currie i.g. fundamental mechanics of fluids (m.dekker, 2003)

Basic 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..

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ISBN: 0 - 8247- 0886 -5

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This book covers the fundamental mechanics of fluids as they are treated atthe senior level or in first graduate courses Many excellent books exist thattreat special areas of fluid mechanics such as ideal-fluid flow or boundary-layer theory However, there are very few books at this level that sacrifice anin-depth study of one of these special areas of fluid mechanics for a briefertreatment of a broader area of the fundamentals of fluid 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 fluid 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, flow kinematics,and some basic theorems of fluid mechanics Part II is entitled ‘‘Ideal-FluidFlow,’’ and it covers two-dimensional potential flows, three-dimensionalpotential flows, 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-drivenflows The final part of the book is entitled ‘‘Compressible Flow of InviscidFluids,’’ and this part contains chapters that deal with shock waves, one-dimensional flows, and multidimensional flows 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 fluids while providing techniquesfor solving specific classes of fluid-flow 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 cliche´s 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 may

be completely covered and even augmented Parts II, III, and IV areessentially independent so that they may be interchanged or any one or more

of them may be omitted This permits a high degree of teaching flexibility,and allows the instructor to include or substitute material which is notcovered in the text Such additional material may include free convection,density stratification, 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 tions to this text I had the privilege of taking lectures from F E Marble,

contribu-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 anddeficiencies in the material content of the draft of this text

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

Part I Governing Equations 1

1 Basic Conservation Laws 31.1 Statistical and Continuum Methods

1.2 Eulerian and Lagrangian Coordinates

1.9 Discussion of Conservation Equations

1.10 Rotation and Rate of Shear

2.2 Circulation and Vorticity

2.3 Stream Tubes and Vortex Tubes

2.4 Kinematics of Vortex Lines

3 Special Forms of the Governing Equations 553.1 Kelvin’s Theorem

3.2 Bernoulli Equation

3.3 Crocco’s Equation

3.4 Vorticity Equation

Part II Ideal-Fluid Flow 69

4 Two-Dimensional Potential Flows 734.1 Stream Function

4.2 Complex Potential and Complex Velocity

4.3 Uniform Flows

4.4 Source, Sink, and Vortex Flows

4.5 Flow in a Sector

4.6 Flow Around a Sharp Edge

4.7 Flow Due to a Doublet

4.8 Circular Cylinder Without Circulation

4.9 Circular Cylinder With Circulation

4.10 Blasius’ Integral Laws

4.11 Force and Moment on a Circular Cylinder

4.12 Conformal Transformations

4.13 Joukowski Transformation

4.14 Flow Around Ellipses

4.15 Kutta Condition and the Flat-Plate Airfoil

4.16 Symmetrical Joukowski Airfoil

4.17 Circular-Arc Airfoil

4.18 Joukowski Airfoil

4.19 Schwarz-Christoffel Transformation

4.20 Source in a Channel

4.21 Flow Through an Aperture

4.22 Flow Past a Vertical Flat Plate

5 Three-Dimensional Potential Flows 1615.1 Velocity Potential

5.2 Stokes’ Stream Function

5.3 Solution of the Potential Equation

5.4 Uniform Flow

5.5 Source and Sink

5.6 Flow Due to a Doublet

5.7 Flow Near a Blunt Nose

5.8 Flow Around a Sphere

5.9 Line-Distributed Source

5.10 Sphere in the Flow Field of a Source

5.11 Rankine Solids

5.12 D’Alembert’s Paradox

5.13 Forces Induced by Singularities

5.14 Kinetic Energy of a Moving Fluid

5.15 Apparent Mass

6.1 The General Surface-Wave Problem

6.2 Small-Amplitude Plane Waves

6.3 Propagation of Surface Waves

6.4 Effect of Surface Tension

6.5 Shallow-Liquid Waves of Arbitrary Form

6.6 Complex Potential for Traveling Waves

6.7 Particle Paths for Traveling Waves

6.8 Standing Waves

6.9 Particle Paths for Standing Waves

6.10 Waves in Rectangular Vessels

6.11 Waves in Cylindrical Vessels

6.12 Propagation of Waves at an Interface

Part III Viscous Flows of Incompressible Fluids 249

7.1 Couette Flow

7.2 Poiseuille Flow

7.3 Flow Between Rotating Cylinders

7.4 Stokes’ First Problem

7.5 Stokes’ Second Problem

7.6 Pulsating Flow Between Parallel Surfaces

7.7 Stagnation-Point Flow

7.8 Flow in Convergent and Divergent Channels

7.9 Flow Over a Porous Wall

8 Low-Reynolds-Number Solutions 2888.1 The Stokes Approximation

8.2 Uniform Flow

8.3 Doublet

8.4 Rotlet

8.5 Stokeslet

8.6 Rotating Sphere in a Fluid

8.7 Uniform Flow Past a Sphere

8.8 Uniform Flow Past a Circular Cylinder

8.9 The Oseen Approximation

9.7 Flow in a Convergent Channel

9.8 Approximate Solution for a Flat Surface

9.9 General Momentum Integral

9.10 Ka´rma´n-Pohlhausen Approximation

9.11 Boundary-Layer Separation

9.12 Stability of Boundary Layers

10 Buoyancy-Driven Flows 36310.1 The Boussinesq Approximation

10.2 Thermal Convection

10.3 Boundary-Layer Approximations

10.4 Vertical Isothermal Surface

10.5 Line Source of Heat

10.6 Point Source of Heat

10.7 Stability of Horizontal Layers

Part IV Compressible Flow of Inviscid Fluids 395

11.1 Propagation of Infinitesimal Disturbances

11.2 Propagation of Finite Disturbances

11.3 Rankine-Hugoniot Equations

11.4 Conditions for Normal Shock Waves

11.5 Normal Shock-Wave Equations

11.6 Oblique Shock Waves

12 One-Dimensional Flows 43012.1 Weak Waves

12.2 Weak Shock Tubes

12.3 Wall Reflection of Waves

12.4 Reflection and Refraction at an Interface

13.2 Janzen-Rayleigh Expansion

13.3 Small-Perturbation Theory

13.4 Pressure Coefficient

13.5 Flow Over a Wave-Shaped Wall

13.6 Prandtl-Glauert Rule for Subsonic Flow

13.7 Ackert’s Theory for Supersonic Flows

GOVERNING EQUATIONS

In this ¢rst part of the book a su⁄cient set of equations will be derived, based

on 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 from equations

of state Having established a su⁄cient 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 mathematicalequationsthat result from invoking certainphysical lawspostulated tobe validfor a moving £uid These equations may assume di¡erent forms, dependingupon which variables are chosen and upon which simplifying assumptions aremade.The remaining parts of the book are devoted to solving these governingequations for di¡erent classes of £uid £ows and thereby explaining quantita-tively some of the phenomena that are observed in £uid £ow

1

Basic 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.The ¢rst topic of discussion is the two basic ways in whichthe conservation equations may be derived: the statistical method and thecontinuum method 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 Reynolds’transport theorem, is derived, since this theorem relates derivatives in thelagrangian framework to derivatives in the eulerian framework

Having established the basic method 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 AND CONTINUUM METHODS

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 is

in 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 coe⁄cients, such as thecoe⁄cient 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 ofcontinuous matter At each point of this continuous £uid there is supposed to

be 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 di¡erentialequations governing the ¢eld variables The solution to these di¡erentialequations then de¢nes the variation of each ¢eld variable with space and timewhich corresponds to the mean value of the molecular magnitude of that ¢eldvariable at each corresponding position and time

The statistical method is rather elegant, and it may be used to treat gas

£ows in situations where the continuum concept 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 satis¢ed, there no distinction amonglight gases, dense gases, or even liquids
the 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 continuum concept will now be explored in somedetail The ¢eld variables, such as the density r and the velocity vector u, will

in general be functions of the spatial coordinates and time In symbolic formthis is written as r¼ rðx; tÞ and u ¼ uðx; tÞ, where x is the position vectorwhose certesian coordinates are x, y, and z At any particular point in spacethese continuum variables are de¢ned in terms of the properties of the var-ious molecules that occupy a small volume in the neighborhood of that point.Consider a small volume of £uid DV containing a large number ofmolecules LetDm and v be the mass and velocity of any individual moleculecontained within the volumeDV, as indicated in Fig 1.1.The density and thevelocity at a point in the continuum are then de¢ned by the following limits:

r¼ lim

DV !e

PDmDV

where e is a volume which is su⁄ciently small that e1 =3is small compared withthe smallest signi¢cant length scale in the £ow ¢eld but is su⁄ciently largethat it contains a large number of molecules The summations in the aboveexpressions are taken over all the molecules contained within the volumeDV

FIGURE1.1 An individual molecule in a small volumeDV having a mass Dm andvelocity v

The other ¢eld variables may be de¢ned in terms of the molecular properties

a rocket passes through the edge of the atmosphere

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 in which 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 di¡erent £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

In the lagrangian approach, attention is ¢xed on a particular mass of

£uid 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 it

£ows, resulting in a set of conservation equations in lagrangian coordinates

In this reference frame x, y, z, and t are no longer independent variables, since

if it is known that our colored portion of £uid passed through the coordinates

x0, y0, and z0at some time t0, then its position at some later time may be culated if the velocity components u,v, and w are known.That is, as soon as atime interval (t t0) is speci¢ed, 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 z0are the coordinates which a spe-ci¢ed £uid element passed through at time t0 That is, the coordinates x0, y0,and z0identify which £uid element is being considered, and the time t iden-ti¢es its instantaneous location

cal-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 di¡erent £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 justi¢ed 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 di¡erent derivatives will beestablished

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 speci¢c £uid element is observedfor a short period of time dt as it £ows, its position will change by amounts dx,

dy, and dz while its value of a will 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 z0are initial coordinates for the £uidelement Thus, x, y, and z are no longer independent variables but are

functions of t as de¢ned by the trajectory of the element During the time dtthe change in a may be calculated from di¡erential calculus to be

da

dt ¼ @a

@tþ

dxdt

@a

@xþ

dydt

@a

@yþ

dzdt

@a

@zThe 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 becomesthe velocitycomponent inthe xdirection,namely,u Similarly,dy=dt ! v and dz=dt ! w as dt ! 0, the expression for the change in

£uid and is watching a particular mass of the £uid.The entire right-hand side

of Eq (1.1) represents the total change in a expressed in eulerian coordinates.The term ukð@a=@xkÞ expresses the fact that in a time-independent £ow ¢eld

in which the £uid properties depend upon the spatial coordinates only, there

is a change in a due to the fact that a given £uid element changes its positionwith time and therefore assumes di¡erent 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 change Da=Dt of a for a given £uid element interms of the eulerian derivatives@a=@t and @a=@x

1.4 CONTROL VOLUMES

The concept of a control volume, as required to derive the basic 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 from which to choose One of these is

a parallelepiped of sides dx, dy, and dz Each £uid property, such as the 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 di¡erential equation for thatconservation principle is obtained Frequently, shortcuts are taken and thecontrol volume is taken to have sides of length dx, dy, and dz with only the ¢rstterm of theTaylor series being carried out

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

con-Vr dV , where r is the £uiddensity 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-di¡erential equation of the type

ZV

La dV ¼ 0

where L is some di¡erential 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 satis¢ed is by setting La¼ 0,which gives the di¡erential equation

of the conservation law If the integrand in the above equation was not equal

to zero, it would be possible to rede¢ne the control volume V in such a waythat the integral of La was not equal to zero, contradicting the integro-di¡erential equation above

Each of these two types of control volumes has some merit, and in thisbook each will be used at some point, depending upon which 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 two methods are identical

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

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 speci¢c mass of £uid and follow it for a short period of time

dt as it £ows Let a be any property of the £uid such as its mass, momentum insome direction, or energy Since a speci¢c 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, a¼ aðtÞ only and the rate of change of the inte-gral of a will be de¢ned by the following limit:

Z

V ðtþdtÞaðt þ dtÞ dV 

ZVðtÞaðtÞ dV

where VðtÞ is the control volume containing the speci¢ed mass of £uid andwhich may change its size and shape as it £ows The quantity aðt þ dtÞ inte-grated over VðtÞ will now be subtracted, then added again inside the abovelimit

Z

V ðtþdtÞaðt þ dtÞ dV 

ZVðtÞaðt þ dtÞ dV

þ1dt

Z

V ðtÞaðt þ dtÞ dV 

ZVðtÞaðtÞ dV

The ¢rst two integrals inside this limit correspond to holding the integrand

¢xed 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 de¢nition, 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:

ZVðtþdtÞV ðtÞaðt þ dtÞ dV

þZ

V ðtÞ

@a

@t dVThe remaining limit, corresponding to the volume V changing while aremains ¢xed, may be evaluated from geometric considerations

Figure1.2a shows the control volume VðtÞthat encloses the mass of £uidbeing considered both at time t and at time tþ dt.During this time interval thecontrol volume has moved downstream and has changed its size and shape.The surface that encloses VðtÞ is denoted by SðtÞ, and at any point on thissurface the velocity may be denoted by u and the unit outward normal by n.Figure 1.2b shows the control volume Vðt þ dtÞ superimposed on V ðtÞ,and an element of the di¡erence 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 ¼ un dt dS Then the volume integral inside the limit

FIGURE1.2 (a) Arbitrarily shaped control volume at times t and t þ dt, and(b) superposition of the control volumes at these times showing an element dV ofthe volume change

in the foregoing equation may be transformed into a surface integral in which

þZ

V ðtÞ

@a

@t dV

¼ZSðtÞaðtÞun dSþ

Z

V ðtÞ

@a

@t dVHaving 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 As was mentioned 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 becomes

ZSðtÞaðtÞun dS¼

ZVðtÞ=ðauÞ dVSubstituting this result into the foregoing expression and combining the twovolume integrals gives the preferred form of Reynolds’ transport theorem

DDt

ZV

a dV ¼ZV

ZV

a dV ¼ZV

Consider a speci¢c mass of £uid whose volume V is arbitrarily chosen If thisgiven £uid mass is followed as it £ows, its size and shape will be observed to

change but its mass will remain unchanged This is the principle of massconservation which applies to £uids in which no nuclear reactions are takingplace The mathematical equivalence of the statement of mass conservation

is to set the lagrangian derivative D=Dt of the mass of £uid contained in V,which is R

Vr dV , equal to zero That is, the equation that expresses servation of mass is

con-DDt

ZV

r dV ¼ 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 þ @@xk

ðrukÞ ¼ 0 ð1:3aÞ

Equation (1.3a) expresses more than the fact that mass is conserved Since it

is a partial di¡erential equation, the implication is that the velocity is 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 which

con-no 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 of

£uid 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 £ow of liquids In such cases the £uid

is said to be incompressible, which means that as a given mass of £uid is 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

£ows of the type depicted in Fig 1.3 A £uid particle that follows the lines

r¼ r1or r¼ r2will have its density remain ¢xed at r¼ r1or r¼ r2so thatDr=Dt ¼ 0 However, r is not constant everywhere, so that @r=@x 6¼ 0 and

@r=@y 6¼ 0 Such density strati¢cations 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 satis¢ed by the velocity andthe density No dynamical relations have been used to this point, but theconservation-of-momentum principle will utilize dynamics

1.7 CONSERVATION OF MOMENTUM

The principle of conservation of momentum is, in e¡ect, an application ofNewton’s second law of motion to an element of the £uid That is, when con-sidering a given mass of £uid in a lagrangian frame of reference, it is stated thatthe rate at which the momentum of the £uid mass is changing is equal to the netexternal 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 acceleration

of the element) is equal to the net external force acting on the element.The external forces that may act on a mass of the £uid may 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 volume V will beR

Vrf dV Also, if P is a face vector that represents the resultant surface force per unit area, the netexternal surface force acting on the surface S containing V will beR

sur-sPdS.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) The mass per unit volume is rand its momentum is ru, so that the momentum contained in the volume V isR

Vru dV Then, if the mass of the arbitrarily chosen volume V is observed inthe lagrangian frame of reference, the rate of change of momentum of themass contained with V will be ðD=DtÞRVru dV Thus, the mathematical

FIGURE1.3 Flow of a density-stratified fluid in which Dr=Dt ¼ 0 but for which

@r=@x 6¼ 0 and @r=@y 6¼ 0

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

DDt

ZV

ru dV ¼

Zs

P dSþZV

rf dV

In general, there are nine components of stress at any given point, one normalcomponent and two shear components on each coordinate plane These ninecomponents of stress are most 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 in which the stress components will act at a point as thelength of the cube tends to zero In Fig.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 identi¢ed by a double-subscript notation In this nota-tion, a particular component of the stress may be represented by the quantity

sij, 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 point

in a fluid

the plane xi¼ 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 x1¼ 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 ¼ s11n1 Likewise, the forces acting in the x2 direction andthe x3direction are, respectively, P2 ¼ s12n1 and P3¼ s13n1 Then, for anarbitrarily oriented surface whose unit normal has components n1, n2, and

n3, the surface force will be given by Pj¼ sijniin which i is summed from 1

to 3 That is, in tensor notation the equation expressing conservation ofmomentum becomes

DDt

ZV

rujdV ¼

Zs

sijnidSþ

ZV

rfjdV

The left-hand side of this equation may be converted to a volume integral inwhich the integrand contains only eulerian derivatives by use of Reynolds’transport theorem, Eq (1.2), in which the £uid property a here is themomentum per unit volume rujin the xjdirection At the same time the sur-face integral on the right-hand side may be converted into a volume integral

by use of Gauss’ theorem as given in Appendix B In this way the equationthat evolved from Newton’s second law becomes

dV ¼ZV

@sij

@xi

dV þZV

@

@tðrujÞ þ @

@x ðrujukÞ ¼ @sij

@x þ rfj

The left-hand side of this equation may be further simpli¢ed if the two termsinvolved are expanded in which the quantity rujukis considered to be theproduct of rukand uj.

r@uj

@t þ uj@r

@tþ uj @

@xkðrukÞ þ ruk@uj

£ow 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 the mass of the £uid A clear understanding of the physical signi¢cance

of each of the terms in Eq (1.4) is essential when approximations to the fullgoverning equations must be made.The surface-stress tensor sijhas not beenfully explained up to this point, but it will be investigated in detail in a latersection

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 work done on the system during the course of the event and any heat thatwas added Although a speci¢ed mass of £uid in a lagrangian frame of refer-ence may be considered to be a thermodynamic system, it is, in general, never

at rest and therefore never in equilibrium However, in the thermodynamicsense a £owing £uid is seldom far from a state of equilibrium, and the

apparent di⁄culty 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 mass1

2uu In this way the modi¢edform 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 of

£uid of volume V and follow it in a lagrangian frame of reference as it £ows.The total energy of this mass per unit volume is reþ1

2ruu, so that the totalenergy contained in V will beR

£uid is a surface stress whose magnitude per unit area is represented bythe vector P Then the total work done owing to such forces will beR

suP dS,where S is the surface area enclosing V The other type of force that may act

on 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 will

be R

Vurf dV Finally, an expression for the heat added to the £uid isrequired Let the vector q denote the conductive heat £ux leaving the controlvolume.Then the quantity of heat leaving the £uid mass 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 beR

sqn dS

Having evaluated each of the terms appearing in the physical law that is

to 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 aspeci¢c, 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

here the total energy per unit volumeðre þ1

2ruuÞ The resulting di¡erential equation is

uP dSþ

ZV

urf dV 

Zs

qn dSThe next step is to convert the two surface integrals into volume integrals sothat the arbitrariness of V may be exploited to obtain a di¡erential equationonly Using the fact that the force vector P is related to the stress tensor sijbythe equation Pj¼ sijni, as was shown in the previous section, the ¢rst surfaceintegral may be converted to a volume integral as follows:

Zs

uP dS¼

Zs

ujsijnidS¼

ZV

qn dS¼

Zs

qjnjdS¼

ZV

@qj

@xjdV

Since the stress tensor sijhas been brought into the energy equation, it isnecessary to use the tensor notation from this point on Then the expressionfor conservation of energy becomes

@

@xi

ðujsijÞ dV þ

ZV

ujrfjdV 

ZV

@qj

@xjdV

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

V f g dV ¼ 0, where the choice of V isarbitrary.Then the quantity inside the brackets in the integrand must be zero,which results in the following di¡erential equation:

on the left-hand side may be expanded by considered re and12rujujto be theproducts (r)(e) andðrÞð1ujujÞ, respectively.Then

2rujujukto bethe productð1

collectively amount to the product of ujwith the momentum Eq (1.4) Thusthe equation expressing conservation of thermal energy becomes

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 term beingthe 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 As will 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 from outside

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 state may be added to bring 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 ujand 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 sijandthe heat-£ux vector qjmust be further speci¢ed This leads to the so-calledconstitutive equations in which 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

either an intimate knowledge of tensor analysis or a clear understanding ofthe physical interpretation of certain tensor quantities For this reason, prior

to establishing the constitutive relations the tensor equivalents of rotationand rate of shear will be established

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 in¢nitesimal £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 projection

of a three-dimensional element) whose dimensions at time t¼ 0 are dx and dy.The £uid element is rectangular at time t¼ 0, and its centroid coincides withthe origin of a ¢xed-coordinate system For purposes of identi¢cation, thecorners of the £uid element have been labeled A, B,C, and D

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 infinitesimal element of fluid at time t ¼ 0 (indicated by ABCD) and

at time t ¼ dt (indicated by A0B0C0D0)