 Skin Effect in a Circular Wire 76 Introduction . 76 Model Definition . 76 Results. 77 Modeling Using the Graphical User Interface . 78 Modeling a Wire with an Aluminum Core . 81 Quadrupole Lens 83 Introduction . 83 Model Definition . 83 Results and Discussion. 86 Modeling Using the Graphical User Interface . 88 Spherical Capacitor 93 Introduction . 93 Model Definition . 94 Modeling Using the Graphical User Interface . 95 ii | CONTENTS Chapter 5: Equation-Based Models The Black-Scholes Equation 100 Introduction . 100 Model Definition . 101 Modeling in COMSOL Multiphysics . 101 Modeling Using the Graphical User Interface . 102 Optimizing a Flywheel Profile 104 Introduction . 104 Model Definition . 105 Results and Discussion. 108 Reference . 109 Modeling Using the Graphical User Interface . 109 Electrical Signals in a Heart 118 Introduction . 118 Model Definition . 120 Modeling in COMSOL Multiphysics . 121 Results. 122 Modeling Using the Graphical User Interface . 123 References . 128 An Integro-Partial Differential Equation 129 Introduction . 129 Modeling in COMSOL Multiphysics . 130 Results. 131 Comparison with the Full 3D Radiation Model . 132 Reference . 133 Modeling Using the Graphical User Interface . 134 The KdV Equation and Solitons 138 Introduction . 138 Model Definition . 139 Results. 140 References . 140 Modeling Using the Graphical User Interface . 141 CONTENTS | iii Shallow Water Equations 145 Introduction . 145 Artificial Stabilization . 146 Model Definition . 146 Results and Discussion. 149 Modeling in COMSOL Multiphysics . 150 Reference . 150 Modeling Using the Graphical User Interface . 150 Shell Diffusion 153 Introduction . 153 Model Definition . 153 Results. 154 Modeling in COMSOL Multiphysics . 156 Modeling Using the Graphical User Interface . 156 Spherically Symmetric Transport 159 Introduction . 159 Model Definition . 159 Results. 162 Modeling in COMSOL Multiphysics . 164 Modeling Using the Graphical User Interface . 165 The Telegraph Equation 169 Introduction . 169 Model Definition . 169 Results. 170 Modeling in COMSOL Multiphysics . 172 Modeling Using the Graphical User Interface . 173 A Transport Problem 176 Introduction . 176 Model Definition . 176 Streamline Diffusion. 177 Upwinding Using Discontinuous Basis Functions . 177 Results. 178 Modeling in COMSOL Multiphysics . 179 Modeling Using the Graphical User Interface . 179 iv | CONTENTS The Two-Term Boltzmann Equation 183 Introduction . 183 Model Definition . 185 Results and Discussion. 187 Modeling in COMSOL Multiphysics . 190 References . 191 Modeling Using the Graphical User Interface . 191 Chapter 6: Fluid Dynamics Models Backstep with Argyris Element 204 Introduction . 204 Model Definition . 204 Argyris Element Properties . 204 Results. 205 Using the Graphical User Interface . 205 Flow Past a Cylinder 211 Introduction . 211 Model Definition . 211 Results. 213 Reference . 213 Modeling Using the Graphical User Interface . 214 Time-Dependent Simulation . 217 Terminal Falling Velocity of a Sand Grain 220 Introduction . 220 Model Definition . 220 Results. 222 Reference . 225 Modeling Using the Graphical User Interface . 225 Fluid Valve 230 Introduction . 230 Model Definition . 230 Results. 232 CONTENTS | v Modeling in COMSOL Multiphysics . 234 Modeling Using the Graphical User Interface . 234 Micromixer 240 Introduction . 240 Model Definition . 241 Results. 242 Modeling Using the Graphical User Interface . 243 Shock Tube 252 Introduction . 252 Model Definition . 252 Modeling in COMSOL Multiphysics . 253 Results. 254 Modeling Using the Graphical User Interface . 255 Sloshing Tank 259 Introduction . 259 Model Definition . 260 Results. 262 Modeling Using the Graphical User Interface . 264 Chapter.