Chapter 1. introduction to fluid mechanics

1
Introduction to Fluid Mechanics
Chapter 1
Fluid Mechanics I
(MEng 2306)
Technology College
Mechanical Engineering Department
Prepared by: Addisu D. March, 2013

Definition
 Mechanics is the oldest physical science that deals
with both stationary and moving bodies under the
influence of forces.
 The branch of mechanics that deals with bodies at rest
is called statics, while the branch that deals with
 bodies in motion is called dynamics.
 The subcategory fluid mechanics is defined as the
science that deals with the behavior of fluids at rest
(fluid statics) or in motion (fluid dynamics), and the
interaction of fluids with solids or other fluids at the
boundaries.
2

Definition
 The study of fluids at rest is called fluid statics.
 The study of f1uids in motion, where pressure forces are not
considered, is called fluid kinematics and if the pressure
forces are also considered for the fluids in motion. that
branch of science is called fluid dynamics.
 Fluid mechanics itself is also divided into several categories.
 The study of the motion of fluids that are practically
incompressible (such as liquids, especially water, and gases
at low speeds) is usually referred to as hydrodynamics.
 A subcategory of hydrodynamics is hydraulics, which deals
with liquid flows in pipes and open channels.
3

 Gas dynamics deals with the flow of fluids that undergo
significant density changes, such as the flow of gases
through nozzles at high speeds.
 The category aerodynamics deals with the flow of gases
(especially air) over bodies such as aircraft, rockets, and
automobiles at high or low speeds.
 Some other specialized categories such as meteorology,
oceanography, and hydrology deal with naturally
occurring flows.
4
Definition

What is a Fluid?
 A substance exists in three primary phases: solid, liquid,
and gas. A substance in the liquid or gas phase is referred
to as a fluid.
 Distinction between a solid and a fluid is made on the basis
of the substance’s ability to resist an applied shear (or
tangential) stress that tends to change its shape.
 A solid can resist an applied shear stress by deforming,
whereas a fluid deforms continuously under the influence
of shear stress, no matter how small.
 In solids stress is proportional to strain, but in fluids stress
is proportional to strain rate.
5

What is a Fluid?
 When a constant
shear force is
applied, a solid
eventually stops
deforming, at some
fixed strain angle,
whereas a fluid
never stops
deforming and
approaches a certain
rate of strain.
6
Figure.
Deformation of a rubber eraser
placed between two parallel plates
under the influence of a shear force.

 In a liquid, molecules can move
relative to each other, but the volume
remains relatively constant because of
the strong cohesive forces between the
molecules.
 As a result, a liquid takes the shape of
the container it is in, and it forms a
free surface in a larger container in a
gravitational field.
 A gas, on the other hand, expands until
it encounters the walls of the container
and fills the entire available space.
 This is because the gas molecules are
widely spaced, and the cohesive forces
between them are very small.
 Unlike liquids, gases cannot form a
free surface7
What is a Fluid?

Application areas of Fluid Mechanics
 Mechanics of fluids is extremely important in many areas
of engineering and science. Examples are:
 Biomechanics
 Blood flow through arteries and veins
 Airflow in the lungs
 Flow of cerebral fluid
 Households
 Piping systems for cold water, natural gas, and sewage
 Piping and ducting network of heating and air-
conditioning systems
 Meteorology and Ocean Engineering
 Movements of air currents and water currents
 Chemical Engineering
 Design of chemical processing equipment8

Application areas of Fluid Mechanics
 Mechanical Engineering
 Design of pumps, turbines, air-conditioning equipment,
pollution-control equipment, etc.
 Design and analysis of aircraft, boats, submarines,
rockets, jet engines, wind turbines, biomedical devices,
the cooling of electronic components, and the
transportation of water, crude oil, and natural gas.
 Civil Engineering
 Transport of river sediments
 Pollution of air and water
 Design of piping systems
 Flood control systems
9

Classification of Fluid Flows
 There is a wide variety of fluid flow problems encountered
in practice, and it is usually convenient to classify them on
the basis of some common characteristics to make it
feasible to study them in groups.
Viscous versus Inviscid Regions of Flow
 When two fluid layers move relative to each other, a
friction force develops between them and the slower layer
tries to slow down the faster layer.
 This internal resistance to flow is quantified by the fluid
property viscosity, which is a measure of internal stickiness
of the fluid.
 Viscosity is caused by cohesive forces between the
molecules in liquids and by molecular collisions in gases.
10

Viscous versus Inviscid Regions of Flow…
 There is no fluid with zero viscosity, and thus all fluid
flows involve viscous effects to some degree.
 Flows in which the frictional effects are significant are
called viscous flows.
 However, in many flows of practical interest, there are
regions (typically regions not close to solid surfaces) where
viscous forces are negligibly small compared to inertial or
pressure forces.
 Neglecting the viscous terms in such inviscid flow regions
greatly simplifies the analysis without much loss in
accuracy.
11

Internal versus External Flow
 A fluid flow is classified as being internal or external,
depending on whether the fluid is forced to flow in a
confined channel or over a surface.
 The flow of an unbounded fluid over a surface such as a
plate, a wire, or a pipe is external flow.
 The flow in a pipe or duct is internal flow if the fluid is
completely bounded by solid surfaces.
 Water flow in a pipe, for example, is internal flow, and
airflow over a ball or over an exposed pipe during a windy
day is external flow .
12

Compressible versus Incompressible Flow
 A flow is classified as being compressible or
incompressible, depending on the level of variation of
density during flow.
 Incompressibility is an approximation, and a flow is said to
be incompressible if the density remains nearly constant
throughout.
 Therefore, the volume of every portion of fluid remains
unchanged over the course of its motion when the flow (or
the fluid) is incompressible.
 The densities of liquids are essentially constant, and thus
the flow of liquids is typically incompressible. Therefore,
liquids are usually referred to as incompressible substances.
13

Compressible versus Incompressible Flow…
 A pressure of 210 atm, for example, causes the density of
liquid water at 1 atm to change by just 1 percent.
 Gases, on the other hand, are highly compressible.
 A pressure change of just 0.01 atm, for example, causes a
change of 1 percent in the density of atmospheric air.
 Gas flows can often be approximated as incompressible if
the density changes are under about 5 percent.
 The compressibility effects of air can be neglected at
speeds under about 100 m/s.
14

Laminar versus Turbulent Flow
 Some flows are smooth and orderly
while others are rather chaotic.
 The highly ordered fluid motion
characterized by smooth layers of
fluid is called laminar.
 The flow of high-viscosity fluids
such as oils at low velocities is
typically laminar.
 The highly disordered fluid motion
that typically occurs at high
velocities and is characterized by
velocity fluctuations is called
turbulent .
15

Laminar versus Turbulent Flow
 The flow of low-viscosity fluids
such as air at high velocities is
typically turbulent.
 A flow that alternates between
being laminar and turbulent is
called transitional.
16

Natural (or Unforced) versus Forced Flow
 A fluid flow is said to be natural or forced, depending on
how the fluid motion is initiated.
 In forced flow, a fluid is forced to flow over a surface or in
a pipe by external means such as a pump or a fan.
 In natural flows, any fluid motion is due to natural means
such as the buoyancy effect, which manifests itself as the
rise of the warmer (and thus lighter) fluid and the fall of
cooler (and thus denser) fluid .
 In solar hot-water systems, for example, the
thermosiphoning effect is commonly used to replace pumps
by placing the water tank sufficiently above the solar
collectors.
17

Steady versus Unsteady Flow
 The terms steady and uniform are used frequently in
engineering, and thus it is important to have a clear
understanding of their meanings.
 The term steady implies no change at a point with time.
 The opposite of steady is unsteady.
 The term uniform implies no change with location
over a specified region.
18

Properties of Fluids
 Any characteristic of a system is called a property.
 Some familiar properties are pressure P, temperature T,
volume V, and mass m.
 Other less familiar properties include viscosity, thermal
conductivity, modulus of elasticity, thermal expansion
coefficient, electric resistivity, and even velocity and
elevation.
 Properties are considered to be either intensive or extensive.
Intensive properties are those that are independent of the mass
of a system, such as temperature, pressure, and density.
 Extensive properties are those whose values depend on the
size—or extent—of the system. Total mass, total volume V,
and total momentum are some examples of extensive
properties.
19

 An easy way to determine
whether a property is
intensive or extensive is to
divide the system into two
equal parts with an imaginary
partition.
 Each part will have the same
value of intensive properties
as the original system, but
half the value of the extensive
properties.
20

Density or Mass Density.
 Density or mass density of a fluid is defined as the ratio of
the mass of a f1uid to its volume. Thus mass per unit
volume of a fluid is called density. It is denoted the symbol
ρ (rho). The unit of mass density in SI unit is kg per cubic
meter, i.e ., kg/m3.
 The density of liquids may be considered as constant while
that of gases changes with the variation of pressure and
temperature.
 Mathematically mass density is written as.
 The value of density of water is 1 gm/cm3 or 1000 kg/m3.
21
fluidofVolume
fluidofMass


Density or Mass Density.
 The density of a substance, in general, depends on
temperature and pressure.
 The density of most gases is proportional to pressure and
inversely proportional to temperature.
 Liquids and solids, on the other hand, are essentially
incompressible substances, and the variation of their
density with pressure is usually negligible.
22

Specific weight or Weight Density.
 Specific weight or weight density of a fluid is the ratio
between the weight of a fluid to its volume.
 Thus weight per unit volume of a fluid is called weight
density and it is denoted by the symbol w.
 Mathematically,
23
gw
w






gx
fluidofVolume
gxfluidofMass
fluidofVolume
gravitytodueonAcceleratixfluid)of(Mass
fluidofVolume
fluidofWeight

Specific Volume.
 Specific volume of a fluid is defined as the volume
of a fluid occupied by a unit mass or volume per
unit mass of a fluid is called specific volume.
 Mathematically, it is expressed as
 Thus specific volume is the reciprocal of mass
density. It is expressed as m3/kg.
 It is commonly applied to gases.24

1
Volume
fluidofMass
1
fluidofMass
fluidofVolume
volumeSpecific 

Specific Gravity.
 Specific gravity is defined as the ratio of the weight density (or
density) of a fluid to the weight density (or density) of a standard
fluid.
 For liquids, the standard fluid is taken water and for gases, the
standard fluid is taken air. Specific gravity is also called relative
density. It is dimensionless quantity and is denoted by the symbol S.
25
3
3
kg/m1000x
waterofx DensityliquidaofdensityThus
N/m9.81x1000x
waterofdensityWeightxliquidaofdensityweightThus
airof(density)densityWeight
gasof(density)densityWeight
gases)(for
waterof(density)densityWeight
liquidof(density)densityWeight
liquids)(for
S
S
S
S
S
S







Specific Gravity.
 If the specific gravity of a
fluid is known, then the
density of the fluid will be
equal to specific gravity of
fluid multiplied by the
density of water.
 For example the specific
gravity of mercury is 13.6,
hence density of mercury
= 13.6 x 1000 = 13600
kg/m3.
26

27
Example 1.
Calculate the specific weight, density and specific gravity of one
liter of a liquid which weighs 7 N.

28
Example 2. Calculate the density, specific weight and weight of
one liter of petrol of specific gravity = 0.7

Viscosity
 Viscosity is defined as the property of a fluid which offers
resistance to the movement of one layer of fluid over another
adjacent layer of the fluid.
 When two layers of a fluid, a distance 'dy' apart move one over
the other at different velocities say u and u+ du as shown in Fig.
1.1 , the viscosity together with relative velocity causes a shear
stress acting between the fluid layers:
29

Viscosity
 The top layer causes a shear stress on the adjacent lower
layer while the lower layer causes a shear stress on the
adjacent top layer.
 This shear stress is proportional to the rate of change of
velocity with respect to y. It is denoted by symbol τ called
Tau.
 Mathematically,
 or
30
(1.2)
dy
du
 

 where μ (called mu) is the constant of proportionality
and is known as the coefficient of dynamic viscosity or
only viscosity.
 represents the rate of shear strain or rate of shear
deformation or velocity gradient.
 From equation (1.2) we have
 Thus viscosity is also defined as the shear stress
required to produce unit rate of shear strain.
31
dy
du
(1.3)
dy
du

 

Unit of Viscosity.
 The unit of viscosity is obtained by putting the
dimension of the quantities in equation ( 1.3)
32
22
m
Ns
m
secondNewton
viscosityofunitSI 

Kinematic Viscosity.
 It is defined as the ratio between the dynamic viscosity and
density of fluid.lt is denoted by the Greek symbol (ν) called
'nu' . Thus, mathematically,
 The SI unit of kinematic viscosity is m2/s.
Newton's Law of Viscosity.
 It states that the shear stress (τ) on a fluid element layer is
directly proportional to the rate of shear strain. The constant
of proportionality is called the co-efficient viscosity.
Mathematically, it is expressed as given by equation (1 . 2).
33


 
Density
Viscosity

 Fluids which obey the above relation are known as
Newtonian fluids and the fluids which do not obey the
above relation are called Non-newtonian fluids.
Variation of Viscosity with Temperature
 Temperature affects the viscosity.
 The viscosity of liquids decreases with the increase of
temperature while the viscosity of gases increases with
increase of temperature. This is due to reason that the
viscous forces in a fluid are due to cohesive forces and
molecular momentum transfer.
 In liquids the cohesive forces predominates the molecular
momentum transfer due to closely packed molecules and
with the increase in temperature, the cohesive forces
decreases with the result of decreasing viscosity.
34

 But in the case of gases the cohesive force are small and
molecular momentum transfer predominates. With the
increase in temperature, molecular momentum transfer
increases and hence viscosity increases. The relation between
viscosity and temperature for liquids and gases are:
35
9-
o
2
o
3-
o
o
o
2
o
2o
10x0.11896,0.0000000517,0000.0airforwhere
gas,aFor(ii)
0.000221and0.03368poise,10x1.79μFor water,
liquidfor theconstantsare,
poisein,0atliquidofViscosity
10
1
poise1poiseinC,at tliquidofViscositywhere
1
1
liquids,For(i)






















tt
C
m
Ns
tt

Types of Fluids
1. Ideal Fluid. A fluid, which is incompressible and is
having no viscosity, is known as an ideal fluid. Ideal
fluid is only an imaginary fluid as all the fluids, which
exist, have some viscosity.
2. Real fluid. A fluid, which possesses viscosity, is known as
real fluid. All the fluids: in actual practice, are real fluids.
3. 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 a Newtonian fluid.
4. Non-Newtonian fluid. A real fluid, in which shear stress
is not proportional to the rate of shear strain (or velocity
gradient), known as a Non-Newtonian fluid.
36

Types of Fluids
5. Ideal Plastic Fluid.
A fluid, in which shear
stress is more than the
yield value and shear
stress is proportional
to the rate of shear
strain (or velocity
gradient), is known as
ideal plastic fluid.
37

Example 3
38
If the velocity distribution over a plate is given by
in which u is velocity in metre per second at a distance y
metre above the plate, determine the shear stress at y = 0
and y= 0.15 m. Take dynamic viscosity of fluid as 8.63
poises.
2
3
2
yyu 

Example 4
40
Calculate the dynamic viscosity of an oil, which is used for
lubrication between a square plate of size 0.8 m x 0.8 m and an
inclined plane with angle of inclination 30o as shown in Fig. 1.4.
The weight of the square plate is 300 N and it slides down the
inclined plane with a uniform velocity of 0.3 m/s. The thickness
of oil film is 1.5 mm.
Fig.1.4

Example 5
42
The space between two square flat parallel plates is filled with
oil. Each side of the plate is 60 cm. The thickness of the oil
film is 12.5 mm. The upper plate, which moves at 2.5 metre per
sec requires a force of 98.1 N to maintain the speed.
Determine : ·
i.the dynamic viscosity of the oil, and
ii.the kinematic viscosity of the oil if the specific gravity of the
oil is 0.95.
Solution. Given:
Each side of a square plate = 60 cm = 0.6 m
Area A= 0.6 x 0.6 = 0.36 m2
Thickness of oil film dy = 12.5 mm = 12.5 x 10-3 m
Velocity of upper plate u = 2.5 m/s

Thermodynamic Properties
 Fluids consist of liquids or gases. But gases are compressible
fluids and hence thermodynamic properties play an important
role.
 With the change of pressure and temperature, the gases undergo
large variation in density.
 The relationship between pressure (absolute), specific volume
and temperature (absolute) of a gas is given by the equation of
state as
44

 The value of gas constant R is
 Isothermal Process. If the changes in density occurs at
constant temperature, then the process is called isothermal
and relationship between pressure (p) and density (ρ) is
given by
 Adiabatic Process. If the change in density occurs with no
heat exchange to and from the gas, the process is called
adiabatic. And if no heat is generated within the gas due to
friction, the relationship between pressure and density is
given by
45
Kkg
J
R
.
287
constant
ρ
p

constant
ρ
p
k


 where k = Ratio of specific heat of a gas at constant
pressure and constant volume.
 k = 1.4 for air
46

Compressibility and Bulk Modulus
 Compressibility is the reciprocal of the bulk modulus of
elasticity, K which is defined as the ratio of compressive
stress to volumetric strain.
 Consider a cylinder fitted with a piston as shown in the Fig.
 Let V= Volume of a gas enclosed in the cylinder
p =Pressure of gas when volume is V
 Let the pressure is increased to p+ dp, the volume of gas
decreases from V to V – dV.
 Then increase in pressure = dp
 Decrease in volume = dV
 Volumetric strain = - dV/V
47

 - ve sign means the volume
decreases with increase of pressure.
 Compressibility is given by = 1/K
48
V
dV
dp


V
dV-
dp
strainVolumetric
pressureofIncrease
KmodulesBulk
Compressibility and Bulk Modulus

Surface Tension and Capillarity
 Surface tension is defined as the tensile force acting on the
surface of a liquid in contact with a gas or on the surface
between two immiscible liquids such that the contact
surface behaves like a membrane under tension.
 Surface tension is created due to the unbalanced cohesive
forces acting on the liquid molecules at the fluid surface.
 Molecules in the interior of the fluid mass are surrounded
by molecules that are attracted to each other equally.
 However, molecules along the surface are subjected to a net
force toward the interior.
 The apparent physical consequence of this unbalanced
force along the surface is to create the hypothetical skin or
membrane.
49

 A tensile force may be
considered to be acting in the
plane of the surface along any
line in the surface.
 The intensity of the molecular
attraction per unit length along
any line in the surface is
called the surface tension.
 It is denoted by Greek letter σ
(called sigma).
 The SI unit is N/m.
50

Surface Tension on liquid Droplet and
Bubble
 Consider a small spherical droplet of a
liquid of radius ‘R'. On the entire
surface of the droplet, the tensile force
due to surface tension will be acting.
 Let σ = surface tension of the liquid
 P= Pressure intensity inside the
droplet (in excess of the outside
pressure intensity)
 R= Radius of droplet.
 Let the droplet is cut into two halves.
The forces acting on one half (say left
half) will be
51

 (i) tensile force due to
surface tension acting
around the circumference
of the cut portion as
shown and this is equal to
= σ x Circumference
= σ x 2πR
 (ii) pressure force on the
area (π/4)d2 and
 = P x πR2 as shown
52

 These two forces will be equal and opposite under
equilibrium conditions, i.e.,
 A hollow bubble like a soap bubble in air has two surfaces
in contact with air, one inside and other outside. Thus two
surfaces are subjected surface tension.
53

Surface Tension……. Example 1
 Find the surface tension in a soap bubble of 40 mm
diameter when the inside pressure is 2.5 N/m2 above
atmospheric pressure.
54

Surface Tension……. Example 2
 The pressure outside the droplet of water of diameter
0.04 mm is 10.32 N/cm2 (atmospheric pressure).
Calculate the pressure within the droplet if surface
tension is given as 0.0725 N/m of water.
55

Capillarity
 Capillarity is defined as a phenomenon of rise or fall of a
liquid surface in a small tube relative to the adjacent general
level of liquid when the tube is held vertically in the liquid.
 The rise of liquid surface is known as capillary rise while
the fall of the liquid surface is known as capillary
depression.
 The attraction (adhesion) between the wall of the tube and
liquid molecules is strong enough to overcome the mutual
attraction (cohesion) of the molecules and pull them up the
wall. Hence, the liquid is said to wet the solid surface.
 It is expressed in terms of cm or mm of liquid. Its value
depends upon the specific weight of the liquid, diameter of
the tube and surface tension of the liquid.
56

Expression for Capillary Rise
 Consider a glass tube of small
diameter ‘d’ opened at both ends
and is inserted in a liquid, say water.
 The liquid will rise in the tube
above the level of the liquid.
 Let h = the height of the liquid in
the tube . Under a state of
equilibrium, the weight of the liquid
of height h is balanced by the force
at the surface of the liquid in the
tube. But the force at the surface of
the liquid in the tube is due to
surface tension.57

Expression for Capillary Rise…
 Let σ = Surface tension of liquid
θ = Angle of contact between the liquid and glass tube
 The weight of the liquid of height h in the tube
= (Area of the tube x h) x ρ x g
58

Expression for Capillary Rise…
 The value of θ between water and clean glass tube is
approximately equal to zero and hence cos θ is equal to
unity. Then rise of water is given by
 Contact angle depends on both the liquid and the solid.
 If θ is less than 90o, the liquid is said to "wet" the solid.
However, if θ is greater than 90o, the liquid is repelled by
the solid, and tries not to "wet" it.
 For example, water wets glass, but not wax. Mercury on the
other hand does not wet glass.
59

Capillarity
Expression for Capillary Fall
 lf the glass tube is dipped in mercury, the revel of
mercury in the tube will be lower than the general level
of the outside liquid as shown above.
60

Capillarity
 Let h = Height of depression in
tube.
 Then in equilibrium, two forces
arc acting on the mercury inside
the tube.
 First one is due to surface tension
acting in the downward direction
and is equal to σ x πd x cos θ.
 Second force is due to hydrostatic
force acting upward and is equal
to intensity of pressure at a depth
'h' x Area
61

62
Capillarity
Value of θ for mercury and glass tube is 128o

Capillarity…Example 1
 Calculate the capillary rise in a glass tube of 2.5 mm
diameter when immersed vertically in (a) water and (b)
mercury. Take surface tensions σ = 0.0725 N/m for
water and σ = 0.52 N/m for mercury in contact with
air. The specific gravity for mercury is given as 13.6
and angle of contact = 130o.
63

 Find out the minimum size of glass tube that can be used to
measure water level if the capillary rise in the tube is to be
restricted to 2 mm. Consider surface tension of water in
contact with air as 0.073575 N/m.
65
Capillarity…Example 2

Dimensions and Units
 Fluid mechanics deals with the measurement of many
variables of many different types of units. Hence we
need to be very careful to be consistent.
Dimensions and Base Units
 The dimension of a measure is independent of any
particular system of units. For example, velocity may
be in metres per second or miles per hour, but
dimensionally, it is always length per time, or L/T =
LT−1 .
 The dimensions of the relevant base units of the
Système International (SI) system are:
66

67
Dimensions and Units
Derived Units

Unit Table
Quantity SI Unit English Unit
Length (L) Meter (m) Foot (ft)
Mass (m) Kilogram (kg) Slug (slug) =
lb*sec2/ft
Time (T) Second (s) Second (sec)
Temperature ( ) Celcius (oC) Farenheit (oF)
Force Newton
(N)=kg*m/s2
Pound (lb)

69

Dimensions and Units…
 1 Newton – Force required to accelerate a 1 kg of mass
to 1 m/s2
 1 slug – is the mass that accelerates at 1 ft/s2 when acted
upon by a force of 1 lb
 To remember units of a Newton use F=ma (Newton’s 2nd
Law)
 [F] = [m][a]= kg*m/s2 = N
 To remember units of a slug also use F=ma => m = F / a
 [m] = [F] / [a] = lb / (ft / sec2) = lb*sec2 / ft
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Chapter 1. introduction to fluid mechanics

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Chapter 1. introduction to fluid mechanics