MECHANICAL , MAINTENANCE AND CIVIL ENGINEERING DEPARTMENTS
FLUID MECHANICS 1-MENG 225
Time: Mondays-11:00 am -1:00 pm
Tuesdays - 2:00 pm – 3:00 pm
Lecturer/s: Ing. Sahr Tamba Nyalloma
Ing. Martin Sankoh
Texts/ Learning resources:
 Fluid mechanics by John F. Douglas, Janusz M. Gasiorek, John A. Swaffield.
 Solutions to problems in Fluid mechanics 1 by John F. Douglas
 Fluid mechanics by John Cimbala and Yunus A. Cengel
 A textbook of Fluid Mechanics and hydraulics by R.J.Rajput
FLUID MECHANICS I-MENG 225/FBC/USL/STN
 Assessment
• Attendance -5%
• Tests assignments, quizzes, and presentations -25%
• End of semester examination-70%
COURSE OUTLINE-SECOND SEMESTER 2023/2024
 INTRODUCTIONS TO FLUID MOTION-HYDRODYNAMICS
 FLUID KINEMATICS
o Introduction
o Description of fluid motion
o Types of fluid flow
o Types of flow lines
o Rate of fluid flow-discharge
 FLUID DYNAMICS-KINETICS
 Introduction
 Different types of heads (energies) of fluids in motion
 Bernoulli’s equation
 Euler’s equations
 Application of Bernoulli’s equations to flowmeters (venturimeters and pitot-tubes)
 LIQUID IN RELATIVE EQUILIBRIUM
 IMPULSE-MOMENTUM EQUATION
FLUID MECHANICS I-MENG 225
LECTURE 1
INTRODUCTION TO HYDRODYNAMICS
INTRODUCTION TO HYDRODYNAMICS
Introdction
 Hydrodynamics is a branch of Fluid Mechanics that studies how
energy and forces interact with fluids, including gases and liquids..
 Hydrodynamics is the study of liquids in motion. Specifically, it looks at
the ways different forces affect the movement of liquids.
 Aerodynamics is a further subset of hydrodynamics that specifically
examines gases in motion, while
 The aspect of hydrodynamic that focuses on the spacial condition
and rate of change of such positions without consideration for the
associated forces and energies of the fluid is referred to Fluid
kinematic.
FLUID MECHANICS I-MENG 225/FBC/USL/STN
INTRODUCTION TO HYDRODYNAMICS
 Fluid kinematics deals basically with the mathematical description
or specification of a flow field, divorced from any account of the
forces and energies/conditions that might actually create such a
flow
 Fluid dynamics/kinetics deals basically with the forces,
energies/conditions that create/are associated with flow
fields/regimes.
FLUID MECHANICS I-MENG 225/FBC/USL/STN
INTRODUCTION TO HYDRODYNAMICS
FLUID KINEMATICS
Fluid kinematics can be defined again as follows;
This is a branch of Fluid mechanics that is concerned with the study of the
distribution of the particles of fluids in space, their change, and rate of
change with time ;as in velocity and acceleration with out recourse for
any force or energy.
 The motion of fluids can be fully described by an expression describing
the location of a fluid particle in space at any instant in time. This makes
for the computation of the magnitude and directions of velocity and
acceleration in a flow field.
FLUID MECHANICS I-MENG 225/FBC/USL/STN
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DESCRIPTION OF FLUID MOTION
Two methods have been put forward for the description of the motion of
fluid particles. These methods are;
 The Lagrangian method
 The Eulerian method
The Lagrangian method.
In this method the observer focuses on the movement of a single particle.
The path traced by the particle and the variations in its velocity and
acceleration are studied
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INTRODUCTION TO HYDRODYNAMICS
 A Cartesian frame of reference is used to establish the position of a
fluid particle in space at any instant in time.
𝑧 − 𝑎𝑥𝑖𝑠
𝑧
𝑥, 𝑦, 𝑧
𝑦
𝑥
𝑥 − 𝑎𝑥𝑖𝑠
𝑦 − 𝑎𝑥𝑖𝑠
 Other coordinate systems
include
 Polar coordinate systems
 Cylindrical coordinate
system
 Spherical coordinate system.
𝑥, 𝑦
Fig 1: The Cartesian frame of reference
In the cartesisan coordinate system, the position of a particle in space
(𝒙, 𝒚, 𝒛) at any instant in time 𝒕 from a position (𝒂, 𝒃, 𝒄) is given by
FLUID MECHANICS I-MENG 225/FBC/USL/STN
INTRODUCTION TO HYDRODYNAMICS
𝑥 = 𝑎, 𝑏, 𝑐, 𝑡
𝑦 = 𝑎, 𝑏, 𝑐, 𝑡
𝑧 = 𝑎, 𝑏, 𝑐, 𝑡
If we denote the velocity components of the fluid particle at coordinate
𝑥, 𝑦, 𝑧 as 𝑢, 𝑣, 𝑤 respectively, we can now determine the velocity and
acceleration components in the three orthogonal directions by obtaining the
derivatives of the coordinates with respect to time ;
The velocity components will be given by;
𝜕𝑥
𝜕𝑦
𝜕𝑧
𝑢=
,
𝑣=
,
𝑤=
𝜕𝑡
𝜕𝑡
𝜕𝑡
The resultant velocity shall be the resultant of the three components of
velocity.
FLUID MECHANICS I-MENG 225/FBC/USL/STN
INTRODUCTION TO HYDRODYNAMICS
𝑽=
𝒖𝟐 + 𝒗𝟐 + 𝒘𝟐
The acceleration components will be given by;
𝜕2𝑥
𝜕2𝑦
𝜕2𝑧
𝑎𝑥 = 2 ,
𝑎𝑦 = 2 ,
𝑎𝑧 = 2
𝜕𝑡
𝜕𝑡
𝜕𝑡
Similarly, the resultant acceleration shall be the resultant of the three
components.
𝑎=
𝒂𝒙 + 𝒂𝒚 + 𝒂𝒛
 Other quantities like pressure, density etc. can be found in a similar
manner.
 This approach however is very complex and the resulting equations of
motion are difficult to solve.
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The Eulerian method
 In this method we assess fluid motion by focusing on a point in the fluid
system. The velocity, acceleration and other features of the fluid at
that particular point are studied.
 This method has proven to be relatively easier to analyze fluid motions
mathematically.
 Here the velocity components in the 𝑥, 𝑦 and 𝑧 direction
𝑢, 𝑣, 𝑤 respectively are expressed as a function of position and time 𝑡 at
the point 𝑥, 𝑦, 𝑧 .
𝑢 = 𝑓1 𝑥, 𝑦, 𝑧. 𝑡
𝑣 = 𝑓2 𝑥, 𝑦, 𝑧. 𝑡
𝑤 = 𝑓1 𝑥, 𝑦, 𝑧. 𝑡
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 The components of acceleration of the fluid particle can expressed with the
help of partial derivatives as follows;
Given that in the 𝑥 −direction the total derivative of 𝑢 is;
𝜕𝑢
𝜕𝑢
𝜕𝑢
𝜕𝑢
𝑑𝑢 =
. 𝑑𝑥 +
. 𝑑𝑦 +
. 𝑑𝑧 +
. 𝑑𝑡
𝜕𝑥
𝜕𝑦
𝜕𝑧
𝜕𝑡
𝑑𝑢
𝜕𝑢 𝑑𝑥 𝜕𝑢 𝑑𝑦 𝜕𝑢 𝑑𝑧
𝜕𝑢 𝑑𝑡
= 𝑎𝑥 =
.
+
.
+
.
+
.
𝑑𝑡
𝜕𝑥 𝑑𝑡 𝜕𝑦 𝑑𝑡 𝜕𝑧 𝑑𝑡
𝜕𝑡 𝑑𝑡
For
𝑑𝑥
= 𝑢,
𝑑𝑡
𝑑𝑦
= 𝑣,
𝑑𝑡
𝑑𝑧
=𝑤
𝑑𝑡
𝝏𝒖
𝝏𝒖
𝝏𝒖
𝝏𝒖
𝒂𝒙 = 𝒖
+𝒗
+𝒘
+
𝝏𝒙
𝝏𝒚
𝝏𝒛
𝝏𝒕
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Similarly in the 𝑦 −direction the total derivative of 𝑣 is;
𝜕𝑣
𝜕𝑣
𝜕𝑣
𝜕𝑣
𝑑𝑣 =
. 𝑑𝑥 +
. 𝑑𝑦 +
. 𝑑𝑧 +
. 𝑑𝑡
𝜕𝑥
𝜕𝑦
𝜕𝑧
𝜕𝑡
𝑑𝑣
𝜕𝑣 𝑑𝑥 𝜕𝑣 𝑑𝑦 𝜕𝑣 𝑑𝑧
𝜕𝑣 𝑑𝑡
= 𝑎𝑦 =
.
+
.
+
.
+
.
𝑑𝑡
𝜕𝑥 𝑑𝑡 𝜕𝑦 𝑑𝑡 𝜕𝑧 𝑑𝑡
𝜕𝑡 𝑑𝑡
For
𝑑𝑥
= 𝑢,
𝑑𝑡
𝑑𝑦
= 𝑣,
𝑑𝑡
𝑑𝑧
=𝑤
𝑑𝑡
𝝏𝒗
𝝏𝒗
𝝏𝒗
𝝏𝒗
𝒂𝒚 = 𝒖
+𝒗
+𝒘
+
𝝏𝒙
𝝏𝒚
𝝏𝒛
𝝏𝒕
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Similarly in the 𝑧 −direction the total derivative of 𝑤 is;
𝜕𝑤
𝜕𝑤
𝜕𝑤
𝜕𝑤
𝑑𝑤 =
. 𝑑𝑥 +
. 𝑑𝑦 +
. 𝑑𝑧 +
. 𝑑𝑡
𝜕𝑥
𝜕𝑦
𝜕𝑧
𝜕𝑡
𝑑𝑤
𝜕𝑤 𝑑𝑥 𝜕𝑤 𝑑𝑦 𝜕𝑤 𝑑𝑧
𝜕𝑤 𝑑𝑡
= 𝑎𝑧 =
.
+
.
+
.
+
.
𝑑𝑡
𝜕𝑥 𝑑𝑡 𝜕𝑦 𝑑𝑡 𝜕𝑧 𝑑𝑡
𝜕𝑡 𝑑𝑡
For
𝑑𝑥
= 𝑢,
𝑑𝑡
𝑑𝑦
= 𝑣,
𝑑𝑡
𝑑𝑧
=𝑤
𝑑𝑡
𝝏𝒘
𝝏𝒘
𝝏𝒘
𝝏𝒘
𝒂𝒛 = 𝒖
+𝒗
+𝒘
+
𝝏𝒙
𝝏𝒚
𝝏𝒛
𝝏𝒕
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The resultant velocity is given by;
𝑽=
𝒖𝟐 + 𝒗𝟐 + 𝒘𝟐
The resultant acceleration is given by;
𝒂=
𝒂𝟐𝒙 + 𝒂𝟐𝒚 + 𝒂𝟐𝒛
In vector notation we can write the velocity and acceleration as follows;
𝑉 = 𝑢𝒊 + 𝑣𝒋 + 𝑤𝒌
𝑎 = 𝑎 𝑥 𝒊 + 𝑎𝑦 𝒋 + 𝑎𝑧 𝒌
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𝜕𝑉
𝑎 = 𝑉. 𝛻 𝑉 +
𝜕𝑡
The velocity and acceleration of the particle are function of space (𝑠) and
time (𝑡), that is;
𝑉 = 𝑓 𝑥, 𝑦, 𝑧, 𝑡 = 𝑠, 𝑡
𝑑𝑉 𝜕𝑉 𝑑𝑠 𝜕𝑉 𝑑𝑡
𝑎=
=
. +
.
𝑑𝑡
𝜕𝑠 𝑑𝑡 𝜕𝑡 𝑑𝑡
𝝏𝑽
𝝏𝑽
𝒂= 𝑽
+
= 𝒂𝒄𝒐𝒏 + 𝒂𝒍𝒐𝒄
𝝏𝒔 𝒄𝒐𝒏
𝝏𝒕 𝒍𝒐𝒄
The acceleration consists of two parts;
FLUID MECHANICS I-MENG 225/FBC/USL/STN
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𝝏𝑽
(i) 𝒂𝒄𝒐𝒏 = 𝑽 ∶ This component is the convective acceleration and is due
𝝏𝒔
to the change in position or movement of the fluid particle.
𝝏𝑽
(ii) 𝒂𝒍𝒐𝒄 = :This component is the local (temporal) acceleration and is
𝝏𝒕
due to the change in velocity with respect to time at a given location.
Tangential and normal acceleration
When the motion of a fluid particle is
curvilinear, the acceleration is in two
parts;
 A tangential component 𝑎𝑠
 A normal component 𝑎𝑛
Curved path
𝑎𝑠
𝑉2
𝑎𝑛 =
𝑟
𝑉 = 𝑓(𝑠, 𝑡)
P
Fig 2: Curvilinear motion of fluid particles
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𝒂 = 𝒂𝒔 + 𝒂𝒏
Types of fluid flow
Fluid flows may be classified as follows;






Steady and unsteady flows
Uniform and non-uniform flows
One-, two- and three-dimensional flows
Rotational and irrotational flows
Lamina and turbulent flows
Compressible and incompressible flows
FLUID MECHANICS I-MENG 225/FBC/USL/STN
INTRODUCTION TO HYDRODYNAMICS
Steady and unsteady flows.
Steady flow is a type of flow in which the fluid characteristics like velocity,
pressure, density etc. at any point in the flow do not change with time.
Mathematically, we have:
𝜕𝑢
𝜕𝑣
𝜕𝑤
= 0;
= 0;
=0
𝜕𝑡 𝑥 ,𝑦 ,𝑧
𝜕𝑡 𝑥 ,𝑦 ,𝑧
𝜕𝑡 𝑥 ,𝑦 ,𝑧
0
0 0
0
0 0
0
0 0
𝜕𝑝
𝜕𝜌
= 0;
=0
𝜕𝑡 𝑥 ,𝑦 ,𝑧
𝜕𝑡 𝑥 ,𝑦 ,𝑧
0
0 0
0
0 0
Where 𝑥0 , 𝑦0 , 𝑧0 is a fixed point in fluid field where the variables are being
measured with time. Example. Flow through a prismatic or non-prismatic
duct at a constant discharge.
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Unsteady flow is a type of flow in which the fluid characteristics like
velocity, pressure, density etc. at any point in the flow do change with time.
Mathematically, we have:
𝜕𝑢
𝜕𝑣
𝜕𝑤
≠ 0;
≠ 0;
≠0
𝜕𝑡 𝑥 ,𝑦 ,𝑧
𝜕𝑡 𝑥 ,𝑦 ,𝑧
𝜕𝑡 𝑥 ,𝑦 ,𝑧
0
0 0
0
0 0
0
0 0
𝜕𝑝
𝜕𝜌
≠ 0;
≠0
𝜕𝑡 𝑥 ,𝑦 ,𝑧
𝜕𝑡 𝑥 ,𝑦 ,𝑧
0
0 0
0
0 0
Example. Flow through a prismatic or non-prismatic duct at a constant
discharge. Velocity components are function of time.
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Uniform and non-uniform flows.
Uniform flow is a type of flow in which the velocity do not change with
space at any instant in time. Mathematically, we have:
𝜕𝑉
= 0;
𝜕𝑠 𝑡=𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡
Where,
𝜕𝑉 = 𝐶ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦, 𝑎𝑛𝑑
𝜕𝑠 = 𝐷𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑚𝑒𝑛𝑡 𝑖𝑛 𝑎𝑛𝑦 𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜𝑛
. Example. Flow through a prismatic or non-prismatic duct at a constant
discharge.
FLUID MECHANICS I-MENG 225/FBC/USL/STN
Non-uniform flow is a type of flow in which the velocity does change with
space at any instant in time. Mathematically, we have:
𝜕𝑉
≠ 0;
𝜕𝑠 𝑡=𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡
Where,
. Example. Flow through a non-prismatic duct.
Flow around a uniform diameter pipe-bend or a canal bend.
One-,two-, and three-dimensional flows.
A flow is considered as one-dimensional if the factors or parameters such
as velocity, pressure, elevation which define the flow at any given instant
vary only in one-direction (the direction of flow and not across the crosssection at any point). Flow can be described along either the 𝒙 −, or 𝒚 − or
𝒛 −direction, but not in any two- or all three-direction.
For this type of flow,
𝑢 = 𝑓 𝑥, 𝑡 ,
𝑣 = 0,
𝑤=0
𝑢 = 0,
𝑣 = 𝑓 𝑦, 𝑡 ,
𝑤=0
𝑢 = 0,
𝑣 = 0,
Or
Or
𝑤 = 𝑓(𝑧, 𝑡)
𝑦
𝑥
Fig 3: One-dimensional flow model
Two dimensional flow is a type of flow in which the velocity is a function
of time and two rectangular space coordinates. Mathematically;
𝑢 = 𝑓1 𝑥, 𝑦, 𝑡 ,
𝑣 = 𝑓2 𝑥, 𝑦, 𝑡 ,
𝑤=0
Examples. Flow round a curved surface
Flow in a tapering duct
Flow at a bend in a duct
Three dimensional flow is a type of flow in which the velocity is a function
of time and three mutually perpendicular directions in space.
Mathematically;
𝑢 = 𝑓1 𝑥, 𝑦, 𝑧, 𝑡 ,
𝑣 = 𝑓2 𝑥, 𝑦, 𝑧, 𝑡 ,
𝑤 = 𝑓3 𝑥, 𝑦, 𝑧, 𝑡
Examples. Flow round a curved surface
Flow in a tapering duct (convergent or divergent)
Flow at a bend in a duct
𝒚
𝒚
𝒙
𝒙
Fig 3. 2-dimensional flow ( Flow round a cylinder)
𝒚
Fig 4. 2-dimensional flow (Flow over a weir)
𝒙
Fig 5. 2-dimensional flow Flow in a duct with varying elevation and section
𝑣
𝑢
𝑤
Fig 6: three-dimensional flow models
Rotational and irrotational flows
Rotational flows are flows in which the fluid particles while flowing along
streamlines, rotate about their centre of mass/own axes. Examples of
rotational flows include;
 Flow of liquids in rotating cylinder (forced vortex)
 Flow in floor drains (drain holes of tanks, wash basins etc.)
INTRODUCTION TO HYDRODYNAMICS
Irrotational flows are flows in which fluid particles while flowing along
streamlines, do not rotate about their own axes.
Lamina and turbulent flows
Lamina flows are flows in which the paths taken by the individual fluid
particles do not intersect/cross one another and move along well defined
paths. This type of flow is also called stream-line or viscous flow.
Examples include the following;
 Flow through a capillary tube
 Flow of blood in veins and arteries.
 Ground water flow.
Turbulent flows are flows in which fluid particles move in a zig-zag way.
Almost all fluid flow problems in engineering practice possess some
turbulent character. High velocity flow in ducts of large size.
FLUID MECHANICS I-MENG 225/FBC/USL/STN
INTRODUCTION TO HYDRODYNAMICS
Fig 7: Lamina flow
Fig 8: Turbulent flow
Lamina and turbulent flows are characterized on the basis of Reynolds
number as follows;
 For Reynolds number (𝑹𝒆) < 𝟐𝟎𝟎𝟎
……flow in pipes is lamina
 For Reynolds number (𝑹𝒆) > 𝟒𝟎𝟎𝟎
……flow in pipes is turbulent
 For Reynolds number 2000< (𝑹𝒆) < 4000 …flow in pipes is
transitional
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Compressible and incompressible flows.
Compressible flows are flows in which the fluid exhibits changes in volume
and hence density with time and position along the flow.
Mathematically:
𝜌 ≠ 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡
Examples. Flow of gases through orifices, nozzles, gas turbines etc.
Incompressible flows on the other hand is a flow in which the fluid exhibits
constant density at different sections of the flow duct with time. In reality, a
rigorous incompressible fluid does not exist.
Mathematically:
𝜌 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡
Examples. Subsonic aerodynamics
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Types of flow lines
 Whenever a fluid is in motion, it numerous particles move along certain
lines depending upon the condition of flow (lamina, transitional or
turbulent).
 Some important forms of flow lines are as follows.
Path line: This is the locus of points or the path followed by a fluid particle
in motion. It can one, two or three-dimensional as the nature of the flow
dictates.
Fig 9: Path lines
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Streamlines: These are imaginary lines within a flow which are such that
the tangent at any point on them gives the velocity of the flow at that point.
Equation of a streamline in a three-dimensional flow is given by;
𝑑𝑥 𝑑𝑦 𝑑𝑧
=
=
𝑢
𝑣
𝑤
Fig 10a: streamlines
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Fig 10b: streamlines
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Let us note the features of streamlines
 A streamline cannot cross itself, nor can any two streamlines cross
each other.
 There cannot be any movement of the fluid across the streamlines.
 Streamline spacing varies inversely with the velocity, so
convergence of streamlines in any particular directions is indicative of
accelerated flow in that direction and vice versa.
 A streamline indicates the direction of a number of fluid particles at
the same time.
 A series of streamlines represents the flow pattern at an instant.
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Stream Tube: This is a mass of fluid bounded by a group of streamlines.
The content of a stream tube is also know as “current filament” Examples of
stream tubes are pipes and nozzles.
Fig 11: stream Tube
Streak Line: This is the locus of points which gives the instantaneous
picture of the location of a fluid particle. Examples include smoke
particles, injection of colour dyes into a fluid stream etc.
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INTRODUCTION TO HYDRODYNAMICS
Fig 12: streakline by dye introduction
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Discharge, mean velocity, and continuity
 The total quantity of fluid flowing per unit time past any particular
cross-section of a stream is called the discharge or flow at that section.
 When discharge is measured in terms of mass, it is referred to as the
ሶ
mass flow rate (𝒎).
 When discharge is measured in terms of volume, it is referred to as the
volume flow rate (𝑸).
 In an ideal fluid in which there is no friction the velocity adjacent to the
solid boundary will be zero or equal to the wall velocity in the flow
direction. This is a condition known as ‘no slip’ which holds for as long
as the flow is in contact with the wall.
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 For such a flow of sectional area 𝑨 and velocity 𝒖, the discharge is given by
𝑸.
𝑄 = 𝐴𝑢
𝐴
𝑢
Fig 13:Discharge in Ideal fluid
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 With the exception of nuclear reaction where the law of conservation of
mass does not hold, it can be applied to flowing fluids. Considering any
fixed region in a flow constituting a control volume; a mass balance
equation for such a system will be as follows:
𝑀𝑎𝑠𝑠 𝑜𝑓 𝑓𝑙𝑢𝑖𝑑 = 𝑚𝑎𝑠𝑠 𝑜𝑓 𝑓𝑙𝑢𝑖𝑑 𝑙𝑒𝑎𝑣𝑖𝑛𝑔 +𝐼𝑛𝑐𝑟𝑒𝑎𝑠𝑒 𝑜𝑓 𝑚𝑎𝑠𝑠
𝑜𝑓 𝑓𝑙𝑢𝑖𝑑 𝑖𝑛 𝑡ℎ𝑒 𝑐𝑜𝑛𝑡𝑟𝑜𝑙
𝑒𝑛𝑡𝑒𝑟𝑖𝑛𝑔 𝑝𝑒𝑟 𝑢𝑛𝑖𝑡 𝑝𝑒𝑟 𝑢𝑛𝑖𝑡 𝑡𝑖𝑚𝑒
𝑣𝑜𝑙𝑢𝑚𝑒 𝑝𝑒𝑟 𝑢𝑛𝑖𝑡 𝑡𝑖𝑚𝑒
𝑡𝑖𝑚𝑒
Mass of fluid
entering
Control
volume
Fig 14:Control volume
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Mass of fluid
leaving
INTRODUCTION TO HYDRODYNAMICS
For steady flow, the mass of fluid in the control volume remains constant
and the mass balance equation reduces to
𝑀𝑎𝑠𝑠 𝑜𝑓 𝑓𝑙𝑢𝑖𝑑
= 𝑚𝑎𝑠𝑠 𝑜𝑓 𝑓𝑙𝑢𝑖𝑑 𝑙𝑒𝑎𝑣𝑖𝑛𝑔
𝑒𝑛𝑡𝑒𝑟𝑖𝑛𝑔 𝑝𝑒𝑟 𝑢𝑛𝑖𝑡
𝑝𝑒𝑟 𝑢𝑛𝑖𝑡 𝑡𝑖𝑚𝑒
𝑡𝑖𝑚𝑒
Applying this principle to a steady flow in a streamtube as shown in
Fig. 10, having an elemental cross-sectional area (𝜹𝑨) small enough to
consider the velocity across it as constant; then for two points (1) and
(2) along the stream tube.
𝑀𝑎𝑠𝑠 𝑜𝑓 𝑓𝑙𝑢𝑖𝑑
= 𝑚𝑎𝑠𝑠 𝑜𝑓 𝑓𝑙𝑢𝑖𝑑 𝑙𝑒𝑎𝑣𝑖𝑛𝑔
𝑝𝑒𝑟 𝑢𝑛𝑖𝑡 𝑡𝑖𝑚𝑒 𝑎𝑡 𝑠𝑒𝑐𝑡𝑖𝑜𝑛 (2)
𝑒𝑛𝑡𝑒𝑟𝑖𝑛𝑔 𝑝𝑒𝑟 𝑢𝑛𝑖𝑡
𝑡𝑖𝑚𝑒 𝑎𝑡 𝑠𝑒𝑡𝑖𝑜𝑛 (1)
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𝑀𝑎𝑠𝑠 𝑒𝑛𝑡𝑒𝑟𝑖𝑛𝑔 𝑝𝑒𝑟 𝑢𝑛𝑖𝑡 𝑡𝑖𝑚𝑒 𝑎𝑡 1 = 𝜌1 𝛿𝐴1 𝑢1
𝑀𝑎𝑠𝑠 𝑙𝑒𝑎𝑣𝑖𝑛𝑔 𝑝𝑒𝑟 𝑢𝑛𝑖𝑡 𝑡𝑖𝑚𝑒 𝑎𝑡 2 = 𝜌2 𝛿𝐴2 𝑢2
2
1
𝜌2 𝛿𝐴2 𝑢2
𝜌1 𝛿𝐴1 𝑢1
For steady flow;
Fig 10:Flow through a sreamtube
𝜌1 𝛿𝐴1 𝑢1 = 𝜌2 𝛿𝐴2 𝑢2
This is the equation of continuity for the flow of a compressible fluid
through a streamtube for which 𝑢1 and 𝑢2 are the velocities measure
normal to the section 𝛿𝐴1 and 𝛿𝐴2 respectively.
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For flow of a real fluid through a pipe or other conduits, the velocity will
vary from wall to wall. However the mean velocity (ഥ
𝒖) can be considered
ሶ as
for the continuity equation to determine the mass flow rate (𝒎)
follows.
ഥ 𝟏 = 𝝆𝟐 𝑨𝟐 𝒖
ഥ 𝟐 … … … . . (𝒊)
𝒎ሶ = 𝝆𝟏 𝑨𝟏 𝒖
 If the fluid is incompressible, so that the density is constant at every
section of the flow (𝝆𝟏 = 𝝆𝟐 = 𝝆), then equation (i) becomes;
ഥ 𝟏 = 𝝆𝑨𝟐 𝒖
ഥ 𝟐 … … … . . (𝒊′)
𝒎ሶ = 𝝆𝑨𝟏 𝒖
ഥ 𝟏 = 𝑨𝟐 𝒖
ഥ 𝟐 … … … … . (𝒊𝒊)
𝑸 = 𝑨𝟏 𝒖
 The continuity equation is one of the major form of the law of mass
conservation that is used to calculate velocities at different points in a
system.
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END OF LECTURE 5
QUESTIONS
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TUTORIALS
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Example 1. In a fluid, the velocity field is given by;
𝑽 = 3𝑥 + 2𝑦 𝒊 + 2𝑧 + 3𝑥 2 𝒋 + 2𝑡 − 3𝑧 𝒌
Determine:
(i)The velocity components, 𝒖, 𝒗, 𝒘 at any point in the flow field.
(ii)The speed at point (𝟏, 𝟏, 𝟏);
(iii)The speed at time 𝒕 = 𝟐𝒔 at point (𝟎, 𝟎, 𝟐)
Also classify the velocity field as steady, or unsteady, uniform, or nonuniform, and one-, two-, or three-dimensional.
Solution.
For the velocity field given by;
𝑽 = 3𝑥 + 2𝑦 𝒊 + 2𝑧 + 3𝑥 2 𝒋 + 2𝑡 − 3𝑧 𝒌
(i) This can be written as;
𝑽 = 𝑢𝒊 + 𝑣𝒋 + 𝑤𝒌
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The velocity components are:
𝒖 = 𝟑𝒙 + 𝟐𝒚, 𝒗 = 𝟐𝒛 + 𝟑𝒙𝟐 , 𝒘 = 𝟐𝒕 − 𝟑𝒛
(ii) Speed at point 1,1,1 , 𝑉 1,1,1
Substituting 𝑥 = 1, 𝑦 = 1, 𝑧 = 1 in the expressions for 𝑢, 𝑣, 𝑤, we have:
𝑢 = 3 1 + 2 1 = 5, 𝑣 = 2 1 + 3 1 2 = 5, 𝑤 = 2𝑡 − 3 1
But
𝑉 = 𝑢2 + 𝑣 2 + 𝑤 2 = (5)2 +(5)2 +(2𝑡 − 3 )2
𝑽 𝟏,𝟏,𝟏 = 𝟒𝒕𝟐 − 𝟏𝟐𝒕 + 𝟓𝟗
(iii) Speed at 𝑡 = 2𝑠 at point (0,0,2):
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= 2𝑡 − 3
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Substituting t = 2, 𝑥 = 0, 𝑦 = 0, 𝑧 = 2 in the expressions for 𝑢, 𝑣, 𝑤, we have:
𝑢 = 3 0 +2 0
But
= 0, 𝑣 = 2 2 + 3 0 2 = 4, 𝑤 = 2(2) − 3 2
𝑉=
02 + 42 + (−2)2 =
= −2
0 + 16 + 4
𝑽 𝟎,𝟎,𝟐 = 𝟐𝟎
 Since 𝑉 at given (𝑥, 𝑦, 𝑧) depends on 𝑡, the flow is an unsteady flow.
 Since at given 𝑡 velocity changes with space 𝑆, it is a non-uniform flow
 Since 𝑉 depends on 𝑥, 𝑦, 𝑧, the flow is a three- dimensional flow.
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Example 2. In a fluid, the velocity field is given by;
𝑽 = 3 + 2𝑥𝑦 + 4𝑡 2 𝒊 + 𝑥𝑦 2 + 3𝑡 𝒋
Determine the velocity and acceleration at a point (𝟏, 𝟐) after 𝟐 𝒔𝒆𝒄
Solution
For the velocity field given by;
𝑽 = 3 + 2𝑥𝑦 + 4𝑡 2 𝒊 + 𝑥𝑦 2 + 3𝑡 𝒋
(i) This can be written as;
𝑽 = 𝑢𝒊 + 𝑣𝒋
𝑢 = 3 + 2𝑥𝑦 + 4𝑡 2 , 𝑣 = 𝑥𝑦 2 + 3𝑡
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Speed at point 1,2 , 𝑉 1,2
Substituting 𝑥 = 1, 𝑦 = 2, 𝑡 = 2 in the expressions for 𝑢, 𝑣 we have:
𝑢 = 3 + 2(1)(2) + 4(2)2 = 23, 𝑣 = (1) 2 2 + 3(2) = 10
But
𝑉=
(23)2 +(10)2
𝑽 𝟏,𝟏,𝟏 = 𝟐𝟓. 𝟎𝟖 𝒖𝒏𝒊𝒕𝒔
(ii) For acceleration at point (1,2), 𝑎(1,2) :
We know that:
𝑑𝑉
𝜕𝑉
𝜕𝑉
𝜕𝑉
𝑎=
= 𝑢 +𝑣
+
𝑑𝑡
𝜕𝑥
𝜕𝑦
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𝜕𝑡
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𝑉 = 3 + 2𝑥𝑦 + 4𝑡 2 𝑖 + 𝑥𝑦 2 + 3𝑡 𝑗
𝜕𝑉
= 2𝑦𝑖 + 𝑦 2 𝑗,
𝜕𝑥
𝜕𝑉
= 2𝑥𝑖 + 2𝑥𝑦𝑗,
𝜕𝑦
𝜕𝑉
= 8𝑡𝑖 + 3𝑗
𝜕𝑡
But;
𝑑𝑉
𝜕𝑉
𝜕𝑉
𝜕𝑉
𝑎=
= 𝑢
+𝑣
+
𝑑𝑡
𝜕𝑥
𝜕𝑦
𝜕𝑡
3 + 2𝑥𝑦 + 4𝑡 2 2𝑦𝑖 + 𝑦 2 𝑗 + (𝑥𝑦 2 + 3𝑡)(2𝑥𝑖 + 2𝑥𝑦𝑗) +
(8𝑡𝑖 + 3𝑗)
At point (1,2) and 𝑡 = 2 we have,
3 + 2(1)(2) + 4(2)2 2(2)𝑖 + (2)2 𝑗 +
𝑎=
+ (8(2)𝑖 + 3𝑗)
2
((1) 2 + 3(2))(2(1)𝑖 + 2(1)(2)𝑗)
𝑎=
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= 23 4𝑖 + 4𝑗 + 10 2𝑖 + 4𝑗 + 16𝑖 + 3𝑗
= 128𝑖 + 135𝑗
𝒂(𝟏,𝟐) =
1282 + 1352 = 𝟏𝟖𝟔. 𝟎𝟑 𝒖𝒏𝒊𝒕𝒔
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Example 3:A sphere of diameter 300mm falls axially down a 305mm
diameter vertical cylinder which is closed at its lower end and contains
water as shown in Fig. E3. If the sphere falls at a speed of 150mm/s, what
is the mean velocity relative to the cylinder walls of the water in the gap
surrounding midsection of the sphere.
Data
Diameter of sphere 𝒅𝒔 = 𝟑𝟎𝟎𝒎𝒎 = 𝟎. 𝟑𝒎
Diameter of cylinder 𝒅𝒄 = 𝟑𝟎𝟓𝒎𝒎 = 𝟎. 𝟑𝟎𝟓𝒎
Rate of fall of sphere 𝒖𝒔 = 𝟏𝟓𝟎𝒎𝒎/s
Fig. E3
Required
 Mean velocity of water relative to cylinder
walls 𝒖𝒘 =?
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 As the sphere falls through the water, the water will
rush past the it relative to the cylinder walls.
𝑢𝑤
𝑢𝑤
𝑄𝑠 = 𝑄𝑤
𝜋𝑑𝑠2
𝑄𝑠 = 𝐴𝑠 𝑢𝑠 =
𝑢𝑠
4
𝑢𝑠
𝐴𝑠
 The rate of volume of water displace by the section
of the sphere is equal to the rate of volume of water
rushing past the sides of the cylinder.
𝐴𝑤
2 − 𝑑2
𝜋 𝑑𝑤
𝑠
𝑄𝑤 = 𝐴𝑤 𝑢𝑤 =
𝑢𝑤
4
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Now ;
2
𝜋 𝑑𝑤
− 𝑑𝑠2
𝜋𝑑𝑠2
𝑢𝑤 =
𝑢𝑠
4
4
𝑑𝑠2
0.32
𝒖𝒘 =
2 = 0.3052 − 0.32 × 0.15 = 𝟒. 𝟒𝟔𝟕𝒎/𝒔
2
𝑑𝑤 − 𝑑𝑠
Examples 4: A conical pipe shown in Fig. E4 diverges uniformly from
𝟏𝟎𝟎𝒎𝒎 to 𝟐𝟎𝟎𝒎𝒎 diameter over a length of 1𝑚.
(i) Show that for a steady flow with a constant discharge of
𝑸 = 𝟎. 𝟏𝟐𝒎𝟑 𝒔−𝟏 , the velocity at any section 𝒙 m from the inlet is given by:
𝑸
𝒖 = 𝒖𝒙 =
𝟐
𝟎.𝟎𝟎𝟕𝟖𝟓 𝟏+𝒙
(ii)Hence determine the local and convective acceleration at the
midsection of the pipe.
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Fig. E4
Solution
Data
Inlet diameter 𝐷1 = 100𝑚𝑚
Exit diameter 𝐷2 = 200𝑚𝑚
Length
𝑙 = 1𝑚
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𝑙, 𝐷2
𝑦
0, 𝐷1
𝑥, 𝐷𝑥
𝑥
(i) Flow is steady and one-dimensional. Using the principle of continuity
𝑄 = 𝐴1 𝑢1 = 𝐴𝑥 𝑢𝑥 = 𝐴1 𝑢2
At any section 𝑥 of the duct;
𝑄
𝑢𝑥 =
𝐴𝑥
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𝜋𝐷𝑥2
𝐴𝑥 =
4
We can determine 𝐷𝑥 at 𝑥 in terms of the dimensions of the tapering pipe
by considering the slope of one side of the duct in the 𝑥𝑦 −plane.
Now:
𝐷𝑥 − 𝐷1 𝐷2 − 𝐷1
𝐷2 − 𝐷1
=
 𝐷𝑥 = 𝐷1 +
𝑥
𝑥−0
𝑙−0
𝑙
Hence;
0.2 − 0.1
0.1
𝐷𝑥 = 0.1 +
𝑥 = 0.1 +
𝑥 = 0.1(1 + 𝑥)
𝑙
1
𝜋𝐷𝑥2 𝜋 0.1(1 + 𝑥) 2
𝐴𝑥 =
=
= 0.00785 1 + 𝑥 2
4
4
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Recall;
𝑄
𝑸
𝒖𝒙 =
=
𝐴𝑥 𝟎. 𝟎𝟎𝟕𝟖𝟓 𝟏 + 𝒙 𝟐
(ii) The acceleration of the fluid in the duct is given by:
𝜕𝑢 𝜕𝑢
𝑎 = 𝑎𝑐𝑜𝑛𝑣 + 𝑎𝑙𝑜𝑐 = 𝑢
+
𝜕𝑥 𝜕𝑡
For steady flow velocity is not a function of time 𝑡; hence:
𝜕𝑢
𝑎𝑙𝑜𝑐 =
=0
𝜕𝑡
𝜕𝑢
𝑄
𝜕
𝑄
𝑎𝑐𝑜𝑛𝑣 = 𝑢
=
𝜕𝑥 0.00785 1 + 𝑥 2 𝜕𝑥 0.00785 1 + 𝑥 2
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−2𝑄 2
=
0.00785 2 1 + 𝑥 5
For a given constant discharge 𝑄 = 0.12𝑚3 𝑠 −1 , at the mid section of the
pipe 𝑥 = 0.5𝑚.
−2(0.12)2
−𝟐
𝒂𝒄𝒐𝒏𝒗 =
=
−𝟔𝟏.
𝟓𝒎𝒔
0.00785 2 1 + 0.5 5
Note the negative sign.
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Examples 5: A pipe (1) 𝟒𝟓𝟎𝒎𝒎 in diameter branches into two pipes (2 and
3) of diameters 𝟑𝟎𝟎𝒎𝒎 and 𝟐𝟎𝟎𝒎𝒎 respectively as shown in Fig. E5. If the
mean velocity in the 𝟒𝟓𝟎𝒎𝒎 diameter pipe is 𝟑𝒎 𝒔−𝟏 find:
(i)Discharge through the 𝟒𝟓𝟎𝒎𝒎 diameter pipe;
(ii) Velocity in 𝟐𝟎𝟎𝒎𝒎 diameter pipe if the mean velocity in the 𝟑𝟎𝟎𝒎𝒎
pipe is 𝟐. 𝟓𝒎𝒔−𝟏
Fig. E5.
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(i) Discharge through the 450mm diameter duct is given by;
𝜋
𝑸𝟏 = 𝐴1 𝑉1 = 0.45 2 3 = 𝟎. 𝟒𝟕𝟕𝒎𝟑 𝒔−𝟏
4
(ii) By the principle of continuity:
𝑄1 = 𝑄2 + 𝑄3
𝑄1 = 𝐴2 𝑉2 + 𝐴3 𝑉3
𝜋
𝜋
𝜋
2
2
0.45 3 = 0.30 2.5 + 0.20 2 𝑉3
4
4
4
0.45 2 (3) − (0.30)2 (2.5)
−𝟏
𝑽𝟑 =
=
𝟗.
𝟓𝟔𝒎𝒔
0.202
FLUID MECHANICS I-MENG 225/FBC/USL/STN
INTRODUCTION TO HYDRODYNAMICS
PRACTICE
EXERCISE
FLUID MECHANICS I-MENG 225/FBC/USL/STN