1) A fluid is defined as a substance that permanently deforms under an infinitesimal force. Liquids and gases are considered fluids. 2) There are two approaches to studying fluid motion - the Lagrangian approach which follows individual particles, and the Eulerian approach which examines properties at fixed points in space through which fluid moves. 3) Kinematics is the study of motion without considering forces, and examines streamlines, pathlines, streaklines and timelines to understand fluid behavior.

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Definition of a fluid:
Liquids and gases are called fluids. A fluid id defined as a substance that under
the action of an infinitesimal force deforms permanently and continuously.
h
Fluid
tº t1 t2
F
tº<t1<t2
Continuum Hypothesis:
A Criterion used to evaluate the validity of the continuum approach is based on
Knudsen number (mean free path of molecules / characteristic length of flow):
Kn < 0.01 continuum approach is valid
Kn > 0.1 must use statistical approach
m
V


mass,
volume,
Elemental
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ining flui
gion conta
Re
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m
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lim
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10 mm
V 



V


1
2

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Compressibility:
Compressibility defines the ability of a fluid to change its density under the
action of pressure. It is defined as the inverse of the bulk modulus, that is:
No-slip Boundary Condition
Experiments have shown that a fluid adjacent to a solid interface cannot slip
relative to the surface:
fluids.
ible
incompress
as
treated
are
liquids
flows,
of
analysis
In
.
negligible
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pressure
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density wi
of
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kPa),
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and
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specific
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ility,
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the
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Where
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~
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1
1





C
dP
d
dP
d
E









wall
fluid
u
u 
Surface Tension:
For molecules in the interior (bulk), interactions are isotropic and the net
force on each liquid molecule vanishes. This is not the case for
molecules at the interface. These are attracted more in the interior of
the liquid than by gas molecules such that a nonzero net force results.
F=0
 F≠0
3
4

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R
Δ P
or
R
R
P




2
2
2



This pressure difference is called the capillary pressure, which is due to
the surface tension. Note that the pressure inside a drop is greater than
the pressure outside the drop.
For 2-dimensional surfaces










2
1
1
1
R
R
σ
P
Δ
Where R1 and R2 are the two principal radii of curvature of the 2-dimensional
surface (Young-Laplace equation of capillary).
R
(1) For a plane surface: R1=R2=
B
A

jump)
pressure
no
is
there
interfaces
plane
(For
P
P
or
0
P A B



(2) For a sphere: R1=R2=R
B
A outside).
to
inside
the
from
jump
pressure
a
is
there
case
(For this
R
/
2
P
P
or
R
/
2
P A
B 
 



(3) For a cylinder: R1= and R2=R
outside).
to
inside
the
from
jump
pressure
a
also
is
there
interfaces
l
cylindrica
(For
R
/
P
-
P A
B 

B
A

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6

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cos
2
or
cos
2
2
ρ R g
θ
σ
h
R
h
R
g 
 




recovery
oil
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and
Formation
-
bubbles
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films,
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flow
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other
and
soil
through
liquids
of
Movement
-
:
in
role
t
significan
a
plays
tension
Surface
tension.
surface
the
determine
to
method
a
as
equation
this
use
may
one
,
and
,
Measuring 
h
h

h
R
ρ 2


 R
2
Kinematics:
Kinematics is defined as the science that deals with the study of motion
without making references to the force that cause motion. It is essential for:
-The development of a quantitative theory of Fluid Mechanics.
-The interpretation of data obtaining using various visualization
experimental methods.
Four different types of curves are considered in the study of fluid motion:
-Streamlines
-Pathlines
-Streaklines
-Timelines
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Streamline:
It is a line in space that is everywhere tangent to the velocity vector at every
instant of time.
The velocity (v) is parallel to ds, so that: v  ds = 0
Then:
z
y
x v
dz
v
dy
v
dx


ds
V

V

Pathline:
It is the actual path traversed by a given fluid particle. A pathline may be
identified by a fluid with a luminous dye injected instantaneously at one point
and take a long exposure photograph.
The position of this line depends on the particle selected and the time interval
over which this line is traversed by the particle:
  3
2
1
,
, 3
2
1 ,
,
i
for
x
x
x
u
dt
dx
i
i


)
3
3
2
2
1
1 (t
x
x
(t)
x
x
(t)
x
x 


:
pathline
for the
equations
parametric
obtain the
may
one
equations
these
g
Integratin
0
t 
t
t 

9
10

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Streakline:
It is the line joining the temporary location of all particles that have passed
through a given point in a flow field. A plume of smoke or dye injected at one
point gives a streakline.
Figure below illustrates pathlines and streaklines for an unsteady flow. Note that
for a steady state flow all streamlines, pathlines and streaklines coincide.
Timeline: At time t=t0 a set of fluid particles is marked and the subsequent
behavior of the lines thus formed is monitored.
timelines at different times
t0
t1
t2
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Eulerian approach versus Lagrangian approach:
Two approaches are possible for the study of fluid motion:
-the Lagrangian approach
-the Eulerian approach
The Lagrangian approach is based on an analysis of the motion of a particular
collection of particles. For each of the particles the following two fundamental
principles can be applied:
This description is not very convenient to analyze fluid motion and it is used
mainly in particle mechanics.
i
F
dt
dV
m
dt
dm



motion
of
law
second
s
Newton'
-
0
mass
of
on
Conservati
-
The Eulerian Approach is more convenient where our focus of interest is
generally a fixed region of space through which the material moves, rather
than a particular body of material.
We are interested to determine , v, T and P at various positions in the space.
To transform the conservation of mass and momentum equations from the
Lagrangian to the Eulerian approach, we need two tools:
1. The material or substantial derivative operator
2. The Reynolds transport theorem
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Material Derivative
d𝜑
d𝑡
𝑣 . ∇φ
𝜕𝜑
𝜕𝑡
Material Derivative
Convective acceleration
Local acceleration
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𝐴 → 𝐴
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Stream Functions:
For two dimensional and axisymmetric flows, the continuity can be used to show
that the complete velocity field can be described in terms of a single, scalar field
variable, which is called stream function.
Two-dimensional flows:
y
x
vx
vy
dy
dx P (x,y)
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