1.0. Introduction
FLUID
MECHANICS
1. INTRODUCTION
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ME301 -- FLUID MECHANICS
1.0. Introduction
UNIVERSITY OF TURKISH AERONAUTICAL ASSOCIATION
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DEPARTMENT OF AERONAUTICAL ENGINEERING
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ME301 -- FLUID MECHANICS
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ME301 -- FLUID MECHANICS
1.0. Introduction
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ME301 -- FLUID MECHANICS
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1.0. Introduction
1.1. Solids and fluids
Visualization of flow around
the F-18 in NASA's water
tunnel
MATTER
Solid
Liquid
• Strong
intermolecular
attraction forces
• Relative positions
of the molecules
are fixed
• Definite shapes and
definite volumes
• Medium
intermolecular
attraction forces
• Quite free to
change their
relative positions
• Indefinite shapes
but definite
volumes
Wingtip vortex of an agricultural plain
obtained with colored smoke
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Gas
• Very weak
intermolecular
attraction forces
• Practically
unrestricted
• Indefinite shapes
and indefinite
volumes
FLUIDS
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1.2. The concept of continuum
1.2. The concept of continuum
Fluids are composed of molecules that are in constant motion and
colliding with one another.  Can we keep track of each molecule?
Fluids are composed of molecules that are in constant motion and
colliding with one another.  Can we keep track of each molecule?
1m3 of air
~2 × 10 molecules
A glass of water
~2 × 10 molecules
Statistical approach:
● Equations of motion are written for each separate molecule
o Too cumbersome, and even impossible for practical calculations
o Intermolecular attraction forces are not known
● Used in statistical mechanics
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1m3 of air
~2 × 10 molecules
A glass of water
~2 × 10 molecules
Continuum approach:
● Used in most engineering problems
● Principle interest is the gross behavior of the fluid as a continuous
material
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1.2. The concept of continuum
Continuum approach (cont’d):
● Valid whenever the smallest volume of fluid, known as a fluid
particle, contains enough number of molecules for making statistical
averages.
How many is «enough»?
Depends on Knudsen Number (Kn)
𝐾𝑛 =
𝜆
molecular mean free path
=
𝐿 characteristic linear dimension of the flow field
• Molecular mean free path is the average distance travelled by the
molecules between two consecutive collisions.
• Continuum approach is valid for 𝐾𝑛 < 0.01.
• In this course, fluid is treated as continuum.
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1.4. The properties of
continuum
Mass Density, 𝝆 [kg/m3]
Domain of
continuum
P
’
Asymptotic line that
defines the density

●Δ𝑚⁄Δ∀ is the average density.
● The average density approaches an asymptotic value as fluid becomes more
and more homogeneous.
● When Δ∀ becomes very small, it contains only few molecules and density
fluctuates as molecules pass into and out of this volume
● The minimum volume Δ∀′ satisfies the continuum postulate.
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The description can be made in two ways:
a) Material (Lagrangian) Description:
An identified fluid particle is followed in the course of time and
variation of its properties are described.
b) Spatial (Eulerian) Description:
Attention is focused on a fixed point in space and variation of fluid
properties is observed at that point.
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Mass Density (cont’d)
Domain of
molecular
effects

Continuum hypothesis assumes that, when a fluid particle moves in a
flow field, its properties (density, velocity, pressure, temperature…)
change
i. from point to point
ii. from time to time
continuously.
1.4. The properties of
continuum
m/

1.3. Description of continuum
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The mass density at point P is given by
Δ𝑚
𝜌 = lim
∀→ ∀ Δ∀

● The density of a fluid is a function of
P
temperature and pressure.
o The density of liquids is slightly
affected from these properties so that

the density of liquids is almost constant
and they are known as incompressible
fluids.
o The density of gases is strongly affected from these properties so
that they are called compressible fluids.
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1.4. The properties of
continuum
1.4. The properties of
continuum
Relative Density, s
Mass Density (cont’d)
𝑠=
For a homogeneous fluid, the average density is
𝑚
𝜌=
∀
The density has units of kg/m3 in SI system of units.
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The relative density is sometimes referred to as the specific gravity.
Specific weight, 𝜸 [N/m3]
Force exerted by the gravity field on a unit volume of the fluid, or the
weight of the fluid per unit volume.
𝛾 = 𝜌𝑔
𝑔: gravitational acceleration
• Not as useful as density since gravitational acceleration changes
with location and altitude
• Mostly used in hydraulic engineering.
• Has units of N/m3 in SI units.
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1.4. The properties of
continuum
𝑉=
∑ 𝑚 𝑉
∑ 𝑚
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No-slip condition
’
The fluid velocity 𝑉(𝑥, 𝑦, 𝑧, 𝑡) at a point
𝑃(𝑥, 𝑦, 𝑧) is defined as the instantaneous
velocity of the center of gravity of the
volume 
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1.4. The properties of
continuum
z
Fluid velocity, 𝑽 [m/s]
Density of any substance
𝜌
=
Density of water
𝜌
𝑉(𝑥, 𝑦, 𝑧, 𝑡)
𝑃
r
y
It is an experimental fact that the fluid particles which are in direct
contact with the solid boundary, has the same velocity as the boundary
itself.
In other words, the fluid sticks to the solid boundaries.
x
𝑁 : number of molecules inside 
𝑚 : mass of the 𝑖 th molecule
𝑉 : volume of the 𝑖 th molecule
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1.4. The properties of
continuum
Couette flow
ℎ
𝑦
𝑈
𝑈
𝑢(𝑦)
1.4. The properties of
continuum
Forces Acting on a Body of Fluid, 𝑭 [N = kg m/s2]
𝑈
𝑢=
𝑦
ℎ
𝑣=0
𝑤=0
Body forces
● Distributed over the entire volume
of fluid.
𝑥
𝑢=0
Hagen-Poiseuille flow
● Defined as the force per unit mass
of the fluid
𝑢=0
𝑟
𝑅
𝑧
𝑉 𝑟
𝑉 =0
𝑉 =0
𝑟
1−
𝑅
𝑉 =𝑉
𝑢=0
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1.4. The properties of
continuum
Δ𝐴
𝑃
Δ𝐹
Δ𝐹
𝐹
● Gravitational force per unit mass:
gravitational acceleration, 𝑔⃗
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Stress field at a point is a tensor quantity.
𝜏
◦ Nine components
𝜏
Cartesian stress tensor:
𝜎
𝜏
𝜏
𝜏
𝜎
𝜏
𝜏
𝜏
𝜎
lim
→
dirn. of
surface normal
|
𝑃
𝐹
𝑦
→
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𝑛
𝐹 :normal force
𝐹 : tangential (shear) force
lim
Shear stress, 𝜏, at point P
𝜏=
𝑃
Stress (cont’d)
𝑛
Δ𝐹⃗
Normal stress, 𝜎, at point P
𝐹⃗
𝐹⃗
● Caused by gravitational fields,
magnetic field and electrostatic
fields
1.4. The properties of
continuum
Stress, 𝝈, 𝝉 [Pa = N/m2]
𝜎=
Surface forces
● Forces exerted on the
boundaries of a fluid element by
its surroundings via direct contact
at the surface.
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𝜎
𝑥
𝑧
dirn. of
force
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1.4. The properties of
continuum
1.4. The properties of
continuum
Pressure [Pa = N/m2]
Pressure (cont’d)
𝑧
● Is the normal component of a
force acting on an area divided by
that area.
𝑧
𝑝 𝐴
𝑛
● Has the same magnitude as the
𝑝 𝐴
normal stress, but it acts in a
direction opposite to the unit normal
vector of this area.
● Pressure at a point is independent
of direction, whenever the fluid is at
rest.
𝑝 𝐴
𝑝 𝐴 cos 𝛼

𝑘
𝑗
𝑖
𝑥
𝑦
𝑦
● The prism is small enough so that
the pressure on each face may be
assumed constant.
● For a fluid at rest, sum of the
forces acting on the body of fluid
must be equal to zero.
𝑧
𝑝 𝐴
𝑛
𝑝 𝐴
𝑝 𝐴 cos 𝛼

𝑗
𝑖
𝑥
𝑦
𝑦
𝑝 𝐴
𝑥
or in scalar form
𝑝 𝐴
𝑘
𝐹⃗ = 0
𝑝 𝐴
𝑥
𝑧
∑ 𝐹 = 0, ∑ 𝐹 = 0, ∑ 𝐹 = 0
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1.4. The properties of
continuum
Pressure (cont’d)
∀ of the prism
𝐹 = 𝑝 𝐴 − 𝑝 𝐴 cos 𝛼 + 𝑓
𝐴 𝑦
3
body force
in 𝑦-dirn
Pressure (cont’d)
𝑧
𝑧
𝜌=0
𝑝 𝐴
𝑛
𝑝 𝐴
𝐴 : area of the prism in the 𝑥𝑧 plane
𝑥
𝑝 : average pressure on the area 𝐴
𝑥
𝐴 : area of the inclined surface
𝑝 : average pressure on the area 𝐴
𝛼: angle between the normals of the surfaces 𝐴 and 𝐴
𝑦 : intercept of the prism faces on the 𝑦-axis
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𝑝 𝐴 cos 𝛼

𝑗
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𝑦
𝑦
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𝑧
𝜌=0
𝑓 𝜌𝑦
3
𝑝 𝐴
𝑛
𝑝 𝐴
𝐴 = 𝐴 cos 𝛼  𝐴 =
𝑝 −𝑝 +
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𝐴 𝑦
3
From the geometry of the prism:
𝐴 𝑦
𝑝 𝐴 −𝑝 𝐴 +𝑓 𝜌
3
𝑝 𝐴
𝑧
𝐹 = 𝑝 𝐴 − 𝑝 𝐴 cos 𝛼 + 𝑓
𝑝 𝐴
𝑘
𝑖
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1.4. The properties of
continuum
mass inside
the prism
● Hence, in 𝑦-dirn:
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𝑝 𝐴
𝑝 𝐴 cos 𝛼

𝑘
𝑗
𝑖
=0
𝑥
𝑥
𝑦
𝑝 𝐴
=0
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𝑦
1.4. The properties of
continuum
Pressure (cont’d)
𝑝 −𝑝 +
𝑓 𝜌𝑦
3
𝑧
𝑧
=0
As 𝑦 → 0, in the limit, eqn. becomes
𝑝 =𝑝
𝑛
𝑝 𝐴
𝑥
∑𝐹 = 0  𝑝 = 𝑝
𝑥
Hence,
𝑝 =𝑝 =𝑝 =𝑝 =𝑝
𝑝 𝐴 cos 𝛼

𝑗
𝑖
∑𝐹 = 0  𝑝 = 𝑝
𝑝 𝐴
𝑘
Similarly;
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𝑝 𝐴
𝑦
𝑝 𝐴
the pressure at a point is equal in all
directions for a fluid at rest
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𝑦