Fluid Mechanics

Richard Fitzpatrick

Professor of Physics

The University of Texas at Austin

Contents

1 Overview 7

1.1 Intended Audience............................................7

1.2 Major Sources..............................................7

1.3 To Do List................................................7

2 Mathematical Models of Fluid Motion 9

2.1 Introduction...............................................9

2.2 What is a Fluid?.............................................9

2.3 Volume and Surface Forces.......................................10

2.4 General Properties of Stress Tensor...................................11

2.5 Stress Tensor in a Static Fluid......................................12

2.6 Stress Tensor in a Moving Fluid.....................................13

2.7 Viscosity.................................................14

2.8 Conservation Laws...........................................15

2.9 Mass Conservation...........................................15

2.10 Convective Time Derivative.......................................16

2.11 MomentumConservation........................................16

2.12 Navier-Stokes Equation.........................................18

2.13 Energy Conservation..........................................18

2.14 Equations of Incompressible Fluid Flow................................20

2.15 Equations of Compressible Fluid Flow.................................21

2.16 Dimensionless Numbers in Incompressible Flow............................22

2.17 Dimensionless Numbers in Compressible Flow............................23

2.18 Fluid Equations in Cartesian Coordinates................................25

2.19 Fluid Equations in Cylindrical Coordinates...............................26

2.20 Fluid Equations in Spherical Coordinates................................27

2.21 Exercises.................................................28

3 Hydrostatics 31

3.1 Introduction...............................................31

3.2 Hydrostatic Pressure...........................................31

3.3 Buoyancy................................................31

3.4 Equilibriumof Floating Bodies.....................................32

3.5 Vertical Stability of Floating Bodies..................................33

3.6 Angular Stability of Floating Bodies..................................34

3.7 Determination of Metacentric Height..................................35

3.8 Energy of a Floating Body.......................................38

3.9 Curve of Buoyancy...........................................38

3.10 Rotational Hydrostatics.........................................42

2 FLUIDMECHANICS

3.11 Equilibriumof a Rotating Liquid Body.................................44

3.12 Maclaurin Spheroids..........................................46

3.13 Jacobi Ellipsoids.............................................49

3.14 Roche Ellipsoids.............................................51

3.15 Exercises.................................................57

4 Surface Tension 61

4.1 Introduction...............................................61

4.2 Young-Laplace Equation........................................61

4.3 Spherical Interfaces...........................................63

4.4 Capillary Length.............................................63

4.5 Angle of Contact............................................64

4.6 Jurin's Law...............................................65

4.7 Capillary Curves.............................................66

4.8 Axisymmetric Soap-Bubbles......................................70

4.9 Exercises.................................................75

5 Incompressible Inviscid Fluid Dynamics 77

5.1 Introduction...............................................77

5.2 Streamlines,StreamTubes,and StreamFilaments...........................77

5.3 Bernoulli's Theorem...........................................77

5.4 Vortex Lines,Vortex Tubes,and Vortex Filaments...........................79

5.5 Circulation and Vorticity........................................80

5.6 Kelvin Circulation Theorem.......................................80

5.7 Irrotational Flow.............................................81

5.8 Two-Dimensional Flow.........................................83

5.9 Two-Dimensional UniformFlow....................................85

5.10 Two-Dimensional Sources and Sinks..................................86

5.11 Two-Dimensional Vortex Filaments...................................87

5.12 Two-Dimensional Irrotational Flow in Cylindrical Coordinates....................90

5.13 Inviscid Flow Past a Cylindrical Obstacle................................91

5.14 Inviscid Flow Past a Semi-Innite Wedge................................94

5.15 Inviscid Flow Over a Semi-Innite Wedge...............................95

5.16 Velocity Potentials and StreamFunctions................................97

5.17 Exercises.................................................98

6 2D Potential Flow 101

6.1 Introduction...............................................101

6.2 Complex Functions...........................................101

6.3 Cauchy-Riemann Relations.......................................102

6.4 Complex Velocity Potential.......................................102

6.5 Complex Velocity............................................103

6.6 Method of Images............................................104

6.7 Conformal Maps.............................................109

6.8 Complex Line Integrals.........................................113

6.9 Theoremof Blasius...........................................114

6.10 Exercises.................................................118

7 Incompressible Boundary Layers 121

7.1 Introduction...............................................121

7.2 No Slip Condition............................................121

7.3 Boundary Layer Equations.......................................121

CONTENTS 3

7.4 Self-Similar Boundary Layers......................................125

7.5 Boundary Layer on a Flat Plate.....................................128

7.6 Wake Downstreamof a Flat Plate....................................132

7.7 Von K´arm´an MomentumIntegral....................................136

7.8 Boundary Layer Separation.......................................137

7.9 Criterion for Boundary Layer Separation................................140

7.10 Approximate Solutions of Boundary Layer Equations.........................142

7.11 Exercises.................................................147

8 Incompressible Aerodynamics 149

8.1 Introduction...............................................149

8.2 Theoremof Kutta and Zhukovskii...................................149

8.3 Cylindrical Airfoils...........................................151

8.4 Zhukovskii's Hypothesis........................................153

8.5 Vortex Sheets..............................................158

8.6 Induced Flow..............................................159

8.7 Three-Dimensional Airfoils.......................................159

8.8 Aerodynamic Forces...........................................162

8.9 Ellipsoidal Airfoils...........................................165

8.10 Simple Flight Problems.........................................167

8.11 Exercises.................................................168

9 Incompressible Viscous Flow 171

9.1 Introduction...............................................171

9.2 Flow Between Parallel Plates......................................171

9.3 Flow Down an Inclined Plane......................................172

9.4 Poiseuille Flow.............................................174

9.5 Taylor-Couette Flow...........................................174

9.6 Flow in Slowly-Varying Channels....................................175

9.7 Lubrication Theory...........................................177

9.8 Stokes Flow...............................................179

9.9 Axisymmetric Stokes Flow.......................................180

9.10 Axisymmetric Stokes Flow Around a Solid Sphere...........................181

9.11 Axisymmetric Stokes Flow In and Around a Fluid Sphere.......................185

9.12 Exercises.................................................188

10 Waves in Incompressible Fluids 191

10.1 Introduction...............................................191

10.2 Gravity Waves..............................................191

10.3 Gravity Waves in Deep Water......................................193

10.4 Gravity Waves in Shallow Water....................................194

10.5 Energy of Gravity Waves........................................195

10.6 Wave Drag on Ships...........................................196

10.7 Ship Wakes...............................................198

10.8 Gravity Waves in a Flowing Fluid....................................202

10.9 Gravity Waves at an Interface......................................203

10.10 Steady Flow over a Corrugated Bottom.................................205

10.11 Surface Tension.............................................205

10.12 Capillary Waves.............................................206

10.13 Capillary Waves at an Interface.....................................207

10.14 Wind Driven Waves in Deep Water...................................208

10.15 Exercises.................................................209

4 FLUIDMECHANICS

11 Equilibriumof Compressible Fluids 211

11.1 Introduction...............................................211

11.2 Isothermal Atmosphere.........................................211

11.3 Adiabatic Atmosphere..........................................212

11.4 Atmospheric Stability..........................................213

11.5 Eddington Solar Model.........................................213

11.6 Exercises.................................................219

A Vectors and Vector Fields 223

A.1 Introduction...............................................223

A.2 Scalars and Vectors...........................................223

A.3 Vector Algebra..............................................223

A.4 Cartesian Components of a Vector...................................225

A.5 Coordinate Transformations.......................................226

A.6 Scalar Product..............................................227

A.7 Vector Area...............................................228

A.8 Vector Product..............................................229

A.9 Rotation.................................................231

A.10 Scalar Triple Product..........................................233

A.11 Vector Triple Product..........................................234

A.12 Vector Calculus.............................................234

A.13 Line Integrals..............................................235

A.14 Vector Line Integrals..........................................237

A.15 Surface Integrals.............................................237

A.16 Vector Surface Integrals.........................................239

A.17 Volume Integrals.............................................239

A.18 Gradient.................................................240

A.19 Grad Operator..............................................243

A.20 Divergence................................................243

A.21 Laplacian Operator...........................................246

A.22 Curl...................................................247

A.23 Useful Vector Identities.........................................250

A.24 Exercises.................................................250

B Cartesian Tensors 253

B.1 Introduction...............................................253

B.2 Tensors and Tensor Notation......................................253

B.3 Tensor Transformation.........................................255

B.4 Tensor Fields..............................................257

B.5 Isotropic Tensors............................................259

B.6 Exercises.................................................261

C Non-Cartesian Coordinates 265

C.1 Introduction...............................................265

C.2 Orthogonal Curvilinear Coordinates..................................265

C.3 Cylindrical Coordinates.........................................268

C.4 Spherical Coordinates..........................................270

C.5 Exercises.................................................272

D Calculus of Variations 273

D.1 Euler-Lagrange Equation........................................273

D.2 Conditional Variation..........................................275

CONTENTS 5

D.3 Multi-Function Variation........................................276

D.4 Exercises.................................................277

E Ellipsoidal Potential Theory 279

6 FLUIDMECHANICS

Overview 7

1 Overview

1.1 Intended Audience

This book presents a single semester course on uid mechanics that is intended primarily for advanced undergraduate

students majoring in physics.A thorough understanding of physics at the lower-division level,including a basic

working knowledge of the laws of mechanics,is assumed.It is also taken for granted that students are familiar

with the fundamentals of multi-variate integral and diﬀerential calculus,complex analysis,and ordinary diﬀerential

equations.On the other hand,vector analysis plays such a central role in the study of uid mechanics that a brief,

but fairly comprehensive,review of this subject area is provided in Appendix A.Likewise,those aspects of cartesian

tensor theory,orthogonal curvilinear coordinate systems,and the calculus of variations,that are required in the study

of uid mechanics are outlined in Appendices B,C,and D,resp ectively.

1.2 Major Sources

The material appearing in Appendix A is largely based on the author's recollections of a vector analysis course given

by Dr.Stephen Gull at the University of Cambridge.Major sources for the material appearing in other chapters and

appendices include:

Statics,Including Hydrostatics and the Elements of the Theory of Elasticity H.Lamb,3rd Edition (Cambridge Uni-

versity Press,Cambridge UK,1928).

Hydrodynamics H.Lamb,6th Edition (Dover,New York NY,1945).

Theoretical Aerodynamics L.M.Milne-Thomson,4th Edition,Revised and enlarged (Dover,New York NY,1958).

Ellipsoidal Figures of Equilibrium S.Chandrasekhar (Yale University Press,New Haven CT,1969).

Boundary Layer Theory H.Schlichting,7th Edition (McGraw-Hill,New York NY,1970).

Mathematical Methods for the Physical Sciences K.F.Riley (Cambridge University Press,Cambridge UK,1974).

Fluid Mechanics L.D.Landau,and E.M.Lifshitz,2nd Edition (Butterworth-Heinemann,Oxford UK,1987).

Physical Fluid Dynamics D.J.Tritton,2nd Edition (Oxford University Press,Oxford UK,1988).

Fluid Dynamics for Physicists T.E.Faber,1st Edition (Cambridge University Press,Cambridge UK,1995).

Schaum's Outline of Fluid Dynamics W.Hughes,and J.Brighton,3rd Edition (McGraw-Hill,NewYork NY,1999).

An Introduction to Fluid Dynamics G.K.Batchelor (Cambridge University Press,Cambridge UK,2000).

Theoretical Hydrodynamics L.M.Milne-Thomson,5th Edition (Dover,New York NY,2011).

1.3 To Do List

1.Add chapter on vortex dynamics.

2.Add chapter on 3D potential ow.

3.Add appendix on group velocity and Fourier transforms.

4.Add chapter on incompressible ow in rotating systems.

5.Add chapter on instabilities.

8 FLUIDMECHANICS

6.Add chapter on turbulence.

7.Add chapter on 1D compressible ow.

8.Add chapter on sound waves.

9.Add chapter on compressible boundary layers.

10.Add chapter on supersonic aerodynamics.

11.Add chapter on convection.

MathematicalModelsofFluidMotion 9

2 Mathematical Models of Fluid Motion

2.1 Introduction

In this chapter,we set forth the mathematical models commonly used to describe the equilibrium and dynamics of

uids.Unless stated otherwise,all of the analysis is perfo rmed using a standard right-handed Cartesian coordinate

system:x

1

,x

2

,x

3

.Moreover,the Einstein summation convention is employed (so repeated roman subscripts are

assumed to be summed from1 to 3see Appendix B).

2.2 What is a Fluid?

By denition,a solid material is rigid.Now,although a rigid material tends to shatter when subjected to very large

stresses,it can withstand a moderate shear stress (i.e.,a stress that tends to deformthe material by changing its shape,

without necessarily changing its volume) for an indenite p eriod.To be more exact,when a shear stress is rst applied

to a rigid material it deforms slightly,but then springs back to its original shape when the stress is relieved.

A plastic material,such as clay,also possess some degree of rigidity.However,the critical shear stress at which it

yields is relatively small,and once this stress is exceeded the material deforms continuously and irreversibly,and does

not recover its original shape when the stress is relieved.

By denition,a uid material possesses no rigidity at all.In other words,a small uid element is unable to

withstand any tendency of an applied shear stress to change its shape.Incidentally,this does not preclude the possibility

that such an element may oﬀer resistance to shear stress.However,any resistance must be incapable of preventing

the change in shape from eventually occurring,which implies that the force of resistance vanishes with the rate of

deformation.An obvious corollary is that the shear stress must be zero everywhere inside a uid that is in mechanical

equilibrium.

Fluids are conventionally classied as either liquids or gases.The most important diﬀerence between these two

types of uid lies in their relative compressibility:i.e.,gases can be compressed much more easily than liquids.Con-

sequently,any motion that involves signicant pressure va riations is generally accompanied by much larger changes

in mass density in the case of a gas than in the case of a liquid.

Of course,a macroscopic uid ultimately consists of a huge n umber of individual molecules.However,most

practical applications of uid mechanics are concerned wit h behavior on length-scales that are far larger than the

typical intermolecular spacing.Under these circumstances,it is reasonable to suppose that the bulk properties of a

given uid are the same as if it were completely continuous in structure.A corollary of this assumption is that when,

in the following,we talk about innitesimal volume element s,we really mean elements which are suﬃciently small

that the bulk uid properties (such as mass density,pressur e,and velocity) are approximately constant across them,

but are still suﬃciently large that they contain a very great number of molecules (which implies that we can safely

neglect any statistical variations in the bulk properties).The continuumhypothesis also requires innitesimal volu me

elements to be much larger than the molecular mean-free-path between collisions.

In addition to the continuumhypothesis,our study of uid me chanics is premised on three major assumptions:

1.Fluids are isotropic media:i.e.,there is no preferred direction in a uid.

2.Fluids are Newtonian:i.e.,there is a linear relationship between the local shear stress and the local rate of strain,

as rst postulated by Newton.It is also assumed that there is a linear relationship between the local heat ux

density and the local temperature gradient.

3.Fluids are classical:i.e.,the macroscopic motion of ordinary uids is well-describe d by Newtonian dynamics,

and both quantumand relativistic eﬀects can be safely ignored.

It should be noted that the above assumptions are not valid for all uid types ( e.g.,certain liquid polymers,which

are non-isotropic;thixotropic uids,such as jelly or pain t,which are non-Newtonian;and quantum uids,such as

liquid helium,which exhibit non-classical eﬀects on macroscopic length-scales).However,most practical applications

10 FLUIDMECHANICS

of uid mechanics involve the equilibriumand motion of bodi es of water or air,extending over macroscopic length-

scales,and situated relatively close to the Earth's surfac e.Such bodies are very well-described as isotropic,Newtonian,

classical uids.

2.3 Volume and Surface Forces

Generally speaking,uids are acted upon by two distinct typ es of force.The rst type is long-range in nature

i.e.,such that it decreases relatively slowly with increasing distance between interacting elementsand is capable

of completely penetrating into the interior of a uid.Gravi ty is an obvious example of a long-range force.One

consequence of the relatively slow variation of long-range forces with position is that they act equally on all of the

uid contained within a su ﬃciently small volume element.In this situation,the net force acting on the element

becomes directly proportional to its volume.For this reason,long-range forces are often called volume forces.In the

following,we shall write the total volume force acting at time t on the uid contained within a small volume element

of magnitude dV,centered on a xed point whose position vector is r,as

F(r,t) dV.(2.1)

The second type of force is short-range in nature,and is most convenientlymodeled as momentumtransport within

the uid.Such transport is generally due to a combination of the mutual forces exerted by contiguous molecules,and

momentumuxes caused by relative molecular motion.Suppos e that π

x

(r,t) is the net ux density of x-directed uid

momentumdue to short-range forces at position r and time t.In other words,suppose that,at position r and time t,as a

direct consequence of short-range forces,x-momentumis owing at the rate of |π

x

| newton-seconds per meter squared

per second in the direction of vector π

x

.Consider an innitesimal plane surface element,dS = ndS,located at point

r.Here,dS is the area of the element,and n its unit normal.(See Section A.7.) The uid which lies on tha t side of the

element toward which n points is said to lie on its positive side,and vice versa.The net ux of x-momentumacross the

element (in the direction of n) is π

x

dS newtons,which implies (fromNewton's second law of motion) that the uid

on the positive side of the surface element experiences a force π

x

dS in the x-direction due to short-range interaction

with the uid on the negative side.According to Newton's thi rd law of motion,the uid on the negative side of the

surface experiences a force −π

x

dS in the x-direction due to interaction with the uid on the positive s ide.Short-range

forces are often called surface forces because they are directly proportional to the area of the surface element across

which they act.Let π

y

(r,t) and π

z

(r,t) be the net ux density of y- and z- momentum,respectively,at position r and

time t.By a straightforward extension of above argument,the net surface force exerted by the uid on the positive side

of some planar surface element,dS,on the uid on its negative side is

f = (−π

x

dS,−π

y

dS,−π

z

dS).(2.2)

In tensor notation (see Appendix B),the above equation can be written

f

i

= σ

i j

dS

j

,(2.3)

where σ

11

= −(π

x

)

x

,σ

12

= −(π

x

)

y

,σ

21

= −(π

y

)

x

,etc.(Note that,since the subscript j is repeated,it is assumed to be

summed from1 to 3.Hence,σ

i j

dS

j

is shorthand for

P

j=1,3

σ

i j

dS

j

.) Here,the σ

i j

(r,t) are termed the local stresses

in the uid at position r and time t,and have units of force per unit area.Moreover,the σ

i j

are the components of a

second-order tensor (see Appendix B),known as the stress tensor.[This follows because the f

i

are the components

of a rst-order tensor (since all forces are proper vectors),and the dS

i

are the components of an arbitrary rst-order

tensor (since surface elements are also proper vectorssee Section A.7and (2.3] holds for surface elements whose

normals point in any direction),so application of the quotient rule (see Section B.3) to Equation (2.3) reveals that the

σ

i j

transform under rotation of the coordinate axes as the components of a second-order tensor.] We can interpret

σ

i j

(r,t) as the i-component of the force per unit area exerted,at position r and time t,across a plane surface element

normal to the j-direction.The three diagonal components of σ

i j

are termed normal stresses,since each of themgives

the normal component of the force per unit area acting across a plane surface element parallel to one of the Cartesian

coordinate planes.The six non-diagonal components are termed shear stresses,since they drive shearing motion in

which parallel layers of uid slide relative to one another.

MathematicalModelsofFluidMotion 11

2.4 General Properties of Stress Tensor

The i-component of the total force acting on a uid element consis ting of a xed volume V enclosed by a surface S is

written

f

i

=

Z

V

F

i

dV +

I

S

σ

i j

dS

j

,(2.4)

where the rst term on the right-hand side is the integrated v olume force acting throughout V,whereas the second

termis the net surface force acting across S.Making use of the tensor divergence theorem(see Section B.4),the above

expression becomes

f

i

=

Z

V

F

i

dV +

Z

V

∂σ

i j

∂x

j

dV.(2.5)

In the limit V →0,it is reasonable to suppose that the F

i

and ∂σ

i j

/∂x

j

are approximately constant across the element.

In this situation,both contributions on the right-hand side of the above equation scale as V.Now,according to

Newtonian dynamics,the i-component of the net force acting on the element is equal to the i-component of the rate

of change of its linear momentum.However,in the limit V →0,the linear acceleration and mass density of the uid

are both approximately constant across the element.In this case,the rate of change of the element's linear momentum

also scales as V.In other words,the net volume force,surface force,and rate of change of linear momentum of an

innitesimal uid element all scale as the volume of the elem ent,and consequently remain approximately the same

order of magnitude as the volume shrinks to zero.We conclude that the linear equation of motion of an innitesimal

uid element places no particular restrictions on the stres s tensor.

The i-component of the total torque,taken about the origin O of the coordinate system,acting on a uid element

that consists of a xed volume V enclosed by a surface S is written [see Equations (A.46) and (B.6)]

τ

i

=

Z

V

ǫ

i jk

x

j

F

k

dV +

I

S

ǫ

i jk

x

j

σ

kl

dS

l

,(2.6)

where the rst and second terms on the right-hand side are due to volume and surface forces,respectively.[Here,

ǫ

i jk

is the third-order permutation tensor.See Equation (B.7).] Making use of the tensor divergence theorem (see

Section B.4),the above expression becomes

τ

i

=

Z

V

ǫ

i jk

x

j

F

k

dV +

Z

V

ǫ

i jk

∂(x

j

σ

kl

)

∂x

l

dV,(2.7)

which reduces to

τ

i

=

Z

V

ǫ

i jk

x

j

F

k

dV +

Z

V

ǫ

i jk

σ

k j

dV +

Z

V

ǫ

i jk

x

j

∂σ

kl

∂x

l

dV,(2.8)

since ∂x

i

/∂x

j

= δ

i j

.[Here,δ

i j

is the second-order identity tensor.See Equation (B.9).] Assuming that point O lies

within the uid element,and taking the limit V →0 in which the F

i

,σ

i j

,and ∂σ

i j

/∂x

j

are all approximately constant

across the element,we deduce that the rst,second,and thir d terms on the right-hand side of the above equation scale

as V

4/3

,V,and V

4/3

,respectively (since x ∼ V

1/3

).Now,according to Newtonian dynamics,the i-component of the

total torque acting on the uid element is equal to the i-component of the rate of change of its net angular momentum

about O.Assuming that the linear acceleration of the uid is approx imately constant across the element,we deduce

that the rate of change of its angular momentum scales as V

4/3

(since the net linear acceleration scales as V,so the

net rate of change of angular momentumscales as x V,and x ∼ V

1/3

).Hence,it is clear that the rotational equation of

motion of a uid element,surrounding a general point O,becomes completely dominated by the second term on the

right-hand side of (2.8) in the limit that the volume of the element approaches zero (since this term is a factor V

−1/3

larger than the other terms).It follows that the second term must be identically zero (otherwise an innitesimal uid

element would acquire an absurdly large angular velocity).This is only possible,for all choices of the position of

point O,and the shape of the element,if

ǫ

i jk

σ

k j

= 0 (2.9)

throughout the uid.The above relation shows that the stres s tensor must be symmetric:i.e.,

σ

ji

= σ

i j

.(2.10)

12 FLUIDMECHANICS

It immediately follows that the stress tensor only has six independent components (i.e.,σ

11

,σ

22

,σ

33

,σ

12

,σ

13

,and

σ

23

).

Now,it is always possible to choose the orientation of a set of Cartesian axes in such a manner that the non-

diagonal components of a given symmetric second-order tensor eld are all set to zero at a given point i n space.(See

Exercise B.6.) With reference to such principal axes,the diagonal components of the stress tensor σ

i j

become so-

called principal stresses σ

′

11

,σ

′

22

,σ

′

33

,say.Of course,in general,the orientation of the principal axes varies with

position.The normal stress σ

′

11

acting across a surface element perpendicular to the rst pr incipal axis corresponds

to a tension (or a compression if σ

′

11

is negative) in the direction of that axis.Likewise,for σ

′

22

and σ

′

33

.Thus,the

general state of the uid,at a particular point in space,can be regarded as a superposition of tensions,or compressions,

in three orthogonal directions.

The trace of the stress tensor,σ

ii

= σ

11

+ σ

22

+ σ

33

,is a scalar,and,therefore,independent of the orientation of

the coordinate axes.(See Appendix B.) Thus,it follows that,irrespective of the orientation of the principal axes,the

trace of the stress tensor at a given point is always equal to the sumof the principal stresses:i.e.,

σ

ii

= σ

′

11

+ σ

′

22

+ σ

′

33

.(2.11)

2.5 Stress Tensor in a Static Fluid

Consider the surface forces exerted on some innitesimal cu bic volume element of a static uid.Suppose that the

components of the stress tensor are approximately constant across the element.Suppose,further,that the sides of the

cube are aligned parallel to the principal axes of the local stress tensor.This tensor,which now has zero non-diagonal

components,can be regarded as the sumof two tensors:i.e.,

1

3

σ

ii

0 0

0

1

3

σ

ii

0

0 0

1

3

σ

ii

,(2.12)

and

σ

′

11

−

1

3

σ

ii

0 0

0 σ

′

22

−

1

3

σ

ii

0

0 0 σ

′

33

−

1

3

σ

ii

.(2.13)

The rst of the above tensors is isotropic (see Section B.5),and corresponds to the same normal force per unit

area acting inward (since the sign of σ

ii

/3 is invariably negative) on each face of the volume element.This uniform

compression acts to change the element's volume,but not its shape,and can easily be withstood by the uid within the

element.

The second of the above tensors represents the departure of the stress tensor froman isotropic form.The diagonal

components of this tensor have zero sum,in view of (2.11),and thus represent equal and opposite forces per unit

area,acting on opposing faces of the volume element,which are such that the forces on at least one pair of opposing

faces constitute a tension,and the forces on at least one pair constitute a compression.Such forces necessarily tend to

change the shape of the volume element,either elongating or compressing it along one of its symmetry axes.Moreover,

this tendency cannot be oﬀset by any volume force acting on the element,since such forces become arbitrarily small

compared to surface forces in the limit that the element's vo lume tends to zero (because the ratio of the net volume

force to the net surface force scales as the volume to the surface area of the element,which tends to zero in the limit

that the volume tends to zerosee Section 2.4).Now,we have p reviously dened a uid as a material that is incapable

of withstanding any tendency of applied forces to change its shape.(See Section 2.2.) It follows that if the diagonal

components of the tensor (2.13) are non-zero anywhere inside the uid then it is impossible for the uid at that point to

be at rest.Hence,we conclude that the principal stresses,σ

′

11

,σ

′

22

,and σ

′

33

,must be equal to one another at all points

in a static uid.This implies that the stress tensor takes th e isotropic form (2.12) everywhere in a stationary uid.

Furthermore,this is true irrespective of the orientation of the coordinate axes,since the components of an isotropic

tensor are rotationally invariant.(See Section B.5.)

MathematicalModelsofFluidMotion 13

Fluids at rest are generally in a state of compression,so it is convenient to write the stress tensor of a static uid in

the form

σ

i j

= −p δ

i j

,(2.14)

where p = −σ

ii

/3 is termed the static uid pressure,and is generally a function of r and t.It follows that,in a stationary

uid,the force per unit area exerted across a plane surface e lement with unit normal n is −p n.[See Equation (2.3).]

Moreover,this normal force has the same value for all possible orientations of n.This well-known resultnamely,

that the pressure is the same in all directions at a given point in a static uidis known as Pascal's law,and is a direct

consequence of the fact that a uid element cannot withstand shear stresses,or,alternatively,any tendency of applied

forces to change its shape.

2.6 Stress Tensor in a Moving Fluid

We have seen that in a static uid the stress tensor takes the f orm

σ

i j

= −p δ

i j

,(2.15)

where p = −σ

ii

/3 is the static pressure:i.e.,minus the normal stress acting in any direction.Now,the normal stress at

a given point in a moving uid generally varies with directio n:i.e.,the principal stresses are not equal to one another.

However,we can still dene the mean principal stress as ( σ

′

11

+σ

′

22

+σ

′

33

)/3 = σ

ii

/3.Moreover,given that the principal

stresses are actually normal stresses (in a coordinate frame aligned with the principal axes),we can also regard σ

ii

/3

as the mean normal stress.It is convenient to dene pressure in a moving uid as minus the mean normal stress:i.e.,

p = −

1

3

σ

ii

.(2.16)

Thus,we can write the stress tensor in a moving uid as the sum of an isotropic part,−p δ

i j

,which has the same form

as the stress tensor in a static uid,and a remaining non-iso tropic part,d

i j

,which includes any shear stresses,and also

has diagonal components whose sumis zero.In other words,

σ

i j

= −p δ

i j

+ d

i j

,(2.17)

where

d

ii

= 0.(2.18)

Moreover,since σ

i j

and δ

i j

are both symmetric tensors,it follows that d

i j

is also symmetric:i.e.,

d

ji

= d

i j

.(2.19)

It is clear that the so-called deviatoric stress tensor,d

i j

,is a consequence of uid motion,since it is zero in a static

uid.Suppose,however,that we were to view a static uid bot h in its rest frame and in a frame of reference moving

at some constant velocity relative to the rest frame.Now,we would expect the force distribution within the uid to

be the same in both frames of reference,since the uid does not accelera te in either.However,in the rst frame,

the uid appears stationary and the deviatoric stress tenso r is therefore zero,whilst in the second it has a spatially

uniform velocity eld and the deviatoric stress tensor is also zero ( because it is the same as in the rest frame).We,

thus,conclude that the deviatoric stress tensor is zero both in a stationary uid and in a moving uid possessing no

spatial velocity gradients.This suggests that the deviatoric stress tensor is driven by velocity gradients within the uid.

Moreover,the tensor must vanish as these gradients vanish.

Let the v

i

(r,t) be the Cartesian components of the uid velocity at point r and time t.The various velocity

gradients within the uid then take the form ∂v

i

/∂x

j

.The simplest possible assumption,which is consistent with the

above discussion,is that the components of the deviatoric stress tensor are linear functions of these velocity gradients:

i.e.,

d

i j

= A

i jkl

∂v

k

∂x

l

.(2.20)

14 FLUIDMECHANICS

Here,A

i jkl

is a fourth-order tensor (this follows fromthe quotient rule because d

i j

and ∂v

i

/∂x

j

are both proper second-

order tensors).Any uid in which the deviatoric stress tens or takes the above formis termed a Newtonian uid,since

Newton was the rst to postulate a linear relationship betwe en shear stresses and velocity gradients.

Now,in an isotropic uidthat is,a uid in which there is no preferred direction we would expect the fourth-

order tensor A

i jkl

to be isotropicthat is,to have a formin which all physical d istinction between diﬀerent directions

is absent.As demonstrated in Section B.5,the most general expression for an isotropic fourth-order tensor is

A

i jkl

= αδ

i j

δ

kl

+ βδ

ik

δ

jl

+ γδ

il

δ

jk

,(2.21)

where α,β,and γ are arbitrary scalars (which can be functions of position and time).Thus,it follows from(2.20) and

(2.21) that

d

i j

= α

∂v

k

∂x

k

δ

i j

+ β

∂v

i

∂x

j

+ γ

∂v

j

∂x

i

.(2.22)

However,according to Equation (2.19),d

i j

is a symmetric tensor,which implies that β = γ,and

d

i j

= αe

kk

δ

i j

+ 2 βe

i j

,(2.23)

where

e

i j

=

1

2

∂v

i

∂x

j

+

∂v

j

∂x

i

!

(2.24)

is called the rate of strain tensor.Finally,according to Equation (2.18),d

i j

is a traceless tensor,which yields 3 α =

−2 β,and

d

i j

= 2

e

i j

−

1

3

e

kk

δ

i j

!

,(2.25)

where = β.We,thus,conclude that the most general expression for the stress tensor in an isotropic Newtonian uid

is

σ

i j

= −p δ

i j

+ 2

e

i j

−

1

3

e

kk

δ

i j

!

,(2.26)

where p(r,t) and (r,t) are arbitrary scalars.

2.7 Viscosity

The signicance of the parameter ,appearing in the previous expression for the stress tensor,can be seen from the

form taken by the relation (2.25) in the special case of simple shearing motion.With ∂v

1

/∂x

2

as the only non-zero

velocity derivative,all of the components of d

i j

are zero apart fromthe shear stresses

d

12

= d

21

=

∂v

1

∂x

2

.(2.27)

Thus, is the constant of proportionality between the rate of shear and the tangential force per unit area when parallel

plane layers of uid slide over one another.This constant of proportionality is generally referred to as viscosity.It is a

matter of experience that the force between layers of uid un dergoing relative sliding motion always tends to oppose

the motion,which implies that > 0.

The viscosities of dry air and pure water at 20

◦

C and atmospheric pressure are about 1.8 × 10

−5

kg/(ms) and

1.0×10

−3

kg/(ms),respectively.In neither case does the viscosity exhibit much variation with pressure.However,the

viscosity of air increases by about 0.3 percent,and that of water decreases by about 3 percent,per degree Centigrade

rise in temperature.

MathematicalModelsofFluidMotion 15

2.8 Conservation Laws

Suppose that θ(r,t) is the density of some bulk uid property ( e.g.,mass,momentum,energy) at position r and time t.

In other words,suppose that,at time t,an innitesimal uid element of volume dV,located at position r,contains an

amount θ(r,t) dV of the property in question.Note,incidentally,that θ can be either a scalar,a component of a vector,

or even a component of a tensor.The total amount of the property contained within some xed volume V is

Θ =

Z

V

θ dV,(2.28)

where the integral is taken over all elements of V.Let dS be an outward directed element of the bounding surface of

V.Suppose that this element is located at point r.The volume of uid that ows per second across the element,a nd so

out of V,is v(r,t)dS.Thus,the amount of the uid property under consideration t hat is convected across the element

per second is θ(r,t) v(r,t) dS.It follows that the net amount of the property that is convected out of volume V by uid

ow across its bounding surface S is

Φ

Θ

=

Z

S

θ v dS,(2.29)

where the integral is taken over all outward directed elements of S.Suppose,nally,that the property in question is

created within the volume V at the rate S

Θ

per second.The conservation equation for the uid property takes the form

dΘ

dt

= S

Θ

− Φ

Θ

.(2.30)

In other words,the rate of increase in the amount of the property contained within V is the diﬀerence between the

creation rate of the property inside V,and the rate at which the property is convected out of V by uid ow.The above

conservation law can also be written

dΘ

dt

+ Φ

Θ

= S

Θ

.(2.31)

Here,Φ

Θ

is termed the ux of the property out of V,whereas S

Θ

is called the net generation rate of the property within

V.

2.9 Mass Conservation

Let ρ(r,t) and v(r,t) be the mass density and velocity of a given uid at point r and time t.Consider a xed volume

V,surrounded by a surface S.The net mass contained within V is

M =

Z

V

ρdV,(2.32)

where dV is an element of V.Furthermore,the mass ux across S,and out of V,is [see Equation (2.29)]

Φ

M

=

Z

S

ρv dS,(2.33)

where dS is an outward directed element of S.Mass conservation requires that the rate of increase of the mass

contained within V,plus the net mass ux out of V,should equal zero:i.e.,

dM

dt

+ Φ

M

= 0 (2.34)

[cf.,Equation (2.31)].Here,we are assuming that there is no mass generation (or destruction) within V (since individ-

ual molecules are eﬀectively indestructible).It follows that

Z

V

∂ρ

∂t

dV +

Z

S

ρv dS = 0,(2.35)

16 FLUIDMECHANICS

since V is non-time-varying.Making use of the divergence theorem(see Section A.20),the above equation becomes

Z

V

"

∂ρ

∂t

+ ∇(ρv)

#

dV = 0.(2.36)

However,this result is true irrespective of the size,shape,or location of volume V,which is only possible if

∂ρ

∂t

+ ∇(ρv) = 0 (2.37)

throughout the uid.The above expression is known as the equation of uid continuity,and is a direct consequence of

mass conservation.

2.10 Convective Time Derivative

The quantity ∂ρ(r,t)/∂t,appearing in Equation (2.37),represents the time derivative of the uid mass density at the

xed point r.Suppose that v(r,t) is the instantaneous uid velocity at the same point.It fol lows that the time derivative

of the density,as seen in a frame of reference which is instantaneously co-moving with the uid at point r,is

lim

δt→0

ρ(r + v δt,t + δt) − ρ(r,t)

δt

=

∂ρ

∂t

+ v∇ρ =

Dρ

Dt

,(2.38)

where we have Taylor expanded ρ(r + v δt,t + δt) up to rst order in δt,and where

D

Dt

=

∂

∂t

+ v∇ =

∂

∂t

+ v

i

∂

∂x

i

.(2.39)

Clearly,the so-called convective time derivative,D/Dt,represents the time derivative seen in the local rest frame of

the uid.

The continuity equation (2.37) can be rewritten in the form

1

ρ

Dρ

Dt

=

Dlnρ

Dt

= −∇v,(2.40)

since ∇ (ρv) = v ∇ρ+ρ∇ v [see (A.174)].Consider a volume element V that is co-moving with the uid.In general,

as the element is convected by the uid its volume changes.In fact,it is easily seen that

DV

Dt

=

Z

S

v dS =

Z

S

v

i

dS

i

=

Z

V

∂v

i

∂x

i

dV =

Z

V

∇v dV,(2.41)

where S is the bounding surface of the element,and use has been made of the divergence theorem.In the limit that

V →0,and ∇v is approximately constant across the element,we obtain

1

V

DV

Dt

=

DlnV

Dt

= ∇v.(2.42)

Hence,we conclude that the divergence of the uid velocity a t a given point in space species the fractional rate of

increase in the volume of an innitesimal co-moving uid ele ment at that point.

2.11 MomentumConservation

Consider a xed volume V surrounded by a surface S.The i-component of the total linear momentumcontained within

V is

P

i

=

Z

V

ρv

i

dV.(2.43)

MathematicalModelsofFluidMotion 17

Moreover,the ux of i-momentumacross S,and out of V,is [see Equation (2.29)]

Φ

i

=

Z

S

ρv

i

v

j

dS

j

.(2.44)

Finally,the i-component of the net force acting on the uid within V is

f

i

=

Z

V

F

i

dV +

I

S

σ

i j

dS

j

,(2.45)

where the rst and second terms on the right-hand side are the contributions fromvolume and surface forces,respec-

tively.

Momentum conservation requires that the rate of increase of the net i-momentum of the uid contained within

V,plus the ux of i-momentum out of V,is equal to the rate of i-momentum generation within V.Of course,from

Newton's second law of motion,the latter quantity is equal t o the i-component of the net force acting on the uid

contained within V.Thus,we obtain [cf.,Equation (2.31)]

dP

i

dt

+ Φ

i

= f

i

,(2.46)

which can be written

Z

V

∂(ρv

i

)

∂t

dV +

Z

S

ρv

i

v

j

dS

j

=

Z

V

F

i

dV +

I

S

σ

i j

dS

j

,(2.47)

since the volume V is non-time-varying.Making use of the tensor divergence theorem,this becomes

Z

V

"

∂(ρv

i

)

∂t

+

∂(ρv

i

v

j

)

∂x

j

#

dV =

Z

V

F

i

+

∂σ

i j

∂x

j

!

dV.(2.48)

However,the above result is valid irrespective of the size,shape,or location of volume V,which is only possible if

∂(ρv

i

)

∂t

+

∂(ρv

i

v

j

)

∂x

j

= F

i

+

∂σ

i j

∂x

j

(2.49)

everywhere inside the uid.Expanding the derivatives,and rearranging,we obtain

∂ρ

∂t

+ v

j

∂ρ

∂x

j

+ ρ

∂v

j

∂x

j

!

v

i

+ ρ

∂v

i

∂t

+ v

j

∂v

i

∂x

j

!

= F

i

+

∂σ

i j

∂x

j

.(2.50)

Now,in tensor notation,the continuity equation (2.37) is written

∂ρ

∂t

+ v

j

∂ρ

∂x

j

+ ρ

∂v

j

∂x

j

= 0.(2.51)

So,combining Equations (2.50) and (2.51),we obtain the following uid equation of motion,

ρ

∂v

i

∂t

+ v

j

∂v

i

∂x

j

!

= F

i

+

∂σ

i j

∂x

j

.(2.52)

An alternative formof this equation is

Dv

i

Dt

=

F

i

ρ

+

1

ρ

∂σ

i j

∂x

j

.(2.53)

The above equation describes howthe net volume and surface forces per unit mass acting on a co-moving uid element

determine its acceleration.

18 FLUIDMECHANICS

2.12 Navier-Stokes Equation

Equations (2.24),(2.26),and (2.53) can be combined to give the equation of motion of an isotropic,Newtonian,

classical uid:i.e.,

ρ

Dv

i

Dt

= F

i

−

∂p

∂x

i

+

∂

∂x

j

"

∂v

i

∂x

j

+

∂v

j

∂x

i

!#

−

∂

∂x

i

2

3

∂v

j

∂x

j

!

.(2.54)

This equation is generally known as the Navier-Stokes equation.Now,in situations in which there are no strong

temperature gradients in the uid,it is a good approximatio n to treat viscosity as a spatially uniformquantity,in which

case the Navier-Stokes equation simplies somewhat to give

ρ

Dv

i

Dt

= F

i

−

∂p

∂x

i

+

"

∂

2

v

i

∂x

j

∂x

j

+

1

3

∂

2

v

j

∂x

i

∂x

j

#

.(2.55)

When expressed in vector form,the above expression becomes

ρ

Dv

Dt

= ρ

"

∂v

∂t

+ (v∇) v

#

= F − ∇p +

"

∇

2

v +

1

3

∇(∇v)

#

,(2.56)

where use has been made of Equation (2.39).Here,

[(a∇)b]

i

= a

j

∂b

i

∂x

j

,(2.57)

(∇

2

v)

i

= ∇

2

v

i

.(2.58)

Note,however,that the above identities are only valid in Cartesian coordinates.(See Appendix C.)

2.13 Energy Conservation

Consider a xed volume V surrounded by a surface S.The total energy content of the uid contained within V is

E =

Z

V

ρEdV +

Z

V

1

2

ρv

i

v

i

dV,(2.59)

where the rst and second terms on the right-hand side are the net internal and kinetic energies,respectively.Here,

E(r,t) is the internal (i.e.,thermal) energy per unit mass of the uid.The energy ux acr oss S,and out of V,is [cf.,

Equation (2.29)]

Φ

E

=

Z

S

ρ

E +

1

2

v

i

v

i

!

v

j

dS

j

=

Z

V

∂

∂x

j

"

ρ

E +

1

2

v

i

v

i

!

v

j

#

dV,(2.60)

where use has been made of the tensor divergence theorem.According to the rst law of thermodynamics,the rate of

increase of the energy contained within V,plus the net energy ux out of V,is equal to the net rate of work done on

the uid within V,minus the net heat ux out of V:i.e.,

dE

dt

+ Φ

E

=

.

W −

.

Q,(2.61)

where

.

W is the net rate of work,and

.

Q the net heat ux.It can be seen that

.

W −

.

Q is the eﬀective energy generation

rate within V [cf.,Equation (2.31)].

Now,the net rate at which volume and surface forces do work on the uid within V is

.

W =

Z

V

v

i

F

i

dV +

Z

S

v

i

σ

i j

dS

j

=

Z

V

"

v

i

F

i

+

∂(v

i

σ

i j

)

∂x

j

#

dV,(2.62)

where use has been made of the tensor divergence theorem.

MathematicalModelsofFluidMotion 19

Generally speaking,heat ow in uids is driven by temperature gradients.Let the q

i

(r,t) be the Cartesian com-

ponents of the heat ux density at position r and time t.It follows that the heat ux across a surface element dS,

located at point r,is q dS = q

i

dS

i

.Let T(r,t) be the temperature of the uid at position r and time t.Thus,a general

temperature gradient takes the form∂T/∂x

i

.Let us assume that there is a linear relationship between the components

of the local heat ux density and the local temperature gradi ent:i.e.,

q

i

= A

i j

∂T

∂x

j

,(2.63)

where the A

i j

are the components of a second-rank tensor (which can be functions of position and time).Now,in an

isotropic uid we would expect A

i j

to be an isotropic tensor.(See Section B.5.) However,the most general second-

order isotropic tensor is simply a multiple of δ

i j

.Hence,we can write

A

i j

= −κ δ

i j

,(2.64)

where κ(r,t) is termed the thermal conductivity of the uid.It follows that the most general expression for t he heat

ux density in an isotropic uid is

q

i

= −κ

∂T

∂x

i

,(2.65)

or,equivalently,

q = −κ ∇T.(2.66)

Moreover,it is a matter of experience that heat ows down temperature gradients:i.e.,κ > 0.We conclude that the net

heat ux out of volume V is

.

Q = −

Z

S

κ

∂T

∂x

i

dS

i

= −

Z

V

∂

∂x

i

κ

∂T

∂x

i

!

dV,(2.67)

where use has been made of the tensor divergence theorem.

Equations (2.59)(2.62) and (2.67) can be combined to give t he following energy conservation equation:

Z

V

(

∂

∂t

"

ρ

E +

1

2

v

i

v

i

!#

+

∂

∂x

j

"

ρ

E +

1

2

v

i

v

i

!

v

j

#)

dV

=

Z

V

"

v

i

F

i

+

∂

∂x

j

v

i

σ

i j

+ κ

∂T

∂x

j

!#

dV.(2.68)

However,this result is valid irrespective of the size,shape,or location of volume V,which is only possible if

∂

∂t

"

ρ

E +

1

2

v

i

v

i

!#

+

∂

∂x

j

"

ρ

E +

1

2

v

i

v

i

!

v

j

#

= v

i

F

i

+

∂

∂x

j

v

i

σ

i j

+ κ

∂T

∂x

j

!

(2.69)

everywhere inside the uid.Expanding some of the derivativ es,and rearranging,we obtain

ρ

D

Dt

E +

1

2

v

i

v

i

!

= v

i

F

i

+

∂

∂x

j

v

i

σ

i j

+ κ

∂T

∂x

j

!

,(2.70)

where use has been made of the continuity equation (2.40).Now,the scalar product of v with the uid equation of

motion (2.53) yields

ρv

i

Dv

i

Dt

= ρ

D

Dt

1

2

v

i

v

i

!

= v

i

F

i

+ v

i

∂σ

i j

∂x

j

.(2.71)

Combining the previous two equations,we get

ρ

DE

Dt

=

∂v

i

∂x

j

σ

i j

+

∂

∂x

j

κ

∂T

∂x

j

!

.(2.72)

20 FLUIDMECHANICS

Finally,making use of (2.26),we deduce that the energy conservation equation for an isotropic Newtonian uid takes

the general form

DE

Dt

= −

p

ρ

∂v

i

∂x

i

+

1

ρ

"

χ +

∂

∂x

j

κ

∂T

∂x

j

!#

.(2.73)

Here,

χ =

∂v

i

∂x

j

d

i j

= 2

e

i j

e

i j

−

1

3

e

ii

e

j j

!

=

∂v

i

∂x

j

∂v

i

∂x

j

+

∂v

i

∂x

j

∂v

j

∂x

i

−

2

3

∂v

i

∂x

i

∂v

j

∂x

j

!

(2.74)

is the rate of heat generation per unit volume due to viscosity.When written in vector form,Equation (2.73) becomes

DE

Dt

= −

p

ρ

∇v +

χ

ρ

+

∇ (κ ∇T)

ρ

.(2.75)

According to the above equation,the internal energy per unit mass of a co-moving uid element evolves in time as

a consequence of work done on the element by pressure as its volume changes,viscous heat generation due to ow

shear,and heat conduction.

2.14 Equations of Incompressible Fluid Flow

In most situations of general interest,the owof a conventi onal liquid,such as water,is incompressible to a high degree

of accuracy.Now,a uid is said to be incompressible when the mass density of a co-moving volume element does not

change appreciably as the element moves through regions of varying pressure.In other words,for an incompressible

uid,the rate of change of ρ following the motion is zero:i.e.,

Dρ

Dt

= 0.(2.76)

In this case,the continuity equation (2.40) reduces to

∇v = 0.(2.77)

We conclude that,as a consequence of mass conservation,an incompressible uid must have a divergence-free,or

solenoidal,velocity eld.This immediately implies,from Equation (2.42),that the volume of a co-moving uid

element is a constant of the motion.In most practical situations,the initial density distribution in an incompressible

uid is uniform in space.Hence,it follows from (2.76) that the density distribution remains uniform in space and

constant in time.In other words,we can generally treat the density,ρ,as a uniform constant in incompressible uid

ow problems.

Suppose that the volume force acting on the uid is conservative in nature (see Section A.18):i.e.,

F = −ρ∇Ψ,(2.78)

where Ψ(r,t) is the potential energy per unit mass,and ρΨ the potential energy per unit volume.Assuming that

the uid viscosity is a spatially uniform quantity,which is generally the case (unless there are strong temperature

variations within the uid),the Navier-Stokes equation fo r an incompressible uid reduces to

Dv

Dt

= −

∇p

ρ

− ∇Ψ + ν ∇

2

v,(2.79)

where

ν =

ρ

(2.80)

is termed the kinematic viscosity,and has units of meters squared per second.Roughly speaking,momentumdiﬀuses

a distance of order

√

ν t meters in t seconds as a consequence of viscosity.The kinematic viscosity of water at 20

◦

C

is about 1.0 × 10

−6

m

2

/s.It follows that viscous momentumdiﬀusion in water is a relatively slow process.

MathematicalModelsofFluidMotion 21

The complete set of equations governing incompressible ow is

∇v = 0,(2.81)

Dv

Dt

= −

∇p

ρ

− ∇Ψ + ν ∇

2

v.(2.82)

Here,ρ and ν are regarded as known constants,and Ψ(r,t) as a known function.Thus,we have four equations

namely,Equation (2.81),plus the three components of Equation (2.82)for four unknownsnamely,the pressure,

p(r,t),plus the three components of the velocity,v(r,t).Note that an energy conservation equation is redundant in the

case of incompressible uid ow.

2.15 Equations of Compressible Fluid Flow

In many situations of general interest,the ow of gases is compressible:i.e.,there are signicant changes in the

mass density as the gas ows from place to place.For the case o f compressible ow,the continuity equation (2.40),

and the Navier-Stokes equation (2.56),must be augmented by the energy conservation equation (2.75),as well as

thermodynamic relations that specify the internal energy per unit mass,and the temperature in terms of the density

and pressure.For an ideal gas,these relations take the form

E =

c

V

M

T,(2.83)

T =

M

R

p

ρ

,(2.84)

where c

V

is the molar specic heat at constant volume,R = 8.3145 J K

−1

mol

−1

the molar ideal gas constant,Mthe

molar mass (i.e.,the mass of 1 mole of gas molecules),and T the temperature in degrees Kelvin.Incidentally,1 mole

corresponds to 6.0221 × 10

24

molecules.Here,we have assumed,for the sake of simplicity,that c

V

is a uniform

constant.It is also convenient to assume that the thermal conductivity,κ,is a uniformconstant.Making use of these

approximations,Equations (2.40),(2.75),(2.83),and (2.84) can be combined to give

1

γ − 1

Dp

Dt

−

γ p

ρ

Dρ

Dt

!

= χ +

κ M

R

∇

2

p

ρ

!

,(2.85)

where

γ =

c

p

c

V

=

c

V

+ R

c

V

(2.86)

is the ratio of the molar specic heat at constant pressure,c

p

,to that at constant volume,c

V

.(Incidentally,the result

that c

p

= c

V

+ R for an ideal gas is a standard theorem of thermodynamics.) The ratio of specic heats of dry air at

20

◦

C is 1.40.

The complete set of equations governing compressible ideal gas ow are

Dρ

Dt

= −ρ∇v,(2.87)

Dv

Dt

= −

∇p

ρ

− ∇Ψ +

ρ

"

∇

2

v +

1

3

∇(∇v)

#

,(2.88)

1

γ − 1

Dp

Dt

−

γ p

ρ

Dρ

Dt

!

= χ +

κ M

R

∇

2

p

ρ

!

,(2.89)

where the dissipation function χ is specied in terms of and v in Equation (2.74).Here,,γ,κ,M,and Rare regarded

as known constants,and Ψ(r,t) as a known function.Thus,we have ve equationsnamely,Eq uations (2.87) and

(2.89),plus the three components of Equation (2.88)for v e unknownsnamely,the density,ρ(r,t),the pressure,

p(r,t),and the three components of the velocity,v(r,t).

22 FLUIDMECHANICS

2.16 Dimensionless Numbers in Incompressible Flow

It is helpful to normalize the equations of incompressible uid ow,(2.81)(2.82),in the following manner:

∇ = L∇,

v = v/V

0

,

t = (V

0

/L) t,

Ψ = Ψ/(g L),and

p = p/(ρV

2

0

+ρg L+ρν V

0

/L).Here,L is a typical spatial variation length-

scale,V

0

a typical uid velocity,and g a typical gravitational acceleration (assuming that Ψ represents a gravitational

potential energy per unit mass).All barred quantities are dimensionless,and are designed to be comparable with unity.

The normalized equations of incompressible uid ow take th e form

∇

v = 0,(2.90)

D

v

D

t

= −

1 +

1

Fr

2

+

1

Re

!

∇

p −

∇

Ψ

Fr

2

+

∇

2

v

Re

,(2.91)

where D/D

t = ∂/∂

t +

v

∇,and

Re =

LV

0

ν

,(2.92)

Fr =

V

0

(g L)

1/2

.(2.93)

Here,the dimensionless quantities Re and Fr are known as the Reynolds number and the Froude number,respectively.

The Reynolds number is the typical ratio of inertial to viscous forces within the uid,whereas the square of the Froude

number is the typical ratio of inertial to gravitational forces.Thus,viscosity is relatively important compared to inertia

when Re ≪ 1,and vice versa.Likewise,gravity is relatively important compared to inertia when Fr ≪ 1,and vice

versa.Note that,in principal,Re and Fr are the only quantities in Equations (2.90) and (2.91) that can be signicantly

greater or smaller than unity.

For the case of water at 20

◦

C,located on the surface of the Earth,

Re ≃ 1.0 × 10

6

L(m) V

0

(ms

−1

),(2.94)

Fr ≃ 3.2 × 10

−1

V

0

(ms

−1

)/[L(m)]

1/2

.(2.95)

Thus,if L ∼ 1 mand V

0

∼ 1 ms

−1

,as is often the case for terrestrial water dynamics,then the above expressions sug-

gest that Re ≫1 and Fr ∼ O(1).In this situation,the viscous termon the right-hand side of (2.91) becomes negligible,

and the (unnormalized) incompressible uid owequations r educe to the following inviscid,incompressible,uid ow

equations,

∇v = 0,(2.96)

Dv

Dt

= −

∇p

ρ

− ∇Ψ.(2.97)

For the case of lubrication oil at 20

◦

C,located on the surface of the Earth,ν ≃ 1.0 × 10

−4

m

2

s

−1

(i.e.,oil is about

100 times more viscous than water),and so

Re ≃ 1.0 × 10

4

L(m) V

0

(ms

−1

),(2.98)

Fr ≃ 3.2 × 10

−1

V

0

(ms

−1

)]/[L(m)]

1/2

.(2.99)

Suppose that oil is slowly owing down a narrow lubrication c hannel such that L ∼ 10

−3

m and V

0

≪ 10

−1

ms

−1

.It

follows,from the above expressions,that Re ≪ 1 and Fr ≪ 1.In this situation,the inertial term on the left-hand

side of (2.91) becomes negligible,and the (unnormalized) incompressible uid owequations reduce to the following

inertia-free,incompressible,uid ow equations,

∇v = 0,(2.100)

0 = −

∇p

ρ

− ∇Ψ + ν ∇

2

v.(2.101)

MathematicalModelsofFluidMotion 23

2.17 Dimensionless Numbers in Compressible Flow

It is helpful to normalize the equations of compressible ideal gas ow,(2.87)(2.89),in the following manner:

∇ =

L∇,

v = v/V

0

,

t = (V

0

/L) t,

ρ = ρ/ρ

0

,

Ψ = Ψ/(g L),

χ = (L/V

0

)

2

χ,and

p = (p − p

0

)/(ρ

0

V

2

0

+ ρ

0

g L + ρ

0

ν V

0

/L).

Here,L is a typical spatial variation length-scale,V

0

a typical uid velocity,ρ

0

a typical mass density,and g a typical

gravitational acceleration (assuming that Ψ represents a gravitational potential energy per unit mass).Furthermore,p

0

corresponds to atmospheric pressure at ground level,and is a uniformconstant.It follows that

p represents deviations

from atmospheric pressure.All barred quantities are dimensionless,and are designed to be comparable with unity.

The normalized equations of compressible ideal gas ow take the form

D

ρ

D

t

= −

ρ

∇

v,(2.102)

D

v

D

t

= −

1 +

1

Fr

2

+

1

Re

!

∇

p

ρ

−

∇

Ψ

Fr

2

+

1

Re

ρ

"

∇

2

v −

1

3

∇(

∇

v)

#

,(2.103)

1

γ − 1

"

D

p

D

t

− γ

p

0

+

p

ρ

!

D

ρ

D

t

#

=

χ

1 + Re (1 + 1/Fr

2

)

+

1

Re Pr

∇

2

p

0

+

p

ρ

!

,(2.104)

p

0

=

1

γ M

2

(1 + 1/Fr

2

+ 1/Re)

,(2.105)

where D/D

t ≡ ∂/∂

t +

v

∇,

Re =

LV

0

ν

,(2.106)

Fr =

V

0

(g L)

1/2

,(2.107)

Pr =

ν

κ

H

,(2.108)

M =

V

0

p

γ p

0

/ρ

0

,(2.109)

and

ν =

ρ

0

,(2.110)

κ

H

=

κ M

Rρ

0

.(2.111)

Here,the dimensionless numbers Re,Fr,Pr,and M are known as the Reynolds number,Froude number,Prandtl

number,and Mach number,respectively.The Reynolds number is the typical ratio of inertial to viscous forces within

the gas,the square of the Froude number the typical ratio of inertial to gravitational forces,the Prandtl number the

typical ratio of the momentum and thermal diﬀusion rates,and the Mach number the typical ratio of gas ow a nd

sound propagation speeds.Thus,thermal diﬀusion is far faster than momentum diﬀusion when Pr ≪ 1,and vice

versa.Moreover,the gas ow is termed subsonic when M ≪ 1,supersonic when M ≫ 1,and transonic when

M ∼ O(1).Note that

p

γ p

0

/ρ

0

is the speed of sound in the undisturbed gas.The quantity κ

H

is called the thermal

diﬀusivity of the gas,and has units of meters squared per second.Thus,heat typically diﬀuses through the gas a

distance

√

κ

H

t meters in t seconds.The thermal diﬀusivity of dry air at atmospheric pressure and 20

◦

C is about

κ

H

= 2.1 × 10

−5

m

2

s

−1

.It follows that heat diﬀusion in air is a relatively slow process.The kinematic viscosity of

dry air at atmospheric pressure and 20

◦

C is about ν = 1.5 × 10

−5

m

2

s

−1

.Hence,momentumdiﬀusion in air is also a

relatively slow process.

For the case of dry air at atmospheric pressure and 20

◦

C,

Re ≃ 6.7 × 10

4

L(m) V

0

(ms

−1

),(2.112)

24 FLUIDMECHANICS

Fr ≃ 3.2 × 10

−1

V

0

(ms

−1

)/[L(m)]

1/2

,(2.113)

Pr ≃ 7.2 × 10

−1

,(2.114)

M ≃ 2.9 × 10

−3

V

0

(ms

−1

).(2.115)

Thus,if L ∼ 1 m and V

0

∼ 1 ms

−1

,as is often the case for subsonic air dynamics close to the Earth's surface,the

above expressions suggest that Re ≫ 1,M ≪ 1,and Fr,Pr ∼ O(1).It immediately follows from Equation (2.105)

that

p

0

≫ 1.However,in this situation,Equation (2.104) is dominated by the second term in square brackets on its

left-hand side.Hence,this equation can only be satised if the termin question is small,which implies that

D

ρ

D

t

≪1.(2.116)

Equation (2.102) then gives

∇

v ≪1.(2.117)

Thus,it is evident that subsonic (i.e.,M ≪ 1) gas ow is essentially incompressible.The fact that Re ≫ 1 implies

that such ow is also essentially inviscid.In the incompressible inviscid limit (in which

∇

v = 0 and Re ≫ 1),the

(unnormalized) compressible ideal gas ow equations reduc e to the previously derived inviscid,incompressible,uid

ow equations:i.e.,

∇v = 0,(2.118)

Dv

Dt

= −

∇p

ρ

− ∇Ψ.(2.119)

It follows that the equations which govern subsonic gas dynamics close to the surface of the Earth are essentially the

same as those which govern the ow of water.

Suppose that L ∼ 1 mand V

0

∼ 300 ms

−1

,as is typically the case for transonic air dynamics (e.g.,air owover the

wing of a ghter jet).In this situation,Equations (2.105) a nd (2.112)(2.115) yield Re,Fr ≫1 and M,Pr,

p

0

∼ O(1).

It follows that the nal two terms on the right-hand sides of E quations (2.103) and (2.104) can be neglected.Thus,

the (unnormalized) compressible ideal gas ow equations re duce to the following set of inviscid,adiabatic,ideal gas,

ow equations,

Dρ

Dt

= −ρ∇v,(2.120)

Dv

Dt

= −

∇p

ρ

,(2.121)

D

Dt

p

ρ

γ

!

= 0.(2.122)

In particular,if the initial distribution of p/ρ

γ

is uniformin space,as is often the case,then Equation (2.122) ensures

that the distribution remains uniformas time progresses.In fact,it can be shown that the entropy per unit mass of an

ideal gas is

S =

c

V

M

ln

p

ρ

γ

!

.(2.123)

Hence,the assumption that p/ρ

γ

is uniformin space is equivalent to the assumption that the entropy per unit mass of

the gas is a spatial constant.A gas in which this is the case is termed homentropic.Equation (2.122) ensures that the

entropy of a co-moving gas element is a constant of the motion in transonic ow.A gas in which this is the case is

termed isentropic.In the homentropic case,the above compressible gas ow equ ations simplify somewhat to give

Dρ

Dt

= −ρ∇v,(2.124)

Dv

Dt

= −

∇p

ρ

,(2.125)

p

p

0

=

ρ

ρ

0

!

γ

.(2.126)

MathematicalModelsofFluidMotion 25

Here,p

0

is atmospheric pressure,and ρ

0

is the density of air at atmospheric pressure.Equation (2.126) is known as the

adiabatic gas law,and is a consequence of the fact that transonic gas dynamics takes place far too quickly for thermal

heat conduction (which is a relatively slow process) to have any appreciable eﬀect on the temperature distribution

within the gas.Incidentally,a gas in which thermal diﬀusion is negligible is generally termed adiabatic.

2.18 Fluid Equations in Cartesian Coordinates

Let us adopt the conventional Cartesian coordinate system,x,y,z.According to Equation (2.26),the various compo-

nents of the stress tensor are

σ

xx

= −p + 2

∂v

x

∂x

,(2.127)

σ

yy

= −p + 2

∂v

y

∂y

,(2.128)

σ

zz

= −p + 2

∂v

z

∂z

,(2.129)

σ

xy

= σ

yx

=

∂v

x

∂y

+

∂v

y

∂x

!

,(2.130)

σ

xz

= σ

zx

=

∂v

x

∂z

+

∂v

z

∂x

!

,(2.131)

σ

yz

= σ

zy

=

∂v

y

∂z

+

∂v

z

∂y

!

,(2.132)

where v is the velocity,p the pressure,and the viscosity.The equations of compressible uid ow,(2.8 7)(2.89)

(fromwhich the equations of incompressible uid ow can eas ily be obtained by setting Δ = 0),become

Dρ

Dt

= −ρΔ,(2.133)

Dv

x

Dt

= −

1

ρ

∂p

∂x

−

∂Ψ

∂x

+

ρ

∇

2

v

x

+

1

3

∂Δ

∂x

!

,(2.134)

Dv

y

Dt

= −

1

ρ

∂p

∂y

−

∂Ψ

∂y

+

ρ

∇

2

v

y

+

1

3

∂Δ

∂y

!

,(2.135)

Dv

z

Dt

= −

1

ρ

∂p

∂z

−

∂Ψ

∂z

+

ρ

∇

2

v

z

+

1

3

∂Δ

∂z

!

,(2.136)

1

γ − 1

Dρ

Dt

−

γ p

ρ

Dρ

Dt

!

= χ +

κ M

R

∇

2

p

ρ

!

,(2.137)

where ρ is the mass density,γ the ratio of specic heats,κ the heat conductivity,Mthe molar mass,and R the molar

ideal gas constant.Furthermore,

Δ =

∂v

x

∂x

+

∂v

y

∂y

+

∂v

z

∂z

,(2.138)

D

Dt

=

∂

∂t

+ v

x

∂

∂x

+ v

y

∂

∂y

+ v

z

∂

∂z

,(2.139)

∇

2

=

∂

2

∂x

2

+

∂

2

∂y

2

+

∂

2

∂z

2

,(2.140)

χ = 2

∂v

x

∂x

!

2

+

∂v

y

∂y

!

2

+

∂v

z

∂z

!

2

+

1

2

∂v

x

∂y

+

∂v

y

∂x

!

2

26 FLUIDMECHANICS

+

1

2

∂v

x

∂z

+

∂v

z

∂x

!

2

+

1

2

∂v

y

∂z

+

∂v

z

∂y

!

2

.(2.141)

In the above,γ,,κ,and Mare treated as uniformconstants.

2.19 Fluid Equations in Cylindrical Coordinates

Let us adopt the cylindrical coordinate system,r,θ,z.Making use of the results quoted in Section C.3,the components

of the stress tensor are

σ

rr

= −p + 2

∂v

r

∂r

,(2.142)

σ

θθ

= −p + 2

1

r

∂v

θ

∂θ

+

v

r

r

!

,(2.143)

σ

zz

= −p + 2

∂v

z

∂z

,(2.144)

σ

rθ

= σ

θr

=

1

r

∂v

r

∂θ

+

∂v

θ

∂r

−

v

θ

r

!

,(2.145)

σ

rz

= σ

zr

=

∂v

r

∂z

+

∂v

z

∂r

!

,(2.146)

σ

θz

= σ

zθ

=

1

r

∂v

z

∂θ

+

∂v

θ

∂z

!

,(2.147)

whereas the equations of compressible uid ow become

Dρ

Dt

= −ρΔ,(2.148)

Dv

r

Dt

−

v

2

θ

r

= −

1

ρ

∂p

∂r

−

∂Ψ

∂r

+

ρ

∇

2

v

r

−

v

r

r

2

−

2

r

2

∂v

θ

∂θ

+

1

3

∂Δ

∂r

!

,(2.149)

Dv

θ

Dt

+

v

r

v

θ

r

= −

1

ρr

∂p

∂θ

−

1

r

∂Ψ

∂θ

+

ρ

∇

2

v

θ

+

2

r

2

∂v

r

∂θ

−

v

θ

r

2

+

1

3r

∂Δ

∂θ

!

,(2.150)

Dv

z

Dt

= −

1

ρ

∂p

∂z

−

∂Ψ

∂z

+

ρ

∇

2

v

z

+

1

3

∂Δ

∂z

!

,(2.151)

1

γ − 1

Dρ

Dt

−

γ p

ρ

Dρ

Dt

!

= χ +

κ M

R

∇

2

p

ρ

!

,(2.152)

where

Δ =

1

r

∂(r v

r

)

∂r

+

1

r

∂v

θ

∂θ

+

∂v

z

∂z

,(2.153)

D

Dt

=

∂

∂t

+ v

r

∂

∂r

+

v

θ

r

∂

∂θ

+ v

z

∂

∂z

,(2.154)

∇

2

=

1

r

∂

∂r

r

∂

∂r

!

+

1

r

2

∂

2

∂θ

2

+

∂

2

∂z

2

,(2.155)

MathematicalModelsofFluidMotion 27

χ = 2

∂v

r

∂r

!

2

+

1

r

∂v

θ

∂θ

+

v

r

r

!

2

+

∂v

z

∂z

!

2

+

1

2

1

r

∂v

r

∂θ

+

∂v

θ

∂r

−

v

θ

r

!

2

+

1

2

∂v

r

∂z

+

∂v

z

∂r

!

2

+

1

2

∂v

θ

∂z

+

1

r

∂v

z

∂θ

!

2

.(2.156)

2.20 Fluid Equations in Spherical Coordinates

Let us,nally,adopt the spherical coordinate system,r,θ,φ.Making use of the results quoted in Section C.4,the

components of the stress tensor are

σ

rr

= −p + 2

∂v

r

∂r

,(2.157)

σ

θθ

= −p + 2

1

r

∂v

θ

∂θ

+

v

r

r

!

,(2.158)

σ

φφ

= −p + 2

1

r sinθ

∂v

φ

∂φ

+

v

r

r

+

cot θ v

θ

r

!

,(2.159)

σ

rθ

= σ

θr

=

1

r

∂v

r

∂θ

+

∂v

θ

∂r

−

v

θ

r

!

,(2.160)

σ

rφ

= σ

φr

=

1

r sin θ

∂v

r

∂φ

+

∂v

φ

∂r

−

v

φ

r

!

,(2.161)

σ

θφ

= σ

φθ

=

1

r sin θ

∂v

θ

∂φ

+

1

r

∂v

φ

∂θ

−

cot θ v

φ

r

!

,(2.162)

whereas the equations of compressible uid ow become

Dρ

Dt

= −ρΔ,(2.163)

Dv

r

Dt

−

v

2

θ

+ v

2

φ

r

= −

1

ρ

∂p

∂r

−

∂Ψ

∂r

+

ρ

∇

2

v

r

−

2v

r

r

2

−

2

r

2

∂v

θ

∂θ

−

2 cot θ v

θ

r

2

−

2

r

2

sinθ

∂v

φ

∂φ

+

1

3

∂Δ

∂r

!

,(2.164)

Dv

θ

Dt

+

v

r

v

θ

− cot θ v

2

φ

r

= −

1

ρr

∂p

∂θ

−

1

r

∂Ψ

∂θ

+

ρ

∇

2

v

θ

+

2

r

2

∂v

r

∂θ

(2.165)

−

v

θ

r

2

sin

2

θ

−

2 cot θ

r

2

sin θ

∂v

φ

∂φ

+

1

3r

∂Δ

∂θ

!

,(2.166)

Dv

φ

Dt

+

v

r

v

φ

+ cot θ v

θ

v

φ

r

= −

1

ρr sinθ

∂p

∂φ

−

1

r sin θ

∂Ψ

∂φ

+

ρ

∇

2

v

φ

−

v

φ

r

2

sin

2

θ

+

2

r

2

sin

2

θ

∂v

r

∂φ

+

2 cot θ

r

2

sin θ

∂v

θ

∂φ

+

1

3r sinθ

∂Δ

∂φ

!

,(2.167)

1

γ − 1

Dρ

Dt

−

γ p

ρ

Dρ

Dt

!

= χ +

κ M

R

∇

2

p

ρ

!

,(2.168)

where

Δ =

1

r

2

∂(r

2

v

r

)

∂r

+

1

r sin θ

∂(sinθ v

θ

)

∂θ

+

1

r sinθ

∂v

φ

∂φ

,(2.169)

28 FLUIDMECHANICS

D

Dt

=

∂

∂t

+ v

r

∂

∂r

+

v

θ

r

∂

∂θ

+

v

φ

r sinθ

∂

∂φ

,(2.170)

∇

2

=

1

r

2

∂

∂r

r

2

∂

∂r

!

+

1

r

2

sin θ

∂

∂θ

sin θ

∂

∂θ

!

+

1

r

2

sin

2

θ

∂

2

∂φ

2

,(2.171)

χ = 2

∂v

r

∂r

!

2

+

1

r

∂v

θ

∂θ

+

v

r

r

!

2

+

1

r sin θ

∂v

φ

∂φ

+

v

r

r

+

cot θ v

θ

r

!

2

+

1

2

1

r

∂v

r

∂θ

+

∂v

θ

∂r

−

v

θ

r

!

2

+

1

2

1

r sin θ

∂v

r

∂φ

+

∂v

φ

∂r

−

v

φ

r

!

2

+

1

2

1

r sin θ

∂v

θ

∂φ

+

1

r

∂v

φ

∂θ

−

cot θ v

φ

r

!

2

.(2.172)

2.21 Exercises

2.1.Equations (2.66),(2.75),and (2.87) can be combined to give the following energy conservation equation for a non-ideal

compressible uid:

ρ

DE

Dt

−

p

ρ

Dρ

Dt

= χ − ∇ q,

where ρ is the mass density,p the pressure,E the internal energy per unit mass,χ the viscous energy dissipation rate per unit

volume,and q the heat ux density.We also have

Dρ

Dt

= −ρ∇ v,

q = −κ ∇T,

where v is the uid velocity,T the temperature,and κ the thermal conductivity.Now,according to a standard theorem in

thermodynamics,

T dS = dE −

p

ρ

2

dρ,

where S is the entropy per unit mass.Moreover,the entropy ux densi ty at a given point in the uid is

s = ρ Sv +

q

T

,

where the rst termon the right-hand side is due to direct ent ropy convection by the uid,and the second is the entropy ux

density associated with heat conduction.

Derive an entropy conservation equation of the form

dS

dt

+ Φ

S

= Θ

S

,

where S is the net amount of entropy contained in some xed volume V,Φ

S

the entropy ux out of V,and Θ

S

the net rate of

entropy creation within V.Give expressions for S,Φ

S

,and Θ

S

.Demonstrate that the entropy creation rate per unit volume

is

θ =

χ

T

+

q q

κ T

2

.

Finally,show that θ ≥ 0,in accordance with the second law of thermodynamics.

2.2.The Navier-Stokes equation for an incompressible uid of uniformmass density ρ takes the form

Dv

Dt

= −

∇p

ρ

− ∇Ψ + ν ∇

2

v,

where v is the uid velocity,p the pressure,Ψ the potential energy per unit mass,and ν the (uniform) kinematic viscosity.

The incompressibility constraint requires that

∇ v = 0.

Finally,the quantity

ω ≡ ∇× v

MathematicalModelsofFluidMotion 29

is generally referred to as the uid vorticity.

Derive the following vorticity evolution equation fromthe Navier-Stokes equation:

Dω

Dt

= (ω ∇) v + ν ∇

2

ω.

2.3.Consider two-dimensional incompressible uid ow.Le t the velocity eld take the form

v = v

x

(x,y,t) e

x

+ v

y

(x,y,t) e

y

.

Demonstrate that the equations of incompressible uid ow ( see Exercise 2.2) can be satised by writing

v

x

= −

∂ψ

∂y

,

v

y

=

∂ψ

∂x

,

where

∂ω

∂t

+

∂ψ

∂x

∂ω

∂y

−

∂ω

∂x

∂ψ

∂y

= ν ∇

2

ω,

and

ω = ∇

2

ψ.

Here,∇

2

= ∂

2

/∂x

2

+ ∂

2

/∂y

2

.Furthermore,the quantity ψ is termed a stream function,since v ∇ψ = 0:i.e.,the uid ow is

everywhere parallel to contours of ψ.

2.4.Consider incompressible irrotational ow:i.e.,ow that satises

Dv

Dt

= −

∇p

ρ

− ∇Ψ + ν ∇

2

v,

∇ v = 0,

as well as

∇ × v = 0.

Here,v is the uid velocity,ρ the uniform mass density,p the pressure,Ψ the potential energy per unit mass,and ν the

(uniform) kinematic viscosity.

Demonstrate that the above equations can be satised by writ ing

v = ∇φ,

where

∇

2

φ = 0,

and

∂φ

∂t

+

1

2

v

2

+

p

ρ

+ Ψ = C(t).

Here,C(t) is a spatial constant.This type of ow is known as potential ow,since the velocity eld is derived from a scalar

potential.

2.5.The equations of inviscid adiabatic ideal gas ow are

Dρ

Dt

= −ρ∇ v,

Dv

Dt

= −

∇p

ρ

− ∇Ψ,

D

Dt

p

ρ

γ

!

= 0.

Here,ρ is the mass density,v the ow velocity,p the pressure,Ψ the potential energy per unit mass,and γ the (uniform)

ratio of specic heats.Suppose that the pressure and potent ial energy are both time independent:i.e.,∂p/∂t = ∂Ψ/∂t = 0.

Demonstrate that

H =

1

2

v

2

+

γ

γ − 1

p

ρ

+ Ψ

is a constant of the motion:i.e.,DH/Dt = 0.This result is known as Bernoulli's theorem.

30 FLUIDMECHANICS

2.6.The equations of inviscid adiabatic non-ideal gas ow a re

Dρ

Dt

= −ρ∇ v,

Dv

Dt

= −

∇p

ρ

− ∇Ψ,

DE

Dt

−

p

ρ

2

Dρ

Dt

= 0.

Here,ρ is the mass density,v the ow velocity,p the pressure,Ψ the potential energy per unit mass,and E the internal

energy per unit mass.Suppose that the pressure and potential energy are both time independent:i.e.,∂p/∂t = ∂Ψ/∂t = 0.

Demonstrate that

H =

1

2

v

2

+ E +

p

ρ

+ Ψ

is a constant of the motion:i.e.,DH/Dt = 0.This result is a more general formof Bernoulli's theorem.

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