fluid mechanics for mechanical engineering

Faculty of Engineering and
Technical Studies

Fluid Mechanics for
Mechanical Engineering
TUTORIAL 1 –

UNIT 1:
Chapter 1: Introduction to Fluid
Mechanics
Assoc Prof Dr Shahrir Abdullah
EBMF4103 Fluid Mechanics for Mechanical Engineering
Jan 2005
1
Subject Matter Expert/Author: Assoc Prof Dr Shahrir Abdullah

Copyright © ODL Jan 2005 Open University Malaysia

Technical Studies

SEQUENCE OF CHAPTER 1
Introduction
Objectives
1.1 Fluid Concept
1.2 Units and Dimensions
1.3 Fluid Continuum
1.4 Flow Patterns
1.5 Fluid Density
1.6 Viscosity
1.7 Surface Tension
1.8 Vapour Pressure
Summary

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Introduction
 This chapter will begin with several concepts, definition,
terminologies and approaches which should be understood by
the students before continuing reading the rest of this
module.
 Then, it introduces the student with typical properties of
fluid and their dimensions which are then being used
extensively in the next chapters and units like pressure,
velocity, density and viscosity.
 Some of these can be used to classify type and characteristic
of fluid, such as whether a fluid is incompressible or not or
whether the fluid is Newtonian or non-Newtonian.


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Objectives
At the end of this chapter, you should be able to :
 explain the concept of fluid continuity and flow
representations using streamlines, streaklines and
pathlines,
 Identify and describe typical fluid properties and their
units and dimensions


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1.1 Fluid Concept
• Fluid mechanics is a division in applied mechanics related to
the behaviour of liquid or gas which is either in rest or in
motion.
• The study related to a fluid in rest or stationary is referred
to fluid static, otherwise it is referred to as fluid dynamic.
• Fluid can be defined as a substance which can deform
continuously when being subjected to shear stress at any
magnitude. In other words, it can flow continuously as a
result of shearing action. This includes any liquid or gas.


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1.1 Fluid Concept
• Thus, with exception to solids, any other matters can be
categorised as fluid. In microscopic point of view, this
concept corresponds to loose or very loose bonding between
molecules of liquid or gas, respectively.
• Examples of typical fluid used in engineering applications are
water, oil and air.
• An analogy of how to understand different bonding in solids
and fluids is depicted in Fig. 1.1


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1.1 Fluid Concept
• Thus, with exception to solids, any other matters can be
categorised as fluid. In microscopic point of view, this
concept corresponds to loose or very loose bonding between
molecules of liquid or gas, respectively.
• Examples of typical fluid used in engineering applications are
water, oil and air.
• An analogy of how to understand different bonding in solids
and fluids is depicted in Fig. 1.1


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1.1 Fluid Concept
Free surface

k
k

k
k

(a) Solid

(b) Liquid

(c) Gas

Figure 1.1 Comparison Between Solids, Liquids and Gases

• For solid, imagine that the molecules can be fictitiously
linked to each other with springs.


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1.1 Fluid Concept
 In fluid, the molecules can move freely but are constrained
through a traction force called cohesion. This force is
interchangeable from one molecule to another.
 For gases, it is very weak which enables the gas to
disintegrate and move away from its container.
 For liquids, it is stronger which is sufficient enough to hold
the molecule together and can withstand high compression,
which is suitable for application as hydraulic fluid such as oil.
On the surface, the cohesion forms a resultant force directed
into the liquid region and the combination of cohesion forces
between adjacent molecules from a tensioned membrane
known as free surface.


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• The primary quantities which are also referred to as basic
dimensions, such as L for length, T for time, M for mass and
Q for temperature.
• This dimension system is known as the MLT system where it
can be used to provide qualitative description for secondary
quantities, or derived dimensions, such as area (L), velocity
(LT-1) and density (ML-3).
• In some countries, the FLT system is also used, where the
quantity F stands for force.


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• An example is a kinematic equation for the velocity V of a
uniformly accelerated body,

V = V0 + at
where V0 is the initial velocity, a the acceleration and t the
time interval. In terms for dimensions of the equation, we
can expand that

LT-1 = LT -1 + LT-2 • T


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Example 1.1
 The free vibration of a particle can be simulated by the
following differential equation:

du
m
+ kx = 0
dt
where m is mass, u is velocity, t is time and x is
displacement. Determine the dimension for the stiffness
variable k.


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Example 1.1
 By making the dimension of the first term equal to the
second term:
[m] •

[u]
= [k]•[x]
[t]

Hence,

[m]•[u]
[k] =
=
[t]•[x]

M • LT-1
LT

= MT-2


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1.3 Fluid Continuum
• Since the fluid flows continuously, any method and technique
developed to analyse flow problems should take into
consideration the continuity of the fluid. There are two
types of approaches that can be used:
1.Eulerian approach — analysis is performed by defining a
control volume to represent fluid domain which allows the
fluid to flow across the volume. This approach is more
appropriate to be used in fluid mechanics.
2.Lagrangian approach — analysis is performed by tracking down
all motion parameters and deformation of a domain as it
moves. This approach is more suitable and widely used for
particle and solid mechanics.


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1.3 Fluid Continuum
 The fluid behaviour in which its properties are continuous
field variables, either scalar or vector, throughout the
control volume is known as continuum. From this concept,
several fluid or flow definitions can be made as follows:
 Steady state flow — A flow is said to be in steady state if its
properties is only a function of position (x,y,z) but not time
t:

ρ = ρ (x,y,z), V = V (x,y,z)
An example is the velocity of a steady flow of a river where the upstream
and downstream velocities are different but their values does not change
through time.


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1.3 Fluid Continuum
• Uniform flow — A flow is said to be uniform if its velocity
and all velocity components is only a function of time t:

V = V (t)
An example is the air flow in a constant diameter duct where the velocity
is constant throughout the length of the duct but can be increased
uniformly by increasing the power of the fan.

• Isotropic fluid — A fluid is said to be isotropic if its density is
not a function of position (x,y,z) but may vary with time t:

ρ = ρ (t)
An example is the density of a gas in a closed container where the
container is heated. The density is constant inside the container but
gradually increases with time as the temperature increases.


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1.4 Flow Patterns
• The three ways to represent fluid flow:
1. Streamlines — A streamline is formed by tangents of the
velocity field of the flow.
2. Pathlines — A pathline can be formed from fluid
particles of different colour originated from the same
points, such as a line formed after the introduction of
ink into a shallow water flow.
3. Streaklines — A streakline represents a locus made by a
miniature particles or tracers that passes at a same
point.


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1.5 Density
Density of a fluid, ρ,
Definition: mass per unit volume,
 slightly affected by changes in temperature and pressure.

ρ = mass/volume = m/∀
Units: kg/m3
Typical values:
Water = 1000 kg/m3; Air = 1.23 kg/m3


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1.6 Viscosity
• Viscosity, µ, is a measure of resistance to fluid flow as a
result of intermolecular cohesion. In other words, viscosity
can be seen as internal friction to fluid motion which can
then lead to energy loss.
• Different fluids deform at different rates under the same
shear stress. The ease with which a fluid pours is an
indication of its viscosity. Fluid with a high viscosity such as
syrup deforms more slowly than fluid with a low viscosity
such as water. The viscosity is also known as dynamic
viscosity.
 Units: N.s/m2 or kg/m/s
 Typical values:
Water = 1.14x10-3 kg/m/s;

Air = 1.78x10-5 kg/m/s

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Newtonian and Non-Newtonian Fluid
Fluid

obey

Newton’s law
of viscosity

refer

Newton’s’ law of viscosity is given by;

du
τ=µ
dy

(1.1)

τ
= shear stress
µ
= viscosity of fluid
du/dy = shear rate, rate of strain or velocity gradient

Newtonian fluids
Example:
Air
Water
Oil
Gasoline
Alcohol
Kerosene
Benzene
Glycerine

• The viscosity µ is a function only of the condition of the fluid, particularly its
temperature.
• The magnitude of the velocity gradient (du/dy) has no effect on the magnitude of µ.

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Newtonian and Non-Newtonian Fluid
Fluid

Do not obey

Newton’s law
of viscosity

Non- Newtonian
fluids

• The viscosity of the non-Newtonian fluid is dependent on the
velocity gradient as well as the condition of the fluid.
Newtonian Fluids

 a linear relationship between shear stress and the velocity gradient (rate
of shear),
 the slope is constant
 the viscosity is constant

non-Newtonian fluids

 slope of the curves for non-Newtonian fluids varies

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If the gradient m is constant, the fluid is termed as Newtonian fluid.
Otherwise, it is known as non-Newtonian fluid. Fig. 1.5 shows
several Newtonian and non-Newtonian fluids.

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Kinematic viscosity, ν
Definition: is the ratio of the viscosity to the density;

ν = µ/ρ

• will be found to be important in cases in which significant viscous and
gravitational forces exist.

Units: m2/s
Typical values:
Water = 1.14x10-6 m2/s;

Air = 1.46x10-5 m2/s;

In general,
viscosity of liquids with temperature, whereas
viscosity of gases with

in temperature.


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Specific Weight
Specific weight of a fluid, γ
• Definition: weight of the fluid per unit volume
• Arising from the existence of a gravitational force
• The relationship γ and g can be found using the following:
Since
therefore

ρ = m/∀
γ = ρg

(1.3)

Units: N/m3
Typical values:
Water = 9814 N/m3;


Air = 12.07 N/m3

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Specific Gravity
The specific gravity (or relative density) can be defined in two ways:
Definition 1: A ratio of the density of a liquid to the density of
water at standard temperature and pressure (STP)
(20°C, 1 atm), or
Definition 2: A ratio of the specific weight of a liquid to the
specific weight of water at standard temperature
and pressure (STP) (20°C, 1 atm),

SG =

ρ liquid
ρ water @ STP

=

γ liquid
γ water @ STP

Unit: dimensionless.

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Example 1.2
A reservoir of oil has a mass of 825 kg. The reservoir has a volume
of 0.917 m3. Compute the density, specific weight, and specific
gravity of the oil.

Solution:
ρ oil

mass
m
825
=
= =
= 900kg / m 3
volume ∀ 0.917

γ oil

weight mg
=
=
= ρg = 900 x 9.81 = 8829 N / m 3
volume
∀

SGoil

ρ
900
oil
=
=
=0.9
ρ @ STP
998
w


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1.7 Surface Tension
• Surface tension coefficient s can be defined as the intensity
of intermolecular traction per unit length along the free
surface of a fluid, and its SI unit is N/m.
• The surface tension effect is caused by unbalanced cohesion
forces at fluid surfaces which produce a downward resultant
force which can physically seen as a membrane.
• The coefficient is inversely proportional to temperature and
is also dependent on the type of the solid interface.
• For example, a drop of water on a glass surface will have a
different coefficient from the similar amount of water on a
wood surface.


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1.7 Surface Tension
• The effect may be becoming significant for small fluid system such
as liquid level in a capillary, as depicted in Fig. 1.6, where it will
decide whether the interaction form by the fluid and the solid
surface is wetted or non-wetted.

• If the adhesion of fluid molecules to the adjacent solid surface is
stronger than the intermolecular cohesion, the fluid is said to wet
on the surface. Otherwise, it is a non-wetted interaction.

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1.7 Surface Tension
 The pressure inside a drop of fluid can be calculated using a free-body
diagram of a spherical shape of radius R cut in half, as shown in Fig. 1.7,
and the force developed around the edge of the cut sphere is 2πRσ.

 This force must be balance with the difference between the internal
pressure pi and the external pressure pe acting on the circular area of the
cut. Thus,

2πRσ = ∆pπR2
2σ
∆p = pi –pe =
R

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1.8 Vapour Pressure
• Vapour pressure is the partial pressure produced by fluid vapour
in an open or a closed container, which reaches its saturated
condition or the transfer of fluid molecules is at equilibrium along
its free surface.
• In a closed container, the vapour pressure is solely dependent on
temperature. In a saturated condition, any further reduction in
temperature or atmospheric pressure below its dew point will
lead to the formation of water droplets.
• On the other hand, boiling occurs when the absolute fluid
pressure is reduced until it is lower than the vapour pressure of
the fluid at that temperature.
• For a network of pipes, the pressure at a point can be lower than
the vapour pressure, for example, at the suction section of a
pump. Otherwise, vapour bubbles will start to form and this
phenomenon is termed as cavitation.

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Summary
This chapter has summarized on the aspect below:
 Understanding concept of a fluid
 Ways to visualise the flows and the continuity nature of
the fluid flow
 Fluid properties of density, specific weight, specific
gravity and viscosity were discussed.
 a Newtonian fluid against a non-Newtonian fluid, an inviscid flow against a viscous flow, and a wetted surface
against a non-wetted surface.
 Discussion on the surface tension and vapour pressure


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Tha nk Yo u

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fluid mechanics for mechanical engineering

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