fluid mechanics for engineering student

1. UNIVERSITY OF ENERGY AND NATURAL RESOURCES
(UENR)
INTRODUCTION TO FLUID MECHANICS
MECH 271
LECTURE 1 (UNIT 1)
adupoku2009@yahoo.com
kofi.adu-poku@uenr.edu.gh

2. AIM OF COURSE:
To offer basic knowledge in
fluid mechanics
 To obtain an understanding for
the behaviour of fluids
To solve some simple problems
of the type encountered in
Engineering practice

3. AIM OF FLUID MECHANICS
LECTURES
It is the aim of these lectures to help
students in this process of gaining an
understanding of, and an appreciation
for fluid motion—what can be done
with it, what it might do to you, how to
analyze and predict it.

4. OBJECTIVE OF COURSE
At the end of the course, participants are
expected to be able to:
 Define and use basic fluid properties
 Define and use basic concepts in fluid
mechanics
 Perform simple calculations in hydrostatics
and kinematics
 Make simple designs in hydraulics

5. METHODS TO BE USED
 Lectures
 Workshops (tutorials)
 Laboratory works
Assessment methods
 Class assignments,
 Home assignments
 Laboratory reports
 Examination

6. LITERATURE
➢ Fluid Mechanics (including Hydraulic Machines) – Dr. A. K. Jain,
Khanna Publishers, Delhi, 2003
➢ Fluid Mechanics (6th edition) – Frank M. White; McGraw-Hill 2008
➢ Introduction to Engineering Fluid Mechanics.- J. A. Fox 1985
➢ Fluid Mechanics:- J. F. Douglas; J. M. Gasiorek; J. A. Swaffield
➢ Hydraulics,Fluid Mechanics and Fluid Machines – S. Ramamrutham
➢ Essentials of Engineering Hydraulics – J. M. K. Dake, 1992
➢ Hydrology and Hydraulic Systems – Ram S. Gupta
➢ Mechanics of Fluids – Bernard Massey, revised by john Ward- Smith

7. FLUIDS
 In everyday life, we recognize three states of matter: solid,
liquid and gas. Although different in many respects, liquids
and gases have a common characteristic in which they differ
from solids:
 They are fluids, lacking the ability of solids to offer
permanent resistance to a deforming force.
 Fluids flow under the action of such forces, deforming
continuously for as long as the force is applied.

8.  A fluid is unable to retain any unsupported shape; it flows
under its own weight and takes the shape of any solid body with
which it comes into contact.
 Deformation is caused by shearing forces, i.e. forces such as F as
shown in Fig. 1.1, which act tangentially to the surfaces to which
they are applied and cause the material originally occupying the
space ABCD to deform to AB1C1D.

9. A fluid may be defined in two perspectives:
➢ The form in which it occurs naturally:
▪ A substance that is capable of flowing and has no
definite shape but rather assumes the shape of the
container in which it is placed.
➢ By the deformation characteristics when acted
upon by a shear stress:

10. ▪ A fluid is a substance that deforms continuously under the
action of a shearing stress no matter how small the stress.
➢ Examples of fluid are:
➢ Gases (air, Lpg)
➢ Liquids (water, kerosene, etc).
Conversely, it follows that:
 If a fluid is at rest, there can be no shearing forces acting on
it.
 Therefore, all forces in the fluid must be perpendicular to
the planes upon which they act.

11. SHEAR STRESS IN A MOVING FLUID
 Although there can be no shear stress in a fluid at rest, shear
stresses are developed when the fluid is in motion if the particles
of the fluid move relative to each other so that they have different
velocities, causing the original shape of the fluid to become
distorted.
 If, on the other hand, the velocity of the fluid is the same at every
point, no shear stresses will be produced, since the fluid particles
are at rest relative to each other.
 Usually, we are concerned with flow past a solid boundary. The
fluid in contact with the boundary adheres to it and will,
therefore, have the same velocity as the boundary.

12.  Considering successive layers parallel to the boundary (Fig.
1.2), the velocity of the fluid will vary from layer to layer as
y increases.
Fig 1.2: Variation of velocity with distance from a solid boundary

13.  If ABCD (Fig. 1.1) represents an element in a fluid with
thickness s perpendicular to the diagram, then the force F will
act over an area A equal to BC×s.
 The force per unit area F/A is the shear stress τ and the
deformation, measured by the angle φ (the shear strain), will be
proportional to the shear stress.
 In a solid, Φ will be a fixed quantity for a given value of τ, since
a solid can resist shear stress permanently.
 In a fluid, the shear strain φ will continue to increase with time
and the fluid will flow.

14.  It is found experimentally that, in a true fluid, the rate of
shear strain (or shear strain per unit time) is directly
proportional to the shear stress.
 Suppose that in time t a particle at E (Fig. 1.1) moves
through a distance x. If E is a distance y from AD then, for
small angles:
 Shear strain, φ =x/y
 Rate of shear strain =x/yt=(x/t)/y=u/y
 Where u =x/t is the velocity of the particle at E.

15.  Assuming the experimental results that shear stress is
proportional to shear strain, then
 Τ = constant × u/y
 The term u/y is the change of velocity with y and may be written
in the differential form du/dy.
 The constant of proportionality is known as the dynamic viscosity
μ of the fluid.
 Substituting into equation (1.1),
 τ = μ
d𝑢
d𝑦
 which is Newton’s law of viscosity. The value of μ depends
upon the fluid under consideration.

16. NEWTONIAN AND NON-NEWTONIAN
FLUIDS
 Even among substances commonly accepted as fluids, there
is a wide variation in behaviour under stress.
 Fluids obeying Newton’s law of viscosity (equation (1.2))
and for which μ has a constant value are known as
Newtonian fluids.
 Most common fluids fall into this category, for which shear
stress is linearly related to velocity gradient (Fig. 1.3).
 Fluids which do not obey Newton’s law of viscosity are
known as non- Newtonian and fall into one of the
following groups:

17. FIGURE 1.3: Variation of shear stress with velocity gradient

18.  Plastic for which the shear stress must reach a certain
minimum value before flow commences. Thereafter, shear
stress increases with the rate of shear.
 Pseudo-plastic, for which dynamic viscosity decreases as the
rate of shear increases (e.g. colloidal solutions, clay, milk,
cement).
 Dilatant substances, in which dynamic viscosity increases as
the rate of shear increases (e.g. quicksand).

19. ➢ Classify the substance that has the following rates of
deformation and corresponding shear stresses as shown in
Table 1 and 2 below.
𝑑𝑢
𝑑𝑦
,
𝑟𝑎𝑑
𝑠𝑒𝑐
… …. .
0 0.5 0.7 0.9 1.1 1.3
𝜏, 𝑙𝑏/𝑓𝑡2........ 0 1 3 5 7 9

20. DISTINCTION BETWEEN SOLID AND
FLUID
➢ There are plastic solids which flow under the proper
circumstances and even metals may flow under high
pressures.
➢ On the other hand there are viscous fluids which do not
flow readily and one may easily confuse them with solid
plastics.
➢ The distinction is that any fluid, no matter how viscous will
yield in time to the slightest shear stress.

21. SOLID AND FLUID
➢ But a solid, no matter how plastic, requires a certain
limiting value of stress to be exerted before it will flow.
➢ Also when the shape of a solid is altered (without
exceeding the plastic limit) by external forces, the
tangential stresses between adjacent particles tend to
restore the body to its original shape.
➢ With a fluid, these tangential stresses depend on the
velocity of deformation and vanish as the velocity
approaches zero

22. LIQUID AND GAS
➢ A liquid is composed of relatively closed packed molecules
with strong cohesive forces.
➢ Liquids are relatively incompressible.
➢ As a result, a given mass of fluid will occupy a definite
volume of space if it is not subjected to extensive external
pressures.

23. GAS
➢ Gas molecules are widely spaced with relatively small
cohesive forces.
➢ Therefore if a gas is placed into a container and all external
pressure removed, it will expand until it fills the entire
volume of the container.
➢ Gases are readily compressible. A gas is in equilibrium only
when it is completely enclosed. The volume (or density) of
a gas is greatly affected by changes in pressure or
temperature or both.
➢ It is therefore necessary to take account of changes of
pressure and temperature whenever dealing with gases.

24. FLUID MECHANICS
➢ Fluid mechanics is the science of the mechanics of liquids
and gases and is based on the same fundamental principles
that are employed in solid mechanics.
➢ It studies the behaviour of fluids at rest and in motion.
➢ The study takes into account the various properties of the
fluid and their effects on the resulting flow patterns in
addition to the forces within the fluid and forces interacting
between the fluid and its boundaries

25. FLUID MECHANICS
 The study also includes the mathematical application of
some fundamental laws :
 Conservation of mass - energy
 Newton’s law of motion (force - momentum equation ),
laws of thermodynamics
 Together with other concepts and equations to explain
observed facts and to predict as yet unobserved facts and to
predict as yet unobserved fluid behaviour.

26. FLUID MECHANICS
The study of fluid mechanics subdivides into:
 Fluid statistics
 Fluid kinematics
 Fluid dynamics

27. FLUID STATICS
➢ Fluid statics is the study of the behaviour of fluids
at rest.
➢ Since for a fluid at rest there can be no shearing
forces all forces considered in fluid statics are
normal forces to the planes on which they act.

28. FLUID KINEMATICS/DYNAMICS
➢ Fluid kinematics deals with the geometry (streamlines and
velocities ) of motion without consideration of the forces
causing the motion.
➢ Kinematics is concerned with a description of how fluid
bodies move.
➢ Meanwhile, fluid dynamics is concerned with the relations
between velocities and accelerations and the forces causing
the motion.

29. SYSTEM AND CONTROL VOLUME
In the study of fluid mechanics, we make use of the basic laws
in physics namely:
 The conservation of matter (which is called the continuity
equation)
 Newton’s second law (momentum equation)
 Conservation of energy (1st law of thermodynamics)
 Second law of thermodynamics and
 Numerous subsidiary laws

30. SYSTEM AND CONTROL VOLUME
In employing the basic and subsidiary laws, either one of the
following models of application is adopted:
 The activities of each and every given mass must be such as
to satisfy the basic laws and the pertinent subsidiary laws –
SYSTEM
➢ The activities of each and every volume in space must be
such that the basic and the pertinent subsidiary laws are
satisfied – CONTROL VOLUME

31. SYSTEM & CONTROL VOLUME
➢ A system is a predetermined identifiable quantity of fluid.
It could be a particle or a collection of particles.
➢ A system may change shape, position and thermal
conditions but must always contain the same matter.
➢ A control volume refers to a definite volume designated in
space usually with fixed shape.
➢ The boundary of this volume is known as the control
surface.
➢ A control volume mode is useful in the analysis of
situations where flow occurs into and out of a space

32. FORCES ACTING ON FLUIDS (BODY &
SURFACE FORCES)
➢ Those forces on a body whose distributions act on matter
without the requirement of direct contact are called body
forces (e.g. gravity, magnetic, inertia, etc.
➢ Body forces are given on the basis of the force per unit mass
of the material acted on.
➢ Those forces on a body that arise from direct contact of this
body with other surrounding media are called surface
forces e.g. pressure force, frictional force, surface tension

33. FLUID PROPERTIES
 Property is a characteristic of a substance which is invariant
when the substance is in a particular state.
 In each state the condition of the substance is unique and is
described by its properties.
 The properties of a fluid system uniquely determine the
state of the system.

34. PHYSICAL PROPERTIES OF FLUIDS
➢ Each fluid property is important in a particular field of
application.
➢ Viscosity plays an important role in the problems of
hydraulic friction.
➢ Mass density is important in uniform flow.
➢ Compressibility is a factor in water hammer.
➢ Vapour pressure is a factor in high velocity flow

35. DENSITY
 The density of a substance is that quantity of matter
contained in unit volume of the substance.
 They are important parameters that tend to indicate
heaviness of a substance
 It can be expressed in different ways, which must be clearly
distinguished.
➢ Mass density
➢ Specific weight
➢ Relative density
➢ Specific volume

36.  Mass density is the mass per unit volume usually denoted
by the Greek letter “rho”
ρ=M/V kg/m3
At standard pressure (760 mmHg) and 4o C density of water
= 1000 kg/mm3
➢ Specific volume : Is the reciprocal of the density ie. the
volume occupied per unit mass of fluid.
Vs = 1/ρ = V/M ( m3 / kg)
➢ Specific (unit ) weight: ()- Is the weight per unit volume of
the substance (is and indication of how much a unit volume
of a substance weighs.) = W/V = Mg/V =ρg ( kgm/s2)

37.  Relative density (or specific gravity) σ is defined as the
ratio of the mass density of a substance to some standard
mass density.
 For solids and liquids, the standard mass density chosen is
the maximum density of water (which occurs at 4 °C at
atmospheric pressure): σ = ρsubstanceρH2O at 4 °C.
 For gases, the standard density may be that of air or of
hydrogen at a specified temperature and pressure, but the
term is not used frequently.
 Since relative density is a ratio of two quantities of the
same kind, it is a pure number having no units. Eg. RDH2O 1.0;
oil, 0.9.

41. VISCOSITY
• When two solid bodies in contact move relative to each other, a
friction force develops at the contact surface in the direction
opposite to motion.
• To move a table on the floor, for example, we have to apply a
force to the table in the horizontal direction large enough to
overcome the friction force.
• The magnitude of the force needed to move the table depends
on the friction coefficient between the table and the floor.
• The situation is similar when a fluid moves relative to a solid or
when two fluids move relative to each other.

42. • We move with relative ease in air, but not so in water. Moving
in oil would be even more difficult, as can be observed by the
slower downward motion of a glass ball dropped in a tube
filled with oil.
• It appears that there is a property that represents the internal
resistance of a fluid to motion or the “fluidity,” and that
property is the viscosity.
• Viscosity is a measure of the “stickiness” or “resistance to
deformation” of a fluid.
• It is due to the internal frictional force that develops between
different layers of fluids as they are forced to move relative to
each other.

43. • In general, liquids have higher dynamic viscosities than gases.
• The ratio of viscosity μ to density ρ often appears in the
equations of fluid mechanics, and is defined as the kinematic
viscosity, ν = μ /ρ.
• The force a flowing fluid exerts on a body in the flow direction is
called the drag force and the magnitude of this force depends, in
part, on viscosity.
• To obtain a relation for viscosity, consider a fluid layer.

44. • Consider a fluid layer of thickness ! within a small gap between
two concentric cylinders, such as the thin layer of oil in a
journal bearing.
• The gap between the cylinders can be modeled as two parallel
flat plates separated by a fluid.
• Noting that torque is T=FR (force times the moment arm,
which is the radius R of the inner cylinder in this case), the
tangential velocity is V=wR (angular velocity times the radius),
• Taking the wetted surface area of the inner cylinder to be
A = 2πRL by disregarding the shear stress acting on the two
ends of the inner cylinder, torque can be expressed as:

45. • where L is the length of the cylinder and n is the number of
revolutions per unit time, which is usually expressed in rpm.
• Note that the angular distance traveled during one rotation is
2πrad, and thus the relation between the angular velocity in
rad/min and the rpm is w=2pn
• The Equation above can be used to calculate the viscosity of a
fluid by measuring torque at a specified angular velocity. The
viscosities of some fluids at room temperature are listed in table
below.
• Note that it is more difficult to move an object in a higher-
viscosity fluid such as engine oil than it is in a lower-viscosity
fluid such as water.

48. DYNAMIC & KINEMATIC VISCOSITY
 The coefficient of dynamic viscosity μ can be defined as the shear force
per unit area (or shear stress τ) required to drag one layer of fluid with
unit velocity past another layer a unit distance away from it in the
fluid.
 The constant of proportionality, μ, in the above equation is called the
dynamic viscosity with units Ns./m2
 Units: Newton seconds per square metre (N s m−2) or kilograms per
metre per second (kg m−1s−1).
 (But note that the coefficient of viscosity is often measured in poise (P);
10 P = 1 kg m−1s−1)
 Typical values: water, 1.14 × 10−3 kg m−1 s−1; air, 1.78 × 10−5 kg m−1 s−1

49. ➢ Not all fluids show exactly the same relation between stress and
the rate of deformation.
➢ Newtonian fluids: are fluids for which shear stress is directly
proportional to the rate angular deformation or a fluid for which
the viscosity  is a constant for a fixed temperature and
pressure. eg. Air, water, etc. Petroleum, kerosene, steam.
➢ Non-Newtonian fluids : are fluids which have a variable
proportionality (viscosity  ) between stress and deformation
rate.
➢ In such cases, the proportionality may depend on the length of
time of exposure to stress as well as the magnitude of the stress
eg. Plastics, paint, blood, ink, etc

50. A 50-cm " 30-cm " 20-cm block weighing 150 N is to be moved at a
constant velocity of 0.8 m/s on an inclined surface with a friction
coefficient of 0.27.
Determine
(a) The force F that needs to be applied in the horizontal direction.
(b) If a 0.4-mm-thick oil film with a dynamic viscosity of 0.012 Pa.s
is applied between the block and inclined surface, determine the
percent reduction in the required force.

67. COMPRESSIBLE AND INCOMPRESSIBLE
FLUIDS
➢ Compressible fluids are fluids whose specific volume v or
(density, ρ) is a function of pressure.
➢ An incompressible fluid is a fluid whose density is not
changed by external forces acting on the fluid.
➢ Hydrodynamics is the study of the behaviour of
incompressible fluids whereas gas dynamics is the study of
compressible fluid.

68. COMPRESSIBILITY OF FLUID
 Compressibility of a fluid is a measure of the change in
volume of the fluid when it is subjected to outside force.
 It is defined in terms of an average bulk modulus of
elasticity K.
V
V
p
K


=