KINEMATICS
Kinematics describes fluid flow without analyzing the forces
responsibly for flow generation. Thereby it doesn’t matter
what kind of liquid is in question (water, air, oil).
We already adopted the continuum concept and we are
aware of the fact that real fluid flow reveals changes of
kinematic parameter in time and space. Therefore, one has to
find an adequate way to describe the flow field in a
mathematical manner.
Physical variables used for flow field description are scalars
(pressure, density, temperature, viscosity coefficients),
vectors (velocity, acceleration, pressure gradient) and tensors
(stress tensor, strain velocity tensor).
KINEMATICS
Gradient of scalar
field  is vector:
 x


g
ra
d




y






z


In Cartesian coordinate system the vectors of position x,
velocity v and acceleration a are expressed with three
components:
x

x

t
u

u

t


 



 


x

y v


yt
v a


v

t


 




 


z

z

t
w

w
t


 


The example of total differential for pressure field gives:

p
p
p
p
d
p

d
x

d
y

d
z

d
t

x 
y 
z 
t
and for steady process:

t
0
KINEMATICS
Two different approaches exist when considering the flow of
fluid: according to Lagrange and according to Euler.
In the Lagrangean approach an observer moves together with
the separate fluid particle and “feels” total changes of
kinematic parameters during the time elapsed.
In Eulerian approach the whole flow field is observed and
described from one fixed spatial coordinate.
In order to define the connection between Lagrange and
Euler description of fluid flow we introduce terms as: total
change, convective change and local change.
KINEMATICS
Imagine a situation where we are moving through a
temperature field that exhibits spatial and temporal changes
T(x,y,z,t). After some time elapsed we would experience the
total (substantial) temperature change in time DT/Dt (dT/dt).
We can separate that total change DT/Dt into the partial
change in time T/t (at fixed coordinate in space) and
partial change in space T/s (for one moment in time –
fixed coordinate in time).
D
T
T 
T 
T 
T
 
u 
v 
w
D
t

t

x 
y

z
l
o
k
a
l
n
a
k
o
n
v
e
k
t
i
v
n
a
k
o
m
p
o
n
e
n
t
a
KINEMATICS
During the movement from space coordinate 1 to 2 an
observer will feel total temperature change dT due to partial
change along spatial coordinate T/s·ds along the temporal
coordinate T/t·dt
The same approach can be also
applied at any vector field.
a
D
u
D
t




x
D
u
u

u

u


D
a

a

D
v
D
t


u

u


u
g
r
a
d
u
y

D


t
D
t 
t

t



a
D
w
D
t
z







KINEMATICS
In fluid mechanics we are dealing with relationships between
stresses and strain velocities and not between stresses and
strains as was the case in the study of solid body mechanics.
Strain velocities define the rates of fluid particle rotation and
deformation (includes dilatation and angle deformation).
In case of steady and non-uniform
pressurized pipe flow the fluid
particle will accelerate as the
consequence of decrease in pipe
diameter.
Accordingly, particles of an incompressible fluid will elongate
in flow direction and shrink in perpendicular direction.
KINEMATICS
Three different changes of fluid particle take place:
a) translation,
b) rotation,
c) deformation (dilatation, angle deformation)
Translation is easy to describe with the aim of velocity vector
placed in the center of gravity for the observed fluid particle
(analog as in case of solid body translation).
The part including rotation and deformation is described with
velocity gradient tensor.
The rate of rotation is equal to angular velocity (change of
angle in time di/dt).
KINEMATICS
KINEMATICS
After time increment t fluid particle rotates by average value
of angle X and Y :
Δ
y v
1 v
αx 
 Δ
xΔ
t
 Δ
t
Δ
x x
Δ
x 
x
Δ
y
Δ
x
u
1
u
αy 
 Δ
yΔ
t
 Δ
t
Δ
y
y
Δ
y
y
Δ
x
The net rate of rotation about z axis can be expressed as
algebraic mean of both rates of angle deformation.
It is denoted as angular velocity Z. (analog
1  w v 
for remaining two directions X ,Y)
ωx 

α

α

1


d

v
u
x
y
ω


z

 
d
t

x
y


 22

2  y

z 
1  u w 
ωy  


2  z x 
KINEMATICS
The rate of deformation consists of two components: volume
dilatation (exist only in compressible fluids) and angle
deformation.
When dealing with angle deformation one use the same
procedure as in case of rotation, with the special attention
given on the sign of angle. The rate of angle deformation in
x-y plane is given by equation:
1
u 
v
e
   
1
2
2
y 
x
vj  1
T
1
vi 
e
 

 
v
v

ij
 2
2

x

x
j
i




KINEMATICS
It can be shown that rotation and deformation represent the
components of velocity gradient tensor.
 u

x

 v
grad v v  
x

 w

 x
u
y
v
y
w
y
u 

z

v 
z 

w 

z 
The rate of deformation is described by symmetric part of
tensor, where the diagonal members represent volume
dilatation, and off-diagonal members angle deformation.
The sum of diagonal members defines total volume dilatation
(divergence), that is equal to 0 in case of incompressible fluid.
KINEMATICS
The members of tensor symmetric part in 2D problems are:
(dilatation and angle deformation)





u
1

u
v


  

x
2

y

x
T
1





v


v 
1

2



v 
u

v


  


2

x

y

y






Antisymmetric part of the tensor covers the rotation and
consists of angular velocity vector components (for 2D case):




1

u
v
0


 
2

y

x
0
ω
T


1


z



v


v





ω
0
2


1

v
u
z


0






2

x

y




For 3D case:

 0

 ωz
ω
 y
ωz
0
ωx
ωy 

ωx 
0 

KINEMATICS
The rate of volume change for fluid particle is described by
divergence. If fluid particle elongation occurs only in
xdirection, there is positive change in volume:

u
Δ
V
o
l
Δ
xΔ
t

Δ
yz

Δ

x
The same is valid for the other two directions y and z.
In 3D case the rate of volume change is given by velocity
vector divergence:

V
o
l
1 
u

u

w
d
i
v
v





tΔ
x

Δ
y

Δ
z

x

y

z


For the incompressible fluid holds:

d
ivv 0
KINEMATICS
Pathline – the trace showing the position at successive
intervals of time of a particle which started from a given point
(Lagrange approach).
Streakline – gives an instantaneous picture of the positions of
all the particles which have passed through the particular
point at which the dye is being injected.
Streamline – an imaginary curve in the fluid across which, at a
given instant, there is no flow. Thus, the velocity of every
particle of fluid along the streamline is tangential to it at that
moment.
In case of steady flow all three curves are the same.
Stream tube consists of streamlines. There is no flow through
boundary streamlines.
KINEMATICS
KINEMATICS
The total quantity of fluid-flowing volume in unit time past
any particular cross-section of a stream is called volume
discharge Q. Mass discharge QM is obtained through the
multiplication with density .

3


Q

v

c
o
s

d
A
m
/
s



A


3


QQ


k
g
/
m
m


Applying the law of mass conservation on incompressible
fluid flow through the conservative pipe one gets the
continuity equation, written in simple form as:
Q

Q

v
A

v
A
12
1
12
2
THE CONSERVATION LAWS – transport equation
The changes in physical fields (velocity, concentration) are
carried out by transport process.
Mathematical interpretation of transport process is given
through transport equations, also representing the
conservation law of arbitrary physical field.
We distinguish the intensive fields (independent of mass
amount – scalar field of pressure or temperature ; vector field
of velocity or acceleration) and the extensive fields
(dependent on mass amount – scalar fields of mass, energy,
entropy ; vector field of force or momentum) .
THE CONSERVATION LAWS – transport equation
In analyzing the transport process it is convenient to rely on
volume and not on mass, so one has to introduce the term
density of transport property  = dJ/dV
(transport property / volume; e.g.. transport property is mass
mCO2 in volume V  accompanying density is concentration
with the unit kg/m3).
Integrating over closed volume (indicating m = const.) that
could be time-dependent (V(t) – the so-called material
volume) one reads:
Jt
( )d
V
Vt
()
THE CONSERVATION LAWS – transport equation
Temporal and spatial distribution of fluid properties in flow
field (concentration, temperature, pressure, velocity ...) are
defined by the law of property conservation (mass, energy,
momentum) in the form of its total change in time dJ/dt.
For one moment in time t the material volume is given by V(t).
After the time increment t initial volume V goes to:
V(t+t) = V(t) + V(t+t) = V+V.
and the density of transport property  (t) goes to  (t+t).
As the time increment approaches zero (t 0) the volume
change also approaches zero V 0.
THE CONSERVATION LAWS – transport equation
d
J
Jt
( t)Jt
()
lim


t

0
d
t
t


1
lim (tt)d
V(tt)d
V
()
td
V


t
0
t

V
 V
V


1
lim (tt)()
t d
V(tt)d
V

t
0
t V
V

THE CONSERVATION LAWS – transport equation
Total or substantial change of transport property J(t) in time is
defined with the equation:



d
J
d


d
V

d
V

(
v

d
A
)



d
td
t

t
V
(
t
)
V
(
t
)
A
(
t
)
V(t), A(t) are dependent only on time instant t (not on t+t).
It is possible to replace the material volume V(t) with time
independent volume V(t) = V = konst. that is fixed in space.
Belonging material surface area A(t) is also to be replaced with
stationary surface area A(t) = A = const.
Accordingly, the term “material volume” V(t) is to be replaced
with the term “control volume” (V) and the term “material
surface” A(t) with the term ”control surface” (A).
THE CONSERVATION LAWS – transport equation
Finally, transport equation reads:


d
J 

d
V

(
v

d
A
)


d
t (

t
V
)
(
A
)
If velocity vector v has the same orientation as outer normal n
on any segment of control surface (A), their multiplication
gives the positive value.
When dealing with terms of control volume and control
surface one relies on Euler approach.
The absolute-total change in time dJ/dt is divided in two parts:
the first is partial change in time for fixed spatial coordinate
(local change) and the second is partial spatial change at fixed
moment in time (concoctive change).
THE CONSERVATION LAWS – transport equation
If local component equals 0, the flow is steady.
If convective component equals 0, the flow is uniform.
Applying the GGO theorem:
a

d
A

d
i
v
a
d
V





A
V
on transport equation:


d
J 

d
V

(
v

d
A
)


d
t (

t
V
)
(
A
)
reads:

d
J





d
i
v
(
vd
)
V



d
t (V

t


)
