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- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Sprache: Englisch
INTRODUCTION TO THEORETICAL AND MATHEMATICAL FLUID DYNAMICS A practical treatment of mathematical fluid dynamics In Introduction to Theoretical and Mathematical Fluid Dynamics, distinguished researcher Dr. Bhimsen K. Shivamoggi delivers a comprehensive and insightful exploration of fluid dynamics from a mathematical point of view. The book introduces readers to the mathematical study of fluid behavior and highlights areas of active research in fluid dynamics. With coverage of advances in the field over the last 15 years, this book provides in-depth examinations of theoretical and mathematical fluid dynamics with a particular focus on incompressible and compressible fluid flows. Introduction to Theoretical and Mathematical Fluid Dynamics includes practical applications and exercises to illustrate the concepts discussed within, and real-world examples are explained throughout the text. Clear and explanatory material accompanies the rigorous mathematics, making the book perfect for students seeking to learn and retain this complex subject. The book also offers: * A thorough introduction to the basic concepts and equations of fluid dynamics, including an introduction to the fluid model, the equations of fluid flows, and surface tension effects * Comprehensive explorations of the dynamics of incompressible fluid flows, fluid kinematics and dynamics, the complex-variable method, and three-dimensional irrotational flows * Detailed discussions of the dynamics of compressible fluid flows, including a review of thermodynamics, isentropic fluid flows, potential flows, and nonlinear theory of plane sound waves * Systematic discussions of the dynamics of viscous fluid flows, including shear-layer flow, jet flow and wake flow. Ideal for graduate-level students taking courses on mathematical fluid dynamics as part of a program in mathematics, engineering, or physics, Introduction to Theoretical and Mathematical Fluid Dynamics is also an indispensable resource for practicing applied mathematicians, engineers, and physicists.
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Introduction to Theoretical and Mathematical Fluid Dynamics
Bhimsen K. Shivamoggi University of Central Florida Orlando, Florida, United States
Third Edition
This edition first published 2023
© 2023 John Wiley and Sons, Inc.
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, except as permitted by law. Advice on how to obtain permission to reuse material from this title is available at http://www.wiley.com/go/permissions.
The right of Bhimsen K. Shivamoggi to be identified as the author of this work has been asserted in accordance with law.
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Library of Congress Cataloging-in-Publication Data
Names: Shivamoggi, Bhimsen K., author.Title: Introduction to theoretical and mathematical fluid dynamics / Bhimsen Krishnarao Shivamoggi, University of Central Florida, Orlando, United States.Description: Hoboken, NJ : John Wiley & Sons, 2023. | Includes bibliographical references and index.Identifiers: LCCN 2021041747 (print) | LCCN 2021041748 (ebook) | ISBN 9781119101505 (hardback) | ISBN 9781119101512 (pdf) | ISBN 9781119101529 (epub) | ISBN 9781119765158 (obook)Subjects: LCSH: Fluid dynamics. | Fluid dynamics–Mathematical models.Classification: LCC QA911 .S462 2021 (print) | LCC QA911 (ebook) | DDC 532/.05–dc23LC record available at https://lccn.loc.gov/2021041747LC ebook record available at https://lccn.loc.gov/2021041748
Cover image: © Bocskai Istvan/Shutterstock
Cover design by Wiley
Set in 9.5/12.5pt STIXTwoText by Integra Software Services Pvt. Ltd, Pondicherry, India
In memory of my beloved mother, Pramila.
Contents
Cover
Title page
Copyright
Dedication
Preface to the Third Edition
Acknowledgments
Part I Basic Concepts and Equations of Fluid Dynamics
1 Introduction to the Fluid Model
1.1 The Fluid State
1.2 Description of the Flow-Field
1.3 Volume Forces and Surface Forces
1.4 Relative Motion Near a Point
1.5 Stress–Strain Relations
2 Equations of Fluid Flows
2.1 The Transport Theorem
2.2 The Material Derivative
2.3 The Law of Conservation of Mass
2.4 Equation of Motion
2.5 The Energy Equation
2.6 The Equation of Vorticity
2.7 The Incompressible Fluid
2.8 Boundary Conditions
2.9 A Program for Analysis of the Governing Equations
3 Hamiltonian Formulation of Fluid-Flow Problems
3.1 Hamiltonian Dynamics of Continuous Systems
3.2 Three-Dimensional Incompressible Flows
3.3 Two-Dimensional Incompressible Flows
4 Surface Tension Effects
4.1 Shape of the Interface between Two Fluids
4.2 Capillary Rises in Liquids
Part II Dynamics of Incompressible Fluid Flows
5 Fluid Kinematics and Dynamics
5.1 Stream Function
5.2 Equations of Motion
5.3 Integrals of Motion
5.4 CapillaryWaves on a Spherical Drop
5.5 Cavitation
5.6 Rates of Change of Material Integrals
5.7 The Kelvin Circulation Theorem
5.8 The Irrotational Flow
5.9 Simple-Flow Patterns
(i) The Source Flow
(ii) The Doublet Flow
(iii) The Vortex Flow
(iv) Doublet in a Uniform Stream
(v) Uniform Flow Past a Circular Cylinder with Circulation
6 The Complex-Variable Method
6.1 The Complex Potential
6.2 Conformal Mapping of Flows
6.3 Hydrodynamic Images
6.4 Principles of Free-Streamline Flow
(i) Schwarz-Christoffel Transformation
(ii) Hodograph Method
7 Three-Dimensional Irrotational Flows
7.1 Special Singular Solutions
(i) The Source Flow
(ii) The Doublet Flow
7.2 d’Alembert’s Paradox
7.3 Image of a Source in a Sphere
7.4 Flow Past an Arbitrary Body
7.5 Unsteady Flows
7.6 Renormalized (or Added) Mass of Bodies Moving through a Fluid
8 Vortex Flows
8.1 Vortex Tubes
8.2 Induced Velocity Field
8.3 Biot-Savart’s Law
8.4 von Kármán Vortex Street
8.5 Vortex Ring
8.6 Hill’s Spherical Vortex
8.7 Vortex Sheet
8.8 Vortex Breakdown: Brooke Benjamin’s Theory
9 Rotating Flows
9.1 Governing Equations and Elementary Results
9.2 Taylor-Proudman Theorem
9.3 Propagation of InertialWaves in a Rotating Fluid
9.4 Plane InertialWaves
9.5 ForcedWavemotion in a Rotating Fluid
(i) The Elliptic Case
(ii) The Hyperbolic Case
9.6 Slow Motion along the Axis of Rotation
9.7 RossbyWaves
10 Water Waves
10.1 Governing Equations
10.2 A Variational Principle for SurfaceWaves
10.3 WaterWaves in a Semi-Infinite Fluid
10.4 WaterWaves in a Fluid Layer of Finite Depth
10.5 Shallow-WaterWaves
(i) Analogy with Gas Dynamics
(ii) Breaking ofWaves
10.6 Water Waves Generated by an Initial Displacement over a Localized Region
10.7 Waves on a Steady Stream
(i) One-Dimensional GravityWaves
(ii) One-Dimensional Capillary-GravityWaves
(iii) ShipWaves
10.8 GravityWaves in a Rotating Fluid
10.9 Theory of Tides
10.10 Hydraulic Jump
(i) Tidal Bores
(ii) The Dam-Break Problem
10.11 Nonlinear Shallow-WaterWaves
(i) SolitaryWaves
(ii) Periodic CnoidalWaves
(iii) Interacting SolitaryWaves
(iv) StokesWaves
(v) Modulational Instability and Envelope Solutions
10.12 Nonlinear Capillary-GravityWaves
(i) Resonant Three-Wave Interactions
(ii) Second-Harmonic Resonance
11 Applications to Aerodynamics
11.1 Airfoil Theory: Method of Complex Variables
(i) Force and Moments on an Arbitrary Body
(ii) Flow Past an Arbitrary Cylinder
(iii) Flow Around a Flat Plate
(iv) Flow Past an Airfoil
(v) The Joukowski Transformation
11.2 Thin Airfoil Theory
(i) Thickness Problem
(ii) Camber Problem
(iii) Flat Plate at an Angle of Attack
(iv) Combined Aerodynamic Characteristics
(v) The Leading-Edge Problem of a Thin Airfoil
11.3 Slender-Body Theory
11.4 Prandtl’s Lifting-Line Theory for Wings
11.5 Oscillating Thin-Airfoil Problem: Theodorsen’s Theory
Part III Dynamics of Compressible Fluid Flows
12 Review of Thermodynamics
12.1 Thermodynamic System and Variables of State
12.2 The First Law of Thermodynamics and Reversible and Irreversible Processes
12.3 The Second Law of Thermodynamics
12.4 Entropy
12.5 Liquid and Gaseous Phases
13 Isentropic Fluid Flows
13.1 Applications of Thermodynamics to Fluid Flows
13.2 Linear SoundWave Propagation
13.3 The Energy Equation
13.4 Stream-Tube Area and Flow Velocity Relations
14 Potential Flows
14.1 Governing Equations
14.2 Streamline Coordinates
14.3 Conical Flows: Prandtl-Meyer Flow
14.4 Small Perturbation Theory
14.5 Characteristics
(i) Compatibility Conditions in Streamline Coordinates
(ii) A Singular-Perturbation Problem for Hyperbolic Systems
15 Nonlinear Theory of Plane Sound Waves
15.1 Riemann Invariants
15.2 SimpleWave Solutions
15.3 Nonlinear Propagation of a SoundWave
15.4 Nonlinear Resonant Three-Wave Interactions of Sound Waves
15.5 Burgers Equation
16 Shock Waves
16.1 The Normal ShockWave
16.2 The Oblique ShockWave
16.3 Blast Waves: Taylor’s Self-similarity and Sedov’s Exact Solution
17 The Hodograph Method
17.1 The Hodograph Transformation of Potential Flow Equations
17.2 The Chaplygin Equation
17.3 The Tangent-Gas Approximation
17.4 The Lost Solution
17.5 The Limit Line
18 Applications to Aerodynamics
18.1 Thin Airfoil Theory
(i) Thin Airfoil in Linearized Supersonic Flows
(ii) Far-Field Behavior of Supersonic Flow Past a Thin Airfoil
(iii) Thin Airfoil in Transonic Flows
18.2 Slender Bodies of Revolution
18.3 Oscillating Thin Airfoil in Subsonic Flows: Possio’s Theory
18.4 Oscillating Thin Airfoils in Supersonic Flows: Stewartson’s Theory
Part IV Dynamics of Viscous Fluid Flows
19 Exact Solutions to Equations of Viscous Fluid Flows
19.1 Channel Flows
19.2 Decay of a Line Vortex: The Lamb-Oseen Vortex
19.3 Line Vortex in a Uniform Stream
19.4 Diffusion of a Localized Vorticity Distribution
19.5 Burgers Vortex
19.6 Flow Due to a Suddenly Accelerated Plane
19.7 The Round Laminar Jet: Landau-Squire Solution
19.8 Ekman Layer at a Free Surface in a Rotating Fluid
19.9 Centrifugal Flow Due to a Rotating Disk: von Kármán Solution
19.10 Shock Structure: Becker’s Solution
19.11 Couette Flow of a Gas
20 Flows at Low Reynolds Numbers
20.1 Dimensional Analysis
20.2 Stokes’ Flow Past a Rigid Sphere: Stokes’ Formula
20.3 Stokes’ Flow Past a Spherical Drop
20.4 Stokes’ Flow Past a Rigid Circular Cylinder: Stokes’Paradox
20.5 Oseen’s Flow Past a Rigid Sphere
20.6 Oseen’s Approximation for Periodically Oscillating Wakes
21 Flows at High Reynolds Numbers
21.1 Prandtl’s Boundary-Layer Concept
21.2 The Method of Matched Asymptotic Expansions
21.3 Location and Nature of the Boundary Layers
21.4 Incompressible Flow Past a Flat Plate
(i) The Outer Expansion
(ii) The Inner Expansion
(iii) Flow Due to Displacement Thickness
21.5 Separation of Flow in a Boundary Layer: Landau’s Theor
21.6 Boundary Layers in Compressible Flows
(i) Crocco’s Integral
(ii) Flow Past a Flat Plate: Howarth-Dorodnitsyn Transformation
21.7 Flow in a Mixing Layer between Two Parallel Streams
(i) Geometrical Characteristics of the Mixing Flow
21.8 Narrow Jet: Bickley’s Solution
21.9 Wakes
21.10 Periodic Boundary Layer Flows
22 Jeffrey-Hamel Flow
22.1 The Exact Solution
(i) Only
e
1
Is Real and Positive
(ii)
e
1
,
e
2
, and
e
3
Are Real and Distinct
22.2 Flows at Low Reynolds Numbers
22.3 Flows at High Reynolds Numbers
References
Bibliography
Index
End User License Agreement
List of Illustrations
CHAPTER 01
Figure 1.1 Motion of a...
Figure 1.2 Tetrahedron-shaped fluid element.
Figure 1.3 Deformation near a...
CHAPTER 04
Figure 4.1 Free liquid meeting...
CHAPTER 05
Figure 5.1 Displacement of a...
Figure 5.2 Doubly - connected region.
Figure 5.3 Source flow.
Figure 5.4 Doublet flow.
Figure 5.5 Uniform flow past...
Figure 5.6 Uniform flows past...
CHAPTER 06
Figure 6.1 Flow in a corner...
Figure 6.2 Flow past a...
Figure 6.3 Body-fixed coordinates.
Figure 6.4 Rigid body rotating...
Figure 6.5 Source outside a...
Figure 6.6 Schwarz - Christoffel mapping...
Figure 6.7 Conformal mapping for...
Figure 6.8 Conformal mapping for...
Figure 6.9 Conformal mapping for...
Figure 6.10 Flow enjoining on...
Figure 6.11a Borda Orifice flow.
Figure 6.11b Flow past a...
CHAPTER 07
Figure 7.1 Flow past a body...
Figure 7.2 Comparison of the...
Figure 7.3 Source outside a...
Figure 7.4 Expansion of a...
Figure 7.5 Space-fixed and body...
CHAPTER 08
Figure 8.1 Vortex tube in...
Figure 8.2 A line vortex.
Figure 8.3 A line vortex...
Figure 8.4 von Kármán vortex...
Figure 8.5 A vortex ring.
Figure 8.6 Streamline pattern near...
Figure 8.7 Streamline pattern in...
Figure 8.8 Vortex sheet.
Figure 8.9 Sectional view of...
Figure 8.10 Variation of the...
Figure 8.11 Variation of the...
CHAPTER 09
Figure 9.1 “Taylor column“ experiment
Figure 9.2 Instantaneous streamline pattern...
Figure 9.3 Variation of the...
Figure 9.4 Streamline pattern for...
CHAPTER 10
Figure 10.1 Dispersion curve for...
Figure 10.2 Variation of the...
Figure 10.3 Capillary waves and...
Figure 10.4 Ship-wave pattern.
Figure 10.5 The earth-moon...
Figure 10.6 Tidal bore schemetic.
Figure 10.7 Characteristics for the...
Figure 10.8 The dam break...
Figure 10.9 Burgers shock front.
Figure 10.10 Surface waves on...
Figure 10.11 Numerical solution of...
Figure 10.12 The case with...
Figure 10.13 The case with...
Figure 10.14 One period of...
Figure 10.15 Modulational instability.
Figure 10.16 An envelope solition.
Figure 10.17 Experimental traces of...
Figure 10.18 Recurring modulation and....
CHAPTER 11
Figure 11.1 Force exerted by...
Figure 11.2 Flow with circulation...
Figure 11.3 Conformal mapping of...
Figure 11.4 Forces and moments...
Figure 11.5 Flow near the...
Figure 11.6 Flow aroung a...
Figure 11.7 Conformal mapping of...
Figure 11.8 Flow past a....
Figure 11.9 Conformal mapping of...
Figure 11.10 Conformal mapping of...
Figure 11.11 Conformal mapping of...
Figure 11.12 Airfoil section and...
Figure 11.13 Superposition of a...
Figure 11.14 Flow past a...
Figure 11.15 Uniform flow past...
Figure 11.16 Lifting-line configuration...
Figure 11.17 Flow at sectional...
Figure 11.18 The trailing vortex...
Figure 11.19 Flow situation in...
Figure 11.20 Uniform flow past...
Figure 11.21 Source and sink...
Figure 11.22 Flow generated by...
CHAPTER 12
Figure 12.1 The Carnot engine.
Figure 12.2 Isothermals for.
CHAPTER 13
Figure 13.1 Nozzle flow.
CHAPTER 14
Figure 14.1 The streamline coordinate...
Figure 14.2 The Mach angle.
Figure 14.3 Prandtl-Meyer flow.
Figure 14.4 Flow past a body.
Figure 14.5 Characteristics through a...
Figure 14.6 Characteristics and region...
Figure 14.7 Time-like arc and...
Figure 14.8 Domain of influence...
Figure 14.9 Domain of influence...
Figure 14.10 Radiation problem with...
CHAPTER 15
Figure 15.1 Nonlinear steepening of...
Figure 15.2 Loci of wave...
Figure 15.3 Loci of wave...
Figure 15.4 The second harmonic...
Figure 15.5 Shock structure in...
Figure 15.6 The dispersed shock.
CHAPTER 16
Figure 16.1 Jump discontinuity of...
Figure 16.2 The Hugoniot curve.
Figure 16.3 Piston generated shock...
Figure 16.4 Entropy variation across...
Figure 16.5 The oblique shock.
Figure 16.6 Variation of flow...
Figure 16.7 Variation with time...
Figure 16.8 Variation of flow...
CHAPTER 17
Figure 17.1 The tangent-gas...
Figure 17.2 Incompressible flow past...
CHAPTER 18
Figure 18.1 Supersonic flow past...
Figure 18.2 Influence of the...
Figure 18.3 Uniform flow past...
Figure 18.4 Domain of influence...
Figure 18.5 Uniform flow past...
Figure 18.6 An element of...
CHAPTER 19
Figure 19.1 Couette flow.
Figure 19.2 Poiseuille flow.
Figure 19.3a Circulation distribution for
Figure 19.3b Velocity distribution for...
Figure 19.4 Streamline pattern for...
Figure 19.5 Variation of the...
Figure 19.6 The shock structure.
CHAPTER 20
Figure 20.1 Stokes’ flow past...
Figure 20.2 Drag on a...
Figure 20.3 Variation of drag...
Figure 20.4 Oseen’s flow past...
Figure 20.5 Free shear layer...
CHAPTER 21
Figure 21.1 The inner, outer...
Figure 21.2 The subcharacteristics.
Figure 21.3 Production of a...
Figure 21.4 Theoretical Blasius profile
Figure 21.5 The mixing layer...
CHAPTER 22
Figure 22.1 The Jeffrey-Hamel...
Figure 22.2 Variation of the...
Figure 22.3 Variation of the...
Figure 22.4 Variation of f’....
Figure 22.6 Diverging and converging...
Figure 22.5 Variation of f....
Figure 22.7 Diverging and converging...
Guide
Cover
Title page
Copyright
Dedication
Table of Contents
Preface to the Third Edition
Acknowledgments
Begin Reading
References
Bibliography
Index
End User License Agreement
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Preface to the Third Edition
Though fluid dynamics is primarily concerned with the study of causes and effects of the motion of fluids, it provides a convenient general framework to model systems with complicated interactions among their constituents. This is accomplished via a formalism that uses only a few macroscopic quantities like pressure while ignoring the practically intractable details of these particle interactions. A case in point was the use of a liquid drop model of an atomic nucleus in describing the nuclear fission of heavy elements. In fact, George Uhlenbeck (one of the two famous original proponents of electron spin) pointed out that “every physicist should have some familiarity with the field (fluid dynamics).”
This book is concerned with a discussion of the dynamical behavior of a fluid and seeks to provide the readers with a sound and systematic account of the most important and representative types of fluid flow phenomena. Particular attention has been paid, as with the Second Edition, to emphasize the most generally useful fundamental ideas and formulations of fluid dynamics.
This book is addressed primarily to graduate students and researchers in applied mathematics and theoretical physics. Nonetheless, graduate students and researchers in engineering will find the ideas and formulations in this book to be useful - as confirmed by book reviews on the previous two editions.
The effort to keep the length of the introduction to mainstream fluid dynamics reasonable makes it difficult to provide a thorough treatment of every topic. For instance, engineering details like the skin-friction calculations in boundary layer theory are not dealt with in this book. In the same vein, non-newtonian fluids (for which the coefficient of viscosity is dependent on the rate of deformation of the fluid) have also been excised from the discussion.
As with the Second Edition, flows of an incompressible fluid have been given especially large coverage in the text because of their central place in the subject. In the references provided at the end, I have tried to mention papers connected with the original developments and formulations given in this book. This list is not to be construed as being exhaustive in covering these details. Furthermore, a few exercises have been provided at the end of each chapter. In addition to being essential to understand the ideas and formulations discussed in this book, the solutions to these exercises are intended to encourage further exploration of those topics.
The Third Edition, constitutes an extensive revision and rewrite of the text, and entails the following main features:
major reorganization of several topics to facilitate a smoother reading experience.
refinement and detailed reworking of mathematical developments
addition of several new topics:
Hamiltonian formulation of two-dimensional incompressible flows
Variational principle for surface waves in water
Tidal bores
The dam break problem
The shock tube
Burgers vortex;
addition of several new exercise problems.
Since the discussion of incompressible, compressible and viscous flows constitute the teaching material for three graduate courses, I dropped the discussion of hydrodynamic stability and turbulence from the Third Edition. Special topics like hydrodynamic stability and turbulence are typically not included in the mainstream fluid dynamics graduate course package.
For the readership, an elementary background in fluid dynamics is helpful, but familiarity with the theory and the analytical methods (perturbation methods, in particular) of solutions to differential equations is an essential prerequisite.
In closing, I wish to place on record with gratitude, the immense ingenuity of several applied mathematicians, theoretical physicists, and engineers who developed the theoretical formulations of fluid flows. This great activity is not complete and several major theoretical developments are still to come. However, the most important message from the theoretical accomplishments so far is that analytical formulations are still feasible. Therefore, one may not resort too quickly to exploring on the computer as it may not necessarily provide a proper organization of knowledge and a clear insight into the dynamical behavior of fluids.
Orlando, 2023
Bhimsen K. Shivamoggi
Acknowledgments
I would first like to thank Professor Xin Li, our Chairman and former Graduate Coordinator, for his support in sustaining our Theoretical and Mathematical Fluid Dynamics graduate course sequence. His encouragement and effort to provide resources were crucial in bringing this third edition to fruition. I wish to thank Dr. Nelson Ying, Sr. for a grant that partially supported this book writing project. I wish to express my gratitude to Professor Katepalli Sreenivasan who has been a constant source of encouragement in my academic endeavors. I wish to thank my students who took this course sequence and my colleagues who rendered valuable assistance in this effort. I am thankful to Dr. Leos Pohl for his help with proofreading the book manuscript. My thanks are due to Gayathri Krishnan for her immense help with editing the book manuscript. The responsibility for any remaining defects is however mine alone. As with the previous editions, in preparing the third edition material, I have drawn considerably from ideas and formulations developed by many eminent authors. I have acknowledged the various sources in the Bibliography and References given at the end of this book and wish to apologize to any sources I may have inadvertently missed. I wish to express my gratitude to several publishers for granting permission to reproduce in this book the following figures from the original sources,
8.9-8.11, 9.2-9.4, 10.14, 10.18, 16.7,19.5 Cambridge University Press
10.11, 16.8 Elsevier Publishing Company
20.2, 21.4 Oxford University Press
6.10, 11.21, 11.22 Pearson
10.17 The Royal Society of London
14.6-14.10, 21.3 Springer Publishing Company
11.7, 11.8, 11.10, 11.11, 11.16-11.19 Wiley Publishing Company
My sincere thanks are due to Linda Perez-Rodriguez for the tremendous patience and dedication with which she typed this book and incorporated my numerous changes and to Joe Fauvel for his excellent job in doing the figures. I am very thankful to John Wiley and Sons, and Kimberly Monroe-Hill, the Managing Editor, for their tremendous cooperation in this project. I have tried to do justice to the great intellectual strides made by eminent pioneers in fluids during the nineteenth and twentieth centuries. My immense thanks are due to my wife Jayashree for her understanding and cooperation during this endeavor.
Finally, the writing of this book (or for that matter, even my higher education and creative pursuits) would not have been possible without the inspiration of my beloved late mother, Pramila, who could not attend college but always craved learning and knowledge. This book is dedicated to her memory.
Part I Basic Concepts and Equations of Fluid Dynamics
1 Introduction to the Fluid Model
While dealing with a fluid, in reality, one deals with a system that has many particles which interact with one another. The main utility of fluid dynamics is the ability to develop a formalism which deals solely with a few macroscopic quantities like pressure, while ignoring the details of the particle interactions. Therefore, the techniques of fluid dynamics have often been found useful in modeling systems with complicated interactions (which are either not known or very difficult to describe) between the constituents. Thus, the first successful model of the nuclear fission of heavy elements was the liquid drop model of the nucleus, which treats the nucleus as a fluid. This replaces the many body problem of calculating the interactions of all the protons and neutrons with the much simpler problem of calculating the pressures and surface tension in this fluid.1 Of course, this treatment gives only a very rough approximation to reality, but it is nonetheless a very useful way of approaching the problem.
The primary purpose of fluid dynamics is to study the causes and effects of the motion of fluids. Fluid dynamics seeks to construct a mathematical theory of fluid motion based on the smallest number of dynamical principles, which are adequate to correlate the different types of fluid flow as far as their macroscopic features are concerned. In many circumstances, the incompressible, inviscid fluid model is sufficiently representative of real fluid properties to provide a satisfactory account of a great variety of fluid motions. It turns out that such a model makes accurate predictions for the airflow around streamlined bodies moving at low speeds. While dealing with streamlined bodies (which minimize flow-separation) in flows of fluids of small viscosities, one may divide the flow field into two parts. The first part, where the viscous effects are appreciable consists of a thin boundary layer adjacent to the body and a small wake behind the body. The second part is the rest of the flow field that behaves essentially like an inviscid fluid. Such a division greatly facilitates the mathematical analysis in that the inviscid flow field can first be determined independent of the boundary layer near the body. The pressure field obtained from the inviscid-flow calculation is then used to calculate the flow in the boundary layer.
An attractive feature of fluid dynamics is that it provides ample room for the subject to be expounded as a branch of applied mathematics and theoretical physics.
1.1 The Fluid State
A fluid is a material that offers resistance to attempts to produce relative motions of its different elements, but deforms continually upon the application of surface forces. A fluid does not have a preferred shape, and different elements of a homogeneous fluid may be rearranged freely without affecting the macroscopic properties of the fluid. Fluids, unlike solids, cannot support tension or negative pressure. Thus, the occurrence of negative pressures in a mathematical solution of a fluid flow is an indication that this solution does not correspond to a physically possible situation. However, a thin layer of fluid can support a large normal load while offering very little resistance to tangential motion – a property which finds practical use in lubricated bearings.
A fluid of course, is discrete on the microscopic level, and the fluid properties fluctuate violently, when viewed at this level. However, while considering problems in which the dimensions of interest are very large compared to molecular distances, one ignores the molecular structure and endows the fluid with a continuous distribution of matter. The fluid properties can then be taken to vary smoothly in space and time. The characteristics of a fluid, caused by molecular effects, such as viscosity, are incorporated into the equation of fluid flows as experimentally obtained empirical parameters.
Fluids can exist in either of two stable phases – liquids, and gases. In case of gases, under ordinary conditions, the molecules are so far apart that each molecule moves independently of its neighbors except when making an occasional “collision.” In liquids, on the other hand, a molecule is continually within the strong cohesive force fields of several neighbors at all times. Gases can be compressed much more readily than liquids. Consequently, for a gas, any flow involving appreciable variations in pressure will be accompanied by much larger changes in density. However, in cases where the fluid flows are accompanied by only slight variations in pressure, gases and liquids behave similarly.
In formulations of fluid flows, it is useful to think that a fluid particle is small enough on a macroscopic level that it may be taken to have uniform macroscopic properties. However, it is large enough to contain sufficient number of molecules to diminish the molecular fluctuations. This allows one to associate with it a macroscopic property which is a statistical average of the corresponding molecular property over a large number of molecules. This is the continuum hypothesis.
As an illustration of the limiting process by which the local continuum properties are defined, consider the mass density ρ. Imagine a small volume δV surrounding a point P, let δm be the total mass of material instantaneously in δV . The ratio , as δV reduces to , where is the volume of fluid particle, is taken to give the mass density ρ at P.
1.2 Description of the Flow-Field
The continuum model affords a field description, in that the average properties in the volume surrounding the point P are assigned in the limit to the point P itself. If q represents a typical continuum property, then one has a fictitious continuum characterized by an aggregate of such local values of q, i.e., . This enables one to consider what happens at every fixed point in space as a function of time – the so-called Eulerian description. In an alternative approach, called the Lagrangian description, the dynamical quantities, as in particle mechanics, refer more fundamentally to identifiable pieces of matter, and one looks for the dynamical history of a selected fluid element.
Imagine a fluid moving in a region Ω (see Figure 1.1). Each particle of fluid follows a certain trajectory. Thus, for each point in Ω, there exists a path line given by
Figure 1.1 Motion of a fluid particle.
where the flow mapping , depends continuously on the parameter t. and its inverse are both continuous, so this mapping is one-to-one and onto.2
The velocity of the flow is given by
We then have the following results.
Theorem 1.1 (Existence) Assume that the flow velocity is a C1 function of x and t. Then, for each pair , there exists a unique integral curve – the path line, defined on some small interval in t about t0, such that
Theorem 1.2 (Boundedness) Consider a region Ω with a smooth boundary . If the flow velocity x is parallel to , then the integral curves of i.e., the path lines starting in Ω remain in Ω.
Streamlines are obtained by holding t fixed, say , and solving the differential equation
Streamlines coincide with pathlines if the flow is steady, i.e., if
The transformation from the Eulerian to the Lagrangian description is given by
Since the Lagrangian description makes the formalism cumbersome, we shall instead use the Eulerian description. However, the Lagrangian concepts of material volumes, material surfaces, and material lines which consist of the same fluid particles and move with them are still useful in developing the Eulerian description.
1.3 Volume Forces and Surface Forces
One may think of two distinct kinds of forces acting on a fluid continuum. Long-range forces such as gravity penetrate into the interior of the fluid, and act on all elements of the fluid. If such a force varies smoothly in space, then it acts equally on all the matter within a fluid particle of density ρ and volume δV. The total force acting on the particle is proportional to its mass and is equal to . In this sense, long-range forces are called body forces.
The short-range forces (which have a molecular origin) between two fluid elements, on the other hand, are effective only if they interact through direct mechanical contact. Since the short-range forces on an element are determined by its surface area, one considers a plane surface element of area δA in the fluid. The local short-range force is then specified as the total force exerted on the fluid on one side of δA by the fluid on the other side and is equal to (Cauchy, 1827). The direction of this force is not known a priori for a viscous fluid (unlike in the case of an inviscid fluid). Here, is the unit normal to the surface element δA, and it points away from the fluid on which Σ acts. The total force exerted across δA on the fluid on the side into which points, by Newton’s third law of motion, is , so that Σ is an odd function of . In this sense, short-range forces are called surface forces.
In order to determine the dependence of Σ on , consider all the forces acting instantaneously on the fluid within an element of volume δV shaped like a tetrahedron (Figure 1.2). The three orthogonal faces have areas and unit outward normals , and the fourth inclined face has area δA and unit outward normal . Surface forces will act on the fluid in the tetrahedron, across each of the four faces, and their resultant is
Figure 1.2 Tetrahedron-shaped fluid element.
or
Now, since the body force and inertia force acting on the fluid within the tetrahedron are proportional to the volume δV, they become negligible compared to the surface forces if the linear dimensions of the tetrahedron are made to approach zero without changing its shape. Then, applying Newton’s second law of motion to this fluid element gives
where
which is a second-order tensor called the stress tensor and prescribes the state of stress at a point in the fluid. Thus, the resultant of stress (force per unit area) across an arbitrarily oriented plane surface element with a unit normal is related to the resultant of stress across any three orthogonal plane surface elements at the same position in the fluid as if it were a vector with orthogonal components . Note that , Σ, and τ do not depend at all on the choice of the reference axes.
This leads to the following theorem.
Theorem 1.3 (Cauchy) There exists a matrix function τ, such that
where τij is the i-component of the stress exerted across a plane surface element normal to the j-direction. Thus, the state of stress at a point is characterized by normal stresses and shear stresses acting on three mutually perpendicular planes passing through the point, i.e., by nine cartesian components.
2 Equations of Fluid Flows
The laws governing the motion of fluids are
the law of conservation of mass;
Newton’s laws of motion;
the First Law of thermodynamics.
These laws typically refer to a system, i.e., a collection of matter of fixed identity.
In order to derive the equations governing the motion of fluids embodying these laws, one may consider a material volume W consisting of the same fluid particles, and hence, moving with the fluid (the Lagrangian description). The space coordinates defining W will then be functions of time since they depend on the changing locations of the fluid particles as time progresses. A typical macroscopic property is then represented by a material integral (one that always refers to the same fluid particles),
where a property q has been associated with a fluid particle of fixed identity and mass .
Alternatively, one may consider a fixed region of space called the control volume through which the fluid flows so that different fluid particles occupy this region at different times (the Eulerian description). A typical macroscopic property Q is represented here by an integral over the control volume fixed in space,
where is a scalar field (for example, density or a velocity component) associated with the fluid.
Thus, the rate of change of , namely, associated with the material system instantaneously occupying the control volume receives contributions from the changes of q with time in W and the flux of q out of W.
2.1 The Transport Theorem
Lemma Let be the Jacobian determinant of the flow , i.e.,
Let be invertible as a function of , for , some interval in .
Then,
Proof: Observing that the determinant of a matrix is multilinear in the columns (or rows), we have
Note that,
