The document discusses the principles of fluid mechanics, focusing on the concepts of compressible and incompressible fluids, as well as the definition and significance of viscosity. It explains how liquids are generally treated as incompressible due to negligible density changes, while gases can be either compressible or incompressible depending on velocity and pressure variations. Finally, it introduces Newton's equation of viscosity, which establishes the relationship between shear stress and velocity gradient in fluids, highlighting the differences between Newtonian and non-Newtonian fluids.

1. IntroductionIntroductionIntroductionIntroduction

2. Compressible and Incompressible FluidsCompressible 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.

3. Compressible and Incompressible FluidsCompressible and Incompressible Fluids
• 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, weIn WATER HAMMER problems, we
must consider the compressibility ofmust consider the compressibility of
fluids.fluids.

4. Compressible and Incompressible FluidsCompressible and Incompressible Fluids
• Flow of air in a ventilating system:
• Gas or steam flowing at high velocity through long pipeline:
Gas is treated as
Incompressible
Because, P variation is so small that the change in density is of no
importance.
P variation is great that change in density cannot be ignored.

5. Compressible and Incompressible FluidsCompressible and Incompressible Fluids
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..
An airplane flying
below 250 mph ,
density of air may be
considered as
constant.

6. Compressibility of LiquidsCompressibility 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

7. Compressibility of LiquidsCompressibility of Liquids
• 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.

8. Compressibility of LiquidsCompressibility of Liquids
• 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.

9. IDEAL FLUIDIDEAL 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.

10. ViscosityViscosity
• 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.

11. ViscosityViscosity
• 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

12. ViscosityViscosity
• In Liquids: T inversely related to Viscosity
• In Gases: T directly related to Viscosity
Viscosit
y
Temperatur
e
Liquids Gases

13. ViscosityViscosity
NEWTON’S EQUATION OF VISCOSITY:
• τ = F/A = µ(U/Y) = µ (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.

14. Newton’s Equation of ViscosityNewton’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.

15. Newton’s Equation of ViscosityNewton’s Equation of Viscosity
Y
U
y
dy
du
u
F

16. Newton’s Equation of ViscosityNewton’s Equation of Viscosity
• 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.

17. Newton’s Equation of ViscosityNewton’s Equation of Viscosity
• 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 τ between
any two thin sheets of fluid may be expressed by:
τ = F/A = µ(U/Y) = µ (dU/dY)

18. ViscosityViscosity
From Newton’s equation of viscosity we have,
• µ = τ / (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:

19. ViscosityViscosity
Elasticsolid
Ideal
plastic
N
on
N
ew
tonian
fluid
Newtonian fluid
Ideal fluid
(dU/dY)
τ=Shearstress

20. ViscosityViscosity
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.

21. ViscosityViscosity
• 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.

22. ViscosityViscosity
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.
In BG System:
Dimensions of µ = (lb/ft2
)/(fps/ft) = lb.sec/ft2
In S.I System:
Dimensions of µ = (N/m2
)/(s-1
) = N.s/m2

23. ViscosityViscosity
Kinematic Viscosity = Absolute Viscosity / Density
ν = µ / ƿ
Is called so because force is not involved, the only dimensions being length
and time, as in Kinematics.
UNITS :
In BG: ft2
/sec
In S.I: m2
/s
In Metric system it had units cm2
/s, also
known as STOKE(St).
Name given after Sir George Stoke, an English
Physicist and pioneering investigator of viscosity.
1 cSt = 0.01 St = 10-6
m2
/s

24. ViscosityViscosity
DISTINCTION BETWEEN µ & ν :
• µ 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.

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