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 NavierStokes 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 YoungLaplace 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 SoapBubbles......................................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 TwoDimensional Flow.........................................83
5.9 TwoDimensional UniformFlow....................................85
5.10 TwoDimensional Sources and Sinks..................................86
5.11 TwoDimensional Vortex Filaments...................................87
5.12 TwoDimensional Irrotational Flow in Cylindrical Coordinates....................90
5.13 Inviscid Flow Past a Cylindrical Obstacle................................91
5.14 Inviscid Flow Past a SemiInnite Wedge................................94
5.15 Inviscid Flow Over a SemiInnite 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 CauchyRiemann 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 SelfSimilar 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 ThreeDimensional 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 TaylorCouette Flow...........................................174
9.6 Flow in SlowlyVarying 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 NonCartesian 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 EulerLagrange Equation........................................273
D.2 Conditional Variation..........................................275
CONTENTS 5
D.3 MultiFunction 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 lowerdivision 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 multivariate 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.MilneThomson,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 (McGrawHill,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 (ButterworthHeinemann,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 (McGrawHill,NewYork NY,1999).
An Introduction to Fluid Dynamics G.K.Batchelor (Cambridge University Press,Cambridge UK,2000).
Theoretical Hydrodynamics L.M.MilneThomson,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 righthanded 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 lengthscales 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 meanfreepath 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 welldescribe 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 nonisotropic;thixotropic uids,such as jelly or pain t,which are nonNewtonian;and quantum uids,such as
liquid helium,which exhibit nonclassical eﬀects on macroscopic lengthscales).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 welldescribed 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 longrange 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 longrange force.One
consequence of the relatively slow variation of longrange 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,longrange 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 shortrange 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 xdirected uid
momentumdue to shortrange forces at position r and time t.In other words,suppose that,at position r and time t,as a
direct consequence of shortrange forces,xmomentumis owing at the rate of π
x
 newtonseconds 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 xmomentumacross 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 xdirection due to shortrange 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 xdirection due to interaction with the uid on the positive s ide.Shortrange
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
secondorder tensor (see Appendix B),known as the stress tensor.[This follows because the f
i
are the components
of a rstorder tensor (since all forces are proper vectors),and the dS
i
are the components of an arbitrary rstorder
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 secondorder tensor.] We can interpret
σ
i j
(r,t) as the icomponent of the force per unit area exerted,at position r and time t,across a plane surface element
normal to the jdirection.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 nondiagonal 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 icomponent 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 righthand 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 righthand side of the above equation scale as V.Now,according to
Newtonian dynamics,the icomponent of the net force acting on the element is equal to the icomponent 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 icomponent 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 righthand side are due to volume and surface forces,respectively.[Here,
ǫ
i jk
is the thirdorder 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 secondorder 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 righthand 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 icomponent of the
total torque acting on the uid element is equal to the icomponent 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
righthand 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 secondorder 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 nondiagonal
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 nonzero 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 wellknown 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 noniso 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 socalled 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 fourthorder 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 fourthorder 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 nonzero
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 nontimevarying.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 comoving 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 socalled 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 comoving 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 comoving uid ele ment at that point.
2.11 MomentumConservation
Consider a xed volume V surrounded by a surface S.The icomponent of the total linear momentumcontained within
V is
P
i
=
Z
V
ρv
i
dV.(2.43)
MathematicalModelsofFluidMotion 17
Moreover,the ux of imomentumacross S,and out of V,is [see Equation (2.29)]
Φ
i
=
Z
S
ρv
i
v
j
dS
j
.(2.44)
Finally,the icomponent 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 righthand side are the contributions fromvolume and surface forces,respec
tively.
Momentum conservation requires that the rate of increase of the net imomentum of the uid contained within
V,plus the ux of imomentum out of V,is equal to the rate of imomentum generation within V.Of course,from
Newton's second law of motion,the latter quantity is equal t o the icomponent 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 nontimevarying.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 comoving uid element
determine its acceleration.
18 FLUIDMECHANICS
2.12 NavierStokes 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 NavierStokes 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 NavierStokes 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 righthand 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 secondrank 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 comoving 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 comoving 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 divergencefree,or
solenoidal,velocity eld.This immediately implies,from Equation (2.42),that the volume of a comoving 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 NavierStokes 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 NavierStokes 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 righthand 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 lefthand
side of (2.91) becomes negligible,and the (unnormalized) incompressible uid owequations reduce to the following
inertiafree,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 lengthscale,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
lefthand 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 righthand 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 comoving 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 nonideal
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 righthand 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 NavierStokes 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 NavierStokes equation:
Dω
Dt
= (ω ∇) v + ν ∇
2
ω.
2.3.Consider twodimensional 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 nonideal 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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