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The Method of Weighted Residuals and Variational Principles THE METHOD OF WEIGHTED RESIDUALS AND VARIATIONAL PRINCIPLES WITH APPLICATION IN FLUID MECHANICS, HEAT AND MASS TRANSFER This is Volume 87 in MATHEMATICS IN SCIENCE AND ENGINEERING A series of monographs and textbooks Edited by RICHARD BELLMAN, University of Southern California The complete listing of books in this series is available from the Publisher upon request. The Method of Weighted Residuals and Variational Principles WITH APPLICATION IN FLUID MECHANICS, HEAT AND MASS TRANSFER BRUCE A. FINLAYSON Department of Chemical Engineering University of Washington @ 1972 ACADEMIC PRESS New York and London COPYRIGHT0 1972, BY ACADEMICPRESS, INC. ALL RIGHTS RESERVED NO PART OF THIS BOOK MAY BE REPRODUCED IN ANY FORM, BY PHOTOSTAT, MICROFILM, RETRIEVAL SYSTEM, OR ANY OTHER MEANS, WITHOUT WRITTEN PERMISSION FROM THE PUBLISHERS. ACADEMIC PRESS, INC. 111 Fifth Avenue, New York, New York 10003 United Kingdom Edition Dublished bv ACADEM~CPRESS, INC. (LONDON) LTD. 24/28 Oval Road, London NWl LIBRARYOF CONGRESSCATALOG CARD NUMBER: 74-182607 PRINTED IN THE UNITED STATES OF AMERICA Contents Preface ix AcknowIedgments xiii PART I-THE METHOD OF WEIGHTED RESIDUALS Chapter 1 Introduction 1.1 Basic Equations and Their Classification 4 1.2 Method of Weighted Residuals 7 References 12 Chapter 2 Boundary-Value Problems in Heat and Mass Transfer 2.1 One-Dimensional Heat Conduction 16 2.2 Reduction to Ordinary Differential Equations 20 2.3 Boundary Methods 24 2.4 General Treatment of Steady-State Heat Conduction 28 V vi CONTENTS 2.5 Mass Transfer from a Sphere 30 2.6 Choice of Trial Functions 34 Exercises 36 References 37 Chapter 3 Eigenvalue and Initial-Value Problems in Heat and Mass Transfer 3.1 Eigenvalue Problems 40 3.2 Transient Heat and Mass Transfer 44 3.3 Entry-Length and Initial-Value Problems 48 3.4 Mass Transfer to a Moving Fluid 58 3.5 Heat Transfer Involving a Phase Change 61 Exercises 62 References 65 Chapter 4 Applications to Fluid Mechanics 4.? Laminar Flow in Ducts 68 4.2 Boundary Layer Flow past a Flat Plate 74 4.3 Laminar Boundary Layers 76 4.4 Natural Convection 83 4.5 Coupled Entry-Length Problems 85 4.6 Steady-State Flow Problems 88 Exercises 91 References 92 Chapter 5 Chemical Reaction Systems 5.1 Orthogonal Collocation 97 5.2 Unsteady Diffusion 107 5.3 Reaction and Diffusion in a Catalyst Particle 110 5.4 Tubular Reactor with Axial Dispersion 126 5.5 Packed Bed Reactor with Radial Dispersion 130 5.6 Relation to Other Techniques: Galerkin, Least Squares, Finite Difference, Finite Element Methods 135 Exercises 144 References 146 Chapter 6 Convective Instability Problems 6.1 Choice of Trial Functions 151 6.2 Application of the Galerkin Method 157 6.3 Time-Dependent Motion 176 CONTENTS vii 6.4 Variational Methods 184 6.5 Nonlinear Convective Instability 194 6.6 Hydrodynamic Stability 196 Exercises 202 References 203 PART 11-VARIATIONAL PRINCIPLES Chapter 7 Introduction to Variational Principles 7.1 Calculus of Variations 212 7.2 Steady-State Heat Conduction 22 1 7.3 Laminar Flow through Ducts 225 7.4 Relation to Galerkin and Finite Element Methods 229 7.5 Variational Principles for Eigenvalue Problems 232 7.6 Enclosure Theorems 238 7.7 Least Squares Interpretation of D. H. Weinstein’s Method 240 7.8 Lower Bounds for Eigenvalues 244 Exercises 249 References 250 Chapter 8 Variational Principles in Fluid Mechanics 8.1 Basic Equations 254 8.2 Variational Principles for Perfect Fluids 256 8.3 Magnetohydrodynamics 265 8.4 Non-Newtonian Fluids 270 8.5 Slow Flow past Drops and Particles 278 8.6 Variational Principles for Navier-Stokes Equations 285 8.7 Energy Methods for Stability of Fluid Motion 290 Exercises 294 References 294 Chapter 9 Variational Principles for Heat and Mass Transfer Problems 9.1 Frkchet Derivatives 299 9.2 Variational Principles for Non-Self-Adjoint Equations 307 9.3 Variational Principles for the Transport Equation 312 9.4 Applications to Heat Transfer 317 9.5 Applications to Mass Transfer 321 9.6 Upper Bound for Heat Transport by Turbulent Convection 328 Exercises 332 References 333 ... Vlll CONTENTS Chapter 10 On the Search for Variational Principles 10.1 Introduction 335 10.2 Entropy Production 337 10.3 Heat Transfer 339 10.4 Fluid Mechanics 347 Exercises 349 References 350 Chapter 11 Convergence and Error Bounds I 1.1 Definitions 352 1 1.2 Boundary-Value Problems 357 11.3 Initial-Value Problems 370 11.4 Eigenvalue Problems 378 11.5 Error Bounds Using the Maximum Principle 38 1 1 I .6 Error Bounds Using the Mean-Square Residual 388 References 393 Author Index 397 Subject Index 405 Preface This is a book for people who want to solve problems formulated as differential equations in science and engineering. The subject area is limited to fluid mechanics, heat and mass transfer. While making no pretense at completely covering these subjects and their relationship to variational principles and approximate methods, the book is intended to give the novice an introduction to the subject, and lead him through the difficult research problems being treated in the current literature. The first four chapters give a relatively simple treatment of many classical problems in the field. The literature is full of simple, one-term approximations, but the method of weighted residuals (MWR) can be used to obtain answers of any desired accuracy, and there are several methods specifically adapted to the computer. Tn many test cases MWR compares favorably to finite difference computa- tions in that the MWR results are either more accurate or require less com- putation time to generate or both. Chapter 4 discusses the developments by Professor D. E. Abbott and his students at Purdue University on laminar boundary layer flows. Orthogonal collocation is illustrated in Chapter 5. This method was advanced in 1967 by Professor W. E. Stewart at the Univer- sity of Wisconsin and J. V. Villadsen at Danmarks Tekniske Hsjskole. It drastically reduces the drudgery of setting up the problem, and, when ix X PREFACE applicable, is highly recommended. Chapter 6 studies the Galerkin method as applied to convective instability problems, where it proves effec- tive and accurate. Chapters 5 and 7 relate MWR to finite element methods, which is a promising technique, especially for linear problems with irregular boundaries. The ideas behind the method of weighted residuals are relatively simple and are easily applied. Variational principles are only slightly more compli- cated, but are often irrelevant to applications in engineering. Consequently this material takes a more appropriate back seat. Variational principles in fluid mechanics, heat and mass transfer are presented in Chapters 7-9. Chapter 10 is a short summary of my opinions on the attempt to derive '' variational " principles based on a principle of minimum (or maximum) rate of entropy production. The final chapter gives a summary of results on convergence and error bounds, subjects which are rapidly expanding. While numerical convergence often suffices, the theorems in Chapter 11 give the conditions necessary to ensure convergence in difficult nonlinear problems. The book is intended to be comprehensible to a graduate student with some knowledge of and interest in mathematics through differential equations. I hope it will find use as a reference book for graduate courses discussing approximate and numerical solutions, as well as a convenient reference for those working in the field. Problem sets are included to aid in self-study and course work. One of the pleasurable aspects of writing a book, I have discovered, is that while organizing the material new results are suggested. These are scattered throughout the book so that 1 highlight them here. In the varia- tional method the eigenvalue is stationary to small errors in the approximate eigenfunctions. I show that this is also true of the Galerkin method (Chapter 6). While not a new result, Section 7.8 illustrates two very simple and practi- cal methods of obtaining lower bounds on eigenvalues. A variational prin- ciple is given for a collection of fluid drops suspended in another fluid in slow flow-extending the previous results for Newtonian fluids in both phases to allow non-Newtonian fluids in both phases. The nonexistence of a variational principle for the steady-state Navier-Stokes equations is shown in Section 8.6 using Frechet differentials, making more concise the detailed arguments given in 1930 by C. B. Millikan. Indeed the Frechet derivative proves useful in organizing the many variational principles, or lack of them, in heat and mass transfer (see Chapter 9). The FrCchet derivative is also used to define a variational principle for any differential equation and its " adjoint "-even nonlinear equations-although there appears to be no advantage to do so. One of the difficulties with M WR or variational methods is that the error is often not known. Some of the methods discussed above are specifically designed to allow the results to be carried to numerical convergence. Even PREFACE xi so, it is of interest to have some rigorous definition of the error. The residual -the equation which is solved only approximately-provides such a criteria. Material in Section 11.6, largely developed by one of my students, Noble Ferguson, gives error bounds on the solution in terms of error bounds on the residual. Thus the error can be assessed when the exact solution is unknown. The residual also proves a useful guide in cases where the appropriate theorems have not yet been proved. The review of the literature is complete to October, 1970. This page intentionally left blank Acknowledgments Several teachers have been influential in stimulating my interest in the application of mathematics. My high school physics teacher, Mr. Williams, helped me learn calculus at a time when that subject was not taught in many high schools, especially not in a small town in Oklahoma.
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