Currie I.G. Fundamental Mechanics of Fluids

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Advanced Fluid Mechanics

Transcript of Currie I.G. Fundamental Mechanics of Fluids

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  • Fundamental Mechanics of Fluids Third Edition

    1. G. Currie University of Toronto

    Toronto, Ontario, Canada

    M A R C E L

    MARCEL DEKKER, INC.

    D E K K E R

    NEW YORK . BASEL

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  • To my wife Cathie, our daughter Karen, and our sonsDavid and Brian

  • Preface

    This book covers the fundamental mechanics of uids as they are treated atthe senior level or in rst graduate courses. Many excellent books exist thattreat special areas of uid mechanics such as ideal-uid ow or boundary-layer theory. However, there are very few books at this level that sacrice anin-depth study of one of these special areas of uid mechanics for a briefertreatment of a broader area of the fundamentals of uid mechanics. Thissituation exists despite the fact that many institutions of higher learningoffer a broad, fundamental course to a wide spectrum of their studentsbefore offering more advanced specialized courses to those who are spe-cializing in uid mechanics. This book is intended to remedy this situation.

    The book is divided into four parts. Part I, Governing Equations,deals with the derivation of the basic conservation laws, ow kinematics,and some basic theorems of uid mechanics. Part II is entitled Ideal-FluidFlow, and it covers two-dimensional potential ows, three-dimensionalpotential ows, and surface waves. Part III, Viscous Flows of Incom-pressible Fluids, contains chapters on exact solutions, low-Reynolds-

  • number approximations, boundary-layer theory, and buoyancy-drivenows. The nal part of the book is entitled Compressible Flow of InviscidFluids, and this part contains chapters that deal with shock waves, one-dimensional ows, and multidimensional ows. Appendixes are also inclu-ded which summarize vectors, tensors, the governing equations in thecommon coordinate systems, complex variables, and thermodynamics.

    The treatment of the material is such as to emphasize the phenomenaassociated with the various properties of uids while providing techniquesfor solving specic classes of uid-ow problems. The treatment is notgeared to any one discipline, and it may readily be studied by physicists andchemists as well as by engineers from various branches. Since the book isintended for teaching purposes, phrases such as it can be shown that andsimilar cliches which cause many hours of effort for many students havebeen avoided. In order to aid the teaching process, several problems areincluded at the end of each of the 13 chapters. These problems serve toillustrate points brought out in the text and to extend the material covered inthe text.

    Most of the material contained in this book can be covered in about 50lecture hours. For more extensive courses the material contained here maybe completely covered and even augmented. Parts II, III, and IV areessentially independent so that they may be interchanged or any one or moreof them may be omitted. This permits a high degree of teaching exibility,and allows the instructor to include or substitute material which is notcovered in the text. Such additional material may include free convection,density stratication, hydrodynamic stability, and turbulence with applica-tions to pollution, meteorology, etc. These topics are not included here, notbecause they do not involve fundamentals, but rather because I set up apriority of what I consider the basic fundamentals.

    For the third edition, I redrew all the line drawings, of which there areover 100. The problems have also been reviewed, and some of them havebeen revised in order to clarify and=or extend the questions. Some newproblems have also been included.

    Many people are to be thanked for their direct or indirect contribu-tions to this text. I had the privilege of taking lectures from F. E. Marble,C. B. Millikan, and P. G. Saffman. Some of the style and methods of thesegreat scholars are probably evident on some of the following pages. TheNational Research Council of Canada are due thanks for supplying thephotographs that appear in this book. My colleagues at the University ofToronto have been a constant source of encouragement and help. Finally,sincere appreciation is extended to the many students who have taken mylectures at the University of Toronto and who have pointed out errors anddeciencies in the material content of the draft of this text.

    vi Preface

  • Working with staff at Marcel Dekker, Inc., has been a pleasure. I amparticularly appreciative of the many suggestions given by Mr. John J.Corrigan, Acquisitions Editor, and for the help he has provided in thecreation of the third edition. Marc Schneider provided valuable informationrelating to software for the preparation of the line drawings. Erin Nihill, theProduction Editor, has been helpful in many ways and has converted apatchy manuscript into a textbook.

    I. G. Currie

    Preface vii

  • Contents

    Preface v

    Part I. Governing Equations 1

    1. Basic Conservation Laws 3

    1.1 Statistical and Continuum Methods1.2 Eulerian and Lagrangian Coordinates1.3 Material Derivative1.4 Control Volumes1.5 Reynolds Transport Theorem1.6 Conservation of Mass1.7 Conservation of Momentum1.8 Conservation of Energy

  • 1.9 Discussion of Conservation Equations1.10 Rotation and Rate of Shear1.11 Constitutive Equations1.12 Viscosity Coefcients1.13 Navier-Stokes Equations1.14 Energy Equation1.15 Governing Equations for Newtonian Fluids1.16 Boundary Conditions

    2. FlowKinematics 40

    2.1 Flow Lines2.2 Circulation and Vorticity2.3 Stream Tubes and Vortex Tubes2.4 Kinematics of Vortex Lines

    3. Special Forms of the Governing Equations 55

    3.1 Kelvins Theorem3.2 Bernoulli Equation3.3 Croccos Equation3.4 Vorticity Equation

    Part II. Ideal-Fluid Flow 69

    4. Two-Dimensional Potential Flows 73

    4.1 Stream Function4.2 Complex Potential and Complex Velocity4.3 Uniform Flows4.4 Source, Sink, and Vortex Flows4.5 Flow in a Sector4.6 Flow Around a Sharp Edge4.7 Flow Due to a Doublet4.8 Circular Cylinder Without Circulation4.9 Circular Cylinder With Circulation

    4.10 Blasius Integral Laws4.11 Force and Moment on a Circular Cylinder4.12 Conformal Transformations4.13 Joukowski Transformation4.14 Flow Around Ellipses

    x Contents

  • 4.15 Kutta Condition and the Flat-Plate Airfoil4.16 Symmetrical Joukowski Airfoil4.17 Circular-Arc Airfoil4.18 Joukowski Airfoil4.19 Schwarz-Christoffel Transformation4.20 Source in a Channel4.21 Flow Through an Aperture4.22 Flow Past a Vertical Flat Plate

    5. Three-Dimensional Potential Flows 161

    5.1 Velocity Potential5.2 Stokes Stream Function5.3 Solution of the Potential Equation5.4 Uniform Flow5.5 Source and Sink5.6 Flow Due to a Doublet5.7 Flow Near a Blunt Nose5.8 Flow Around a Sphere5.9 Line-Distributed Source

    5.10 Sphere in the Flow Field of a Source5.11 Rankine Solids5.12 DAlemberts Paradox5.13 Forces Induced by Singularities5.14 Kinetic Energy of a Moving Fluid5.15 Apparent Mass

    6. SurfaceWaves 201

    6.1 The General Surface-Wave Problem6.2 Small-Amplitude Plane Waves6.3 Propagation of Surface Waves6.4 Effect of Surface Tension6.5 Shallow-Liquid Waves of Arbitrary Form6.6 Complex Potential for Traveling Waves6.7 Particle Paths for Traveling Waves6.8 Standing Waves6.9 Particle Paths for Standing Waves

    6.10 Waves in Rectangular Vessels6.11 Waves in Cylindrical Vessels6.12 Propagation of Waves at an Interface

    Contents xi

  • Part III. Viscous Flows of Incompressible Fluids 249

    7. Exact Solutions 253

    7.1 Couette Flow7.2 Poiseuille Flow7.3 Flow Between Rotating Cylinders7.4 Stokes First Problem7.5 Stokes Second Problem7.6 Pulsating Flow Between Parallel Surfaces7.7 Stagnation-Point Flow7.8 Flow in Convergent and Divergent Channels7.9 Flow Over a Porous Wall

    8. Low-Reynolds-Number Solutions 288

    8.1 The Stokes Approximation8.2 Uniform Flow8.3 Doublet8.4 Rotlet8.5 Stokeslet8.6 Rotating Sphere in a Fluid8.7 Uniform Flow Past a Sphere8.8 Uniform Flow Past a Circular Cylinder8.9 The Oseen Approximation

    9. Boundary Layers 313

    9.1 Boundary-Layer Thicknesses9.2 The Boundary-Layer Equations9.3 Blasius Solution9.4 Falkner-Skan Solutions9.5 Flow Over a Wedge9.6 Stagnation-Point Flow9.7 Flow in a Convergent Channel9.8 Approximate Solution for a Flat Surface9.9 General Momentum Integral

    9.10 Karman-Pohlhausen Approximation9.11 Boundary-Layer Separation9.12 Stability of Boundary Layers

    xii Contents

  • 10. Buoyancy-Driven Flows 363

    10.1 The Boussinesq Approximation10.2 Thermal Convection10.3 Boundary-Layer Approximations10.4 Vertical Isothermal Surface10.5 Line Source of Heat10.6 Point Source of Heat10.7 Stability of Horizontal Layers

    Part IV. Compressible Flow of Inviscid Fluids 395

    11. ShockWaves 399

    11.1 Propagation of Innitesimal Disturbances11.2 Propagation of Finite Disturbances11.3 Rankine-Hugoniot Equations11.4 Conditions for Normal Shock Waves11.5 Normal Shock-Wave Equations11.6 Oblique Shock Waves

    12. One-Dimensional Flows 430

    12.1 Weak Waves12.2 Weak Shock Tubes12.3 Wall Reection of Waves12.4 Reection and Refraction at an Interface12.5 Piston Problem12.6 Finite-Strength Shock Tubes12.7 Nonadiabatic Flows12.8 Isentropic-Flow Relations12.9 Flow Through Nozzles

    13. Multidimensional Flows 461

    13.1 Irrotational Motion13.2 Janzen-Rayleigh Expansion13.3 Small-Perturbation Theory13.4 Pressure Coefcient

    Contents xiii

  • 13.5 Flow Over a Wave-Shaped Wall13.6 Prandtl-Glauert Rule for Subsonic Flow13.7 Ackerts Theory for Supersonic Flows13.8 Prandtl-Meyer Flow

    Appendix A.VectorAnalysis 495

    Appendix B.Tensors 500

    Appendix C.Governing Equations 505

    Appendix D.ComplexVariables 510

    Appendix E.Thermodynamics 516

    Index 521

    xiv Contents

  • IGOVERNING EQUATIONS

    In this rst part of the book a sucient set of equationswill be derived,basedon physical laws and postulates, governing the dependent variables of a uidthat is moving. The dependent variables are the uid-velocity components,pressure, density, temperature, and internal energy or some similar set ofvariables. The equations governing these variables will be derived from theprinciples of mass,momentum, and energy conservation and fromequationsof state. Having established a sucient set of governing equations, somepurely kinematical aspects of uid ow are discussed, at which time theconcept of vorticity is introduced. The nal section of this part of the bookintroduces certain relationships that can be derived from the governingequations under certain simplifying conditions.These relationships may beused in conjunction with the basic governing equations or as alternatives tothem.

    Taken as a whole, this part of the book establishes the mathematicalequationsthatresult frominvokingcertainphysical lawspostulatedtobevalidfor a moving uid. These equations may assume dierent forms, dependinguponwhichvariables arechosenanduponwhich simplifyingassumptionsaremade.The remaining parts of thebook aredevoted to solving these governingequations for dierent classes of uid ows and thereby explaining quantita-tively someof thephenomena that areobserved in uidow.

    1

  • 1Basic Conservation Laws

    The essential purpose of this chapter is to derive the set of equations thatresults from invoking the physical laws of conservation of mass,momentum,and energy. In order to realize this objective, it is necessary to discuss certainpreliminary topics.Therst topic of discussion is the twobasicways inwhichthe conservation equations may be derived: the statistical method and thecontinuummethod.Having selected the basic method to be used in derivingthe equations, one is then faced with the choice of reference frame to beemployed, eulerian or lagrangian.Next, a general theorem, called Reynoldstransport theorem, is derived, since this theorem relates derivatives in thelagrangian framework to derivatives in the eulerian framework.

    Having established the basicmethod to be employed and the tools to beused, the basic conservation laws are then derived.The conservation of massyields the so-called continuity equation. The conservation of momentumleads ultimately to the Navier-Stokes equations, while the conservation ofthermal energy leads to the energy equation.The derivation is followed by adiscussion of the set of equations so obtained, and nally a summary of thebasic conservation laws is given.

    3

  • 1.1 STATISTICAL ANDCONTINUUMMETHODS

    There are basically two ways of deriving the equations that govern themotion of a uid. One of these methods approaches the question from themolecular point of view.That is, this method treats the uid as consisting ofmolecules whose motion is governed by the laws of dynamics. The macro-scopic phenomena are assumed to arise from the molecular motion of themolecules, and the theory attempts to predict the macroscopic behavior ofthe uid from the laws of mechanics and probability theory.For a uid that isin a state not too far removed from equilibrium, this approach yields theequations of mass, momentum, and energy conservation. The molecularapproach also yields expressions for the transport coecients, such as thecoecient of viscosity and the thermal conductivity, in terms of molecularquantities such as the forces acting between molecules or molecular dia-meters. The theory is well developed for light gases, but it is incomplete forpolyatomic gas molecules and for liquids.

    The alternative method used to derive the equations governing themotion of a uid uses the continuum concept. In the continuum approach,individual molecules are ignored and it is assumed that the uid consists ofcontinuousmatter.At each point of this continuous uid there is supposed tobe a unique value of the velocity, pressure, density, and other so-called eldvariables. The continuous matter is then required to obey the conservationlaws of mass, momentum, and energy,which give rise to a set of dierentialequations governing the eld variables. The solution to these dierentialequations then denes the variationof each eld variable with space and timewhich corresponds to themean value of themolecularmagnitude of that eldvariable at each corresponding position and time.

    The statistical method is rather elegant, and it may be used to treat gasows in situationswhere the continuumconcept is no longer valid.However,as was mentioned before, the theory is incomplete for dense gases and forliquids. The continuum approach requires that the mean free path of themolecules be very small compared with the smallest physical-length scale ofthe ow eld (such as the diameter of a cylinder or other body about whichthe uid is owing). Only in this way can meaningful averages over themolecules at a point be made and the molecular structure of the uid beignored. However, if this condition is satised, there no distinction amonglight gases, dense gases, or even liquidsthe results apply equally to all.Since the vast majority of phenomena encountered in uid mechanics fallwell within the continuum domain and may involve liquids as well as gases,the continuum method will be used in this book.With this background, themeaning and validity of the continuumconcept will nowbe explored in somedetail.The eld variables, such as the density r and the velocity vector u,will

    4 Chapter1

  • in general be functions of the spatial coordinates and time. In symbolic formthis is written as r rx; t and u ux; t, where x is the position vectorwhose certesian coordinates are x, y, and z. At any particular point in spacethese continuum variables are dened in terms of the properties of the var-iousmolecules that occupy a small volume in the neighborhood of that point.

    Consider a small volume of uid DV containing a large number ofmolecules. Let Dm and v be the mass and velocity of any individual moleculecontained within the volume DV, as indicated in Fig. 1.1.The density and thevelocity at a point in the continuum are then dened by the following limits:

    r limDV!e

    PDmDV

    u limDV!e

    PvDmPDm

    where e is a volumewhich is suciently small that e1=3 is small comparedwiththe smallest signicant length scale in the ow eld but is suciently largethat it contains a large number of molecules. The summations in the aboveexpressions are taken over all themolecules containedwithin the volumeDV.

    FIGURE 1.1 An individual molecule in a small volume DV having a mass Dm andvelocity v.

    Basic Conservation Laws 5

  • The other eld variables may be dened in terms of the molecular propertiesin an analogous way.

    A sucient condition, though not a necessary condition, for the con-tinuum approach to be valid is

    1n e L3

    where n is the number of molecules per unit volume and L is the smallestsignicant length scale in the ow eld, which is usually called the macro-scopic length scale.The characteristicmicroscopic length scale is the mean freepath between collisions of the molecules. Then the above condition statesthat the continuum concept will certainly be valid if some volume e can befound that is much larger than the volume occupied by a single molecule ofthe uid but much smaller than the cube of the smallest macroscopic lengthscale (such as cylinder diameter). Since a cube of gas, at normal temperatureand pressure,whose side is 2 micrometers contains about 2 108 moleculesand the corresponding gure for a liquid is about 2 1011 molecules, thecontinuum condition is readily met in the vast majority of ow situationsencountered in physics and engineering. Itmay be expected to break down insituations where the smallest macroscopic length scale approaches micro-scopic dimensions, such as in the structure of a shock wave, and where themicroscopic length scale approachesmacroscopic dimensions, such aswhena rocket passes through the edge of the atmosphere.

    1.2 EULERIAN AND LAGRANGIAN COORDINATES

    Having selected the continuum approach as the method that will be used toderive the basic conservation laws, one is next faced with a choice of refer-ence frames inwhich to formulate the conservation laws.There are two basiccoordinate systems that may be employed, these being eulerian and lagran-gian coordinates.

    In the eulerian framework the independent variables are the spatialcoordinates x, y, and z and time t. This is the familiar framework in whichmost problems are solved. In order to derive the basic conservation equa-tions in this framework, attention is focused on the uid which passesthrough a control volume that is xed in space. The uid inside the controlvolume at any instant in time will consist of dierent uid particles from thatwhich was there at some previous instant in time. If the principles of con-servation of mass, momentum, and energy are applied to the uid passingthrough the control volume,the basic conservation equations are obtained ineulerian coordinates.

    6 Chapter1

  • In the lagrangian approach, attention is xed on a particular mass ofuid as it ows. Suppose we could color a small portion of the uid withoutchanging its density. Then in the lagrangian framework we follow thiscolored portion as it ows and changes its shape, but we are always con-sidering the same particles of uid.The principles of mass, momentum, andenergy conservation are then applied to this particular element of uid as itows, resulting in a set of conservation equations in lagrangian coordinates.In this reference frame x,y, z, and t are no longer independent variables, sinceif it is known that our colored portion of uid passed through the coordinatesx0, y0, and z0 at some time t0, then its position at some later time may be cal-culated if the velocity components u, v, and w are known.That is, as soon as atime interval (t t0) is specied, the velocity components uniquely deter-mine the coordinate changes (x x0), ( y y0), and (z z0) so that x,y, z, and tare no longer independent.The independent variables in the lagrangian sys-tem are x0, y0, z0, and t,where x0, y0, and z0 are the coordinates which a spe-cied uid element passed through at time t0.That is, the coordinates x0, y0,and z0 identify which uid element is being considered, and the time t iden-ties its instantaneous location.

    The choice of which coordinate system to employ is largely a matter oftaste. It is probably more convincing to apply the conservation laws to acontrol volume that always consists of the same uid particles rather thanone through which dierent uid particles pass. This is particularly truewhen invoking the law of conservation of energy,which consists of applyingthe rst law of thermodynamics, since the same uid particles are morereadily justied as a thermodynamic system. For this reason, the lagrangiancoordinate system will be used to derive the basic conservation equations.Although the lagrangian system will be used to derive the basic equations,the eulerian system is the preferred one for solving the majority of problems.In the next section the relation between the dierent derivatives will beestablished.

    1.3 MATERIAL DERIVATIVE

    Let a be any eld variable such as the density or temperature of the uid.From the eulerian viewpoint, a may be considered to be a function of theindependent variables x, y, z, and t. But if a specic uid element is observedfor a short period of time dt as it ows, its positionwill change by amounts dx,dy, and dzwhile its value of awill change by an amount da.That is, if the uidelement is observed in the lagrangian framework, the independent variablesare x0, y0, z0, and t, where x0, y0, and z0 are initial coordinates for the uidelement. Thus, x, y, and z are no longer independent variables but are

    Basic Conservation Laws 7

  • functions of t as dened by the trajectory of the element. During the time dtthe change in amay be calculated from dierential calculus to be

    @a@t

    dt @a@x

    dx @a@y

    dy @a@z

    dz

    Equating the preceding change in a to the observed change da in the lagran-gian framework and dividing throughout by dt gives

    dadt @a

    @t dx

    dt@a@x

    dydt

    @a@y dzdt

    @a@z

    The left-hand side of this expression represents the total change in a asobserved in the lagrangian framework during the time dt, and in the limit itrepresents the time derivative of a in the lagrangian system, which will bedenoted by Da=Dt. It may be also noted that in the limit as dt ! 0 the ratiodx=dt becomes thevelocitycomponent in the xdirection,namely,u. Similarly,dy=dt ! v and dz=dt ! w as dt ! 0, the expression for the change inabecomes

    DaDt

    @a@t u @a

    @x v @a

    @y w @a

    @z

    In vector form this equation may be written as follows:

    DaDt

    @a@t uHa

    Alternatively, using the Einstein summation convention where repeatedsubscripts are summed, the tensor formmay be written as

    DaDt

    @a@t uk @a

    @xk1:1

    The termDa=Dt in Eq. (1.1) is the so-called material derivative. It representsthe total change in the quantity a as seen by an observer who is following theuid and iswatching a particular mass of the uid.The entire right-hand sideof Eq. (1.1) represents the total change in aexpressed in eulerian coordinates.The term uk@a=@xk expresses the fact that in a time-independent ow eldin which the uid properties depend upon the spatial coordinates only, thereis a change in a due to the fact that a given uid element changes its positionwith time and therefore assumes dierent values of a as it ows. The term@a=@t is the familiar eulerian time derivative and expresses the fact that atany point in space the uid properties may change with time.Then Eq. (1.1)expresses the lagrangian rate of changeDa=Dt of a for a given uid element interms of the eulerian derivatives @a=@t and @a=@xk .

    8 Chapter1

  • 1.4 CONTROLVOLUMES

    The concept of a control volume, as required to derive thebasic conservationequations, has been mentioned in connection with both the lagrangian andthe eulerian approaches. Irrespective of which coordinate system is used,there are two principal control volumes fromwhich to choose.One of these isa parallelepiped of sides dx, dy, and dz. Each uid property, such as the velo-city or pressure, is expanded in aTaylor series about the center of the controlvolume to give expressions for that property at each face of the controlvolume.The conservation principle is then invoked, and when dx, dy, and dzare permitted to become vanishingly small, the dierential equation for thatconservation principle is obtained. Frequently, shortcuts are taken and thecontrol volume is taken tohave sides of lengthdx,dy, anddzwithonly the rstterm of theTaylor series being carried out.

    The second type of control volume is arbitrary in shape, and each con-servation principle is applied to an integral over the control volume. Forexample, the mass within the control volume is

    RV r dV , where r is the uid

    density and the integration is carried out over the entire volumeVof the uidcontained within the control volume.The result of applying each conserva-tion principle will be an integro-dierential equation of the type

    ZVLa dV 0

    where L is some dierential operator and a is some property of the uid. Butsince the control volume V was arbitrarily chosen, the only way this equa-tion can be satised is by settingLa 0,which gives the dierential equationof the conservation law. If the integrand in the above equation was not equalto zero, it would be possible to redene the control volumeV in such a waythat the integral of La was not equal to zero, contradicting the integro-dierential equation above.

    Each of these two types of control volumes has some merit, and in thisbook each will be used at some point, depending uponwhich gives the betterinsight to the physics of the situation under discussion.The arbitrary controlvolume will be used in the derivation of the basic conservation laws, since itseems to detract less from the principles being imposed. Needless to say theresults obtained by the twomethods are identical.

    1.5 REYNOLDS TRANSPORT THEOREM

    The method that has been selected to derive the basic equations fromthe conservation laws is to use the continuum concept and to follow an

    Basic Conservation Laws 9

  • arbitrarily shaped control volume in a lagrangian frame of reference. Thecombination of the arbitrary control volume and the lagrangian coordinatesystem means that material derivatives of volume integrals will be encoun-tered. As was mentioned in the previous section, it is necessary to transformsuch terms into equivalent expressions involving volume integrals ofeulerian derivatives. The theorem that permits such a transformation iscalled Reynolds transport theorem.

    Consider a specic mass of uid and follow it for a short period of timedt as it ows.Let abe any property of the uid such as itsmass,momentum insome direction, or energy. Since a specic mass of uid is being consideredand since x0, y0, z0, and t are the independent variables in the lagrangianframework, the quantity a will be a function of t only as the control volumemoves with the uid.That is, a atonly and the rate of change of the inte-gral of awill be dened by the following limit:

    DDt

    ZV t

    at dV limdt!0

    1dt

    ZV tdt

    at dt dV ZV t

    at dV" #( )

    where V t is the control volume containing the specied mass of uid andwhich may change its size and shape as it ows.The quantity at dt inte-grated over V t will now be subtracted, then added again inside the abovelimit.

    DDt

    ZV t

    at dV limdt!0

    (1dt

    ZV tdt

    at dt dV ZV t

    at dt dV" #

    1dt

    ZV t

    at dt dV ZV t

    at dV" #)

    The rst two integrals inside this limit correspond to holding the integrandxed and permitting the control volume V to vary while the second twointegrals correspond to holding V xed and permitting the integrand a tovary.The latter component of the change is, by denition, the integral of thefamiliar eulerian derivative with respect to time.Then the expression for thelagrangian derivative of the integral of a may be written in the followingform:

    DDt

    ZV t

    at dV limdt!0

    1dt

    ZV tdtV t

    at dt dV" #( )

    ZV t

    @a@t

    dV

    The remaining limit, corresponding to the volume V changing while aremains xed, may be evaluated from geometric considerations.

    10 Chapter1

  • Figure1.2a shows thecontrol volumeV t thatencloses themassof uidbeingconsideredbothat time tandat time t dt.During this time interval thecontrol volume has moved downstream and has changed its size and shape.The surface that encloses V t is denoted by St, and at any point on thissurface the velocity may be denoted by u and the unit outward normal by n.

    Figure1.2b shows the control volumeV t dt superimposed onV t,and an element of the dierence in volumes is detailed. The perpendiculardistance from any point on the inner surface to the outer surface is un dt, sothat an element of surface area dS will correspond to an element of volumechange dV in which dV un dt dS.Then the volume integral inside the limit

    FIGURE 1.2 (a) Arbitrarily shaped control volume at times t and t dt, and(b) superposition of the control volumes at these times showing an element dV ofthe volume change.

    Basic Conservation Laws 11

  • in the foregoing equationmay be transformed into a surface integral inwhichdV is replaced by un dt dS.

    DDt

    ZV t

    at dV limdt!0

    ZSt

    at dtun dS" #( )

    ZV t

    @a@t

    dV

    ZSt

    atun dS ZV t

    @a@t

    dV

    Having completed the limiting process, the lagrangian derivative of a volumeintegral has been converted into a surface integral and a volume integral inwhich the integrands contain only eulerian derivatives. Aswasmentioned inthe previous section, it is necessary to obtain each term in the conservationequations as the volume integral of something. The foregoing form ofReynolds transport theorem may be put in this desired form by convertingthe surface integral to a volume integral by use of Gauss theorem,which isformulated in Appendix A. In this way the surface-integral term becomesZ

    Statun dS

    ZV t=au dV

    Substituting this result into the foregoing expression and combining the twovolume integrals gives the preferred form of Reynolds transport theorem.

    DDt

    ZVa dV

    ZV

    @a@t =au

    dV

    Or, in tensor notation,

    DDt

    ZVa dV

    ZV

    @a@t @@xk

    auk

    dV 1:2

    Equation (1.2) relates the lagrangian derivative of a volume integral of a givenmass toavolume integral inwhich the integrandhaseulerianderivativesonly.

    Having established the method to be used to derive the basic con-servation equations and having established the necessary backgroundmaterial, it remains to invoke the various conservation principles. The rstsuch principle to be treated will be the conservation of mass.

    1.6 CONSERVATIONOFMASS

    Consider a specic mass of uid whose volumeV is arbitrarily chosen. If thisgiven uid mass is followed as it ows, its size and shape will be observed to

    12 Chapter1

  • change but its mass will remain unchanged. This is the principle of massconservationwhich applies to uids inwhich no nuclear reactions are takingplace.The mathematical equivalence of the statement of mass conservationis to set the lagrangian derivative D=Dt of the mass of uid contained inV,which is

    RV r dV , equal to zero. That is, the equation that expresses con-

    servation of mass is

    DDt

    ZVr dV 0

    This equation may be converted to a volume integral in which the integrandcontains only eulerian derivatives by use of Reynolds transport theorem[Eq. (1.2)], in which the uid property a is, in this case, the mass density r.

    ZV

    @r@t @@xk

    ruk

    dV 0

    Since the volumeVwas arbitrarily chosen, the only way in which the aboveequation can be satised for all possible choices ofV is for the integrand to bezero.Then the equation expressing conservation of mass becomes

    @r@t @@xk

    ruk 0 1:3a

    Equation (1.3a) expresses more than the fact that mass is conserved. Since itis a partial dierential equation, the implication is that the velocity is con-tinuous. For this reason Eq. (1.3a) is usually called the continuity equation.The derivation which has been given here is for a single-phase uid in whichno change of phase is taking place. If two phases were present, such as waterand steam, the starting statement would be that the rate at which the mass ofuid 1 is increasing is equal to the rate at which the mass of uid 2 isdecreasing. The generalization to cases of multiphase uids and to cases ofnuclear reactions is obvious. Since such cases cause no changes in the basicideas or principles, they will not be included in this treatment of thefundamentals.

    In many practical cases of uid ow the variation of density of the uidmay be ignored, as for most cases of the owof liquids. In such cases the uidis said to be incompressible, which means that as a given mass of uid is fol-lowed, not only will its mass be observed to remain constant but its volume,and hence its density,will be observed to remain constant. Mathematically,this statement may be written as

    Basic Conservation Laws 13

  • DrDt

    0

    In order to use this special simplication, the continuity equation is rstexpanded by use of a vector identity given in Appendix A.

    @r@t uk @r

    @xk r @uk

    @xk 0

    The rst and second terms in this form of the continuity equation will berecognized as being the eulerian form of the material derivative as given byEq. (1.1).That is, an alternative form of Eq. (1.3a) is

    DrDt

    r @uk@xk

    0 1:3b

    This mixed form of the continuity equation in which one term is given as alagrangian derivative and the other as an eulerian derivative is not useful foractually solving uid-ow problems. However, it is frequently used in themanipulations that reduce the governing equations to alternative forms, andfor this reason it has been identied for future reference. An immediateexample of such a case is the incompressible uid under discussion. SinceDr=Dt 0 for such a uid, Eq. (1.3b) shows that the continuity equationassumes the simpler form r@uk=@xk 0.Since r cannot be zero in general,the continuity equation for an incompressible uid becomes

    @uk@xk

    0 incompressible 1:3c

    It should be noted that Eq. (13c) is valid not only for the special case ofDr=Dt 0 in which r constant everywhere, but also for stratied-uidows of the type depicted in Fig. 1.3. A uid particle that follows the linesr r1 or r r2 will have its density remain xed at r r1 or r r2 so thatDr=Dt 0. However, r is not constant everywhere, so that @r=@x 6 0 and@r=@y 6 0. Such density stratications may occur in the ocean (owing tosalinity variations) or in the atmosphere (owing to temperature variations).However, in the majority of cases in which the uid may be considered to beincompressible, the density is constant everywhere.

    Equation (1.3), in either the general form (1.3a) or the incompressibleform (1.3c), is the rst condition that has to be satised by the velocity andthe density. No dynamical relations have been used to this point, but theconservation-of-momentum principle will utilize dynamics.

    14 Chapter1

  • 1.7 CONSERVATIONOFMOMENTUM

    The principle of conservation of momentum is, in eect, an application ofNewtons second law of motion to an element of the uid.That is,when con-sideringagivenmassof uid ina lagrangian frameof reference, it isstated thatthe rateatwhichthemomentumof theuidmass ischanging isequal to thenetexternal force acting on the mass. Some individuals prefer to think of forcesonly and restate this law in the form that the inertia force (due to accelerationof the element) is equal to the net external force acting on the element.

    The external forces thatmay act on amass of the uidmay be classed aseither body forces, such as gravitational or electromagnetic forces,or surfaceforces, such as pressure forces or viscous stresses. Then, if f is a vector thatrepresents the resultant of the body forces per unit mass, the net externalbody force acting on a mass of volumeVwill be

    RV rf dV . Also, if P is a sur-

    face vector that represents the resultant surface force per unit area, the netexternal surface force acting on the surface S containingVwill be

    Rs PdS.

    According to the statement of the physical law that is being imposed inthis section, the sum of the resultant forces evaluated above is equal to therate of change of momentum (or inertia force).Themass per unit volume is rand its momentum is ru, so that the momentum contained in the volumeV isRV ru dV .Then, if the mass of the arbitrarily chosen volumeV is observed inthe lagrangian frame of reference, the rate of change of momentum of themass contained with V will be D=Dt RV ru dV . Thus, the mathematical

    FIGURE 1.3 Flow of a density-stratied uid in which Dr=Dt 0 but for which@r=@x 6 0 and @r=@y 6 0.

    Basic Conservation Laws 15

  • equation that results from imposing the physical law of conservation ofmomentum is

    DDt

    ZVru dV

    ZsP dS

    ZVrf dV

    In general, there are nine components of stress at any givenpoint,one normalcomponent and two shear components on each coordinate plane.These ninecomponents of stress aremost easily illustrated by use of a cubical element inwhich the faces of the cube are orthogonal to the cartesian coordinates, asshown in Fig.1.4, and inwhich the stress componentswill act at a point as thelengthof the cube tends to zero. InFig.1.4 the cartesian coordinates x,y, and zhave been denoted by x1, x2, and x3, respectively. This permits the compo-nents of stress to be identied by a double-subscript notation. In this nota-tion, a particular component of the stressmay be represented by the quantitysij , in which the rst subscript indicates that this stress component acts on

    FIGURE1.4 Representation of the nine components of stress that may act at a pointin a uid.

    16 Chapter1

  • the plane xi constant and the second subscript indicates that it acts in the xjdirection.

    The fact that the stress may be represented by the quantity sij, inwhich i and j may be 1, 2, or 3, means that the stress at a point may berepresented by a tensor of rank 2. However, on the surface of our controlvolume it was observed that there would be a vector force at each point,and this force was represented by P. The surface force vector P may berelated to the stress tensor sij as follows: The three stress components act-ing on the plane x1 constant are s11, s12, and s13. Since the unit normalvector acting on this surface is n1, the resulting force acting in the x1direction is P1 s11n1. Likewise, the forces acting in the x2 direction andthe x3 direction are, respectively, P2 s12n1 and P3 s13n1. Then, for anarbitrarily oriented surface whose unit normal has components n1, n2, andn3, the surface force will be given by Pj sijni in which i is summed from 1to 3. That is, in tensor notation the equation expressing conservation ofmomentum becomes

    DDt

    ZVruj dV

    Zssijni dS

    ZVrfj dV

    The left-hand side of this equation may be converted to a volume integral inwhich the integrand contains only eulerian derivatives by use of Reynoldstransport theorem, Eq. (1.2), in which the uid property a here is themomentum per unit volume ruj in the xj direction. At the same time the sur-face integral on the right-hand side may be converted into a volume integralby use of Gauss theorem as given in Appendix B. In this way the equationthat evolved fromNewtons second law becomesZ

    V

    @

    @truj @

    @xkrujuk

    dV

    ZV

    @sij@xi

    dV ZVrfj dV

    All these volume integrals may be collected to express this equation in theform

    RV f gdV 0,where the integrand is a dierential equation in eulerian

    coordinates.Asbefore, the arbitrariness of the choice of the control volumeVis now used to show that the integrand of the above integro-dierentialequation must be zero. This gives the following dierential equation to besatised by the eld variables in order that the basic law of dynamics may besatised:

    @

    @truj @

    @xkrujuk @sij

    @xi rfj

    Basic Conservation Laws 17

  • The left-hand side of this equation may be further simplied if the two termsinvolved are expanded in which the quantity rujuk is considered to be theproduct of ruk and uj .

    r@uj@t

    uj @r@t uj @

    @xkruk ruk @uj

    @xk @sij

    @xi rfj

    The second and third terms on the left-hand side of this equation are nowseen to sum to zero, since they amount to the continuity Eq. (1.3a) multipliedby the velocity uj.With this simplication, the equation that expresses con-servation of momentum becomes

    r@uj@t

    ruk @uj@xk

    @sij@xi

    rfj 1:4

    It is useful to recall that this equation came from an application of Newtonssecond law to an element of the uid. The left-hand side of Eq. (1.4) repre-sents the rate of change of momentum of a unit volume of the uid (or theinertia force per unit volume). The rst term is the familiar temporal accel-eration term, while the second term is a convective acceleration andaccounts for local accelerations (such as around obstacles) even when theow is steady.Note also that this second term is nonlinear, since the velocityappears quadratically. On the right-hand side of Eq. (1.4) are the forcescausing the acceleration. The rst of these is due to the gradient of surfaceshear stresses while the second is due to body forces, such as gravity,whichact on themass of the uid.Aclear understanding of the physical signicanceof each of the terms in Eq. (1.4) is essential when approximations to the fullgoverning equationsmust bemade.The surface-stress tensor sijhas not beenfully explained up to this point, but it will be investigated in detail in a latersection.

    1.8 CONSERVATIONOF ENERGY

    The principle of conservation of energy amounts to an application of the rstlaw of thermodynamics to a uid element as it ows.The rst law of thermo-dynamics applies to a thermodynamic system that is originally at rest and,after some event, is nally at rest again. Under these conditions it is statedthat the change in internal energy, due to the event, is equal to the sum of thetotal workdone on the systemduring the course of the event and any heat thatwas added. Although a specied mass of uid in a lagrangian frame of refer-encemay be considered tobe a thermodynamic system, it is, in general, neverat rest and therefore never in equilibrium. However, in the thermodynamicsense a owing uid is seldom far from a state of equilibrium, and the

    18 Chapter1

  • apparent diculty may be overcome by considering the instantaneousenergy of the uid to consist of two parts: intrinsic or internal energy andkinetic energy. That is, when applying the rst law of thermodynamics, theenergy referred to is considered to be the sum of the internal energy per unitmass e and the kinetic energy per unit mass 12 uu. In this way the modiedform of the rst law of thermodynamics that will be applied to an element ofthe uid states that the rate of change of the total energy (intrinsic pluskinetic) of the uid as it ows is equal to the sum of the rate at which work isbeing done on the uid by external forces and the rate at which heat is beingadded by conduction.

    With this basic law in mind, we again consider any arbitrary mass ofuid of volumeVand follow it in a lagrangian frame of reference as it ows.The total energy of this mass per unit volume is re 12 ruu, so that the totalenergy contained inVwill be

    RV re 12ruu dV . As was established in the

    previous section, there are two types of external forces that may act on theuid mass under consideration.The work done on the uid by these forces isgiven by the product of the velocity and the component of each force that iscolinear with the velocity.That is, the work done is the scalar product of thevelocity vector and the force vector. One type of force that may act on theuid is a surface stress whose magnitude per unit area is represented bythe vectorP.Then the total work done owing to such forces will be

    Rs uP dS,

    where S is the surface area enclosingV. The other type of force that may acton the uid is a body force whose magnitude per unit mass is denoted bythe vector f. Then the total work done on the uid due to such forces willbeRV urf dV . Finally, an expression for the heat added to the uid is

    required. Let the vector q denote the conductive heat ux leaving the controlvolume.Then the quantity of heat leaving the uidmass per unit time per unitsurface area will be qn, where n is the unit outward normal, so that the netamount of heat leaving the uid per unit time will be

    Rs qn dS.

    Having evaluated eachof the terms appearing in the physical law that isto be imposed, the statement may now be written down in analytic form. Indoing so, it must be borne in mind that the physical law is being applied to aspecic, though arbitrarily chosen, mass of uid so that lagrangian deriva-tives must be employed. In this way, the expression of the statement that therate of change of total energy is equal to the rate at which work is being doneplus the rate at which heat is being added becomes

    DDt

    ZVre 12ruu dV Rs uP dS RV urf dV Rs qn dS

    This equationmay be converted to one involving eulerian derivatives only byuse of Reynolds transport theorem, Eq. (1.2), in which the uid property a is

    Basic Conservation Laws 19

  • here the total energy per unit volume re 12ruu. The resulting integro-dierential equation isZ

    V

    @

    @t

    re 1

    2ruu

    @@xk

    re 1

    2ruu

    uk

    dV

    ZsuP dS

    ZVurf dV

    Zsqn dS

    The next step is to convert the two surface integrals into volume integrals sothat the arbitrariness of Vmay be exploited to obtain a dierential equationonly.Using the fact that the force vectorP is related to the stress tensor sij bythe equation Pj sijni , aswas shown in the previous section, the rst surfaceintegral may be converted to a volume integral as follows:Z

    suP dS

    Zsujsijni dS

    ZV

    @

    @xiujsij dV

    Here use has been made of Gauss theorem as documented in Appendix B.Gauss theoremmay be applied directly to the heat-ux term to giveZ

    sqn dS

    Zsqjnj dS

    ZV

    @qj@xj

    dV

    Since the stress tensor sij has been brought into the energy equation, it isnecessary to use the tensor notation from this point on.Then the expressionfor conservation of energy becomesZ

    V

    @

    @tre 12rujuj

    @

    @xkre 12rujujuk

    dV

    ZV

    @

    @xiujsij dV

    ZVujrfj dV

    ZV

    @qj@xj

    dV

    Having converted each term to volume integrals, the conservation equationmay be considered to be of the form

    RV f g dV 0,where the choice of V is

    arbitrary.Then thequantity inside thebrackets in the integrandmust be zero,which results in the following dierential equation:

    @

    @tre 12rujuj

    @

    @xkre 12rujujuk

    @

    @xiujsij ujrfj @qj

    @xj

    This equation may be made considerably simpler by using the equationswhich have already beenderived, aswill nowbedemonstrated.The rst termon the left-hand side may be expanded by considered re and 12rujuj to be theproducts (r)(e) and r12ujuj, respectively.Then

    20 Chapter1

  • @@tre 12rujuj r

    @e@t e @r

    @t r @

    @t12ujuj 12ujuj

    @r@t

    Similarly, the second term on the left-hand side of the basic equation may beexpanded by considering reuk to be the product (e)(ruk) and 12rujujuk to bethe product 12ujujruk.Thus,

    @

    @xkre 12rujujuk e

    @

    @xkruk ruk @e

    @xk

    12ujuj@

    @xkruk ruk @

    @xk12ujuj

    In this last equation, the quantity @=@xkruk,which appears in the rst andthird terms on the right-hand side, may be replaced by@r=@t in view of thecontinuity Eq. (1.3a). Hence it follows that

    @

    @xkre 12rujujuk e

    @r@t ruk @e

    @xk 12ujuj

    @r@t ruk @

    @xk12ujuj

    Now when the two components constituting the left-hand side of the basicconservation equation are added, the two terms with minus signs above arecanceled by corresponding terms with plus signs to give

    @

    @tre 12rujuj

    @

    @xkre 12rujujuk

    r @e@t ruk @e

    @xk r @

    @t12ujuj ruk

    @

    @xk12ujuj

    r @e@t ruk @e

    @xk ruj @uj

    @t rujuk @uj

    @xk

    Then, noting that

    @

    @xiujsij uj @sij

    @xi sij @uj

    @xi

    the equation that expresses the conservation of energy becomes

    r@e@t ruk @e

    @xk ruj @uj

    @t rujuk @uj

    @xk uj @sij

    @xi sij @uj

    @xi ujrfj @qj

    @xj

    Now it can be seen that the third and fourth terms on the left-hand side arecanceled by the rst and third terms on the right-hand side, since these terms

    Basic Conservation Laws 21

  • collectively amount to the product of uj with the momentum Eq. (1.4).Thusthe equation expressing conservation of thermal energy becomes

    r@e@t ruk @e

    @xk sij @uj

    @xi @qj

    @xj1:5

    The terms that were dropped in the last simplication were the mechanical-energy terms.The equation of conservation of momentum, Eq. (1.4), may beregarded as an equation of balancing forces with j as the free subscript.Therefore, the scalar product of each force with the velocity vector, or themultiplication by uj, gives the rate of doing work by the mechanical forces,which is the mechanical energy. On the other hand, Eq. (1.5) is a balance ofthermal energy, which is what is left when the mechanical energy is sub-tracted from the balance of total energy, and is usually referred to as simplythe energy equation.

    As with the equation of momentum conservation, it is instructive tointerpret each of the terms appearing in Eq. (1.5) physically. The entire left-hand side represents the rate of change of internal energy, the rst termbeingthe temporal change while the second is due to local convective changescaused by the uid owing from one area to another. The entire right-handside represents the cause of the change in internal energy. The rst of theseterms represents the conversion of mechanical energy into thermal energydue to the action of the surface stresses.Aswill be seen later, part of this con-version is reversible and part is irreversible. The nal term in the equationrepresents the rate at which heat is being added by conduction fromoutside.

    1.9 DISCUSSIONOFCONSERVATION EQUATIONS

    The basic conservation laws, Eqs. (1.3a), (1.4), and (1.5), represent ve scalarequations that the uid properties must satisfy as the uid ows. The con-tinuity and the energy equations are scalar equations,while the momentumequation is a vector equation which represents three scalar equations. Twoequations of statemay be added tobring the number of equations up to seven,but our basic conservation laws have introduced seventeen unknowns.These unknowns are the scalars r and e, the density and the internal energy,respectively; the vectors uj and qj, the velocity and heat ux, respectively,each vector having three components; and the stress tensor sij,which has, ingeneral, nine independent components.

    In order to obtain a complete set of equations, the stress tensor sij andthe heat-ux vector qj must be further specied. This leads to the so-calledconstitutive equations inwhich the stress tensor is related to the deformationtensor and the heat-ux vector is related to temperature gradients. Althoughthe latter relation is very simple, the former is quite complicated and requires

    22 Chapter1

  • either an intimate knowledge of tensor analysis or a clear understanding ofthe physical interpretation of certain tensor quantities.For this reason, priorto establishing the constitutive relations the tensor equivalents of rotationand rate of shear will be established.

    1.10 ROTATION ANDRATEOF SHEAR

    It is the purpose of this section to consider the rotation of a uid elementabout its own axis and the shearing of a uid element and to identify thetensor quantities that represent these physical quantities.This is most easilydone by considering an innitesimal uid element of rectangular cross sec-tion and observing its change in shape and orientation as it ows.

    Figure 1.5 shows a two-dimensional element of uid (or the projectionof a three-dimensional element)whose dimensions at time t 0 are dx and dy.The uid element is rectangular at time t 0, and its centroid coincides withthe origin of a xed-coordinate system. For purposes of identication, thecorners of the uid element have been labeled A,B,C, andD.

    After a short time interval dt, the centroid of the uid element will havemoved downstream to some new location as shown in Fig. 1.5.The distancethe centroid will have moved in the x direction will be given by

    FIGURE1.5 An innitesimal element of uid at time t 0 (indicated by ABCD) andat time t dt (indicated by A0B0C0D0).

    Basic Conservation Laws 23

  • Dx Z dt0

    uxt; yt dt

    Since the values of x and ymust be close to zero for short times such as dt, thevelocity componentumaybeexpanded in aTaylor series about the point (0,0)to give

    Dx Z dt0

    u0; 0 xt @u@x0; 0 yt @u

    @y0; 0

    dt

    where the dots represents terms that are smaller than those presented andthat will eventually vanish as the limit of dt! 0 is taken. Integrating theleading term explicitly gives

    Dx u0; 0dt Z dt0

    xt @u@x0; 0 yt @u

    @y0; 0

    dt

    u0; 0dt

    similarly

    Dy v0; 0dt

    As well as moving bodily, the uid element will rotate and will be dis-torted as indicated by the corners, which are labeled A0;B0;C 0, and D0 torepresent the element at time t dt. The rotation of the side CD to its newposition C 0D0 is indicated by the angle da,where a is positive when measuredcounterclockwise. Similarly, the rotation of the side BC to its new positionB0C 0 is indicated by the angle db, where b is positive when measured clock-wise. Expressions for da and db in terms of the velocity components may beobtained as follows:From the geometry of the element as it appears at time t dt,

    da tan1 y component of D0C 0

    x component of D0C 0

    tan1 v12dx;12dy dt v12dx;12dy dt

    dx

    where v is evaluated rst at the point D, whose coordinates are 12dx;12dy,and secondly at the point C, whose coordinates are 12dx;12dy. Thex component of the side D0C 0 will be only slightly dierent from dx, and

    24 Chapter1

  • it turns out that the precise departure from this value need not be evaluatedexplicitly.

    Expanding the velocity component v in aTaylor series about the point(0,0) results in the following expression for da:

    da tan1 v0; 0 12 dx@v=@x0; 0 12 dy@v=@y0; 0 dt

    dx1

    v0; 0 12 dx@v=@x0; 0 12 dy@v=@y0; 0 dtdx1

    tan1 dx@v=@x0; 0 dtdx1

    tan1 @v=@x0; 0 dt1

    tan1 @v@x0; 0

    dt

    Since the argument of the arctangent is small, the entire right-hand side maybe expanded to give

    da @v@x0; 0

    dt

    dadt @v

    @x0; 0

    This expression represents the change in the angle a per unit time so that inthe limit as dx, dy, and dt all tend to zero, this expression becomes

    _aa @v@x0; 0

    where _aa is the time derivative of the angle a. By an identical procedure it fol-lows that the time derivative of the angle b is given by

    _bb @u@y0; 0

    Basic Conservation Laws 25

  • Recall that a is measured counterclockwise and b is measured clockwise.Thus the rate of clockwise rotation of the uid element about its centroid isgiven by

    12 _bb _aa

    12

    @u@y

    @v@x

    Likewise the shearing action is measured by the rate at which the sides B0C 0

    andD0C 0 are approaching each other and is therefore given by the quantity

    12_bb _aa

    12

    @u@y

    @v@x

    The foregoing analysis was carried out in two dimensions which may beconsidered as the projection of a three-dimensional element on the xy plane.If the analysis is carried out in the other planes, it may be veried that the rateof rotation of the element about its own axes and the rate of shearing aregiven by

    Rate of rotation 12

    @ui@xj

    @uj@xi

    1:6a

    Rate of shearing 12

    @ui@xj

    @uj@xi

    1:6b

    That is, both the rate of rotation and the rate of shearing may be representedby tensors of rank 2. It will be noted that the rate-of-rotation tensor is anti-symmetric and therefore has only three independent components while therate-of-shearing tensor is symmetric and therefore has six independentcomponents. These two quantities are actually the antisymmetric part andthe symmetric part of another tensor called the deformation-rate tensor, asmay be shown as follows:Dene the deformation-rate tensor eij as

    eij @ui@xj

    12

    @ui@xj

    @uj@xi

    12

    @ui@xj

    @uj@xi

    That is, the antisymmetric part of the deformation-rate tensor represents therate of rotationof a uid element in that ow eld about its own axeswhile thesymmetric part of the deformation-rate tensor represents the rate of shearingof the uid element.

    26 Chapter1

  • 1.11 CONSTITUTIVE EQUATIONS

    In this section the nine elements of the stress tensor sij will be related to thenine elements of the deformation-rate tensor ekl by a set of parameters. Allthese parameters except two will be evaluated analytically, and the remain-ing two,which are the viscosity coecients,must be determined empirically.In order to achieve this end, the postulates for a newtonian uid will beintroduced directly. Water and air are by far the most abundant uids onearth, and they behave like newtonian uids, as do many other commonuids. It should be pointed out, however, that some uids do not behave in anewtonian manner, and their special characteristics are among the topics ofcurrent research. One example is the class of uids called viscoelastic uids,whose properties may be used to reduce the drag of a body. Since this book isconcerned with the classical fundamentals only, the newtonian uid will betreated directly. If the various steps are clearly understood, there should beno conceptual diculty in following the details of some of themore complexuids such as viscoelastic uids.

    Certain observations and postulates will now be made concerning thestress tensor.The precise manner in which the postulates are made is largelya mater of taste, but when the newtonian uid is being treated, the resultingequations are always the same. The following are the four conditions thestress tensor is supposed to satisfy:

    1. When the fluid is at rest, the stress is hydrostatic and the pressureexerted by the fluid is the thermodynamic pressure.

    2. The stress tensor sij is linearly related to the deformation-rate ten-sor ekl and depends only on that tensor.

    3. Since there is no shearing action in a solid-body rotation of thefluid, no shear stresses will act during such a motion.

    4. There are not preferred directions in the fluid, so that the fluidproperties are point functions.

    Condition1 requires that the stress tensor sij be of the form

    sij pdij tijwhere tij depends upon the motion of the uid only and is called the shear-stress tensor.The quantity p is the thermodynamic pressure and dij is the Kro-necker delta. The pressure term is negative, since the sign convention beingused here is that normal stresses are positive when they are tensile in nature.

    The remaining unknown in the constitutive equation for stress is theshear-stress tensor tij. Condition 2 postulates that the stress tensor, andhence the shear-stress tensor, is linearly related to the deformation-ratetensor. This is the distinguishing feature of newtonian uids. In general, the

    Basic Conservation Laws 27

  • shear-stress tensor could depend upon some power of the velocity gradientsother than unity, and it could depend upon the velocity itself as well as thevelocity gradient.The condition postulated here can be veried experimen-tally in simple ow elds inmost common uids, and the results predicted formore complex ow elds yield results that agree with physical observations.This is the sole justication for condition 2.

    There are nine elements in the shear-stress tensor tij, and each of theseelements may be expressed as a linear combination of the nine elements inthe deformation-rate tensor ekl ( just as a vector may be represented as a lin-ear combination of components of the base vectors).That is, each of the nineelements of tijwill in general be a linear combination of the nine elements ofekl so that 81parameters are needed to relate tij to ekl.Thismeans that a tensorof rank 4 is required so that the general form of tij will be, according tocondition 2,

    tij aijkl @uk@xl

    It was shown in the previous section that the tensor @uk=@xl, like any othertensor of rank 2, could be broken down into an antisymmetric part and asymmetric part. Here the antisymmetric part corresponds to the rateof rotation of a uid element and the symmetric part corresponds to theshearing rate. According to condition 3, if the ow eld is executing a simplesolid-body rotation, there should be no shear stresses in the uid. But for asolid-body rotation the antisymmetric part of @uk=@xl , namely,12@uk= @xl @ul=@xk,will not be zero.Hence, in order that condition 3maybe satised, the coecients of this part of the deformation-rate tensor mustbe zero.That is, the constitutive relation for stress must be of the form

    tij 12bijkl@uk@xl

    @ul@xk

    The 81 elements of the fourth-rank tensor bijkl are still undertermined, butcondition 4 has yet to be imposed.This condition is the so-called conditionof isotropy,which guarantees that the results obtained should be independentof the orientation of the coordinate system chosen. In Appendix B, the sum-mary of some useful tensor relations, it is pointed out that the most generalisotropic tensor of rank 4 is of the form

    bijkl ldijdkl mdikdjl dildjk gdikdjl dildjk

    where l, m, and g are scalars.The proof of this is straightforward but tedious.The general tensor is subjected to a series of coordinate rotations and

    28 Chapter1

  • inections, and the condition of invariance is applied. In this way the 81quantities contained in the general tensor are reduced to three independentquantities in the isotropic case. In the case of the fourth-rank tensor relatingthe shear-stress tensor to the deformation-rate tensor, namely, bijkl, not onlymust it be isotropic but it must be symmetric in view of condition 3.That is,the coecient gmust be zero in this case so that the expression for the shearstress becomes

    tij 12ldijdkl mdikdjl dildjk@uk@xl

    @ul@xk

    Using the fact that dkl 0 unless lk shows that12ldijdkl

    @uk@xl

    @ul@xk

    ldij @uk

    @xk

    in which l has been replaced by k. Likewise, replacing k by i and l by j showsthat

    12mdikdjl

    @uk@xl

    @ul@xk

    12m

    @ui@xj

    @uj@xi

    and replacing l by i and k by j shows that

    12mdildjk

    @uk@xl

    @ul@xk

    12m

    @uj@xi

    @ui@xj

    Hence the expression for the shear-stress tensor becomes

    tij ldij @uk@xk

    m @ui@xj

    @uj@xi

    Thus the constitutive relation for stress in a newtonian uid becomes

    sij pdij ldij @uk@xk

    m @ui@xj

    @uj@xi

    1:7

    which shows that the stress is represented by a second-order symmetrictensor.

    The nine elements of the stress tensor sij have now been expressed interms of the pressure and the velocity gradients, which have all been pre-viously introduced, and two coecients l and m.These coecients cannot bedetermined analytically and must be determined empirically. Up to thispoint both l and m are just coecients but their nature and physical sig-nicance will be discussed in the next section.

    Basic Conservation Laws 29

  • The second constitutive relation involves the heat-ux vector qj,whichis due to conduction alone. Fouriers law of heat conduction states that theheat ux by conduction is proportional to the negative temperature gradientso that

    qj k @T@xj

    1:8

    This is the constitutive equation for the heat ux,where the proportionalityfactor k in Fouriers law is the thermal conductivity of the uid. In usingEq. (1.8), it is implicitly assumed that the concept of temperature, asemployed in equilibrium thermodynamics, also applies to a moving uid.

    1.12 VISCOSITYCOEFFICIENTS

    It was pointed out in the previous section that the parameters l and m,whichappear in the constitutive equations for stress, must be determined experi-mentally. It is the purpose of this section to establish a physical interpreta-tion of these two parameters and thus show themanner inwhich they may beevaluated.

    Consider a simple shear ow of an incompressible uid in which thevelocity components are dened by

    u uyv w 0

    That is, only the x component of velocity is nonzero, and that component is afunction of y only.From the denition of this ow eld the components of thestress tensor may be evaluated from Eq. (1.7) to give

    s12 s21 m dudys11 s22 s33 ps13 s31 s23 s32 0

    That is, the normal components of the stress are dened by the thermo-dynamic pressure, and the nonzero shear components of the stress areproportional to the velocity gradient with the parameter m as the pro-portionality factor. But, from Newtons law of viscosity, the proportionalityfactor between the shear stress and the velocity gradient in a simple shearow is the dynamic viscosity. Hence the quantity m that appears in the con-stitutive equation for stress is thedynamic viscosityof the uid.Frequently the

    30 Chapter1

  • kinematic viscosity, dened by v m=r, is used instead of the dynamicviscosity.

    The parameter l in Eq. (1.7) is usually referred to as the second viscositycoecient. In order to establish its signicance, the average normal stresscomponent ppwill be calculated.

    pp 13s11 s22 s33

    This average normal stress is the mechanical pressure in the uid and it isequal to one-third of the trace of the stress tensor. Since the mechanicalpressure is either purely hydrostatic or hydrostatic plus a componentinduced by the stresses that result from the motion of the uid, it will, ingeneral, be dierent from the thermodynamic pressure p.Using Eq. (1.7), themechanical pressure ppmay be evaluated as follows:

    pp 13

    p l @uk@xk

    2m @u@x

    p l @uk

    @xk 2m @v

    @y

    p l @uk@xk

    2m @w@z

    p l @uk@xk

    23m@uk@xk

    p l 23m @uk

    @xk

    That is, the dierence between the thermodynamic pressure and themechanical pressure is proportional to the divergence of the velocity vector.The proportionality factor is usually referred to as the bulk viscosity and isdenoted by K.That is,

    p pp K @uk@xk

    where K l 23 m. Of the three viscosity coecients m, l, and K, only twoare independent and the third is dened by the above equation. For pur-poses of physical interpretation of these viscosity coecients it is preferredto discuss m (which has already been done) and K, leaving l to be dened byl K 23m.

    In order to identify the physical signicance of the bulk viscosity, someof the results of the kinetic theory of gaseswill be used.Themechanical pres-sure is ameasureof the translationalenergyof themoleculesonly,whereas the

    Basic Conservation Laws 31

  • thermodynamic pressure is a measure of the total energy, which includesvibrational and rotational modes of energy as well as the translational mode.For liquids, other forms of energy are also included such as intermolecularattraction.These dierent modes of molecular energy have dierent relaxa-tion times, so that in a ow eld it is possible to have energy transferredfrom one mode to another.The bulk viscosity is a measure of this transfer ofenergy from the translational mode to the other modes, as may be seenfrom the relation p pp K@uk=@xk. For example, during the passagethrough a shock wave the vibrational modes of energy are excited at theexpense of the translationalmodes, so that the bulk viscosity will be nonzeroin this case.

    The above discussion has been for a polyatomic molecule of a liquid ora gas. If the uid is amonatomic gas, the onlymode of molecular energy is thetranslational mode. Then, for such a gas the mechanical pressure and thethermodynamic pressure are the same, so that the bulk viscosity is zero.That is,

    l 23m

    which is called Stokes relation, so that there is only one independent viscos-ity coecient in the case of monatomic gases. For polyatomic gases and forliquids the departure from K 0 is frequently small, and many authorsincorporate Stokes relation in the constitutive relation (1.7) for stress. In anycase, for incompressible uids Eq. (1.7) shows that it is immaterial whetherl 23m or not, for then the term involving l is zero by virtue of the con-tinuity equation.

    1.13 NAVIER-STOKES EQUATIONS

    The equation of momentumconservation (1.4) together with the constitutiverelation for a newtonian uid [Eqs. (1.7)] yield the famous Navier-Stokesequations, which are the principal conditions to be satised by a uid as itows. Having obtained an expression for the stress tensor, the term @sij=@xiwhich appears in Eq. (1.4) may be evaluated explicitly as follows:

    @sij@xi

    @@xi

    pdij ldij @uk@xk

    m @ui@xj

    @uj@xi

    @p@xj

    @@xj

    l@uk@xk

    @@xi

    m@ui@xj

    @uj@xi

    32 Chapter1

  • where, in the rst two terms, i has been replaced by j, since it is only when i jthat these terms are nonzero. Substituting this result into Eq. (1.4) gives

    r@uj@t

    ruk @uj@xk

    @p@xj

    @@xj

    l@uk@xk

    @@xi

    m@ui@xj

    @uj@xi

    rfi 1:9a

    Equations (1.9a) are known as the Navier-Stokes equations, and they repre-sent three scalar equations corresponding to the three possible values of thefree subscript j. In the most frequently encountered situations the uid maybe assumed to be incompressible and the dynamic viscosity may beassumed to be constant. Under these conditions the second term on theright-hand side of Eqs. (1.9a) is identically zero and the viscous-shear termbecomes

    @

    @xim

    @ui@xj

    @uj@xi

    m @

    @xj

    @ui@xi

    @

    2uj@xi@xi

    m @

    2uj@xi@xi

    That is, the viscous-shear term is proportional to the laplacian of the velocityvector, and the constant of proportionality is the dynamic viscosity.Then theNavier-Stokes equations for an incompressible uid of constant densitybecome

    r@uj@t

    ruk @uj@xk

    @p@xj

    m @2uj

    @xi@xi rfj 1:9b

    In the special case of negligible viscous eects, Eqs. (1.9a) become

    r@uj@t

    ruk @uj@xk

    @p@xj

    rfi 1:9c

    Equations (1.9c) are known as Euler equations and the uid is called inviscid.

    1.14 ENERGY EQUATION

    The term sij@uj=@xiwhich appears in the equation of energy conservation(1.5) may now be evaluated explicitly by use of Eq. (1.7).

    sij@uj@xi

    pdij ldij @uk@xk

    m @ui@xj

    @uj@xi

    @uj@xi

    Basic Conservation Laws 33

  • Using the fact that in the rst two terms of the stress tensor i j for the non-zero elements, this expression becomes

    sij@uj@xi

    p @uk@xk

    l @uk@xk

    2m @ui

    @xj @uj

    @xi

    @uj@xi

    It will be recalled that the term sij@uj=@xi represents the work done by thesurface forces. The rst term in the expression for this work done, namely,p@uk=@xk, represents the reversible transfer of energy due to compres-sion.The remaining two terms are collectively called the dissipation functionand are denoted by f.That is,

    F l @uk@xk

    2m @ui

    @xj @uj

    @xi

    @uj@xi

    1:10

    The reason F is called the dissipation function is that it is a measure of therate at whichmechanical energy is being converted into thermal energy.Thismay be readily veried by considering an incompressible uid in a cartesian-coordinate system.Then

    F m @ui@xj

    @uj@xi

    @uj@xi

    m @ui@xj

    @uj@xi

    12

    @uj@xi

    @ui@xj

    12

    @uj@xi

    @ui@xj

    12m@ui@xj

    @uj@xi

    2

    which is a positive denite quantity.This shows that the dissipation functionalways works to increase irreversibly the internal energy of an incompres-sible uid.

    In terms of the dissipation function, the total work done by the surfacestresses is given by

    sij@uj@xi

    p @uk@xk

    F

    Using this result and the constitutive relation for the heat ux [Eq. (1.8)] inthe equationof conservationof energy,Eq. (1.5),yields the energy equation fora newtonian uid.

    34 Chapter1

  • r@e@t ruk @e

    @xk p @uk

    @xk @@xj

    k@T@xj

    F 1:11

    whereF is dened by Eq. (1.10).

    1.15 GOVERNING EQUATIONS FOR NEWTONIANFLUIDS

    The equations that govern themotion of a newtonian uid are the continuityequation (1.3a), the Navier-Stokes equations (1.9a), the energy equation(1.11), and equations of state. For purposes of summary and discussion theseequations will be repeated here.