SECOND EDITIONAnalyticalFluidDynamics 9114-FM-Frame Page 3 Thursday, November 2, 2000 11:46 PMSECOND EDITIONBoca Raton London New York Washington, D.C.CRC PressAnalyticalFluidDynamicsGeorge EmanuelProfessorDepartment of Mechanical and Aerospace EngineeringUniversity of Texas, ArlingtonArlington, Texas This book contains information obtained from authentic and highly regarded sources. Reprinted material isquoted with permission, and sources are indicated. A wide variety of references are listed. Reasonable effortshave been made to publish reliable data and information, but the author and the publisher cannot assumeresponsibility for the validity of all materials or for the consequences of their use.Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic ormechanical, including photocopying, microlming, and recording, or by any information storage or retrievalsystem, without prior permission in writing from the publisher.The consent of CRC Press LLC does not extend to copying for general distribution, for promotion, for creatingnew works, or for resale. Specic permission must be obtained in writing from CRC Press LLC for suchcopying.Direct all inquiries to CRC Press LLC, 2000 N.W. Corporate Blvd., Boca Raton, Florida 33431. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are usedonly for identication and explanation, without intent to infringe. 2000 by CRC Press LLCNo claim to original U.S. Government worksInternational Standard Book Number 0-8493-9114-9Library of Congress Card Number 99-089453Printed in the United States of America 1 2 3 4 5 6 7 8 9 0Printed on acid-free paper Library of Congress Cataloging-in-Publication Data Emanuel, GeorgeAnalytical uid dynamics / George Emanuel.-2nd ed.p. cm.Includes bibliographical references and index.ISBN 0-8493-9114-9 (alk paper)1. Fluid dynamics. I. Title.[DNLM: 1. Hepatitis B virus. QW 710 G289h]QA911 .E43 2000.05-dc21 99-089453 CIP532 9114-FM-Frame Page 4 Thursday, November 2, 2000 11:46 PM Dedication Dedicated with love to my wife and companion, Lita 9114-FM-Frame Page 5 Thursday, November 2, 2000 11:46 PM 9114-FM-Frame Page 6 Thursday, November 2, 2000 11:46 PM Preface The objectives of this edition remain the same as the rst. The analysis and formulation are providedfor a variety of selected topics in inviscid and viscous uid dynamics, it is hoped with physicalinsight. In part, this means formulating the appropriate equations and then transforming them intoa suitable form for the specic ow under scrutiny. The approach is applied to viscous boundarylayers, shock waves, PrandtlMeyer ow, etc. Sometimes a solution is obtained; other times a nalanswer requires numerical computation. Of crucial interest, however, is the analytical process itselfand the coinciding physical interpretation.A more in-depth coverage of topics is favored compared to a broad one that bypasses crucial ordifcult details. At the graduate level, I believe an intensive approach is preferable. The book triesto avoid too much repetition of undergraduate course material. Of course, some repetition is bothuseful and unavoidable. When it occurs, however, the level and manner of treatment are different,often markedly so, from those at the undergraduate level. I have attempted whenever possible topoint out the assumptions and limitations of the topic under discussion. Conversely, an attempt ismade to discuss why a particular topic is worthy of study. For instance, a solution may be useful asa rst (or initial value) estimate for CFD calculations. The rate of convergence is usually acceleratedby having a reasonable initial ow eld. Analytical solutions, such as those provided by the substi-tution principle, can be used to verify Euler or NavierStokes codes. An analytical approach oftenyields suitable rst estimates for parameters of interest. In this regard, some of the homeworkproblems are designed to give the student practice in obtaining back-of-the-envelope solutions. Mypersonal motivation, however, still remains the beauty and elegance of analytical uid dynamics(AFD).As mentioned in the preface to the rst edition, much of the material in that edition was unique.This is even truer for this edition, where all of the added material is unique to this text. The chapterscovering a calorically imperfect gas ow, sweep, shock wave interference with an expansion,unsteady one-dimensional ow, and the force and moment analysis are new. In addition, thethermodynamic chapter is largely new as are Appendices B and C. The chapters that remain fromthe rst edition have been revised to improve the clarity of the presentation.When appropriate, topics where future research is warranted are pointed out. Fluid dynamics,including the AFD specialty, is very much alive and growing. Consequently, not everything in thistext is complete or polished. A variety of major topics are not discussed. These topics includeturbulent ow, CFD, experimental methods, etc., that are major subjects in themselves.I owe a debt of gratitude to the many friends and colleagues who have contributed to thisundertaking, especially past and present students. It is indeed a pleasure to acknowledge theircomments and assistance. I particularly thank Dr. Jose Rodriguez, Professor Frank K. Lu, andProfessor Milton Van Dyke for his comments on Chapter 23. I am especially in debt to SusanHouck for her superb typing and preparation of the manuscript. 9114-FM-Frame Page 7 Thursday, November 2, 2000 11:46 PM 9114-FM-Frame Page 8 Thursday, November 2, 2000 11:46 PM Contents Part I: Basic Concepts Outline of Part I .................................................................................................................................3 Chapter 1 Background Discussion 1.1 Preliminary Remarks.................................................................................................................51.2 Euler and Lagrange Formulations ............................................................................................51.3 The Stress Tensor....................................................................................................................161.4 Relation between Stress and Deformation-Rate Tensors .......................................................191.5 Constitutive Relations .............................................................................................................221.6 Integral Relations ....................................................................................................................25References ........................................................................................................................................29Problems...........................................................................................................................................29 Chapter 2 The Conservation Equations 2.1 Preliminary Remarks...............................................................................................................332.2 Mass Equation.........................................................................................................................332.3 Transport Theorem..................................................................................................................342.4 Linear Momentum Equation...................................................................................................352.5 Inertial Frame..........................................................................................................................362.6 Angular Momentum Equation ................................................................................................392.7 Energy Equation......................................................................................................................412.8 Viscous Dissipation.................................................................................................................442.9 Alternate Forms for the Energy Equation ..............................................................................46Reference..........................................................................................................................................48Problems...........................................................................................................................................49 Chapter 3 Classical Thermodynamics 3.1 Preliminary Remarks...............................................................................................................533.2 Combined First and Second Laws..........................................................................................533.3 Potential Functions..................................................................................................................563.4 Open System...........................................................................................................................583.5 Coupling to Fluid Dynamics ..................................................................................................633.6 Compressible Liquid or Solid.................................................................................................733.7 Second Law.............................................................................................................................75References ........................................................................................................................................83Problems...........................................................................................................................................84 Chapter 4 Kinematics 4.1 Preliminary Remarks...............................................................................................................894.2 Denitions ...............................................................................................................................89 9114-FM-Frame Page 9 Thursday, November 2, 2000 11:46 PM 4.3 Kelvins Equation and Vorticity..............................................................................................944.4 Helmholtz Vortex Theorems ...................................................................................................97Reference........................................................................................................................................100Problems.........................................................................................................................................100 Part II: Advanced Gas Dynamics Outline of Part II............................................................................................................................105 Chapter 5 Euler Equations 5.1 Preliminary Remarks.............................................................................................................1075.2 Equations Initial and Boundary Conditions ....................................................................1075.3 Bernoullis Equations............................................................................................................1105.4 Vorticity.................................................................................................................................1145.5 Steady Flow...........................................................................................................................1185.6 Two-Dimensional or Axisymmetric Flow............................................................................1205.7 Natural Coordinates ..............................................................................................................129References ......................................................................................................................................134Problems.........................................................................................................................................135 Chapter 6 Shock Wave Dynamics 6.1 Preliminary Remarks.............................................................................................................1436.2 Jump Conditions ...................................................................................................................1446.3 Steady, Two-Dimensional or Axisymmetric Flow ...............................................................1556.4 Coordinate Transformation ...................................................................................................1656.5 Tangential Derivatives...........................................................................................................1756.6 Normal Derivatives ...............................................................................................................180References ......................................................................................................................................184Problems.........................................................................................................................................184 Chapter 7 The Hodograph Transformation and Limit Lines 7.1 Preliminary Remarks.............................................................................................................1917.2 Two-Dimensional, Irrotational Flow.....................................................................................1927.3 Ringlebs Solution.................................................................................................................2027.4 Limit Lines............................................................................................................................2137.5 General Solution ...................................................................................................................2147.6 Rotational Flow.....................................................................................................................223References ......................................................................................................................................228Problems.........................................................................................................................................229 Chapter 8 The Substitution Principle 8.1 Preliminary Remarks.............................................................................................................2338.2 Transformation Equations.....................................................................................................2338.3 Parallel Flow .........................................................................................................................2418.4 PrandtlMeyer Flow..............................................................................................................2438.5 Rotational Solutions in the Hodograph Plane......................................................................249References ......................................................................................................................................252Problems.........................................................................................................................................252 9114-FM-Frame Page 10 Thursday, November 2, 2000 11:46 PM Chapter 9 Calorically Imperfect Flows 9.1 Preliminary Remarks.............................................................................................................2579.2 Thermodynamics...................................................................................................................2589.3 Isentropic Streamtube Flow..................................................................................................2619.4 Planar Shock Flow................................................................................................................2749.5 PrandtlMeyer Flow..............................................................................................................2829.6 TaylorMaccoll Flow............................................................................................................287References ......................................................................................................................................295Problems.........................................................................................................................................295 Chapter 10 Sweep 10.1 Preliminary Remarks.............................................................................................................29710.2 Oblique Shock Flow .............................................................................................................29710.3 PrandtlMeyer Flow..............................................................................................................307References ......................................................................................................................................318Problems.........................................................................................................................................319 Chapter 11 Interaction of an Expansion Wave with a Shock Wave 11.1 Preliminary Remarks.............................................................................................................32211.2 Flow Topology ......................................................................................................................32311.3 Solution for Regions I, II, and III ........................................................................................32611.4 Curvature Singularity............................................................................................................32811.5 Numerical Procedure.............................................................................................................331References ......................................................................................................................................335Problems.........................................................................................................................................336 Chapter 12 Unsteady, One-Dimensional Flow 12.1 Preliminary Remarks.............................................................................................................33712.2 Incident Normal Shock Waves..............................................................................................33712.3 Reected Normal Shock Waves............................................................................................34312.4 Characteristic Theory............................................................................................................34812.5 Rarefaction Waves.................................................................................................................35412.6 Compression Waves ..............................................................................................................37412.7 Internal Ballistics ..................................................................................................................37812.8 Nonsimple Wave Region.......................................................................................................386References ......................................................................................................................................411Problems.........................................................................................................................................411 Part III: Viscous/Inviscid Fluid Dynamics Outline of Part III ..........................................................................................................................419 Chapter 13 Coordinate Systems and Related Topics 13.1 Preliminary Remarks.............................................................................................................42113.2 Orthogonal Coordinates ........................................................................................................42113.3 Similarity Parameters............................................................................................................427 9114-FM-Frame Page 11 Thursday, November 2, 2000 11:46 PM 13.4 Bulk Viscosity .......................................................................................................................43013.5 Viscous Flow in a Heated Duct ............................................................................................432References ......................................................................................................................................440Problems.........................................................................................................................................441 Chapter 14 Force and Moment Analysis 14.1 Preliminary Remarks ...........................................................................................................44714.2 Momentum Theorem...........................................................................................................44714.3 Surface Integral....................................................................................................................45014.4 Angular Momentum ............................................................................................................45514.5 Hydrostatics .........................................................................................................................45614.6 Flow in a Duct.....................................................................................................................45814.7 Acyclic Motion....................................................................................................................46014.8 Jet-Plate Interaction.............................................................................................................46114.9 Syringe with a Hypodermic Needle....................................................................................46314.10 Shock-Expansion Theory ....................................................................................................46614.11 Forces on a Particle .............................................................................................................47314.12 Entropy Generation .............................................................................................................47814.13 Forces and Moments on a Supersonic Vehicle...................................................................485References ......................................................................................................................................491Problems.........................................................................................................................................492 Part IV: Exact Solutions for a Viscous Flow Outline of Part IV..........................................................................................................................499 Chapter 15 Rayleigh Flow 15.1 Preliminary Remarks.............................................................................................................50115.2 Solution .................................................................................................................................503References ......................................................................................................................................508Problems.........................................................................................................................................508 Chapter 16 Couette Flow 16.1 Preliminary Remarks.............................................................................................................51116.2 Solution .................................................................................................................................51216.3 Adiabatic Wall.......................................................................................................................517Reference........................................................................................................................................518Problems.........................................................................................................................................518 Chapter 17 Stagnation Point Flow 17.1 Preliminary Remarks.............................................................................................................52117.2 Formulation ...........................................................................................................................52217.3 Velocity Solution...................................................................................................................52617.4 Temperature Solution............................................................................................................530Reference........................................................................................................................................533Problems.........................................................................................................................................533 9114-FM-Frame Page 12 Thursday, November 2, 2000 11:46 PM Part V: Laminar Boundary-Layer Theory forSteady Two-Dimensional or Axisymmetric Flow Outline of Part V............................................................................................................................537References ......................................................................................................................................538 Chapter 18 Incompressible Flow over a Flat Plate 18.1 Preliminary Remarks.............................................................................................................53918.2 Derivation of the Boundary-Layer Equations ......................................................................53918.3 Similarity Solution................................................................................................................542References ......................................................................................................................................547Problems.........................................................................................................................................549 Chapter 19 Large Reynolds Number Flow 19.1 Preliminary Remarks.............................................................................................................54919.2 Matched Asymptotic Expansions .........................................................................................56019.3 Governing Equations in Body-Oriented Coordinates ..........................................................562References ......................................................................................................................................563Problems.........................................................................................................................................563 Chapter 20 Incompressible Boundary-Layer Theory 20.1 Preliminary Remarks.............................................................................................................56520.2 Primitive Variable Formulation.............................................................................................56620.3 Solution of the Boundary-Layer Equations..........................................................................568References ......................................................................................................................................574Problems.........................................................................................................................................575 Chapter 21 Compressible Boundary-Layer Theory 21.1 Preliminary Remarks ...........................................................................................................57721.2 Boundary-Layer Equations..................................................................................................57821.3 Solution of the Similarity Equations...................................................................................58421.4 Solution of the Energy Equation.........................................................................................58821.5 The and g w Parameters.....................................................................................................58921.6 Local Similarity...................................................................................................................59221.7 Boundary-Layer Parameters................................................................................................59521.8 Comprehensive Tables.........................................................................................................60321.9 Adiabatic Wall .....................................................................................................................61521.10 Critique of the Prandtl Number and ChapmanRubesin Parameter Assumptions ............61721.11 Nonsimilar Boundary Layers I ......................................................................................62521.12 Nonsimilar Boundary Layers II .....................................................................................627References ......................................................................................................................................638Problems.........................................................................................................................................640 9114-FM-Frame Page 13 Thursday, November 2, 2000 11:46 PM Chapter 22 Supersonic Boundary-Layer Examples 22.1 Preliminary Remarks.............................................................................................................64522.2 Thin Airfoil Theory...............................................................................................................64522.3 Compressive Ramp ...............................................................................................................65122.4 Zero Displacement Thickness Wall Shape ...........................................................................65622.5 Performance of a Scramjet Propulsion Nozzle ....................................................................660References ......................................................................................................................................664Problems.........................................................................................................................................664 Chapter 23 Second-Order Boundary-Layer Theory 23.1 Preliminary Remarks.............................................................................................................66923.2 Inner Equations .....................................................................................................................67423.3 Outer Equations.....................................................................................................................68023.4 Boundary and Matching Conditions.....................................................................................68623.5 Decomposition of the Second-Order Boundary-Layer Equations .......................................69123.6 Example: First-Order Solution..............................................................................................69823.7 Example: Second-Order Outer Solution...............................................................................70323.8 Example: Second-Order Inner Equations.............................................................................707References ......................................................................................................................................720Problems.........................................................................................................................................721 Appendices A Summary of Equations from Vector and Tensor Analysis...................................................725B Jacobian Theory ....................................................................................................................739C Oblique Shock Wave Angle..................................................................................................751D Conditions on the Downstream Side of a Steady Shock Wave in a Two-Dimensional or Axisymmetric Flow of a Perfect Gas.................................................755E Method-of-Characteristics for a Single, First-Order PDE...................................................759F Tangential Derivatives on the Downstream Side of a Shock in the 1 and 2 Directions....................................................................................................765G Conservation and Vector Equations in Orthogonal Curvilinear Coordinates i ..................767H Conservation Equations in Body-Oriented Coordinates ......................................................769I Summary of Compressible, Similar Boundary-Layer Equations.........................................773J Second-Order Boundary-Layer Equations for Supersonic, Rotational Flow over a Flat Plate.............................................................................................................779 Index ..............................................................................................................................................783 9114-FM-Frame Page 14 Thursday, November 2, 2000 11:46 PM Part I Basic Concepts 9114-ch01-Frame Page 1 Thursday, November 2, 2000 10:58 PM 9114-ch01-Frame Page 2 Thursday, November 2, 2000 10:58 PM 3 Outline of Part I We embark on an in-depth study of uid dynamics by discussing a variety of topics in a moregeneral manner than encountered at the undergraduate level. Some of the topics are familiar toyou, e.g., the Euler and NavierStokes equations, and the rst and second laws of thermodynamics.One purpose of this text is to prepare you for intensive courses in computational uid dynamics,turbulence, high-speed ow, rareed gas dynamics, and so on. A second objective is to help youunderstand the uid dynamic journal literature. Last, but not least, I hope to convey some of theintellectual fascination that abounds in our subject.In this text, we often are not concerned with solutions to specic ow problems, although suchsolutions are often used to illustrate the theory. Specic ows also regularly appear in the homeworkproblems and represent an essential element of this text. Nevertheless, we are primarily concernedwith general features of inviscid and viscous uid ows.This is especially true for Part I, which provides many of the basic concepts. The rst chapteris primarily concerned with establishing the Eulerian formulation, the constitutive relations, andseveral integral relations that are needed in later chapters. The conservation equations for mass,momentum, and energy are derived in the second chapter, while a general formulation for thermo-dynamics is provided in Chapter 3. The nal chapter of Part I discusses general properties of auid ow that are not based on the conservation equations or the second law of thermodynamics.Such properties are referred to as kinematic and they include Kelvins equations and the Helmholtzvortex theorems.While some of the topics in Part I date from the very origin of uid mechanics, much of thecontents have a more recent origin. Indeed, since uid dynamics is still evolving, some of thematerial is the result of recent research. Even topics of some antiquity, such as the second law, willappear new to you.One reason well-known topics may appear different is our systematic use of vector and tensoranalysis. Some background in these topics is presumed. A summary of the pertinent vector andtensor equations is provided in Appendix A. The trend toward an ever greater reliance on these andother analytical tools has been evident for some time. This trend stems from the need for a moreexible mathematical language for the increasing complexity of uid dynamics. Once you becomefamiliar with these topics, their utilization for our subject becomes indispensable.Many scientists, mathematicians, and engineers have contributed to uid dynamics over itslong history. The amount of material that could be covered far exceeds my grasp of it or what canbe covered in this text. (The following remarks are not limited to Part I, but hold for the overalltext.) Self-imposed limitations are therefore essential. The rst of these is that the uid, gas orliquid, is easily deformable. We, therefore, deal with that branch of continuum mechanics that doesnot include solids. As a rule we shall assume the uid is(i) isotropic in its properties; uids with polymers, rheological uids, etc. are excluded;(ii) not ionized, chemically reacting, diffusionally mixing, or a multi-phase uid; and(iii) not close to its critical point.[In Chapter 3, when discussing thermodynamics, we are more general and do not always assumeitems (ii) and (iii).]Another major restriction is that the uid behaves as a continuous medium. This implies thatthe mean-free path of the molecules in a gas, or the mean distance between molecules in a liquid, 9114-ch01-Frame Page 3 Thursday, November 2, 2000 10:58 PM 4 Analytical Fluid Dynamics is many orders of magnitude smaller than the smallest characteristic length of physical interest.Under a wide variety of conditions of practical importance, this assumption is fully warranted.Our nal assumptions are that relativistic effects and quantum mechanics can be safely ignored.This would not be the case, for instance, with liquid helium, which is a quantum uid, or in jetsemanating from astrophysical bodies.All of the above assumptions, at one time or another, would require reconsideration. Forinstance, when a meteor is entering the atmosphere the surrounding air is chemically reacting andionized during part of its downward trajectory. Similarly, an orbiting satellite, at a relatively lowaltitude, experiences the drag of a free-molecular ow. Nevertheless, the vast majority of applica-tions that uid dynamicists deal with still adhere to the foregoing assumptions.The above exclusions are usually treated in more advanced courses, like those dealing with thedynamics of real gases or raried ows. This is certainly true for turbulence; hence, we will notbe concerned with turbulent ows. Our discussion, however, will not be restricted to incompressibleuid dynamics, since compressible ows, including those with shock waves, are of fundamentalimportance. We shall also often focus on vorticity for both incompressible and compressible ows. 9114-ch01-Frame Page 4 Thursday, November 2, 2000 10:58 PM 5 1 Background Discussion 1.1 PRELIMINARY REMARKS As always in engineering, we need to reduce the subject to quantiable terms. This means thatsolvable equations need to be established. The relevant equations can be subdivided into threecategories. In the rst, we have the mechanical equations that express conservation of massand the momentum equation, which is based on Newtons second law. In the second category wehave the rst and second laws of thermodynamics. The rst law expresses conservation of energy,while the second law is a constraint on any physically realizable process.The foregoing laws are of great power and generality. (Nevertheless, they do not always hold,e.g., when nuclear reactions occur as in ssion or fusion. In this circumstance, conservation ofmass holds in a modied form.) The nal group of relations is not nearly as general. They arereferred to as constitutive equations. For example, Fouriers heat conduction equation and the perfectgas thermal state equation are in this category. The relation between stress and the rate of defor-mation is similarly a constitutive relation. These relations are not universal but provide the propertiesfor a specic class of substances and hold for a specic class of physical processes. At any rate,they are essential; without them the more general laws do not constitute a closed mathematicalsystem. Closure of the system thus requires a proper number of consistent constitutive relations.Taken as a whole, the complete set of equations is referred to as the governing equations. By wayof contrast, the three equations dealing with mass, momentum, and energy are referred to asconservation equations.This chapter is devoted to a discussion of the Euler and Lagrange formulations in uid dynamics.We then consider the stress tensor and the relation between this tensor and the rate-of-deformationtensor. We conclude by discussing a Newtonian uid, Fouriers equation, the constitutive relations,and certain useful integral relations. 1.2 EULER AND LAGRANGE FORMULATIONSE ULERIAN F ORMULATION There are two ways to formulate the equations of uid dynamics: the Eulerian and Lagrangianapproaches. In the Eulerian formulation, which we discuss rst, the position vector and time t are the independent variables. Thus, any scalar, such as the pressure, can be written as( 1.1 )while a vector, e.g., the uid velocity , becomes( 1.2 )The Eulerian approach provides a eld representation, in terms and t , for any variable of interest.For example, a differential change in the pressure is provided by( 1.3 )rp p r t , ( ) =ww w r t , ( ) =rdp p r------- dr pt------ dt + = 9114-ch01-Frame Page 5 Thursday, November 2, 2000 10:58 PM 6 Analytical Fluid Dynamics where the rst term on the right side is the directional derivative of p in the direction . Supposewe introduce Cartesian coordinates x i and their corresponding orthonormal basis . Then and are given by( 1.4 )( 1.5 )where the repeated index summation convention is used. We also adhere to the convenient conven-tion of not writing Cartesian coordinates as x i , which would be the proper contravariant tensornotation. With Equation (1.5), we can write dp asor as( 1.6 )The velocity is given by( 1.7 )where the w i are the Cartesian velocity components, while the gradient of the pressure is providedby the del operator( 1.8 )Hence, Equation (1.6) reduces to( 1.9 )We shall utilize a notation, rst introduced by George Stokes, to dene the operator( 1.10 )which is called the substantial or material derivative. This denition is independent of any speciccoordinate system. With tensor analysis, the del operator can be dened for any general curvilinearcoordinate system; it is not restricted to Cartesian coordinates as in Equation (1.8). The substantialderivative also can be applied to vector quantities. For instance, when applied to the position vector,we havedr|i r drr xi |i=dr dxi |i=dp pxi------- dxipt------ dt + =dpdt------ pt ------ dxidt------- p xi------- + =w d rdt------- dxidt------- |wi |i = = =p p xi------- |i=dpdt------ p t ------ w p + =DDt------ t ---- w + =D rDt-------- r t ------- w r + = 9114-ch01-Frame Page 6 Thursday, November 2, 2000 10:58 PM Background Discussion 7 where( 1.11 )since and t are independent variables. The gradient of is( 1.12 )where ij is the Kronecker delta. Thus, is a dyadic; in fact, it is the unit dyadic . We therebyobtain (see Appendix A)and D / Dt becomes( 1.13 )As a second example, let us determine the acceleration , which is given by( 1.14 )The dot product on the right side can be interpreted as ( ), which involves the dyadic , oras ( ) , which does not involve a dyadic. With tensor analysis one can show that bothinterpretations yield the same result; the second one is usually preferred because of its greatersimplicity. In Cartesian coordinates, e.g., we have( 1.15 )An alternate expression for , of considerable utility, is based on the vector identity (see Appendix A,Table 5)( 1.16 )where and are arbitrary vectors. We set

, to obtainorr t ------- 0 =r rr |jr xj------- |j|ixixj------- |j|iij|i|i I= = = = =r Iw r w I w = =rD rDt-------- w =aa DwDt-------- wt------- w w + = =w w ww ww ( )w wi|i |kxk--------( ,j \wj |j wixi-------( ,j \wj|j wiwjxi--------- |j= = =a A B ( ) A B B A A B ( ) B A ( ) + + + =A B B A A A ( ) 2A A 2A A ( ) + =A A 12--- A2( ,j \A A ( ) =9114-ch01-Frame Page 7 Thursday, November 2, 2000 10:58 PM8 Analytical Fluid DynamicswhereWe now utilizeand(1.17)where is the vorticity, to obtainThe acceleration is therefore given by(1.18)in any coordinate system.The substantial derivative has an important physical interpretation. It provides the time rate ofchange of any uid quantity, scalar or vector, following a uid particle. This viewpoint is apparentin Equation (1.13), where the time rate of change of the position of a uid particle equals itsvelocity. Thus, the pressure of a given uid particle changes in accordance with Equation (1.9).The substantial derivative consists of two terms. The rst of these, ( )/t, provides the changes ata xed position due to any unsteadiness in the ow. For a steady ow, this term is zero. The secondterm, , is referred to as the convective derivative. It represents the changes that occur withposition at a xed time. This term is generally nonzero in a steady or unsteady ow.LAGRANGIAN FORMULATIONAs mentioned, the Eulerian formulation provides a eld description of a ow. The Lagrangeformulation provides a particle description. Suppose a uid particle has the location at t to.In the Lagrangian approach the independent variables are and t. Thus, the position of the uid particleat time t is given by(1.19)where is the particles position at time toA2A A =A w = w =w w 12---w2( ,j \ w + =a wt------- 12---w2( ,j \ w + + =wr roror r ro, t ( ) =roro r ro, to( ) =9114-ch01-Frame Page 8 Thursday, November 2, 2000 10:58 PMBackground Discussion 9and is a xed label on the particle as it moves. In this formulation, the velocity and acceleration arewhere is kept xed in both derivatives.The two formulations can be related by assuming we know ( ,t) in the Eulerian description.We then integrate Equation (1.13) subject to the initial condition(1.20)The solution is then the Lagrangian description, Equation (1.19).The Lagrangian approach is widely used in mechanics; e.g., consider a marble rolling downan inclined plane under the inuence of gravity. The problem is solved by rst establishing adifferential equation for the motion of the marble. The solution of this equation provides the positionof the marble as a function of time and its initial position.There are several reasons for not utilizing the Lagrangian description. First, we generally arenot interested in the actual location of a uid particle, whereas, as engineers, we are interested inthe pressure and velocity, since these provide the pressure and shear stress forces on a body. Second,obtaining ( ,t) represents a greater effort than is required for obtaining p and . Finally, theLagrangian approach is cumbersome for a viscous ow. We, therefore, follow a well-establishedtradition and hereafter focus on the Eulerian description.Before leaving this topic, recall that the substantial derivative follows a uid particle. Whilethe concept is Lagrangian, the derivative itself is Eulerian, since and t, not and t, are theindependent variables.PATHLINES AND STREAMLINESThe trajectory of a uid particle is called a pathline or particle path. This is found by integratingEquation (1.13) subject to the initial condition, Equation (1.20). We shall not discuss a differenttype of curve called a streakline. (This is a particle path that originates at a xed position.) Moreimportant than either pathlines or streaklines are the streamlines. Streamlines are curves, which ata given instant are tangent to the velocity eld. In an unsteady ow, pathlines, streaklines, andstreamlines are all different. In a steady ow, however, they all coincide.Let d be tangent to the velocity and therefore tangent to a streamline. Then d satises(1.21a)or with Cartesian coordinatesOn expanding this relation, we obtainrow rt-------, a 2rt2--------- = =row rr roat t to= =r ro wr ror rd r w 0 =|1|2|3dx1 dx2 dx3w1 w2 w30 =w3dx2 w2dx3 ( )|1 w3dx1 w1dx3 ( )|2 w2dx1 w1dx2 ( )|3+ 0 =9114-ch01-Frame Page 9 Thursday, November 2, 2000 10:58 PM10 Analytical Fluid Dynamicsor, in scalar form,(1.21b)The solution of these two ordinary differential equations provides the streamline curves, subject toa given initial condition. Recall that the streamlines are tangent to the velocity eld at a giveninstant of time. Thus, if the wi are time-dependent, the t variable is treated as a xed parameterduring the integration of these equations.Illustrative ExampleAs an example, we rst determine the streamline equation for steady, two-dimensional cross owabout a circular cylinder of radius a, as sketched in Figure 1.1(a). (Later, the unsteady ow path-lines are found.) In addition, we assume a uniform freestream, with speed U and an incompressible,inviscid ow without circulation. Hence, the cylinder is not subjected to either a lift or drag force.From elementary aerodynamic theory, we obtain the x and y velocity components as(1.22)where X (x/a) and Y (y/a). Since the ow is two dimensional, we need to integrate only oneof the equations in (1.21b), written asFIGURE 1.1 Coordinate systems associated with ow about a circular cylinder; (a) and (b) are for steadyow; (c) and (d) are for unsteady ow.dx1w1-------- dx2w2-------- dx3w3-------- = =uU---- 1 Y2X2X2Y2+ ( )2-------------------------, vU---- + 2XYX2Y2+ ( )2------------------------- = =dxu------ dyv------ =9114-ch01-Frame Page 10 Thursday, November 2, 2000 10:58 PMBackground Discussion 11to obtain the equation for the streamlines. The equations in (1.22) are substituted into this differentialequation, with the resultTo separate variables, cylindrical coordinates, shown in Figure 1.1(b), are introduced asto obtainThe method of partial fractions is now used for the left side, with the resultwhere a point Yo on the Y axis is used for the lower limit and, at this point, /2. As a result ofthe integration, we obtainBy returning to X,Y coordinates, the streamline equation simplies to(1.23a)where is the streamline ordinate at X . Figure 1.2(a) shows a typical streamline pattern.The two special Y values are related by(1.23b)where Xo 0 and Yo 1 for any streamline outside the cylinder. (There is a related streamlinepattern inside the cylinder.)The solution, Equation (1.23a), can also be obtained, with negligible effort, from the streamfunction (dened in Chapter 5) equationwhere , and from the fact that a stream function is constant along streamlinesin a steady ow. Only in special cases, however, is a stream function available, whereas our purposeis to illustrate how Equations (1.21b) are generally utilized.dXdY------- Y2X2 X2Y2+ ( )2+2XY-------------------------------------------------- =X R cos , Y R sin = =R21 +R 1 ( ) R 1 + ( )R----------------------------------------dR d cot =12--- R2R 1 ------------12---1R 1 ------------12--- R2R 1 +-------------12---1R 1 +------------- R 1R--- + + +( ,j \R dYoR cot d2 =YoYo21 --------------- R21 R--------------1 sin----------- =X2Y2+ YY Y---------------- =YY Yo1Yo----- = Uy 1 a2x2y2+---------------- ( ,j \=Y ,Y ( )/ aU ( ) =9114-ch01-Frame Page 11 Thursday, November 2, 2000 10:58 PM12 Analytical Fluid DynamicsThe determination of the pathlines in an unsteady ow is more difcult. Moreover, the physicalinterpretation of a pathline solution is far from trivial. As indicated in Figure 1.1(c), the sameproblem is considered, but now the cylinder is moving to the left, with a speed U, into a uidthat is quiescent far from the cylinder. A prime is used to denote unsteady variables, and our goalis to determine the trajectory of a uid particle. It is analytically convenient to x the initial conditionfor the particle directly over the center of the cylinder with t 0 and y yo, as shown inFigure 1.1(c). Consequently, a full trajectory requires the particles position for both positive andnegative time. The initial condition phrase therefore does not refer to the particles position whent .This ow is essentially the same as the steady ow case; only our viewpoint is different. In asteady ow, we move with the cylinder, whereas in the unsteady case we have a xed (laboratory)coordinate system. It is convenient to again introduce nondimensional variables FIGURE 1.2 Streamlines (a) and pathlines, (b) and (c), are for ow about a circular cylinder.X xa----, Y ya----, T Ua----t = = =9114-ch01-Frame Page 12 Thursday, November 2, 2000 10:58 PMBackground Discussion 13and use a Galilean transformationto convert the steady ow velocity eld into the unsteady one. Equations (1.22) thus becomeThe center of the cylinder is at x y 0, orHence, the initial condition for a uid particle iswith Yo 1. The X,Y coordinate system is therefore shifted to the left or right until the positionof the particle of interest is located at X 0 when T 0. When T is sufciently negative, theparticle is upstream of the center of the cylinder, which is at a positive X value. Remember thatwhen the particle is above the cylinders center, T X 0. Similarly, when T is sufcientlypositive, X is negative. This behavior is illustrated in Figure 1.2(b), where point a is the locationof a particle when T , while point e is the location when T +. In this gure, the centerof the cylinder moves from X , T to X , T , whereas the lateral motion ofa particle is nite. The one exception is a particle with Y 0; this particle wets the cylinder.At its initial location, when T 0, the velocity components of the particle areThus, the particle, at this time, is moving in the positive X direction, as indicated by point c inFigure 1.2(b). For a particle far upstream of the cylinder, we haveWhen the particle is far downstream of the cylinder, we haveand the cylinder is to the left of the particle. Far from the cylinder, in either X direction, the particlemoves in the negative X direction. The sign change in u, which occurs when the particle is nearthe cylinder, is discussed shortly. Note that Y and Yo are still related by Equation (1.23b).We are now ready to utilize Equation (1.13), written asx x Ut = , y y = , t t = , u u U = , v v =uU---- Y2X T ( )2X T + ( )2Y2+ [ ]2---------------------------------------------, vU----2 X T + ( )YX T + ( )2Y2+ [ ]2--------------------------------------------- = =X T + 0, = Y 0 =X 0 = , Y Yo= when T 0 =uU----( ,j \o1Yo2------, vU----( ,j \o0 = =X 0 > , T < X 0 < , T >> 0, Y Y, uU---- 0, vU---- 0 < < dxdt------- u, dydt------- v = =9114-ch01-Frame Page 13 Thursday, November 2, 2000 10:58 PM14 Analytical Fluid Dynamicsfor the particle paths. In contrast to the streamline situation, we have one additional differentialequation to solve. In terms of nondimensional variables these equations become(1.24)After Equation (1.23a) is transformed, it also represents a particle path. In other words,(1.25)is a rst integral of Equations (1.24). This can be demonstrated by differentiating this equationwith respect to T and eliminating dX/dT and dY/dT to obtain an identity. We next utilize thisequation to eliminate X + T from the dY/dT equation, with the resultwhere a sign is introduced when the square root of (X + T )2 is taken. The plus sign holds whenT < 0, while the minus sign holds when T > 0.The above differential equation is integrated from the initial condition, Y Yo when T 0, toobtainSincethe integral can be written in a standard form asThis quadrature can be evaluated in terms of elliptic integrals of the rst, F, and second, E, kinds,dened as (Handbook of Mathematical Functions, 1972)dXdT-------- Y2X T + ( )2X T + ( )2Y2+ [ ]2---------------------------------------------, dYdT-------- 2 X T + ( )YX T + ( )2Y2+ [ ]2--------------------------------------------- = =X T + ( )2Y2+ YY Y----------------- =dTdT--------2 1 YY Y2 + ( )12Y Y ( )32Y12------------------------------------------------------------------------------ =T12--- Y1 YY Y2 +-----------------------------------( ,j \12 Y dY Y ( )32----------------------------YYo =1 YY Y2 + Yo Y ( ) Y1Yo----- +( ,j \=T12--- YYo Y ( ) Y Y ( )3Y1Yo----- +( ,j \----------------------------------------------------------------------12Y dYYo =F ( ) d1 sin2 sin2 ( )12---------------------------------------------------0=E ( ) 1 sin2 sin2 ( )12 d0=9114-ch01-Frame Page 14 Thursday, November 2, 2000 10:58 PMBackground Discussion 15where is a dummy integration variable. With the aid of a table of elliptic integrals (Gradshteynand Ryzhik, 1980; No. 47, p. 272), one can show that the nal form for T is thenwhere and are given byThis relation, in conjunction with Equation (1.25), represents the pathlines in an implicit form. Inother words, given Yo (or Y) and Y, these two equations determine X and T.Figure 1.2 shows, to scale, the expected streamline pattern in (a) and the pathline pattern in(b) and (c), where all patterns are symmetric about the Y or Y axis. The arrows on the streamlinesand pathlines indicate increasing time or the direction of the velocity.Along a b c in Figure 1.2(b), T is negative, and the center of the cylinder is at the originwhen the uid particle is at point c, where T is zero. At point a, T equals , while at point e,T is +. (The value of Xa is the subject of Problem 1.7.) For any other point on a b c, thecenter of the cylinder is to the right of the uid particle. In this regard, it is useful to note that aparticle is upstream of the cylinders center when X + T < 0 and downstream otherwise. Thisresult stems from the Galilean transformation, X X + T. At points b and d, u is zero, while atpoint c, v is zero. One exception to part of this discussion is a particle with Y 0 and X > 0,which ultimately wets the cylinders surface. Otherwise, all other uid particles have similartrajectories, including the loop.Along c d e, T 0 and the particle is downstream of the cylinders center. Consequently,along a b the particle is being pushed by the cylinder and u 0, while along d e the particleis being pulled by the cylinder, and again u 0. When the particle is close to the cylinder alongb c d, there is a transition region between the pushing and pulling where u 0. In this region,v changes sign. As evident in Figure 1.2(c), the size of the loop depends on Y (or Yo). Particleswith a small Y value, which initially are close to the X axis, have a relatively large loop. This iscaused by the cylinder imparting a large velocity to the particle as it is shoved aside.A particle experiences a horizontal drift (Darwin, 1953) as a result of the cylinders motion, given byAs shown in Problem 1.7, becomes innite when Y 0 and goes to zero as Y . Thisdisplacement, or drift, also occurs in the steady ow case, since the particles that pass close to thecylinder are retarded more than those that pass at a distance. As shown in Problem 5.22 in Chapter 5,along a given pathline the change in kinetic energy balances the work done in moving a uidparticle. Because viscosity is not present, the work done on adjacent pathlines or streamlines is notrelated. Consequently, the change in displacement with Y does not involve any dissipative work.As you might imagine, the streamlines and pathlines for ow about a sphere are similar to thatof a cylinder. Both types of patterns are also considered in Problems 5.23 and 5.24, where a Galileantransformation is again convenient for the unsteady spherical case.T Yo F ( ) E ( ) YoY Yo Y ( )Y Y ( ) 1 YoY + ( )------------------------------------------------12+ = Y ( ) sin1Yo32 Yo Y 1 YoY +--------------------( ,j \12= sin11Yo2------ = Xa Xe 2Xa = =9114-ch01-Frame Page 15 Thursday, November 2, 2000 10:58 PM16 Analytical Fluid Dynamics1.3 THE STRESS TENSORWe now turn our attention to the two types of forces that can act on an arbitrary innitesimal uidelement or particle. One of these is the body force, e.g., the force due to gravity or an electromagneticeld. By denition, a body force is one that acts throughout a volume, as is the case with gravity,where(1.26)and is the acceleration due to gravity. This force per unit volume is , where is the density.Hence, a per unit volume body force is proportional to the density. There are other apparent oreffective body forces in a coordinate system that is rotating or accelerating, like the centripetal andCoriolis forces that also are proportional to the density. (These forces are discussed inSection 2.5.) The electromagnetic force depends on the net charges, not on the bulk density;however, it is properly treated as a body force, since the charges are usually distributed throughoutthe uid medium. We will not be concerned with this type of force.By denition, a surface force is one that is proportional to the amount of surface area it actsupon. The surface of interest need not be a real surface, such as the surface of a droplet, but aconceptual one, such as that surrounding an innitesimal uid particle. The simplest example of asurface force is the one due to hydrostatic pressure. There are also surface forces that act at realsurfaces, such as an interfacial force at a phase boundary. We will not deal with this type of force.An analytical description of a surface force is not nearly as simple as Equation (1.26). For thisdescription, we utilize a differential surface area ds, whose spatial orientation is provided by a unitnormal vector , as indicated in Figure 1.3. The surface force per unit area, , that acts on ds isgenerally not in the plane of the surface. As indicated in the gure, will have a component along and a tangential component in the plane of the surface. Since is per unit area, the actual forceon ds is ds. We call the stress vector; the component along results in the normal stress, whilethe component in the plane of the surface results in the shear stress.In general, is a function of both position and surface orientation; i.e.,(1.27)The stress vector can be related to a second-order tensor that depends on but not on . To showthis, we need Newtons third law, which states that for every action (force) there is an equal butopposite reaction. Hence, we have(1.28) FIGURE 1.3 Schematic of a surface force.Fb g =g gn n n r n , ( ) =r n r n , ( ) r n , ( ) =9114-ch01-Frame Page 16 Thursday, November 2, 2000 10:58 PMBackground Discussion 17For the subsequent discussion, it will be convenient to introduce orthogonal curvilinear coordi-nates i and the corresponding orthonormal basis i. Consider an innitesimal tetrahedron as shownin Figure 1.4. The outward unit normal vectors to the surfaces coplanar with the i coordinates arei. Let i be the outward facing stress vector on these surfaces; i.e.,(1.29)Note that i is a vector, not a component of ( , ). Shortly, we will relate these two vectors. Byvirtue of Equation (1.28), we have(1.30)For the tetrahedron, let s be the slant face surface area, si the surface area normal to i, andv the volume of the tretrahedron. This volume is given bywhere h is the normal distance from the origin to the slant face. With the aid of vector analysis,the various surface areas can be related by(1.31)Since the basis is orthonormal, we have(1.32)As a consequence, when we multiply Equation (1.31) with j, we obtainor(1.33) FIGURE 1.4 Effect of a stress vector on an innitesimal tetrahedron.i r ei , ( ) = r n r ei , ( ) r ei, ( ) =v 13 ( )hs =siei s ( )n =ei ej ij=siij s ( )ej n =si n ei( )s =9114-ch01-Frame Page 17 Thursday, November 2, 2000 10:58 PM18 Analytical Fluid DynamicsNewtons second law for the mass, v, within the tetrahedron can be written aswhere is the masss acceleration, is the density, and the right side represents the four surfaceforces and the body force that act on the tetrahedron. We now replace si with Equation (1.33),v with hs/3, and ( ,i) with Equations (1.29) and (1.30), to obtainWe assume remains nite as the tetrahedron shrinks to a point. In this limit, h 0, andwe obtain(1.34)where the right side contains three terms since i is summed over.As previously indicated, the stress depends on the force vector and the vector that prescribesthe orientation of the surface area on which acts. For a given coordinate system, this dependencecan be reduced to two sets of vectors, i and i. The stress is therefore a second-order tensor, whichcan be written in dyadic form as(1.35)where a dyadic is just the juxtaposition of two vectors. As a consequence, Equation (1.34) becomes(1.36)The second-order stress tensor is thus related to the force vector and helps provide the explicitdependence of on . In other words, is independent of the orientation of the surface. We nowdene the component form of i and as(1.37a)(1.37b)In a Cartesian coordinate system, ij is written as ij. Also note that the right side of Equation(1.37b) consists of nine terms in contrast to Equation (1.35), which contains only three. Bycomparing these two equations, we obtain(1.38)while Equation (1.37a) yields the contravariant result(1.39)These last equations express the fact that ij represents the stress on an area perpendicular to thei coordinate and in the jth direction. v ( ) a r n , ( )s r ei, ( )si v ( )Fb+ =a r13 ( )h s ( )a s i n ei( )s 13 ( )h s ( )Fb+ =a Fb n ei( )i= n eii= r n , ( ) n r ( ) = n i ijej= ijeiej=ijij=i ijej=9114-ch01-Frame Page 18 Thursday, November 2, 2000 10:58 PMBackground Discussion 19The stress vector at is determined by the nine components of and the normal to thesurface ds. Not all of the components are independent of each other. We have already utilized twoconditions, namely, the action equals reaction principle and Newtons second law. The componentsof , however, are subject to a third condition that requires the resultant moment of these forces,about any point, to vanish. This condition will be examined in Section 2.6, where it results in being a symmetric tensor,(1.40)in which case has only six independent components. In this circumstance, Equation (1.36) canbe written as(1.41)If is not symmetric, then .1.4 RELATION BETWEEN STRESS AND DEFORMATION-RATE TENSORSLet us assume a constant velocity eld and ignore gravity. In this circumstance , and therefore, has no dependence on . Furthermore, the uid possesses no shearing motion and no shearstresses. In a Cartesian coordinate system, ij can be written as(1.42)(As mentioned earlier, we use the covariant component form for vectors and tensors when thecoordinates are Cartesian.) Equation (1.42) guarantees no shear stress; i.e., a nonzero shear stressrequires ij 0 for some i j.Our frequent use of a Cartesian coordinate system requires a word of explanation. One canshow, using the GramSchmidt procedure of vector analysis, that any vector basis can be replacedby an orthonormal basis. This new basis, in turn, can be replaced with a Cartesian one. Thesereplacements are performed when convenient and result in no loss of generality, since the laws ofphysics are independent of the choice of a coordinate system. It will prove convenient for us touse Cartesian coordinates for some of the subsequent derivations. As noted, there is no loss ingenerality in doing this.Equation (1.42) means that the normal stress is independent of the orientation of the surfaceds as given by . This is the case for the stress due to the hydrostatic pressure p, which varies withbut not with . We therefore write this equation as(1.43)where, by convention, a compressive stress is negative, hence producing the minus sign. A uidmotion with a nonzero velocity gradient will possess normal stresses that are not equal to thenegative of the hydrostatic pressure.We now subtract the hydrostatic pressure term from to obtain the viscous stress tensor r nijji= or ij ji= n n = = n n rijconstant ( )ij=nr nij pij =ij ij pij ( ) ij pij+ = =9114-ch01-Frame Page 19 Thursday, November 2, 2000 10:58 PM20 Analytical Fluid Dynamicsor more generally(1.44)The viscous stress tensor is nonzero only if the uid possesses a nonzero relative motion. It is that we relate to the rate of deformation. (In a solid, depends on the deformation itself andnot the rate of deformation. This trivial-sounding difference represents the demarcation betweensolid and uid mechanics.) We further require that the dependence of on the rate of deformationbe independent of the choice of the coordinate system.To help x ideas, we observe that the motion of a uid can be decomposed into four types ofmotion: uniform translation, solid-body rotation, extensional strain or dilatation, and a shear strain.The rst two types of motion produce no relative motion; hence, should depend only on thedilatation and shearing motions. Consequently, cannot depend on or its components, i.e., onthe translational motion; however, can depend on derivatives of the velocity components.We now consider the relative motion of two adjacent uid particles that are separated by asmall distance , as shown in Figure 1.5. At some instant, the particles have velocities and + , where becomes d as d . We evaluate d by writing the Taylor seriesto obtain(1.45)The rightmost term is just the directional derivative of in the d direction, and is the velocitygradient tensor.The evaluation of d requires decomposing d in accordance with the above discussion.It is evident that this quantity does not depend on any uniform translational motion, since appearsonly in the gradient. However, we must still subtract any solid-body rotation from d . To accomplishthis, we observe that any second-order tensor can be written as the sum of symmetric and anti-symmetric tensors. Hence, we write(1.46)The symmetric tensor , called the rate-of-deformation tensor, is given by(1.47a) FIGURE 1.5 Strain rate schematic.rrrWr+ww+ pI + = wr ww w w w r r ww r d r + ( ) w r ( ) d r w ( )+ + =dw w r d r + ( ) w r ( ) d r w ( ) + = =w r ww r wwww + = 1/2 ( ) w w ( )t+ [ ] =9114-ch01-Frame Page 20 Thursday, November 2, 2000 10:58 PMBackground Discussion 21where ( )t denotes the transposition operation. For example, we write in Cartesian coordinates asThenand becomes(1.47b)The antisymmetric part of is the rotation tensor(1.48a)which becomes in Cartesian coordinates(1.48b)Observe that and sum to . With Equation (1.46), we see that d is(1.49)We need to interpret the d term. For this, consider a body whose sole motion is that ofsolid-body rotation with a constant angular velocity rot. From mechanics, the velocity at point of the body is provided by the cross product(1.50)We take the curl of both sides to obtainwhere the rightmost equality stems from the use of standard vector identities in Table 5 of AppendixA. (Although is the vorticity, this observation is irrelevant to the present discussion.) Thisrelation is now multiplied by d , with the result ww wixj-------- |j|i=w ( )t wjxi--------- |j|i=12--- wixj-------- wjxi--------- +( ,j \ |j|i=w12--- w w ( )t [ ] =12--- wixj-------- wjxi--------- ( ,j \ |j|i=w wdw dr dr + = r wrw rot r = w rot r ( ) 2rot= =wrrot d r 12--- w ( ) d r =9114-ch01-Frame Page 21 Thursday, November 2, 2000 10:58 PM22 Analytical Fluid DynamicsWith the aid of Equations (1.5) and (1.7), Cartesian coordinates are used to evaluate the right side asHowever, with the use of Equation (1.48b) we observe that(1.51)The d term in Equation (1.49) is thus associated with a solid-body rotation and does notcontribute to the viscous stress tensor. This means that can only depend on the rate-of-deformationtensor . This tensor, however, is symmetric with six independent components. These componentscan be further subdivided into those producing a shearing motion and those responsible for adilatation or extensional strain. This later strain is given by the trace of ,(1.52)where i is summed over. The three independent off-diagonal components produce only a shearingmotion.1.5 CONSTITUTIVE RELATIONSAs indicated at the end of the last section, we will relate to . There are constitutive equationswhose coefcients are transport properties. By means of these equations, we express the uniquecharacteristics of a gaseous or liquid substance. For our purposes, it sufces to view these equationsas empirical, i.e., based on experiment although they can be justied for a gas by kinetic theory.In this section, we derive equations for the viscous stress tensor and the heat ux vector .For , which is discussed rst, we utilize a Newtonian uid assumption, while Fouriers equationwill be used for the heat ux. We return to only later in the discussion. As will become apparent,both the Newtonian approximation and Fouriers equation are based on the same assumptions,namely, isotropy and a linear relation. For an introductory discussion of non-Newtonian uidmechanics, see DeKee and Wissbrun (1998).NEWTONIAN FLUIDAgain following Stokes, we postulate a linear relation between and . We further assume anisotropic uid in which a coordinate rotation or interchange of the axis leaves the stress, rate-of-deformation relation unaltered. A uid that adheres to both assumptions is called Newtonian. Gases,except under extreme conditions such as a shock wave, and most common liquids very accuratelysatisfy the Newtonian approximation. Liquids containing long-chain polymers do not satisfy thisapproximation as accurately.Each of the above tensors has nine scalar components. The linear assumption means that each component is proportional to the nine components of ; hence, there are 81 scalar coefcientsthat relate the two tensors. These coefcients are the components of a fourth-order tensor, since34 81. With a subscript notation, the linear relation is(1.53)rot d r 12--- wixj-------- wjxi--------- ( ,j \dxj |i=d r rot d r = r iiwixi-------- w = =qij cijmnmn=9114-ch01-Frame Page 22 Thursday, November 2, 2000 10:58 PMBackground Discussion 23where cijmn is called the fourth-order viscosity coefcient tensor. The most general form for anisotropic fourth-order tensor iswhere A, B, and C are the only coefcients remaining out of the original 81. Consequently, we haveSince is symmetric, this further simplies toWe now utilize Equation (1.52) and introduce the notation and for the rst and second viscositycoefcients, respectively:Normally, is simply referred to as the viscosity or shear viscosity. The nal form for the viscousstress tensor is(1.54a)In tensor notation, we have(1.54b)where is the unit dyadic.By incorporating the pressure stresses, we obtain the familiar result for the stress tensorcomponents(1.55a)while in tensor notation, this becomes(1.55b)We emphasize that these equations are restricted to a Newtonian uid.When initially discussing isotropy, we stated that (or ) should be invariant with respect toan interchange of axis. This is accomplished for by interchanging the i and j subscripts in theviscous terms in Equation (1.55a). Observe that the interchange does not alter these terms.cijmn Aijmn Bimjn Cinjm+ + =ij Aijmnmn Bimjnmn Cinjmmn+ + =Aijmm Bij Cji+ + =ij Aijmm B C + ( )ij+ =12--- B C + ( ) = , A =ij2ij ij w + = 2 w ( )I + = Iij pij wixj-------- wjxi--------- +( ,j \ijwkxk--------- + + = p w + ( )I 2 + = 9114-ch01-Frame Page 23 Thursday, November 2, 2000 10:58 PM24 Analytical Fluid DynamicsIn Section 2.2, we show that for an incompressible ow, conservation of mass becomes(1.56)In this circumstance, the term in Equations (1.54) and (1.55) containing is zero; therefore, thesecond viscosity coefcient plays no role in an incompressible ow.To appreciate the role of for a compressible ow, we take the trace of ij and divide by three.An average of these normal stresses is thus obtained:where(1.57)is the bulk viscosity. We replace with b in Equation (1.54b), with the resultwhere the tensor has a zero trace. Although this tensor has a zero trace, its diagonal elementsare generally nonzero with only their sum being zero. This is called the rate-of-shear tensor, sinceit provides the viscous stresses associated only with a shearing motion. Consequently, the bulkviscosity term provides the viscous stresses due to a dilatational motion. The shear and dilatationalstresses are caused by the attractive and repulsive forces between molecules and the collisionalrelaxation of the rotational and vibrational energy modes of polyatomic molecules, respectively.Ultrasonic absorption measurements show b to be zero for a low-density monatomic gas, inaccordance with kinetic theory. For certain polyatomic molecules, such as CO2, b can be muchlarger than . In any case, the second law of thermodynamics requires b 0, as will be shown inChapter 3. Stokes originally hypothesized that(1.58)for all gases. This hypothesis is exact for a monatomic gas (except at a very high density) and canbe used for an incompressible ow when the value of is irrelevant. The approximation, however,is often used for compressible ows of polyatomic gases. For instance, ultrasonic measurementsyield for air at 293 K and 1 atm. This is a very small value for b and frequently canbe neglected. Nevertheless, we will not invoke this hypothesis anywhere in the subsequent analysis.A more comprehensive physical discussion of bulk viscosity can be found in Section 13.4.FOURIERS EQUATIONAs discussed in Section 1.1, we neglect the transport of energy by molecular diffusion, chemicalreactions, or radiative heat transfer. For heat conduction, a linear relation between the heat ux and the temperature gradient is assumed as w 0 =13---ii p 23--- +( ,j \+ p b w + = =b 23--- + = 2 13--- wI ( ,j \b wI + 2 b wI + = = b0, 23--- = =b2/3 ( ) qq T = 9114-ch01-Frame Page 24 Thursday, November 2, 2000 10:58 PMBackground Discussion 25where T is the temperature and is a second-order thermal conductivity tensor. Note that is notnecessarily oriented in the same direction as T. (This difference in orientation occurs in crystals.)Although referred to as the heat ux, actually provides the rate of heat transfer having units ofenergy per unit area per unit time. (The thermal conductivity tensor has units of energy per unitlength per degree Kelvin per unit time.) In Cartesian component form, the heat ux becomesFor an isotropic uid, the ij tensor is given bywhere is the conventional coefcient of thermal conductivity. (For notational consistency, transportcoefcients are denoted with Greek symbols; hence, we use for the thermal conductivity insteadof the more common k symbol.) We thus obtain the standard form for Fouriers equation(1.59)where the minus sign ensures that heat ows from hot to cold when > 0.By way of summary, we observe that we have obtained the constitutive relations, Equations(1.55b) and (1.59), for the stress tensor and heat ux vector. These equations contain only threescalar coefcients, , , and . This is a remarkable simplication since the total number of scalarcoefcients in the two equations has been reduced from 90. These three coefcients are viewed asknown functions of the temperature and density. This dependence is often empirically establishedor it may come from the kinetic theory of gases.1.6 INTEGRAL RELATIONSA number of integral equations will be needed in the subsequent analysis. The rst few of theseare standard vector relations; we state them without proof. The rst one is the Stokes theorem:(1.60)where ( ) is an arbitrary vector function. The surface S is an open surface, or cap, bounded bya simple closed curve C as shown in Figure 1.6. The vector d is tangent to C, while is a unitvector normal to S. The Stokes theorem is useful for converting a line integral into a surface integral,or vice versa.FIGURE 1.6 Cap bordered by a simple closed curve.dsdrSCn^rqqqi ijTxj------- ij ij =q T =A d rC n A ( )S ds =A rr n9114-ch01-Frame Page 25 Thursday, November 2, 2000 10:58 PM26 Analytical Fluid DynamicsWe write the Gauss divergence theorem in a generalized form(1.61)where the volume V is fully enclosed by S, is an outward unit normal vector to S, ( ) is an arbitraryscalar function, and ( ) is again an arbitrary vector function. This result is actually three separateequations combined into a single convenient form. Of these three equations, we shall make explicituse of the rst two. Note that the surface integral is a double integral while the volume integral isa triple integral. We shall also need a dyadic version of the middle equation, given by(1.62)where is an arbitrary dyadic in three-dimensional space.The next standard relation is the integral denition of the divergence operation, given by(1.63)where v is a small volume bounded by s. This relation is easily derived from Equation (1.61).There are various extensions or generalizations to Equations (1.60) and (1.63) (see Appendix A,Tables 4 and 6) that are not considered, since they will not be needed.LEIBNIZS RULESuppose the integrand and one or both integration limits of a one-dimensional integral depend ona parameter t. If the integral is differentiated with respect to t, Leibnizs rule providesIn this mathematical identity t is not necessarily time, but this identication provides us with asuitable physical interpretation. Thus, dxi/dt represents the speed with which the end points move,while the (xi,t)dxi/dt terms represent the ux of across the end points.We will need a three-dimensional version of Leibnizs rule.* For this, we introduce a volumeV of nite magnitude with a surface S that encloses V. Let s( ,t) be the velocity with which Smoves where denotes a point on S, and let be the outward unit normal vector to S. For thedesired generalization, we need to evaluate the amount of that crosses S due to its motion. Theamount that crosses a differential area ds of S, per unit time, is* I am indebted to Professor M. L. Rasmussen for suggesting the Reynolds transport theorem derivation, which starts withLeibnizs three-dimensional rule.AAv dV nAAdsS=n rA r v dV n dsS= A limv 0 1v----- n Adss=ddt----- x t , ( ) x dxi t ( )x2 t ( ) t------- x dx1 t ( )x2 t ( ) x2 t ( ) t , ( ) dx2dt-------- x1 t ( ) t , ( ) dx1dt-------- + =w rr nws n ds9114-ch01-Frame Page 26 Thursday, November 2, 2000 10:58 PMBackground Discussion 27When s > 0, the ux s of leaves V; thus, the net ux of that crosses S isConsequently, the rule in three dimensions is(1.64)where the volume integral on the right represents the change in that occurs within V, while thesurface integral accounts for the transport of across the moving surface S. Note that if s iseverywhere positive on S, then V and S increase with time. In this circumstance, the surface integralis positive when > 0 and contributes toward an increasing value for the volume integral on theleft side. Equation (1.64) does not stem from uid dynamics; it is purely mathematical. In thisregard, observe that s is not necessarily related to a uid velocity and has not been identiedwith any uid property.REYNOLDS TRANSPORT THEOREMWe now assume S(t) moves with the uid velocity(1.65)As a consequence, a uid particle initially within V will remain within V, and a particle outside ofV remains outside. Thus, V(t) contains a xed amount of mass, referred to as a material volume,and is equivalent to a closed thermodynamic system. In this circumstance, it is appropriate toreplace the time derivative on the left side of Equation (1.64) with the substantial derivative. Withthis change, Equation (1.64) becomes(1.66)which is the rst version of the transport theorem. The quantity ( ,t) can be a scalar, vector, orhigher order tensor, and the terms on the right side have the same physical interpretation as thoseon the right side of Equation (1.64).If, instead of Equation (1.65), we set s 0, the volume and bounding surface are usuallyreferred to as a xed control volume (CV) and control surface (CS). Since mass may cross thecontrol surface, this is an open system. Equation (1.64) now reduces to(1.67)where should be continuous within the control volume. The need for this proviso becomesevident by setting for a control volume containing two ows that are separated by a movingshock wave.w n wws n dsSddt----- v dV t ( ) t------- v dV t ( ) ws n dsS t ( )+ =w nwws w =DDt------ v dV t ( ) t------- v dV t ( ) w n s dS t ( )+ =rwddt----- v dCV t------- v dCV=9114-ch01-Frame Page 27 Thursday, November 2, 2000 10:58 PM28 Analytical Fluid DynamicsA relation between an open and closed system is obtained by equating V and S with CV andCS, respectively, at a given instant of time. By subtracting Equation (1.67) from (1.66), we obtain(1.68)where the left side refers to a moving material volume and the right side refers to a xed controlvolume, which is an open system.When is a scalar (or a higher-order tensor; see Problem 1.6), the surface integral in Equation(1.66) can be written aswhere ( ) is a dyadic if is a vector. By means of the divergence theorem, Equation (1.61), weobtainand Equation (1.66) becomes(1.69a)The divergence term can be expanded asto yield(1.69b)Equations (1.66) and (1.69) are alternate forms of the transport theorem, where Equation (1.69b)is utilized in Section 2.3.As an illustration, set 1 and replace V with a small volume v, to obtainfrom Equation (1.69b). The integral on the left side is just v. We thus haveDDt------ v dV ddt----- v dCV w n dsCS+ =w n dsS n w ( )dsS n w ( )S ds = =wwS n ds w ( ) v dV=DDt------ v dV t------- w ( ) + v dV= w ( ) w w + =DDt------ v dV t------- w