Introduction of Fluid Mechanics

INTRODUCTION TO
FLUID DYNAMICS
Peter Huruma Mammba
Department of General Studies
DODOMA POLYTECHNIC OF ENERGY AND EARTH RESOURCES
MANAGEMENT (MADINI INSTITUTE) –DODOMA
peter.huruma2011@gmail.com

 Introduction
(Definitions of fluid, Stresses, Types of fluids,
Newton’s law of viscosity, Laminar flow,
Turbulent flow, Rate flow, Equation of continuity,
Bernoulli's equation, Total energy)
 Where you find Fluids and Fluid-Dynamics?
 Blood flow in arteries and veins
 Interfacial fluid dynamics
 Geological fluid mechanics
 The dynamics of ocean
 Laminar-turbulent transition
 Solidification of fluids

Vortex shedding off
back of Sorrocco Island

 is a sub discipline of fluid mechanics that deals
with fluid flow. i.e. the science
of fluids (liquids and gases) in motion.
 It has several sub disciplines itself,
including aerodynamics (the study of air and other
gases in motion) and hydrodynamics (the study of
liquids in motion)

 First (the fluid is nonviscous)
 Means, there is no internal frictional
force between the adjacent layers of the
fluid

A viscous fluid, such as honey, does not flow readily and is said
to have a large viscosity.
In contrast, water is less viscous and flows more readily; water
has a smaller viscosity than honey.
The flow of a viscous fluid is an energy-dissipating process.
A fluid with zero viscosity flows in an unhindered manner with
no dissipation of energy.
Although no real fluid has zero viscosity at normal temperatures,
some fluids have negligibly small viscosities.
An incompressible, nonviscous fluid is called an ideal fluid.

 Secondly, (the fluid is incompressible)
 This means that the velocity , density and
pressure at each point in the fluid do not
change with time.
 NB.
 Such simplifying assumption s permit us to study
the flow of fluid in a simple way.

Most liquids are nearly incompressible; that is, the density of a
liquid remains almost constant as the pressure changes.
To a good approximation, then, liquids flow in an incompressible
manner.
In contrast, gases are highly compressible. However, there are
situations in which the density of a flowing gas remains constant
enough that the flow can be considered incompressible.

Introduction of Fluid Mechanics

 Substances with no strength
 Deform when forces are applied
 Include water and gases
Solid:
Deforms a fixed amount or breaks completely
when a stress is applied on it.
Fluid:
Deforms continuously as long as any shear stress is
applied.

The study of motion and the forces which cause (or
prevent) the motion.
Three types:
 Kinematics (kinetics): The description of motion:
displacement, velocity and acceleration.
 Statics: The study of forces acting on the particles
or bodies at rest.
 Dynamics: The study of forces acting on the
particles and bodies in motion.

Stress = Force /Area
 Shear stress/Tangential stress:
The force acting parallel to the surface per unit
area of the surface.
 Normal stress:
A force acting perpendicular to the surface per
unit area of the surface.

Basic laws of physics:
 Conservation of mass
 Conservation of momentum – Newton’s second law of
motion
 Conservation of energy: First law of thermodynamics
 Second law of thermodynamics
+ Equation of state
Fluid properties e.g., density as a function of pressure and
temperature.
+ Constitutive laws
Relationship between the stresses and the deformation of the
material.

Example: Density of an ideal gas
Ideal gas equation of state
Newton’s law of viscosity:
2 3
PV=nRT,
P: pressure (N/m ),V: volume(m ),
T:temperature(K),n:number of moles.
mass nM
=
V V
pM
=
RT




Stress α train (deformation)
du du
=
dy dy
   
S
: coefficientof viscosity(Dynamicviscosity)

It is define as the resistance of a fluid which is being deformed
by the application of shear stress.
In everyday terms viscosity is “thickness”. Thus, water is
“thin” having a lower viscosity, while honey is “think”
having a higher viscosity.
 Common fluids, e.g., water, air, mercury obey Newton's
law of viscosity and are known as Newtonian fluid.
 Other classes of fluids, e.g., paints, polymer solution, blood
do not obey the typical linear relationship of stress and strain.
They are known as non-Newtonian fluids.
Unit of viscosity: Ns/m2 (Pa.s)

 Very Complex
 Rheology of blood
 Walls are flexible
 Pressure-wave travels
along the arteries.
 Frequently encounter
bifurcation
 There are vary small veins

 Frequently encounter
 Many complex phenomenon
 Surface tension
 Thermo-capillary flow
 In industries: oil/gas
 Hydrophobic nature
Challenges :
 Interfacial boundary condition.
 Numerical study becomes
computationally very expensive.
On going work at IIT H

When a viscous fluid flows over a solid surface, the fluid
elements adjacent to the surface attend the velocity of the
surface. This phenomenon has been established through
experimental observations and is known as “no-slip”
condition.
Many research work have been conducted to understand the
velocity slip at the wall, and has been continued to be an open
topic of research.

Fluids can move or flow in many ways.
Water may flow smoothly and slowly in a quiet
stream or violently over a waterfall.
The air may form a gentle breeze or a raging
tornado.
To deal with such diversity, it helps to identify
some of the basic types of fluid flow.

 When the pressure is lower on one side of a fluid
than on other side, the fluid will flow toward the
low-pressure region.
 Fluid flow is characterized by two main types;-
(i) Steady flow or streamline flow
(ii) Turbulent flow

 If the flow of fluid is steady, then all
the fluid particles that pass any given
point follow the same path at the
same speed.
i.e… they have the same speed

In steady flow the velocity, density and pressure of the fluid
particles at any point is constant as time passes.
Unsteady flow exists whenever the velocity at a point in the
fluid changes as time passes.

When the flow is steady, streamlines are often used to
represent the trajectories of the fluid particles.
A streamline is a line drawn in the fluid such that a tangent
to the streamline at any point is parallel to the fluid velocity
at that point.
Steady flow is often called streamline flow.

(a) In the steady flow of a liquid, a colored dye reveals the
streamlines. (b) A smoke streamer reveals a streamline
pattern for the air flowing around this pursuit cyclist, as he
tests his bike for wind resistance in a wind tunnel.

 This is the special case of steady flow in
which the velocities of all the particles on
any given streamline are the same through
the particles of different streamlines may be
move at different speed.

 Also known as
streamline flow
 Occurs when the
fluid flows in
parallel layers, with
no disruption
between the layers
 The opposite of
turbulent flow
(rough)

 In fluid dynamics (scientific study of properties of
moving fluids), laminar flow is:
 A flow regime characterized by high momentum
diffusion, low momentum convection, pressure
and velocity independent from time.
*momentum diffusion refers to the spread of momentum
(diffusion) between particles of substances, usually
liquids

 Laminar flow over a flat
and horizontal surface
can be pictured as
consisting of parallel
and thin layers
 Layers slide over each
other, thus the name
‘streamline’ or smooth.
 The paths are regular
and there are no
fluctuations
Laminar Flow
Turbulent
Flow

 3 Conditions
 fluid moves slowly
 viscosity is relatively high
 flow channel is relatively small
 Blood flow through capillaries is laminar flow,
as it satisfies the 3 conditions
 Most type of fluid flow is turbulent
 There is poor transfer of heat energy!

Turbulent flow is an extreme kind of
unsteady flow and occurs when there
are sharp obstacles or bends in the path
of a fast-moving fluid.
In turbulent flow, the velocity at a point
changes erratically from moment to
moment, both in magnitude and
direction.

 Usually occurs when the
liquid is moving fast
 The flow is ‘chaotic’ and
there are irregular
fluctuations
 Includes:
 Low momentum diffusion
 high momentum
convection
 rapid variation of pressure
and velocity of the fluid
 Good way to transfer

 The speed of the fluid at a point is continuously
undergoing changes in both magnitude and
direction.

 Oceanic and atmospheric layers and ocean
currents
 External flow of air/water over vehicles such as
cars/ships/submarines
 In racing cars, e.g. leading car causes understeer
at fast corners
 Turbulence during air-plane’s flight
 Most of terrestrial atmospheric circulation
 Flow of most liquids through pipes

 This is the volume of a liquid that passes the
cross-section per unit time.
 It is denoted by the symbol Q
 𝑅𝑎𝑡𝑒 𝑜𝑓 𝑓𝑙𝑜𝑤, 𝑄 =
𝑉
𝑡
 The SI unit of rate of flow is 𝑚3
𝑠−1

 Consider a pipe of uniform cross-sectional area 𝐴1
as shown in figure below
 If the pipe is running full with liquid at an average
velocity of 𝑣1, then distance 𝑙 through which the
liquid moves in time t, it rate of flow will be;-
 𝑄 =
𝑉
𝑡
=
𝐴1 𝑥 𝑣1 𝑡
𝑡
 𝑄 = 𝐴1 𝑣1 (also called discharge equation)

 The Reynolds number Re is the ratio of the inertia
forces in the flow to the viscous forces in the flow
and can be calculated using:
• If Re < 2000, the flow will be laminar.
• If Re > 4000, the flow will be turbulent.
• If 2000<Re<4000, the flow is transitional
• The Reynolds number is a good guide to the type of flow

 μ is the dynamic viscosity of the fluid
 v is the kinematic velocity of the fluid
 A is the pipe cross-sectional area.
 p is the density of the fluid
 V is the mean fluid velocity
 D is the diameter
 Q is the volumetric flow rate
Dynamic Pressure
Shearing Stress

 The Reynold’s number can be used to
determine if a flow is laminar, transient or
turbulent
 Laminar when Re < 2300
 Turbulent when Re > 4000
 Transient when 2300 < Re < 4000
Spermatozoa 1×10−4
Blood flow in brain 1×102
Blood flow in aorta 1×103

Q: Have you ever used your thumb to control the
water flowing from the end of a hose?

Q: Have you ever used your thumb to control the water flowing from
the end of a hose?
A: When the end of a hose is partially closed off, thus reducing its
cross-sectional area, the fluid velocity increases.
This kind of fluid behavior is described by the equation of
continuity.

Water flows through a horizontal pipe of
varying cross-section at the rate of 10
𝑚3
/minutes. Determine the velocity of water
at point where the radius of the pipe is 10 cm.

Rate of discharge, Q = 10 𝑚3
/minutes = 1
6 𝑚3
/s
Cross-sectional area, A = 𝜋𝑟2
= 𝜋 0.1 2
=
0.0314𝑚2
Q = Av
Velocity of water, v =
𝑄
𝐴
=
1
6
0.0314
= 5.3 m/s

Water flows through a pipe of
internal diameter 20 cm at the
speed of 1 m/s. what should the
diameter nozzle be if the water is
to emerge at the speed of 4 m/s ?

 Here 𝑑1 = 20 cm = 0.2 m; 𝑣1 = 1 m/s; 𝑣2 = 4
m/s; 𝑑2 = ?
𝐴1 𝑣1 = 𝐴2 𝑣2
𝜋
4
𝑑1
2
𝑣1 =
𝜋
4
𝑑2
2
𝑣2
𝑑2
2
𝑑1
2 =
𝑣1
𝑣2
=
1
4
𝑑1
𝑑2
=
1
2
𝑑2 =
𝑑1
2
=
0.2
2
= 0.1 m = 10 cm

Water is flowing through a horizontal
pipe of varying cross-sectional at the
rate of 20 liters per minutes. Find the
velocity of water at a point where
diameter of the pipe is 2 cm.
Answer 10.61 m/s

A water pipe is 10 cm in diameter and has a
construction of 2 cm diameter. If the velocity of flow
in the main pipe is 0.84 m/s, calculate (i) the velocity
of flow in the construction (ii) rate of discharge of
water through the pipeline.
Answer (i) 21 m/s (ii) 0.0066 𝑚3/𝑠

 A moving liquid can possess the following
types of energies;
(i) Kinetic energy due to its motion.
(ii) Potential (gravitational) energy due to its
position .
(iii) Pressure energy due to the pressure of the
liquid.

 Figure 23.6 shows the
flow of liquid of
density 𝜌 from a tank
to a pipe; the water
level AB in the tank
being kept constant

 Le the section a-𝑎′
of the pipe,
 m = mass of the liquid passing
at any instant
 v = velocity of the liquid
 ℎ1 = height of the pipe above
chosen reference level
 P = pressure of liquid = h𝜌𝑔

 At the section a- 𝑎′ of the pipe, the liquid has
various energies as under;
Kinetic energy (K.E.) = 1
2 𝑚𝑣2
Potential energy (P.E.) = mgℎ1
Pressure energy = mgh where (P = h𝜌𝑔)
= 𝑚𝑔 𝑥
𝑃
𝜌𝑔
=
𝑚𝑃
𝜌

∴ 𝑇𝑜𝑡𝑎𝑙 𝑒𝑛𝑒𝑟𝑔𝑦 𝑎𝑡 𝑠𝑒𝑐𝑡𝑖𝑜𝑛 a-𝑎′
=;
1
2 𝑚𝑣2
+ mgℎ1 +
𝑚𝑃
𝜌
∴ 𝑇𝑜𝑡𝑎𝑙 𝑒𝑛𝑒𝑟𝑔𝑦 𝑝𝑒𝑟 𝑢𝑛𝑖𝑡 𝑚𝑎𝑠𝑠 𝑎𝑡 a-𝑎′
=;
1
2 𝑣2
+ gℎ1 +
𝑃
𝜌
∴ 𝑇𝑜𝑡𝑎𝑙 𝑒𝑛𝑒𝑟𝑔𝑦 𝑝𝑒𝑟 𝑢𝑛𝑖𝑡 𝑣𝑜𝑙𝑢𝑚𝑒 𝑎𝑡 a-𝑎′
=;
1
2 𝜌𝑣2
+ 𝜌gℎ1 + 𝑃

For steady flow, the speed, pressure, and elevation of an
incompressible and nonviscous fluid are related by an equation
discovered by Daniel Bernoulli (1700–1782).

In the steady flow of a nonviscous, incompressible fluid of
density , the pressure P, the fluid speed v, and the elevation
y at any two points (1 and 2) are related by

The tarpaulin that covers the cargo is flat when the truck is
stationary but bulges outward when the truck is moving.

In a household plumbing system, a vent is necessary to equalize the
pressures at points A and B, thus preventing the trap from being
emptied. An empty trap allows sewer gas to enter the house.

At what speed will the
velocity head of a stream of
water be equal to 40 cm?

 Velocity head,
𝑣2
2𝑔
= h
v = 2𝑔ℎ
Here g = 9.8 m/s.s; h = 40 cm = 0.4 m
v = 2 𝑥 9.8 𝑥 0.4
v = 2. 8 m/s

A pipe is running full of water. At a
certain point A, it tapers from 60 cm
diameter to 20 cm diameter at B; the
pressure different between A and B is
100 cm of water column. Find the rate of
flow through the pipe.

𝑎 𝐴
𝑎 𝐵
=
𝑑 𝐴
𝑑 𝐵
2
=
0.6
0.2
2
= 9
𝑃𝐴 − 𝑃𝐵 = 𝐻𝜌𝑔 = 1 𝑥 1000 𝑥 9.8 = 9800𝑁/𝑚2
𝑣 𝐴 𝑎 𝐵 = 𝑣 𝐵 𝑎 𝐵
𝑣 𝐵 = 𝑣 𝐴
𝑎 𝐴
𝑎 𝐵
= 9𝑣 𝐴

 Using Bernoulli’s theorem for a horizontal pipe,
𝑃 𝐴
𝜌
+
1
2
𝑣 𝐴
2
=
𝑃 𝐵
𝜌
+
1
2
𝑣 𝐵
2
1
𝜌
𝑃𝐴 − 𝑃𝐵 =
1
2
𝑣 𝐵
2
− 𝑣 𝐴
2
𝑃𝐴 − 𝑃𝐵=
1
2
𝑣 𝐵
2
− 𝑣 𝐴
2

9800 =
100
2
81𝑣 𝐴
2
− 𝑣 𝐴
2
∴ 𝑣 𝐵 = 9𝑣 𝐴
9800 = 40000𝑣 𝐴
2
𝑣 𝐴 =
9800
40000
= 0.495 m/s
Rate of discharge, Q = 𝑣 𝐴 𝑎 𝐴 = 0.495 x
𝜋
4
0.6 2
= 0.14 𝑚3
/𝑠

Introduction of Fluid Mechanics

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Introduction of Fluid Mechanics