2. introduction

INTRODUCTION (Contd…)
Lecture # 02

CONTENTS OF TODAY’S LECTURE:
 Compressibility
 Viscosity
 Newton’s equation of viscosity
 Units of Viscosity
FLUID MECHANICS-I
CE-224

Engr. Fazal-E-Jalal
Prepared by: Engr. Fazal-E-Jalal Fluid Mechanics-I 1

Compressible and Incompressible Fluids

• Fluid mechanics deals with both compressible
and in compressible fluids, that is with liquids
and gases, of either constant or variable
density.
• No such thing in reality as “Incompressible
fluid”, the term is used when the change in
density with pressure is NEGLIGIBLE.
AND THIS IS THE CASE WITH “LIQUIDS”. We may also consider
GASES as incompressible when P variation is small compared with
absolute pressure.


• Evidence of Elasticity of fluids is that sound
waves (which really are pressure waves) travel
through liquids. Ordinarily liquids are
considered to be incompressible fluids.
In WATER HAMMER problems, we must consider the
compressibility of fluids.



• Flow of air in a ventilating system:
Gas is treated as Incompressible

Because, P variation is so small that the change in density is of no importance.

• Gas or steam flowing at high velocity through
long pipeline:
P variation is great that change in density cannot be ignored.



High up in air!!!

An airplane flying below
250 mph , density of air
may be considered as
constant.

But an object moving at 760 mph
(approaching velocity of sound), then P &
Density adjacent to body is different from
distant air. TREAT AIR AS COMPRESSIBLE
FLUID..

Compressibility of Liquids
• Compressibility is the change in volume due to
change in pressure.
• The compressibility of liquid is inversely
related to its volume modulus of elasticity
(also known as bulk modulus).
• Eν = - ν(dp/dν) = - (ν/dν)dp
Where; ν = Specific Volume.
(ν/dν) = Dimensionless ratio


• In most engineering problems, the bulk
modulus at or near atmospheric pressure is
one of the interest.
The BULK MODULUS is a property of fluid.
And for liquids, is a function of temperature and pressure.

Eν is directly related to temperature. It
is maximum at 50 ͦC. Thus water has
minimum compressibility at this
temperature.


• We often specify applied pressures in terms of
absolute terms, because atmospheric pressure
varies.
• Absolute pressure is the actual pressure on
fluid relative to absolute zero.
• The standard atmospheric pressure at sea
level is about 14.7 psia or 101.3 kn/m2 abs or
1013 mb.


• Bars and milli bars were previously used in
metric systems to express pressure.
• 1 mb = 100 N/m2
• Most pressures are measured relative to the
atmosphere and are known as GAUGE
PRESSURES.


Eν =  ν/ ν = p/ Eν
Eν = Mean value of
At any modulus for
temperature, pressure range
Volume
the bulk modulus of mild
modulus of steel is about (1,2 refer to before
water does 26,000,000 psi. and after
And that of cold condition)
not vary water is
great deal 320,000 psi.
for moderate WESEE THAT
WATER IS
range in
ABOUT 80
pressure.
TIMES AS
COMPRESSIBLE
AS STEEL.
Mercury is Compressibility
approximately of Nitric acid is
8% as 6
nearly times
compressible greater than
as water. that of water.

• Assuming Eν to have value of 32,000 psi, we
see the increasing pressure of water by 1000
psi will compress it only 0.3% of it’s original
volume.
Therefore, we find that the usual assumption
regarding water as being INCOMPRESSIBLE is
justified.
Sample problem 2.3
Page 18
Chapter. 2 Properties of
Prepared by: Engr. Fazal-E-Jalal Fluid Mechanics-I
fluids) 11

Practice Problems
• 2.5.1
• 2.5.2 Fluid Mechanics
• 2.5.3 with Engineering
• 2.5.4
• 2.5.5
Applications


IDEAL FLUID
• The fluid in which there is no friction; it is
INVISCID (it’s viscosity is zero).
• The internal forces at any section within it are
always normal to the section, even during
motion.
• So, these forces are purely pressure forces.
This does not exist in reality, many fluids approximate frictionless flow at
sufficient distances from solid boundaries and hence we can analyze their
behavior by assuming an ideal fluid.


Viscosity
• In real fluids, either liquid or gas, tangential or
shearing forces are developed always
whenever there is motion relative to a
body, thus creating fluid friction, because
these forces oppose the motion of one
particle past another.
• These frictional forces give rise to a fluid
property called Viscosity.


Viscosity
• The viscosity of a fluid is a measure of its
resistance to shear or angular deformation.
• The friction forces in flowing fluid result from
cohesion and momentum interchange
between molecules.
For Example Motor oil has high viscosity, and
resistance to shear, and feels “sticky” whereas
gasoline has low viscosity.


Viscosity
• In Liquids: T inversely related to Viscosity
• In Gases: T directly related to Viscosity

Liquids Gases
Viscosity

Temperature


Viscosity
NEWTON’S EQUATION OF VISCOSITY:
•  = F/A = (U/Y) =  (dU/dY)
• This was first suggested by Sir Isaac Newton
(1642-1727) first suggested it.
He is better known for his formulation of the fundamental
laws of motion and gravity and for the development of
differential calculus, NEWTON, an English mathematician
and natural philosopher, also made many pioneering
studies in FLUID MECHANICS.


Newton’s Equation of Viscosity
• Consider two plates, sufficiently large so that
end conditions may be neglected, placed on
small distance Y apart, the space between
being filled with the fluid.
• The lower surface is assumed to be
stationary, while the upper one is moved
parallel to it with a velocity U by the
application of force F corresponding to some
area A of the moving plate.


..
U
F
dy
Y y u du


• Particles in the fluid in contact with each plate
will adhere to it.
• And, if Y is not too great or the velocity U too
high, the velocity gradient will be a straight
line.
• The action is much as if the fluid were made
up of a series of thin sheets, each of which
would slip a little relative to the next.


• Experiment has shown that for a large class of
fluids:
F α (A.U)/Y
• It may be seen from similar triangles in figure that
U/Y can be replaced by the velocity gradient
du/dy.
• If a constant of proportionality μ is now
introduced, the shear stress  between any two
thin sheets of fluid may be expressed by:
 = F/A = (U/Y) =  (dU/dY)


Viscosity
From Newton’s equation of viscosity we have,
•  =  / (dU/dY)
• This is known as Coefficient of viscosity, the
absolute viscosity, the dynamic viscosity or
simply the viscosity of fluid.
The distinction between solids and fluid lies in the
manner in which each can resist SHEARING STRESS.
Further distinction among various kinds of fluids and
solids is as:

Viscosity
… Elastic solid
 = Shear stress

Ideal fluid

(dU/dY)


Viscosity
• In case of solids, shear stress depends on
magnitude of deformation but according to
Newton’s equation of viscosity the shear
stress is proportional to time rate of (angular)
deformation.
A fluid for which absolute viscosity does not change with rate of
deformation is called NEWTONIAN FLUID.
The slope of this line is “Absolute Viscosity”

A fluid for which absolute viscosity changes with rate of deformation is
called NON-NEWTONIAN FLUID.


Viscosity
• Non Newtonian fluids are relatively
uncommon in engineering use (examples are
paints, printer’s ink, gels and emulsions,
sludges and slurries, and certain plastics).
• So, we will use fluids that obey Newton’s
equation of viscosity under normal conditions.


Viscosity
In BG System:
• Dimensions of  = (lb/ft2)/(fps/ft) = lb.sec/ft2
In S.I System:
• Dimensions of  = (N/m2)/(s-1) = N.s/m2
Poise (P) is a widely used unit for viscosity in Metric system.
•IT IS NAMED AFTER JEAN LOUIS POISEILLE (1799-1869).
•HE WAS ONE OF THE FIRST INVESTIGATORS OF VISCOSITY.
1 poise = 0.10 N.s/m2
1 Cp = 0.01 poise = 1 Mn.s/m2 (Frequently a more Convenient unit)
•VISCOSITY OF WATER AT 68.4 ͦF IS 1 Cp. So viscosity of fluid in cPs is
viscosity of fluid relative to that of water at 68.4 ͦF.


Viscosity
Kinematic Viscosity = Absolute Viscosity / Density

•ν =  / ƿ
• Is called so because force is not involved, the only
dimensions being length and time, as in Kinematics.
UNITS:
2/sec
In Metric system it had units
In BG: ft cm2/s, also known as STOKE(St).
In S.I: m2/s Name given after Sir George Stoke, an English
Physicist and pioneering investigator of viscosity.
1 cSt = 0.01 St = 10-6 m2/s


Viscosity
DISTINCTION BETWEEN  & ν :
•  of most fluids is virtually INDEPENDENT of
pressures encountered ordinarily in
engineering work.
• ν of gases varies strongly with pressure
because of change in density.
Sample problem 2.8
Page 34
Chapter. 2 (Properties of
fluids)


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