Chapter 2
Kinematics
Kinematics is a study of the geometry of motion independently of applied
forces.It consists of descriptive tools and various constraints that limit in
important ways how the ﬂuid can respond to the application of forces.
We will learn about vectors,with velocity as the foremost example;about
various algebraic,diﬀerential and integral operations on vectors;about such
vectors as attributes of a particle (Lagrangian description) or of a point in
space (Eulerian description);about rateofstrain and vorticity,maybe the
most important new concept in the whole course;about mass balance and
its consequences;and about ﬂow decompositions,vortices as ﬂow structures,
and much more.
Particularly relevant movies are:Kline’s Flow visualization,as back
ground for all others;Lumley’s Eulerian and Lagrangian descriptions,and
Shapiro’s Vorticity.
2.1 Vectors
Classical mechanics are ﬁrmly grounded in 3dimensional space,in which
vectors are ubiquitous objects.Sure enough,scalar quantities such as energy
and pressure are important,as seen in Bernoulli’s equation;but mechanics
is dominated by the use of vectors for position,momentum,rotation,etc.
Vectors are mathematical objects endowed with a direction and a mag
nitude — they may also have a point or line of application.In this course,
we will use three equivalent notations for vectors,each with advantages de
pending on the problem at hand.It is important that the student learn to
35
36 CHAPTER 2.KINEMATICS
use these notations interchangeably:practice,practice...
2.1.1 Intrinsic notations
The ﬁrst notation,used in earlier chapters,makes no reference to any co
ordinate system:the vectors exist independently of how they are described.
This is the notation used in previous chapters,by default.A vector is de
noted by an underline,as in velocity u
.Operations on vectors include the
multiplication by a number (scalar),e.g.
ρu
= u
ρ,(2.1)
the dot product,which makes a scalar from two vectors taken in any order
u
2
= u
· u
(2.2)
and the cross product,which makes a vector from two vectors,but reverses
sign if the order of vectors is changed
u
= Ω
×r
= −r
×Ω
.(2.3)
The student is expected to knowthese operations fromundergraduate classes,
this presentation being included for the sake of notations and as a point of
reference for alternative notations (below).
Of a diﬀerent nature,the vectoroperator ∇ is deﬁned by its properties,
although in many ways the component notation may be easiest to grasp.
Assuming some background again,we simply note the gradient of a scalar
function
grad f = ∇f,(2.4)
the divergence of a vector
div u
= ∇· u
,(2.5)
the curl of a vector
curl u
= ∇×u
,(2.6)
and the Laplacian
∇
2
= ∇· ∇ (2.7)
(this last deﬁnition holds regardless of what the operator is applied to,vector
or scalar).As an operator,∇ only acts on the variables that follow it,
therefore
u
· ∇ ̸= ∇· u
.(2.8)
2.1.VECTORS 37
Without implication about a particular system of axes,this notation will
be used by default,but is sometimes awkward when tensors of orders larger
than vectors are being used,e.g.
∇u
or ∇· (a
b
),(2.9)
for which alternative notations are clearer.
2.1.2 Component notations
Component notations are most popular at the undergraduate level,but suf
fer from runaway bulk!We will use them in this course only when diﬀerent
directions contain diﬀerent physics that should be reﬂected in the notations,
for example in boundary layers where the streamwise (along the wall),trans
verse (normal to the wall) and spanwise (along the wall,normal to the other
two) directions are very distinct.
The use of nonCartesian coordinates (polar,cylindrical,spherical,etc.)
is less popular for beginners,but is actually implied by the above reason
ing:boundary conditions may well dictate which coordinate system to use
for easier handling!We will use nonCartesian coordinates as needed,and
encourage the students to overcome any reluctance in this regard as soon
as possible.Pointers will be included at appropriate places throughout the
chapters.
Coordinate axes may be Cartesian (e
x
,e
y
,e
z
),equivalent to (i
,j
,k
);or
cylindricalpolar (e
r
,e
θ
,e
z
);among many options.The component of a vector
along such an axis is the projection of the vector on the axis,e.g.
u
x
= u
· e
x
.(2.10)
Then,we can write
u
= u
x
e
x
+u
y
e
y
+u
z
e
z
.(2.11)
It is suﬃcient to write the vector as the ordered set of its components,for
example (u
x
,u
y
,u
z
) for which the alternative notation (u,v,w) is also common.
Multiplication by the unit vectors is implied,and should be restored when
in doubt.This notation can then be used to evaluate vector operations.The
dot product reduces to
u
· v
= u
x
v
x
+u
y
v
y
+u
z
v
z
.(2.12)
38 CHAPTER 2.KINEMATICS
Figure 2.1:Vector notations
2.1.VECTORS 39
For the cross product u
= Ω
×r
,its xcomponent is
u
x
= Ω
y
r
z
−Ω
z
r
y
,(2.13)
and similar expressions are obtained by even permutation of the subscripts.
The ∇ vectoroperator is best expressed in Cartesian coordinates.Then,
its components are (∂
x
,∂
y
,∂
z
) and we can write (in a convenient mix of
notations)
∇· u
= ∂
x
u
x
+∂
y
u
y
+∂
z
u
z
.(2.14)
Similarly,the xcomponent of ω
= ∇×u
is
ω
x
= ∂
y
u
z
−∂
z
u
y
,(2.15)
and the Laplacian is the familiar
∇
2
= ∂
2
xx
+∂
2
yy
+∂
2
zz
.(2.16)
There is no general expression for ∇ in nonCartesian coordinates;the con
text of gradient,divergence,curl or Laplacian is necessary for speciﬁc terms
to make sense,and the corresponding expressions can be found in many
reference books.
2.1.3 Index notations
A more concise version of component notations is possible when the Carte
sian axes are assumed,but without any particular orientation.Then,the
subscripts x,y and z are just an ordered triad,and 1,2 and 3 turns out to
facilitate the sums.With unit vectors of the form e
i
for i = 1,2,3,we have
u
=
∑
u
i
e
i
(2.17)
and the triad of components can be written simply as (u
i
).Then for a dot
product
u
· v
=
∑
i
u
i
v
i
= u
i
v
i
(2.18)
where the summation over repeated indices is implied (Einstein’s convention).
Some textbooks and websites use ‘index notations’ while keeping explicitly
the unit vectors e
i
:this defeats the purpose of compact notations.Any
component with a ’free index’ is understood to multiply the corresponding
40 CHAPTER 2.KINEMATICS
unit vector,with summation implied:u
i
stands for all components and unit
vectors together.The use of e
i
will be penalized in this course.
Concise notations can save work,but require neat handling.Students
should be careful to adhere to the following rules:
• any repeated index in a product implies summation;therefore,that in
dex pair should be used only once in an expression,but can be renamed
at will (dummy index) to avoid conﬂicts with other pairs.Thus
u
i
v
i
= u
k
v
k
;(2.19)
• no index may appear 3 or more times in a product;in rare instances
when this occurs,the use of more detailed notations is required,and
summation is usually restored explicitly to avoid confusion
• an index appearing once denotes the component of the vector (’free
index’);it must be the same for all terms in a sum,since they all are
factors of the same implied e
i
;thus
a
i
= b
i
+c
k
(2.20)
is meaningless.
They can also be generalized very usefully to higherorder objects:while
a vector requires a single index,a matrix requires 2,etc.The economy of
notation is most obvious in such context,as we will see in later chapters.
Two symbols are particularly important:the Kronecker symbol
δ
ij
(2.21)
has value 1 if i = j and 0 otherwise.It is a representation of the identity
matrix,since
δ
ij
u
j
= u
i
.(2.22)
Note also that,in 3 dimensions,
δ
ii
= 3.(2.23)
More complicated,the permutation symbol
ϵ
ijk
(2.24)
2.1.VECTORS 41
has value 1 if (i,j,k) is an even permutation of (1,2,3),1 if an odd permu
tation,and zero otherwise (i.e.if any two indices are equal).ϵ is used to
represent the cross product:
u
i
= (Ω
×r
)
i
= ϵ
ijk
Ω
j
r
k
.(2.25)
This should be veriﬁed component by component (yes,do it!) One notewor
thy relation,used for handling double cross products,is
ϵ
ijk
ϵ
ilm
= δ
jl
δ
km
−δ
jm
δ
kl
.(2.26)
Note that the order of the indices of ϵ can be rearranged based on
ϵ
ijk
= ϵ
kij
= −ϵ
jik
(2.27)
and similar alternatives
With these tools,we can write the diﬀerential operators as
(∇f)
i
= ∂
i
f (2.28)
for the gradient;the divergence takes the form
∇· u
= ∂
i
u
i
.(2.29)
The Laplacian is
∇
2
= ∂
2
ii
.(2.30)
Finally,for the curl
ω
i
= ϵ
ijk
∂
j
u
k
.(2.31)
The order of indices of ϵ is important:ﬁrst index for the free index,second
for the derivative,third for the vector of which the curl is taken (and you
can rotate the indices afterward,of course).
Using index notations requires practice (beyond assigned problems),but
is well worth the eﬀort when mastered.In this course,all three notations
will be used depending on the context:the idea is to save work and facilitate
formal manipulations.
42 CHAPTER 2.KINEMATICS
J.L.Lumley's movie (excerpt):Eulerian
vs.Lagrangian
• No dynamics!This is kinematics
• Velocity of particle,velocity at a point
• Convective derivative
• Pathlines,streaklines,timelines,streamlines
• Unsteady ﬂow:wave ﬁeld
Figure 2.2:Eulerian and Lagrangian descriptions,from Lumley’s movie.
Tensor notations
In the case of nonCartesian coordinates,index notations can be generalized
as tensor notations
1
.Added complications include co and contravariance,
and the use of the local metric tensor (responsible for the ‘strange’ factors
in the expression of curl,div and grad).In ﬂuid dynamics,this can be
encountered in some treatments of computational grid generation,in the
study of nonNewtonian ﬂuids,and a few other applications,but will not be
needed in this course.
2.2 Eulerian vs.Lagrangian descriptions
Viewing of Lumley’s movie,as summarized on Fig.2.2.
Newton’s mechanics,with its discovery of the point particles as a useful
concept,traces the motion of material points.These points are endowed
with mass,but otherwise have no properties,and the purpose is to study the
motion of these simple objects.This simple program was expanded later to
include rigid bodies (motion of and around a point,moments of inertia) and
1
Even more elegant notation is made possible by using exterior products and dierential
forms;I am not aware that these have been used in mainstream uid dynamics.
2.2.EULERIAN VS.LAGRANGIAN DESCRIPTIONS 43
continua:this idea will be expanded in the next Chapter.
Consider a traﬃc ﬂow,and you are seated in one particular car,which is
assimilated to a point particle.As the car moves and stops,it experiences
forces that change its speed and direction.The point of application of the
force changes with the location of the car,and the associated accelerations
are those of your car.This is the Lagrangian description of the traﬃc ﬂow,
very close in spirit to Newton’s initial formulation.It deals with the motion
of material points.
An alternative,called Eulerian description,looks at the traﬃc ﬂow from
the viewpoint of the bystander:motion at a geometrical point regardless
of which particle is moving by.If you stand at a traﬃc light,the vehicles
will experience periodic accelerations and decelerations,as the light turns
from red to green and back again.This periodicity,a dominant feature
of what happens at this intersection,is not experienced in the Lagrangian
description.Thus,the two descriptions will give rather diﬀerent views of
the motion of traﬃc.”Steady” or ”periodic” ﬂow are Eulerian properties.
The only steady ﬂow in the Lagrangian description is uniform,i.e.at rest
is a suitable frame of reference,and of no practical interest.Mathematical
description and display of results are equivalent but show a diﬀerent aspect of
the same reality.The choice of formulation may be dictated by convenience
or habit,but the results must be interpreted accordingly and converted if
necessary:experimental ﬂow visualization is naturally Lagrangian,whereas
analytical and/or numerical solutions yield Eulerian properties ﬁrst.In the
case of traﬃc ﬂow,it would be necessary to know the state of (Lagrangian)
motion of all vehicles at various times in order to reconstruct the Eulerian
motion at a given point;and conversely.The purpose of this section is to
develop the required tools and to provide illustrations of the main steps;
only the simplest cases can be treated in closed form,but the procedure
is identical for the numerical processing of more complex experimental or
numerical data.
2.2.1 Lagrangian description of motion
For the purposes of presentation,we will focus on the description of a ve
locity ﬁeld;appropriate changes can be made for other properties such as
acceleration,pressure,temperature,etc.In the Lagrangian description,
u
(t,x
0
,t
0
) =
d
dt
x
(t,x
0
,t
0
) (2.32)
44 CHAPTER 2.KINEMATICS
Figure 2.3:Lagrangian,Eulerian,experimental,numerical
2.2.EULERIAN VS.LAGRANGIAN DESCRIPTIONS 45
describes the velocity of the particle that was at location x
0
at time t = t
0
and is now at x(t).Notations such as x
0
reinforce this idea,but can be
cumbersome;as we omit some details,the student should retain the implied
dependences in their thinking.We can also write
∫
x
x
0
dx = x −x
0
=
∫
t
0
u(τ,x
0
,t
0
)dτ (2.33)
From elementary mechanics,we know that,as time t varies,this is the tra
jectory of the particle,in parametric form as time changes.Elimination of
time between the components of the trajectory gives one (2D) or two (3D)
equations for the trajectory components in terms of each other.Its calcula
tion requires the knowledge of the history of velocity for this particle.In the
example of the traﬃc ﬂow,knowing history of your velocity enables you to
evaluate your current location relative to the point of origin.This is true for
each vector component.Mapping ocean currents can be done by tracking the
trajectory of a buoy.In ﬂuid mechanics,a pathline is equivalent terminology
for trajectory,and its reference to a particle makes it a Lagrangian concept.
Experimentally,you might mark one particle to be very bright,and take a
very long exposure that will record its trajectory:if it is a ﬂuid particle,you
have generated a pathline (but if it is a ﬁreﬂy,you only have a trajectory of
the ﬂy,which is not a picture of the motion of the air).
A diﬀerent concept is generated by the gradual release of particles (e.g.
smoke or dye) from one point (Fig.2.5).The resulting line is not a pathline,
but a streakline (and they coincide if the trajectories are independent of the
time of release t
0
).Its equation should make reference to the release from a
given point x
0
at successive instants in the past
x = x
0
+
∫
t
t
0
u(τ,x
0
,t
0
)dτ (2.34)
Here,the parameter along the streakline is the release time t
0
,with the
material points closer to x
0
having been released most recently.We note
that the equations for pathlines and streaklines are sections through the
same family of solutions,with diﬀerent parameters being held ﬁxed.The
streakline is commonly achieved by the release of dye,bubbles,smoke or
other particles as markers of the ﬂuid particles (question:are they good
markers?)
To illustrate all this,use the ﬂowaround a sphere,in the frame of reference
of the sphere and for a sphere in free fall,from Tritton’s book.See also
example below,and in Currie’s book.
46 CHAPTER 2.KINEMATICS
Figure 2.4:Particle pathline
Figure 2.5:Streaklines
2.2.EULERIAN VS.LAGRANGIAN DESCRIPTIONS 47
S.Kline's movie:Flow vizualization
• Hydrogen bubbles:water electrolysis
• Bubbles:good tracers?
• Lagrangian or Eulerian?
• Types of lines
{ Path lines
{ Streaklines
{ Time lines
{ Streamlines?
{ Steady/unsteady
• Other ﬂow markers:dyes,particles
• Also note:
{ Break of symmetry in cylinder wake,in diﬀuser
{ Instability in free shear layers (lifting airfoil wake,etc)
Figure 2.6:S.Kline’s movie on ﬂow visualization:what do we really see?
48 CHAPTER 2.KINEMATICS
Figure 2.7:Eulerian velocity vectors from particle traces
2.2.2 Eulerian description
It is not uncommon for experimentalists to observe many pieces of trajectories
at once:seeding many bright particles in a ﬂow,and taking an exposure of
short but ﬁnite time δτ,the collection of vectors (for many initial locations
x
0
is
x −x
0
=
∫
t+dt
t
u(t,x
0
,τ)dτ ∼ u(t,x
0
,t)dt (2.35)
for each x
0
gives a ﬁnite diﬀerence approximation of the velocity at each
point,leading to the Eulerian description of the ﬁeld (Fig.2.7).Note that
time t denotes the running time as well as the release time.
In the Eulerian description,the emphasis shifts away from the individual
particles (still necessary to assign a velocity!) in favor of the distribution of
values (e.g.or velocity) in space,i.e.the velocity ﬁeld.This is the approach
taken in PIV and related methods.
Given an experimental sample such as on the ﬁgure,the eye (actually:
our brains,processing visual information and superimposing patterns) in
terpolates lines that are at every point tangent to the velocity vectors:the
streamlines.In the plane,the equation of streamlines,given a velocity ﬁeld
in Cartesian coordinates,is easily constructed:at every point,the slope of
2.2.EULERIAN VS.LAGRANGIAN DESCRIPTIONS 49
Figure 2.8:Direction of a streamline
the line must match the direction of the velocity vector
dy
dx

streamline
=
v
u
(2.36)
In 3D,a line requires 2 equations:it can be seen that the streamline is
determined by
dx
u
=
dy
v
=
dz
w
(2.37)
With the velocity vector tangent at each point to pathlines and to stream
lines,the distinction between the two types of lines is emphasized.For the
streamline,the direction of the line results from a snapshot:simultaneous
velocity vectors at each point.For the pathline,we have the history of mo
tion of a particle.Once again,for steady ﬂow,they coincide;in unsteady
ﬂow,they can be unrecognizable fromeach other (Tritton).When comparing
experimental and numerical results,this can complicate the interpretation.
2.2.3 Pathlines,streaklines and streamlines
The conversion from Lagrangian to Eulerian (and conversely) is a useful
exercise.An example is given in Currie (p3842).Here,we will treat a few
50 CHAPTER 2.KINEMATICS
examples of increasing complexity.Students should carefully mark what is
general and what is speciﬁc to the problemat hand,and not use relations out
of context;this is also a cue about brushing up on diﬀerential equations and
elementary calculus.The pivotal item is velocity:the Eulerian velocity is
also the velocity of a particle that happens to be at the point of observation.
Example 1:Steady ow
Consider the velocity ﬁeld (Eulerian:why?) given by the expressions
u = xy
v = y.(2.38)
which is independent of time.The equations of the (Eulerian:why?) stream
lines is easily obtained
dx
u
=
dx
xy
=
dy
y
(2.39)
and are readily integrated:
y = y
0
+ln(
x
x
0
) (2.40)
Switching to the Lagrangian concepts requires an important change:the
particle trajectories are generated by updating their Eulerian locations to
match the location of each material point.Hence
u =
dx
dt
= x(t)y(t)
v =
dy
dt
= y(t).(2.41)
Because the yequation can be integrated separately,this problemis relatively
simple.We get
y = y
0
e
t−t
0
(2.42)
which is substituted into the xequation,and gives
ln(
x
x
0
) = y
0
(e
t−t
0
−1).(2.43)
There are two parameters (t and t
0
) for the location of a particle associ
ated with x
0
,y
0
.If we eliminate t,we obtain the trajectory
y = y
0
+ln(
x
x
0
) (2.44)
2.2.EULERIAN VS.LAGRANGIAN DESCRIPTIONS 51
identical to the streamlines through the same point.If instead we eliminate
t
0
,thus obtaining the equation of the streakline through the same point at
the current time t,the result is again the same,a feature of steady ﬂows.
The distinction between streamlines,pathlines and streaklines can be
subtle:the correct interpretation of the relations depends on identifying the
relevant parameters.
Example 2:Unsteady ow
This distinction is highlighted in unsteady ﬂows,where the three types of
lines are usually diﬀerent.We start from the Eulerian ﬁeld
u = x sin(t)
v = y (2.45)
Within the Eulerian framework,we obtain the streamlines
ln(
x
x
0
) = sin(t) ln(
y
y
0
),(2.46)
which vary in time.
Switching to the Lagrangian framework,we have for the particle motion
dx
dt
= x(t) sin(t)
dy
dt
= y(t) (2.47)
The two variables are separated,allowing for simple solutions:
ln(
x
x
0
) = cos(t
0
) −cos(t)
ln(
y
y
0
) = t −t
0
(2.48)
Two families of lines are included here:if we eliminate t,we have the path
lines x(y  x
0
,y
0
,t
0
):
ln(
x
x
0
) = cos(t
0
) −cos(t
0
+ln(
y
y
0
));(2.49)
whereas if we eliminate t
0
,we get the streaklines
ln(
x
x
0
) = −cos(t) +cos(t −ln(
y
y
0
));(2.50)
The 3 families of lines (streamlines,pathlines and streaklines) are illus
trated on Fig.2.9
52 CHAPTER 2.KINEMATICS
0
1
2
3
4
5
6
7
8
9
10
0
1
2
3
4
5
6
7
8
9
10
Figure 2.9:Streamlines (solid lines) at times t = 0 (black),π/4 (red),π/3
(magenta) and π/2 (blue);Pathlines (dashed) through the point (1,1) at
times t
0
= 0,π/4,π/3 and π/2;Streaklines (dotted) through the point (1,1)
at times t = 0,π/4,π/3 and π/2.
2.2.EULERIAN VS.LAGRANGIAN DESCRIPTIONS 53
Figure 2.10:The oscillating hose
Example 3:Unsteady ow
In this example,we go from Lagrangian to Eulerian.In an approximation
of an oscillating garden hose (Fig.2.10),the water motion is limited to
the horizontal x − y plane,ignoring the eﬀect of gravity.At time t
0
,the
water leaves the nozzle (origin of coordinates) a constant speed U,but at an
oscillating angle α = α
0
sinωt
0
.If the angular amplitude α
0
of oscillations is
small,we have (at ﬁrst order in α
0
)
u
0
(t
0
) = U (2.51)
and
v
0
(t
0
) = Uα
0
sin(ωt
0
) (2.52)
as the initial conditions for the trajectory of a drop leaving the nozzle at time
t
0
.
In the absence of applied forces,the trajectory of a drop is easily calcu
lated:
x(t,t
0
) = 0 +u
0
(t
0
)(t −t
0
) (2.53)
and
y(t,t
0
) = 0 +v
0
(t
0
)(t −t
0
).(2.54)
54 CHAPTER 2.KINEMATICS
This is the parametric representation of pathlines for this particular problem,
with t as the parameter and t
0
as an additional label (note the diﬀerent role
played by t and t
0
below).We can eliminate the parameter t easily:
t −t
0
=
x
u
0
=
y
v
0
(2.55)
or
y =
v
0
u
0
x = xα
0
sin(ωt
0
) (2.56)
The pathlines are straight lines issuing from the origin,with a slope depend
ing on t
0
.
Streaklines are more diﬃcult to obtain:the collection of drops that left
the nozzle at diﬀerent times in the past.This requires the elimination of t
0
from the individual trajectories.We have
x = (t −t
0
)U −→ t
0
= t −
x
U
(2.57)
which is substituted in the equation for y
y = xα
0
(t −
x
U
)sin(ω(t −
x
U
)).(2.58)
This is the equation of streaklines at a given time t.
Note which substitution changes the formulae from a particle trajectory
into a streakline!Obtaining the streakline in the form y(x;t) could be quite
diﬃcult without the initial linearization.Yet,taking a snapshot of the water
drops would achieve the same result.This illustrates how seemingly simple
experimental results can be analytically diﬃcult to reproduce.The concep
tual key is in the role of the various parameters.
Finally,the streamlines are obtained easily:at any instant,the velocity
vector is radial from the origin,so that the streamlines are indistinguishable
from the pathlines:
dx
u
=
dy
v
(2.59)
gives again
y =
v
u
x (2.60)
The superposition of pathlines and streamlines arises because the Eulerian
velocity ﬁeld is independent of time.Similarly,if the Lagrangian velocity is
timeindependent,the pathlines and streaklines coincide.All three lines are
identical in steady ﬂow,as seen above.
2.2.EULERIAN VS.LAGRANGIAN DESCRIPTIONS 55
Figure 2.11:Pathlines,streaklines and streamlines:the parameters
56 CHAPTER 2.KINEMATICS
Stagnation point...
x
y
O
S
3
2
1
0
1
2
3
3
2
1
0
1
2
3
...is not Galileaninvariant
x
y
O
S
3
2
1
0
1
2
3
3
2
1
0
1
2
3
Figure 2.12:Stagnation point (ﬁrst frame) is modiﬁed by the superposition
of a uniform speed (second frame).Streamlines are blue.
2.2.4 Caution!
The streamlines,pathlines and streaklines are very dependent on the frame
of observation.From a moving reference frame even in uniform motion
(Galilean invariance:see Ch.10 for rotating systems with ﬁctitious forces),
the appearance of the lines can be dramatically aﬀected.
For example,consider the simple potential ﬂow (See Ch.5) with a stag
nation point at the origin.Froma reference point moving at uniformvelocity
at 45 degrees to the axes,the addition of the corresponding velocity changes
the streamlines as shown on Fig 2.12.The problem of describing ﬂow topol
ogy in a way that is independent of the point of observation goes beyond the
scope of this course.
2.3 Dierential concepts and their geometry
2.3.1 Dierentials and vectors
The gradient of a function f requires no explanation at this level.It is a
vector (direction,magnitude),represented by ∇f which has various repre
sentations depending on the coordinate system (Cartesian,polar,etc.) In
two dimensions,think of a pressure ﬁeld p(x,y).The gradient will have a
direction normal to the isolines p = C This interpretation translates easily
2.3.DIFFERENTIAL CONCEPTS AND THEIR GEOMETRY 57
Figure 2.13:Gradient is normal to isolines of a function
to 3D.Turn your attention to one component,e.g.in the ydirection,of this
gradient:the projection of the gradient on the unit vector e
y
,it measures the
rate of increase of p in the y direction.This is the basis for the deﬁnition of
the directional derivative.Given a unit vector e
λ
in some direction denoted
by λ,the directional derivative of p in the λdirection is naturally
∂
λ
p = e
λ
· ∇p,(2.61)
or,regardless of the ﬁeld to which it is applied
∂
λ
= e
λ
· ∇.(2.62)
Learn to recognize the combinations vector
· ∇,whenever it occurs.
Now consider a function f(r
,t).The vector position has coordinates
r
= (x(t),y(t),z(t)) that move with the ﬂuid.If we look at the evolution of
f,its time derivative is obtained through the chain rule
d
t
f = ∂
t
f +∂
x
fd
t
x +∂
y
fd
t
y +∂
z
fd
t
z
= ∂
t
f +u∂
x
f +v∂
y
f +w∂
z
f
= ∂
t
f +u
· ∇f (2.63)
This ‘material derivative’,i.e.time derivative following a ﬂuid particle,is
sometimes denoted as D
t
for emphasis.The alert student has spotted the di
58 CHAPTER 2.KINEMATICS
Figure 2.14:Deﬁnition sketches for Gauss’ and Stokes’ theorems
rectional derivative u
·∇in the direction of streamlines,also called ‘convective
derivative’.
We already saw Gauss’s divergence theorem:
∫
∇· F
dV =
∮
F
· dA
(2.64)
or,in index notations
∫
∂
i
F
i
dV =
∮
F
i
dA
i
(2.65)
Another important theorem is due to Stokes
∮
F
· dr
=
∫
(∇×F
) · dA
(2.66)
In the case where the vector ﬁeld is velocity,Stokes’ theorem introduces two
important concepts.Vorticity is deﬁned as
ω
= ∇×u
.(2.67)
Vorticity will be one of dominant new concepts throughout this course.Re
lated to vorticity by Stokes’ theorem is the concept of circulation
Γ =
∮
u
· dr
(2.68)
2.4.MASS BALANCE 59
2.4 Mass balance
We start from the integral representation
∂
t
∫
ρ dV +
∮
ρ(U
· dA
) = 0,(2.69)
assume that the control volume is ﬁxed in space,and make use of Gauss’
divergence theorem to obtain
∫
∂
t
ρ dV +
∫
∇· (ρU
) dV =
∫
(∂
t
ρ +∇· (ρU
)) dV = 0 (2.70)
Since this holds for any ﬁxed control volume,the integrand must vanish at
every point:therefore
∂
t
ρ +∇· (ρU
) = 0 (2.71)
Note how much simpler this derivation is,compared to the use of material
derivatives following a ﬂuid particle.Then
d
t
ρ ̸= 0 (2.72)
because the density changes as the particle’s volume does:the divergence of
velocity accounts for this.
Since we limit ourselves to the study of incompressible ﬂows,where den
sity can be treated as constant,the continuity equation reduces to
∇· U
= 0 or ∂
i
u
i
= 0 (2.73)
This relation is a kinematic constraint on changes in the velocity compo
nents.For example,as a unidirectional ﬂow approaches stagnation point and
its velocity decreases,it must develop a component of velocity normal to the
upstream motion for mass to be conserved.It is beneﬁcial to compare the
results of control volume analysis and continuity equation in this regard.
One of the great simpliﬁcations in incompressible ﬂow,relative to com
pressible ﬂows,is related to the use of the simple (linear) continuity equation
just derived.It is interesting to note (proof in Ch.3) that,in the case of
natural convection,the continuity equation can be applicable even though
the variations in density drive the motion.This nontrivial result is known
as the Boussinesq approximation.
60 CHAPTER 2.KINEMATICS
2.4.1 Vector potential and stream function
It is known from vector calculus that any divergencefree vector can be ex
pressed as the curl of another vector (and conversely).Therefore,whenever
the continuity equation applies,there is a vector A
(called the vector poten
tial) such that
U
= ∇×A
(2.74)
The vector potential will be used extensively later.Note that arbitrary con
stants,or even the gradient of any smooth ﬁeld,can be added to each com
ponent of A
without aﬀecting the velocity ﬁeld.This ﬂexibility is used to
ensure that ∇· A
= 0 without loss of generality (gage invariance).
A particular case of considerable importance relates to the 2D ﬂows (in
the xy plane;axisymmetric versions can be found below).Then,inspection
of the components of ∇×A
shows that only the zcomponent of A
is nonzero.
In this case,we denote
A
z
= ψ(x,y) (2.75)
and we have
u = ∂
y
ψ (2.76)
and
v = −∂
x
ψ.(2.77)
Substitution into the continuity equation shows that mass balance is satisﬁed.
Lines of constant ψ are of particular interest.It so happens that the
directional derivative of ψ along streamlines is zero:
U
· ∇ψ = u∂
x
ψ +v∂
y
ψ = −uv +vu = 0.(2.78)
This means that lines of constant ψ are the streamlines.For incompressible
ﬂow,ﬁnding the streamfunction and its contour lines is the fastest way to plot
streamlines.Alternatively,the above relation can be read as the projection
of ∇ψ on the velocity vector,and they are orthogonal.In other words again,
it means that there is no mass ﬂux across lines of constant ψ,which can
be used as impermeable boundaries as long as no viscous eﬀects need to be
described (slip allowed along the boundary).Furthermore,we see that
dψ = ∂
x
ψdx +∂
y
ψdy = −vdx +udy = U
· dA
(2.79)
so that the diﬀerence in streamfunction value is proportional to the mass
ﬂux through the streamtube.Or again,we can say that the velocity in
a streamtube is inversely proportional to its width,and velocity can only
vanish if the streamtube expands into a reservoir of inﬁnite size.
2.5.FLOWABOUT A POINT:HELMHOLTZ 61
Figure 2.15:Streamfunction and 2D velocity
2.5 Flow about a point:Helmholtz
Consider a velocity ﬁeld (the following applies to any vector ﬁeld) u
i
(x
k
)
(the use of a free index for x
k
should convey that any velocity component
can depend on all coordinates.) In the vicinity of this point (Fig.2.16),say
at x
k
+δx
k
,a Taylor series will show the variations in ﬁeld values.Keeping
only ﬁrst order terms,we have
u
i
(x
k
+δx
k
) = u
i
(x
k
) +δx
j
∂
j
u
i

x
k
+...
= u
i

x
k
+
1
2
δx
j
(∂
j
u
i
+∂
i
u
j
) +
1
2
δx
j
(∂
j
u
i
−∂
i
u
j
) +...
= u
i

x
k
+δx
j
s
ij

x
k
+δx
j
r
ji

x
k
+...(2.80)
which can be interpreted as a sum of three contributions:a translation with
the point of reference,a straining motion,and a rotation:this is shown in
the next few subsections.
2.5.1 Rate of strain
With the understanding that all properties are evaluated at x
k
,we have
introduced the symmetric part of the velocity derivative
s
ij
=
1
2
(∂
j
u
i
+∂
i
u
j
) (2.81)
62 CHAPTER 2.KINEMATICS
Figure 2.16:Flow about a point
which is called the rate of strain.(This is similar to the strain in solid me
chanics,except this applies to velocities instead of displacements.) Because
of incompressibility,its trace vanishes
s
ii
= 0 (2.82)
The eigenvalues and eigenvectors of a matrix are scalars s and unit vectors
e
i
,respectively,such that
s
ij
e
j(k)
= s
(k)
e
i(k)
(2.83)
The parenthesis indicates that k is a label distinct from the indices and
not subject to implied summation.Interpreting the deﬁnition,applying the
matrix to the vector does not change its direction,but can change its mag
nitude by a factor s.A 3x3 matrix has 3 such combinations of directions
and stretching,some of which can be complexvalued.But since s
ij
is real
and symmetric,we know from linear algebra that its eigenvalues and eigen
vectors are all real.Combining this with the vanishing trace,which is the
sum of the eigenvalues,we see that at least one eigenvalue of rate of strain
is positive,corresponding to an extensional axis,and at least one value is
negative,corresponding to a compression axis,while the middle eigenvalue
adjusts the sum of all eigenvalues to be zero.
2.5.FLOWABOUT A POINT:HELMHOLTZ 63
6
?
L

U




e
(1)
@
@
@
@I
e
(2)
Figure 2.17:Couette ﬂow and principal axes
The eigenvectors are orthogonal to each other and can be adopted as local
coordinates:they are called principal axes.In such a coordinate system,the
matrix is diagonal,i.e.
s
ij
= δ
ij
s
(i)
(2.84)
Example:Couette ow
In the Couette ﬂow shown on the ﬁgure,the velocity ﬁeld is given by
u(x,y) =
U
L
y and v(x,y) = 0 (2.85)
You can verify that this 2D ﬂow is incompressible.The successive matrices
is easily constructed from their deﬁnitions:
(∂
j
u
i
) =
U
L
(
0 1
0 0
)
(2.86)
Then
(s
ij
) =
U
2L
(
0 1
1 0
)
(2.87)
It is then a simple matter to obtain the two eigenvalues
s = ±
U
2L
(2.88)
64 CHAPTER 2.KINEMATICS
corresponding to the eigenvectors
e
=
1
√
2
(
±1
1
)
(2.89)
Rate of strain takes a square (cube in 3D) with diagonals along the princi
pal axes and makes it into a diamond with its long diagonal on the extensional
axis.For inﬁnitesimal stretching,this conserves area (incompressible).
2.5.2 Local rotation and vorticity
The antisymmetric part of the velocity derivative is
r
ij
=
1
2
(∂
i
u
j
−∂
j
u
i
),(2.90)
in which only 3 components are independent.r
ij
could therefore be arranged
as a vector.In fact,it can be seen that
r
ij
=
1
2
ϵ
ijk
ω
k
(2.91)
where
ω
i
= ϵ
ijk
∂
j
u
k
or ω
= ∇×U
(2.92)
is the vorticity vector.The corresponding contribution to the motion around
a point (Helmholtz above) is then written as
u
i
(x
k
+δx
k
) = u
i

x
k
+δx
j
s
ij

x
k
+
1
2
ϵ
ijk
ω
j

x
k
δx
k
(2.93)
In the +
1
2
ω
× dr
term,we recognize the expression for rigidbody rotation
around the (local) origin,with angular velocity −
1
2
ω
.Pay attention to the
factors 1/2 in the rateofstrain and rotation rate;pay attention to signs,
they matter.
Example:Couette ow
Pursuing the example of the Couette ﬂow,the vorticity vector is seen to have
only one component normal to the plane of the ﬂow,with magnitude
ω = −∂
y
u = −
U
L
(2.94)
uniform throughout the ﬂow ﬁeld.Note that vorticity and the implied rota
tion do not imply curved streamlines or trajectories!It is about the relative
motion of nearby points.
2.5.FLOWABOUT A POINT:HELMHOLTZ 65
A.H.Shapiro's movie:Vorticity
• Kinematics
{ Deﬁnition,vorticity meter
{ Potential vortex
{ Vorticity and circulation:Stokes’ theorem
∗ Startup vortex
• Dynamics
{ Bernoulli’s equation:conditions for B = p +
1
2
ρV
2
+ρg z to
be constant
{ Crocco’s equation:how B changes
{ Kelvin’s theorem:inviscid,Lagrangian,vortex stretching
• Potential ﬂow,singular vortices
{ Model (ν = 0)
{ Motion of vortex pair,induced velocity
• Helmholtz’s theorems (material lines,vortex tubes,vortex lines
end only at boundary)
• How to introduce vorticity
{ Singularities
{ ∇p ×∇ρ and interface
{ unsteadiness
{ rotational forces
{ pressure gradient along boundary (viscous eﬀect)
{ See Ch.9
Figure 2.18:Kinematics and dynamics of vorticity and circulation,A.
Shapiro’s movie.
66 CHAPTER 2.KINEMATICS
Figure 2.19:Rotation in Couette ﬂow
2.6 Kinematic decompositions of a velocity
eld
Also due to Helmholtz,the kinematic decomposition of velocity is frequently
used:
u
= u
ϕ
+u
ω
(2.95)
The rotational part u
ω
is obtained fromthe vorticity distribution through the
BiotSavart relation (below).Then,boundary conditions must be adjusted
by the uniquely deﬁned potential ﬂow u
ϕ
(see Ch.5).Helmholtz proved
that such a decomposition is always possible for incompressible ﬂow.Hodge
further proved that the decomposition is unique,and that the potential and
rotational components are orthogonal in a mathematical sense (see Chorin
& Marsden,or Majda and Bertozzi).
For an alternative decomposition of velocity,due to Clebsch,see e.g.
Panton Section 14.4 p.355.
2.7.VORTICITY,∇×,∇
−2
,ETC.67
2.7 Vorticity,∇×,∇
−2
,etc.
2.7.1 BiotSavart relation
The BiotSavart relation ﬁrst came up in relation with the magnetic ﬁeld
induced by a current:it turns out the equations are identical to ours,with
vorticity instead of current and velocity instead of magnetic induction.The
most common form for 3D ﬁelds is (e.g.Panton Section 14.2 p.351)
u
ω
= −
1
4π
∫
r
×ω
′
r
3
dV
′
.(2.96)
Here,
r
= x
−x
′
 (2.97)
is the distance between the vorticity element at x’ and the point x where it
induces velocity.(Note that ∇f(r) = −∇
′
f(r) for any function of r = r

2
).
The velocity ﬁeld can be rewritten in equivalent forms,by integration
by parts,provided the boundary terms vanish fast enough at inﬁnity (see
Batchelor,section 2.9 for the eﬀect of boundaries).First we get
u
ω
=
1
4π
∫
∇
1
r
×ω
′
dV
′
= ∇×
1
4π
∫
ω
′
r
dV
′
= ∇×A
(2.98)
where we recognize the vector potential for which ∇
2
A
= −ω
.Also,inte
grating by parts
u
ω
= −
1
4π
∫
∇
′
1
r
×ω
′
dV
′
=
1
4π
∫
1
r
∇
′
×ω
′
dV
′
= −
1
4π
∫
α
′
r
dV
′
.(2.99)
Here
α
= ∇
2
u
= −∇×ω
(2.100)
2
In these notes,the notation r will be used for the radial distance in po
lar/spherical/cylindrical coordinates,or for the distance between two points:the student
should pay attention to the context.
68 CHAPTER 2.KINEMATICS
deﬁnes ﬂexion.So,the BiotSavart formula reconstructs velocity from
its Laplacian (for incompressible ﬁelds),or the velocity potential from its
Laplacian:it gives one explicit form of the inverse Laplacian,since
u
ω
= ∇
−2
α
.(2.101)
This allows us to view the BiotSavart integration kernels
G(x,x
′
) = −
1
4πr
(3D) and
lnr
2π
(2D) (2.102)
as Green’s functions for the Poisson equation.The Poisson equation is of
the form
∇
2
f = s,(2.103)
where s is a source term.Then,Green’s function gives
f =
∫
G(x,x
′
)s
′
dx
′
(2.104)
the solution f in terms of the superposition of eﬀects of localized source terms
distributed throughout the ﬁeld.Green’s function is solution of the equation
∇
2
G(x,x
′
) = δ(x
′
),(2.105)
where δ(x) is Dirac’s distribution.
For example,a 2D potential vortex (see Ch.5 for details) is given by
ω(x,y) = δ(x,y) (2.106)
which is inﬁnite at the origin,zero everywhere else,and integrates to 1.You
can think of the 2D version of δ as the limit of a normalized Gaussian bell
shape of unit integral as its scale becomes smaller (and its magnitude larger):
δ(x,y) = lim
r→0
1
4πr
e
x
2
+y
2
4r
2
(2.107)
Then the streamfunction is the z component of the vector potential (∇
2
ψ =
−ω),so that we reconstruct ψ by using Green’s function:
ψ(x,y) = −
1
2π
∫
ω
′
lnr δ(x −x
′
,y −y
′
) dx
′
dy
′
= −
1
2π
ln
√
x
2
+y
2
(2.108)
2.7.VORTICITY,∇×,∇
−2
,ETC.69
(avoid confusion with 2 expressions of r:distance between x and x’ or polar
distance...)
Green’s function (and the BiotSavart relation in particular) introduces
remote eﬀect of sources in the solution to a problem.While the use of Green’s
function to solve problems goes beyond the scope of this course,it is impor
tant to realize the nonlocal properties it introduces in the ﬂows.Vorticity
induces velocity at large distances because of the nonlocal dependence of
solutions of the Laplace and Poisson equations;other examples include the
pressure (Ch.3),and eﬀects of diﬀusion (Ch.8).
2.7.2 Vorticity,streamfunctions,and more
As seen above,it is a mathematical fact that
∇· u
= 0 ⇔u
= ∇×A
(2.109)
It can also be shown that there is no loss of generality in selecting the vector
potential to be incompressible (gage invariance).
3
Then,the vector identity
∇×(∇×·) = −∇
2
(·) (2.110)
(for incompressible vector ﬁelds only!) (check the general relation in your
favorite reference book).In particular:
∇
2
A
= −ω
(2.111)
In conjunction with the BiotSavart equation,this is the motif in a pattern
of relations between
• vectors:the vector potential A
,velocity U
,vorticity ω
,and ﬂexion
α
= ∇
2
U
• diﬀerential operators:the curl ∇×,the Laplacian ∇
2
,and their inverses
The pattern is summarized on the ﬁgure.It is good to remember that the
inverse curl and inverse Laplacian are nonlocal,i.e.they induce ﬁeld prop
erties far away from the sources.
3
Suppose that A
is a vector potential,with ∇· A
= b ̸= 0.Adding a gradient does not
change the resulting velocity eld,so we can consider any A
′
= A
+∇B as equivalent.We
then adjust B so that ∇· A
′
= b +∇
2
B = 0,i.e.B = −∇
−2
b,a familiar operation.
70 CHAPTER 2.KINEMATICS
Figure 2.20:Kinematic relations
2.7.VORTICITY,∇×,∇
−2
,ETC.71
2.7.3 Vorticity and boundary conditions
Consider a conﬁguration with solid boundaries,and a certain distribution
of vorticity (respectively,ﬂexion).The BiotSavart relation gives an induced
velocity ﬁeld,which will not,in general,satisfy the boundary conditions.Let
us call δU
the error on the velocity at the boundary.If we add to the velocity
ﬁeld a potential component u
ϕ
(resp.:a component of zero Laplacian) which
has value −δU
at the boundary,we have a well deﬁned problem of solving
the Laplace equation with Neumann (resp.,Dirichlet) boundary conditions,
and a unique solution with zero vorticity (resp.,ﬂexion).Thus,the addition
of a potential component is related to the presence of boundaries and the
enforcement of boundary conditions.
Part of Ch.6 (Potential ﬂows) could actually be moved here:assuming
irrotational ﬂow(zero vorticity) is a kinematic condition.That corresponding
material is conventionally coupled with inviscid ﬂow,and this Chapter is
bulky enough as it is.
2.7.4 Discrete vortices vs.continuous vorticity
This section can be omitted without loss of continuity in the text.It has a
bearing on vortexbased numerical methods,as well as on various modeling
approaches in turbulence (see Saﬀman’s book for vorticity,Chorin’s book for
applications to turbulence).The point is made here in 2D,the same applies
in 3D.
Consider a collection of point vortices
ω
i
= Γ
i
δ(x
i
) (2.112)
The overall streamfunction is obtained by summation
ψ =
∑
i
Γ
i
2π
∫
δ(x
′
i
) lnr dA
′
=
∑
i
Γ
i
2π
lnr
i
=
∑
ψ
i
(2.113)
which obeys the Laplace equation except at the singular points.So the
induced velocity ﬁeld is irrotational.
The same is not true,in general,for continuous distributions of vorticity
Γ(x) over a ﬁnite region.Indeed,for
ψ =
∫
Γ(x
′
)
2π
lnr dA
′
,(2.114)
72 CHAPTER 2.KINEMATICS
the vorticity is
ω = −∇
2
ψ =
∫
Γ(x
′
)
2π
∇
2
lnr dA
′
,(2.115)
which is,in general,nonzero even in the region where Gamma vanishes.So,
for continuous distributions of vorticity,the induced velocity is not irrota
tional.
2.7.5 Vorticity and circulation
Vorticity and circulation are not equivalent concepts,but are obviously re
lated through Stokes’ theorem:
Γ =
∫
ω
· dA
(2.116)
where the integral is over any simple surface supported by a closed contour.
We can also write
Γ = ω
av
n
A (2.117)
for some suitable average vorticity normal to the surface.This latter presen
tation is most useful in plane (2D) ﬂows,since the contour would normally
be in plane of the ﬂow,and the surface would normally be the same plane
normal to the vorticity vector.The average is not necessarily representative,
as the potential vortex makes clear.In 3D,it is good to remember that ω
n
can change sign over the surface.
It should be clear at this point that,without any dynamics involved yet
(and more kinematics coming up),vorticity is related to many other concepts:
it could be argued that it is central to ﬂuid mechanics.At this point,the
web of relations can be summarized (students:to build on this as the course
goes on!)
2.7.6 The Helmholtz theorems
The Helmholtz theorems on vorticity (see Panton Section 13.8 p.338,Acheson
Section 5.3 p.162) are very important.
1.vortex lines cannot end in the ﬂuid,they can only end at a boundary.
2.in inviscid incompressible ﬂow,ﬂuid elements that lie on a vortex line
remain on vortex line (this means that vorticity is a marker just as
dye).
2.7.VORTICITY,∇×,∇
−2
,ETC.73
Figure 2.21:Vorticity at the center of its web
74 CHAPTER 2.KINEMATICS
3.circulation is the same for all sections of a vortex tube.
Theorem 2 relies on dynamical considerations,and we will return to it in Ch.
5.Theorems 1 and 3 have a counterpart for streamlines:the key element is
that the ﬁeld (vorticity,velocity,whatever) is incompressible.The proofs are
based on the divergence theorem.
2.7.7 Helicity,Lamb vector
Two combination of velocity and vorticity can be introduced here.First,
helicity is deﬁned as
h = ω
i
u
i
.(2.118)
Since it vanishes identically in 2D ﬂows (where vorticity is perpendicular to
the plane of the ﬂow),it is used as an indicator of 3dimensionality,useful in
turbulence.Some care needs to be taken about its lack of Galilean invariance.
Also important (see next chapter) is the Lamb vector,deﬁned as
ℓ
i
= ϵ
ijk
ω
j
u
k
(2.119)
ℓ
is perpendicular to both u
and ω
.See Ch.3 for the role it plays in
Dynamics.
2.7.8 Famous vortices
This subsection is a list of classic vortices.The dynamical equations of which
some of them are solutions,will be presented in later chapters:for now,a
simple description is suﬃcient.
2D potential vortex.
See also Ch.5
ω = Γδ(x,y) (2.120)
ψ =
Γ
2π
lnr (2.121)
v
θ
=
Γ
2π
1
r
(2.122)
2.7.VORTICITY,∇×,∇
−2
,ETC.75
Figure 2.22:Rankine vortex
Rankine vortex
Also 2dimensional (plane ﬂow),the Rankine vortex is a simple model cor
responding to a potential ﬂow surrounding a core of ﬁnite size,in which
solidbody rotation reﬂects the dominance of viscous forces (Fig.2.22).
ω =
Γ
πR
2
for r ≤ R
= 0 for r > R (2.123)
v
θ
= ωr for r ≤ R
= ω
R
2
r
for r ≥ R (2.124)
The model can be useful in some applications in spite of the abrupt transition
between the viscous core and the induced potential ﬂow This is not quite
realistic,but is promptly smoothed by viscous diﬀusion.
Oseen vortex
See Panton Section 11.6 p.283.The Oseen vortex is an exact 2D unsteady
solution to the equations of motion,with the axisymmetry eliminating the
nonlinearities.The vortex results from the viscous relaxation of a (singular)
potential vortex (see Ch.5).The vorticity distribution is Gaussian (Fig.
76 CHAPTER 2.KINEMATICS
Figure 2.23:Oseen vortex and asymptotes
2.23),and the corresponding tangential velocity (polar coordinates,of course)
is
v
θ
=
Γ
2πr
(1 −e
−
r
2
4t
) (2.125)
It should be noted that the ‘size’ of the vortex is timedependent;it is also of
interest to note the limits or solidbody rotation and potential ﬂow for very
small and large r,respectively.It can be shown (endofchapter problem) that
the exponential distribution of vorticity yields the correct induced velocity
for the 2D problem;however,3D conﬁgurations based on distortions of the
Oseen vortex are not likely to be kinematically possible.
Burgers'3D vortex
Burgers’ vortex is one of those rare solutions of the exact equations that
combines nonlinear and viscous terms in a stationary solution.A vortex core
along one axis is stretched by an axisymmetric rateof strain (Fig.2.24).
See Acheson Section 5.9 p.187.
Hill's spherical vortex
Also an exact solution (Panton Section 13.6 p.332),but with pressure jump
at the interface (see Saﬀman’s book).
2.7.VORTICITY,∇×,∇
−2
,ETC.77
Figure 2.24:Burgers’ vortex
Figure 2.25:Hill vortex
78 CHAPTER 2.KINEMATICS
Figure 2.26:Ring vortex
Ring vortices
See back cover of Van Dyke’s book (Fig.2.26).Stability analysis by Widnall
and others.
Taylor vortices
Result of ﬂow instability,for the Couette ﬂow between coaxial cylinders
with the inner cylinder spinning.As viscous forces set the ﬂuid in mo
tion (Poiseuille ﬂow),centrifugal forces will tend to expel the moving ﬂuid
to the outside,with the motionless ﬂuid driven inward by pressure forces.
Viscous forces will tend to oppose such motion at low angular speeds,but
for suﬃciently rapid rotation a convective pattern develops,made of a stack
of donutshaped vortices rotating in alternating directions (Fig.2.27).This
case of ﬂow instability (see Ch.11) was ﬁrst studied by G.I.Taylor.
See Tritton Section 17.5 p 258267,Van Dyke p76.
Karman vortices
Counterrotating staggered vortices form as the result of the boundary layer
separation from alternate sides of bluﬀ bodies in 2D ﬂows (Fig.2.28).The
telltale pattern is a classic example of ﬂowstructure interaction:oscillating
forces are applied to the obstacle as well.For a detailed analysis,see e.g.
Saﬀman’s book.
2.7.VORTICITY,∇×,∇
−2
,ETC.79
Figure 2.27:Taylor vortices
Figure 2.28:K`arm`an vortex street
80 CHAPTER 2.KINEMATICS
Figure 2.29:Necklace or horseshoe vortex
Necklace or horseshoe vortices
When an obstacle protrudes normally from the wall in a boundary layer,
vortex lines bunch up near the upstream stagnation wall and are stretched
downstream on both sides of the obstacle (Fig.2.29).The resulting ﬂow
patterns,also called horseshoe vortices,can aﬀect bridge pilings,aircraft
wings and engine pilons,and many similar conﬁgurations.
2.8 Advanced topics and ideas for further read
ing
This chapter contains a lot of information,and will require persistence from
the student.Every later use of the notations and concepts developed here
should be an opportunity to review this material,to relearn it.
It might be a good idea to reviewlinear algebra for principal axes;see your
advanced math class (or standard references) for more on Green’s functions:
the important here is to understand the general idea behind them.
Kinematics does not usually provide solutions to problems:to be a solu
tion,a ﬂow conﬁguration should also satisfy Newton’s second law,the object
of the next Chapter.But kinematics establishes constraints that a dynamical
solution must satisfy.
Stokes’ streamfunction applies to axisymmetric ﬂows.See e.g.Currie,
2.8.ADVANCED TOPICS AND IDEAS FOR FURTHER READING 81
p.145,for relevant equations.
Problems
1.Express the velocity ﬁeld for Couette ﬂow in principal axes,and calcu
late the rate of strain,eigenvalues and eigenvectors in this coordinate
system.
2.Using em only index notations,show the following identities
• ∇· (F
×G
) = G
· (∇×F
) −F
· (∇×G
)
• ∇×(F
×G
) = (G
· ∇)F
−(F
· ∇)G
+F
(∇· G
) −G
(∇· F
)
3.Consider the potential ﬂow for deepwater waves:
ϕ = Ce
−ky
sin(kx−ωt) with the ﬂow ﬁeld u = ∂
x
ϕ and v = ∂
y
ϕ.Is this
Eulerian or Lagrangian?Derive the equations of streamlines,pathlines
and streaklines:carry out the analysis as far as possible,then restrict
it to motion of small amplitude to complete the problem.Compare the
solution to the ﬂow vizualization in Van Dyke’ book p.110111.
4.By integration,construct the streamlines corresponding to the velocity
ﬁeld u = Ωy,v = −Ωx and w = w
0
.Describe their shape.
5.Consider the Oseen 2D vortex;using Maple or Mathematica for the
manipulations,calculate the induced velocity by integration,then take
the curl and compare the induced vorticity with the original ‘source’.
6.(Adapted from Problem 23 p.475 from Tritton.) Consider one of the
following expressions:
(a) ϕ = c (x
2
+y
2
)
(b) ϕ = c (x
2
−y
2
)
(c) ϕ = c xy
(d) ϕ = c
√
x
2
+y
2
(e) ϕ =
c
√
x
2
+y
2
(f) ψ = c (x
2
+y
2
)
(g) ψ = c (x
2
−y
2
)
82 CHAPTER 2.KINEMATICS
(h) ψ = c xy
(i) ψ =
c
√
x
2
+y
2
(j) ψ = c
√
x
2
+y
2
calculate corresponding velocity components and list the implied as
sumptions;verify if the ﬂow is incompressible and/or irrotational;and
give the analytical expressions and plots of streamlines and potential
lines,if they exist,or explain why they do not exist.
7.Consider a 2D ﬂow ﬁeld with velocity components u = 1,v = u ∗ f(t)
for some function f.Find the equations of the streamlines,pathlines
and streaklines with the relevant parameters.(Adapted from Currie,
p.48.)
8.Show that streamlines and pathlines coincide for the unstready ﬂow
u
i
= x
i
/f(t).Are the streaklines diﬀerent?(Adapted from Currie,
p.48.)
9.(Adapted fromTritton,problem10 p.470.) You are given a Lagrangian
representation of motion as x = x
0
e
−2t/s
,y = y
0
e
t/s
,z = z
0
e
t/s
.(s is a
positive constant,other notations selfexplanatory.)
(a) write the equations for the path of a particle,and plot them
(b) write the Eulerian equations for this motion
(c) is the ﬂow steady?incompressible?
(d) write the equations of streamlines,and plot them
10.(Adapted fromTritton,problem10 p.470.) You are given a Lagrangian
representation of motion as x = x
0
e
−2t/s
,y = y
0
(1+t/s)
2
,z = z
0
e
t/s
(1+
t/s)
2
.(s is a positive constant,other notations selfexplanatory.)
(a) write the equations for the path of a particle,and plot them
(b) write the Eulerian equations for this motion
(c) is the ﬂow steady?incompressible?
(d) write the equations of streamlines,and plot them
2.8.ADVANCED TOPICS AND IDEAS FOR FURTHER READING 83
11.(Adapted from Tritton,problem 9 p.470.) You are given the equa
tions of motion u = −Ay,v = Ax and w = B (A and B are positive
constants,other notations selfexplanatory.) Is this Lagrangian or Eu
lerian?Is the ﬂow steady?incompressible?Write the equations of
pathlines and streamlines,and describe them.
12.(Adapted fromTritton,problem11 p.471.) You are given the equations
describing a ﬂow as u = u
0
,v = v
0
cos(κx −ωt) (u
0
,v
0
,κ and ω are
constants.) Is this Eulerian or Lagrangian?Write the equations of
pathlines and streamlines.What other questions can you answer about
this ﬂow?
84 CHAPTER 2.KINEMATICS
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