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• Fluid Mechanics Second Edition

• Joseph H. Spurk · Nuri Aksel

Fluid Mechanics

Second Edition

123

• Professor Dr. Joseph H. Spurk (em.) TU Darmstadt Institut für Technische Strömungslehre Petersenstraße 30 64287 Darmstadt Germany

Professor Dr. Nuri Aksel Universität Bayreuth Lehrstuhl für Technische Mechanik und Strömungsmechanik Universitätsstraße 30 95447 Bayreuth Germany

ISBN 978-3-540-73536-6

DOI 10.1007/978-3-540-73537-3

e-ISBN 978-3-540-73537-3

Library of Congress Control Number: 2007939489

© 2008, 1997 Springer-Verlag Berlin Heidelberg

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 for 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.

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• Preface to the First English Edition

This textbook is the translation of the fourth edition of Strömungslehre, Ein- führung in die Theorie der Strömungen. The German edition has met with a favorable reception in German-speaking countries, showing that there was a demand for a book that emphazises the fundamentals. In the English lit- erature there are books of the same nature, some excellent, and these have indeed influenced me to write this book. However, they cover different ground and are not aimed primarily at mechanical engineering students, which this book is. I have kept the original concept throughout all editions and there is little to say that has not been said in the preface to the first German edition. There is now a companion volume Solved Problems in Fluid Mechanics, which alleviates the drawback of the first German edition, namely the absence of problem exercises.

The book has been translated by Katherine Mayes during her stay in Darmstadt, and I had the opportunity to work with her daily. It is for this reason that I am solely responsible for this edition, too. My thanks also go to Prof. L. Crane from Trinity College in Dublin for his assistance with this book. Many people have helped, all of whom I cannot name, but I would like to express my sincere thanks to Ralf Münzing, whose dependable and unselfish attitude has been a constant encouragement during this work.

Darmstadt, January 1997 J. H. Spurk

Preface to the Second English Edition

The first English edition was the translation of the fourth German edition. In the meantime the textbook has undergone several additions, mostly stimu- lated by consulting activities of the first author. Since the textbook continues to receive favourable reception in German speaking countries and has been translated in other languages as well, the publisher suggested a second English edition. The additions were translated for the most part by Prof. L. Crane from Trinity College in Dublin, who has accompanied this textbook from the very beginning. Since the retirement of the first author, Prof. N. Aksel from the University of Bayreuth, Germany, the second author, was actively engaged in the sixth and the seventh edition. The additions were written by the first author who accepts the responsibility for any mistakes or omissions in this book.

• Contents

1 The Concept of the Continuum and Kinematics . . . . . . . . . . 1 1.1 Properties of Fluids, Continuum Hypothesis . . . . . . . . . . . . . . . 1 1.2 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.2.1 Material and Spatial Descriptions . . . . . . . . . . . . . . . . . . 7 1.2.2 Pathlines, Streamlines, Streaklines . . . . . . . . . . . . . . . . . 10 1.2.3 Differentiation with Respect to Time . . . . . . . . . . . . . . . 14 1.2.4 State of Motion, Rate of Change of Line, Surface

and Volume Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 1.2.5 Rate of Change of Material Integrals . . . . . . . . . . . . . . . 29

2 Fundamental Laws of Continuum Mechanics . . . . . . . . . . . . . . 35 2.1 Conservation of Mass, Equation of Continuity . . . . . . . . . . . . . . 35 2.2 Balance of Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.3 Balance of Angular Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . 44 2.4 Momentum and Angular Momentum

in an Accelerating Frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 2.5 Applications to Turbomachines . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 2.6 Balance of Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 2.7 Balance of Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 2.8 Thermodynamic Equations of State . . . . . . . . . . . . . . . . . . . . . . . 71

3 Constitutive Relations for Fluids . . . . . . . . . . . . . . . . . . . . . . . . . 75

4 Equations of Motion for Particular Fluids . . . . . . . . . . . . . . . . 95 4.1 Newtonian Fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

4.1.1 The Navier-Stokes Equations . . . . . . . . . . . . . . . . . . . . . . 95 4.1.2 Vorticity Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 4.1.3 Effect of Reynolds’ Number . . . . . . . . . . . . . . . . . . . . . . . 100

4.2 Inviscid Fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 4.2.1 Euler’s Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 4.2.2 Bernoulli’s Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 4.2.3 Vortex Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 4.2.4 Integration of the Energy Equation . . . . . . . . . . . . . . . . . 138

4.3 Initial and Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . 141 4.4 Simplification of the Equations of Motion . . . . . . . . . . . . . . . . . . 145

• VIII Contents

5 Hydrostatics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 5.1 Hydrostatic Pressure Distribution . . . . . . . . . . . . . . . . . . . . . . . . 151 5.2 Hydrostatic Lift, Force on Walls . . . . . . . . . . . . . . . . . . . . . . . . . . 156 5.3 Free Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

6 Laminar Unidirectional Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 6.1 Steady Unidirectional Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

6.1.1 Couette Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 6.1.2 Couette-Poiseuille Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 6.1.3 Flow Down an Inclined Plane . . . . . . . . . . . . . . . . . . . . . . 171 6.1.4 Flow Between Rotating Concentric Cylinders . . . . . . . . 174 6.1.5 Hagen-Poiseuille Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 6.1.6 Flow Through Noncircular Conduits . . . . . . . . . . . . . . . . 180

6.2 Unsteady Unidirectional Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 6.2.1 Flow Due to a Wall Which Oscillates

in its Own Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 6.2.2 Flow Due to a Wall Which is Suddenly Set in Motion 186

6.3 Unidirectional Flows of Non-Newtonian Fluids . . . . . . . . . . . . . 188 6.3.1 Steady Flow Through a Circular Pipe . . . . . . . . . . . . . . 188 6.3.2 Steady Flow Between a Rotating Disk

and a Fixed Wall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 6.3.3 Unsteady Unidirectional Flows

of a Second Order Fluid . . . . . . . . . . . . . . . . . . . . . . . . . . 191 6.4 Unidirectional Flows of a Bingham Material . . . . . . . . . . . . . . . 197

6.4.1 Channel Flow of a Bingham Material . . . . . . . . . . . . . . . 197 6.4.2 Pipe Flow of a Bingham Material . . . . . . . . . . . . . . . . . . 202

7 Fundamentals of Turbulent Flow . . . . . . . . . . . . . . . . . . . . . . . . . . 205 7.1 Stability and the Onset of Turbulence . . . . . . . . . . . . . . . . . . . . . 205 7.2 Reynolds’ Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 7.3 Turbulent Shear Flow Near a Wall . . . . . . . . . . . . . . . . . . . . . . . . 213 7.4 Turbulent Flow in Smooth Pipes and Channels . . . . . . . . . . . . . 223 7.5 Turbulent Flow in Rough Pipes . . . . . . . . . . . . . . . . . . . . . . . . . . 226

8 Hydrodynamic Lubrication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 8.1 Reynolds’ Equation of Lubrication Theory . . . . . . . . . . . . . . . . . 229 8.2 Statically Loaded Bearing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232

8.2.1 Infinitely Long Journal Bearing . .

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