Session 02
Basic Concepts in Fluid Mechanics - 2
Content
2.1 Types of Fluids 15
2.2 Dimensions 17
2.3 Units 18
2.4 Dimensional Homogeneity 19
2.5 Applications of fluid mechanics in different areas 19
Summary 22
Learning Outcomes 22
2.1 Types of Fluids
There are five types of fluids.
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•
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Ideal Fluid
Real Fluid
Newtonian Fluid
Non Newtonian Fluid
Ideal Plastic Fluid
Ideal Fluid
In the preceding sections we have reviewed the principal physical properties of
fluids. It is, however, rare for a specific engineering problem to be solved after
taking into consideration all the properties of fluids. The mathematics and
physical arguments become too complicated if all the properties of fluid is
considered. In fact, the numerical accuracy achieved by considering real fluid
behavior is often not necessary. An approximate solution obtained by making
simplifying assumptions is usually adequate. The simplification often made is to
assume the fluid to be ideal. The ideal fluid is assumed,
(a)
to have zero viscosity and, hence, cannot develop and shear stresses;
(b)
to be incompressible;
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(c)
to have no surface tension; and
(d)
not to vaporise.
With these assumptions, many problems can now be solved, although there is
certain unreality when compared with real fluids.
Experimental evidence can be used to convert the predicted result, and the value
of empirical coefficients used is an indication of the accuracy of assuming an
ideal fluid.
Real fluid
A fluid which possesses as real fluid. All the fluids in actual practice are real
fluids.
Newtonian fluid
A real fluid, in which the shear stress is directly, proportional to the rate of shear
strain (or velocity gradient), is known as Newtonian fluid. The graphs plotting
shear stress against rate of shear or velocity gradient is called rheograms. The
slope of the straight line in any rheogram of a Newtonian fluid represents the
viscosity (or more specifically, the dynamic viscosity) of the fluid. The higher
viscosity of a fluid becomes the steeper the slope in the rheogram. Fluids such as
air, water, oil, glycerin, and honey are described as Newtonian fluids.
Figure 2.1: Rheogram of various Newtonian fluids
Non-Newtonian fluid
A real fluid, in which the shear stress is not proportional to the rate of shear strain
(or velocity gradient), is known as Non-Newtonian fluid. For Non-Newtonian
fluids, the line in the rheogram either is curved or does not pass through the
origin, or both. Figure (3.2) shows the rheograms of various types of Non16
Newtonian fluids. Note that while the rheogram of a pseudoplastic fluid curves
downward (i.e., having decreasing slope with increased shear), for a dilatant fluid
the rheogram curves upward (i.e. having increasing slope with increased shear).
Otherwise, the two are similar: they both pass through the origin of the rheogram,
as is the case with Newtonian fluids.
As shown in Fig. (1.6), a Bingham plastic (or simply Bingham) fluid is
represented by a straight line in rheograms, but the line does not pass through the
origin. It takes a certain minimum shear stress, called the yield stress to cause a
Bingham fluid to behave like a fluid.
Figure 2.3: Rheogram of various Non-Newtonian fluids
Ideal plastic fluid
A real fluid, in which the shear stress is more that the yield value and the shear
stress is proportional to the rate of shear strain (or velocity gradient), is known as
ideal plastic fluid.
2.2 Dimensions
A dimension is a category that represents a physical quantity such as mass, length,
time, momentum, force, acceleration, and energy. To simplify matters, engineers
express dimensions using a limited set that are called primary dimensions. Table
2.1 lists one common set of primary dimensions.
Table 2.1: Primary Dimensions
Dimension
Symbol
Unit (SI)
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Length
L
meter (m)
Mass
M
kilogram (kg)
Time
T
second (s)
Temperature
θ
kelvin (K)
Electric current
i
ampere (A)
Amount of light
C
candela (cd)
Amount of matter
N
mole (mol)
Secondary dimensions such as momentum and energy can be related to primary
dimensions by using equations. For example, the secondary dimension “force” is
expressed in primary dimensions by using Newton’s second law of motion, 𝐹 =
𝑚𝑎. The primary dimensions of acceleration are 𝐿/𝑇 2 , Therefore
𝐹 = 𝑚𝑎 =
𝑀𝐿
𝑇2
2.3 Units
While a dimension expresses a specific type of physical quantity, a unit assigns a
number so that the dimension can be measured. For example, measurement of
volume (a dimension) can be expressed using units of liters. Similarly,
measurement of energy (a dimension) can be expressed using units of joules.
Most dimensions have multiple units that are used for measurement. For example,
the dimension of “force” can be expressed using units of newtons, pounds-force,
or dynes. In practice, there are several unit systems in use. The International
System of Units (abbreviated SI from the French “Le Système International
d'Unités”) is based on the meter, kilogram, and second. Although the SI system is
intended to serve as an international standard, there are other systems in common
use in the United States. The U.S. Customary System (USCS), sometimes called
English Engineering System, uses the pound-mass (lbm) for mass, the foot (ft) for
length, the pound-force (lbf) for force, and the second (s) for time.
Table 2.2: Units for different quantities
QUANTITY
SI UNITS
Mass
kilogram (kg)
Length
metre (m)
Time
second (s)
Density
kilogrames/metre3 (kg/m3)
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Force
Newton (N)
Velocity
metre/Sec (m/s)
Acceleration
metre/Sec2 (m/s2)
Pressure
Pascal (N/m2)
Surface tension
Newton/metre (N/m)
Work
Joules (J)
Power
kilowatt (kw)
Volume
metre3 (m3)
2.4 Dimensional Homogeneity
When the primary dimensions on each term of an equation are the same, the
equation is dimensionally homogeneous. For example, consider the equation for
vertical position 𝑠 of an object moving in a gravitational field:
𝑠=
𝑔𝑡 2
2
+ 𝑣0 𝑡 + 𝑠0
In the equation, 𝑔 is the gravitational constant, 𝑡 is time, 𝑣𝑜 is the magnitude of
the vertical component of the initial velocity, and 𝑠𝑜 is the vertical component of
the initial position. This equation is dimensionally homogeneous because the
primary dimension of each term is length 𝐿.
2.5 Applications of fluid mechanics in different areas
Hydraulics
Hydraulics is the study of the flow of water through pipes, rivers, and openchannels.
This includes pumps and turbines and applications such as hydropower.
Hydraulics is important for ecology, policymaking, energy production, recreation,
fish and game resources, and water supply.
Hydrology
Hydrology is the study of the movement, distribution, and quality of water
throughout
the earth. Hydrology involves the hydraulic cycle and water resource issues. Thus,
hydrology provides results that are useful for environmental engineering and for
policymaking. Hydrology is important nowadays because of global challenges in
providing water for human societies.
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Aerodynamics
Aerodynamics is the study of air flow. Topics include lift and drag on objects
(e.g., airplanes, automobiles, birds), shock waves associated with flow around a
rocket, and the flow through a supersonic or deLaval nozzle. Aerodynamics is
important for the design of vehicles, for energy conservation, and for
understanding nature.
Bio-fluid mechanics
Bio-fluid mechanics is an emerging field that includes the study of the lungs and
circulatory system, blood flow, micro-circulation, and lymph flow. Bio-fluids also
includes development of artificial heart valves, stents, vein and dialysis shunts,
and artificial organs. Bio-fluid mechanics is important for advancing health care.
Acoustics
Acoustics is the study of sound. Topics include production, control, transmission,
reception of sound, and physiological effects of sound. Since sound waves are
pressure waves in fluids, acoustics is related to fluid mechanics. In addition, water
hammer in a piping system, which involves pressure waves in liquids, involves
some of the same knowledge that is used in acoustics.
Microchannel flow
Microchannel flow is an emerging area that involves the study of flow in tiny
passages. The typical size of a microchannel is a diameter in the range of 10 to
200 micrometers. Applications that involve microchannels include
microelectronics, fuel cell systems, and advanced heat sink designs.
Computational fluid dynamics (CFD)
CFD is the application of numerical methods implemented on computers to model
and solve problems that involve fluid flows. Computers perform millions of
calculations per second to simulate fluid flow. Examples of flows that are
modeled by CFD include water flow in a river, blood flow in the abdominal aorta,
and air flow around an automobile.
Petroleum engineering
Petroleum engineering is the application of engineering to the exploration and
production of petroleum. Movement of oil in the ground involves flow through a
porous medium. Petroleum extraction involves flow of oil through passages in
wells. Oil pipelines involve pumps and conduit flow.
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Atmospheric science
Atmospheric science is the study of the atmosphere, its processes, and the
interaction
of the atmosphere with other systems. Fluid mechanics topics include flow of the
atmosphere and applications of CFD to atmospheric modeling. Atmospheric
science is important for predicting weather and is relevant to current issues
including acid rain, photochemical smog, and global warming.
Electrical engineering
Electrical engineering problems can involve knowledge from fluid mechanics. For
example, fluid mechanics is involved in the flow of solder during a manufacturing
process, the cooling of a microprocessor by a fan, sizing of motors to operate
pumps, and the production of electrical power by wind turbines.
Environmental engineering
Environmental engineering involves the application of science to protect or
improve
the environment (air, water, and or land resources) or to remediate polluted sites.
Environmental engineers design water supply and wastewater treatment systems
for communities. Environmental engineers are concerned with local and
worldwide environmental issues such as acid rain, ozone depletion, water
pollution, and air pollution.
S.A.Q. 1:
Show that the Reynolds number (𝑅𝑒) given by 𝑅𝑒 =
𝜌𝑣𝐷
𝜇
, is a dimensionless
group, where 𝜌 is the density, 𝑣 is the velocity, 𝐷 is the diameter and 𝜇 is the
viscosity.
S.A.Q. 2:
Derive the units of viscosity in terms of force, time and length in the SI system
Answer
In the SI system, Force – Newton (N)
Time
– seconds (s)
Length – metres (m)
τ = Force/Area = N/m2
𝑑𝑢 = 𝑚⁄𝑠
𝑑𝑦 = 𝑚
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μ=
τ
du
( )
dy
𝜇=
(𝑁⁄ 2 )
𝑚
(𝑚⁄𝑠)
𝑚
= 𝑁𝑠/𝑚2 = 𝑃𝑎
Summary
Fluid mechanics involves the application of scientific concepts such as position,
force, velocity, acceleration, and energy to materials that are in the liquid or gas
states. A fluid is defined as a substance that continually deforms (flows) under an
applied shear stress. Basically, the fluids are classified into five types and these
are;
1. Ideal fluid
2. Real fluid
3. Newtonian fluid
4. Non-Newtonian fluid
5. Ideal plastic fluid.
In fluid mechanics, there are four primary dimensions: mass, length, time, and
temperature. Primary dimensions are defined as independent dimensions, from
which all other dimensions can be obtained. Dimensions have no numbers
associated with them. A unit is a way to assign a number or measurement to a
dimension. Units must always have numbers associated with them.
Learning Outcomes
At the end of this lesson the student should be able to
•
identify different types of fluids, be familiar with some of its basic
properties
•
be able to deal with the systems of units and dimensions used in measuring
different fluid quantities.
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