19 pius augustine fluid flow and viscocity

1
Viscosity and Bernoulli’s Equation (Fluid Flow)

Dr. Pius Augustine, SH College, Kochi

Stream line or steady flow of a
liquid is a steady flow in which
each layer of liquid follows the
same path and has the same
velocity as that of its
predecessor.
Stream line or Steady Flow

When the velocity of a point in the liquid
changes with time the flow is called unsteady
flow.
Unsteady flow is called turbulent flow, when
there are bents in the path of a fast moving
liquid.
Velocity of liquid change continuously and
haphazardly both in magnitude and direction
Turbulent flow

The path followed by an element of a
moving fluid is called line of flow.
A collection of identical streamlines is
called a tube of flow.
Infinitesimally small volume element of
the liquid is called particle of liquid
Line of Flow, Tube of Flow, Particle of a Liquid

Steady flow: Particular point -velocity of
fluid particle is same
Volume of liquid crossing any section per
second is same
Velocity high at narrow region

A stream line may be defined as a curve, the tangent
to which at any point gives the direction of the flow of
liquid at that point.
v1 = constant, v2 = constant , v3 = constant
v1 ≠ v2 ≠ v3
V1
V2
V3

Two stream lines never cross ?
if intersect, there will be two tangents and
two different velocity at same point which
is not possible.

Laminar flow
Liquid flow in which different layers or
laminae glide over one another at a slow and
steady velocity, without intermixing, is
called laminar or viscous flow.

Magnitude of velocities of various layers is
represented by the length of arrowed lines
Dotted curve : velocity profile or velocity
shape
Velocity profile is parabolic for tube of flow
Laminar flow

velocity gradient
Velocity of layers increase from zero at the
walls (bottom) to maximum along the axis.
Rate of change of velocity with distance is
called velocity gradient.

Just for information….
Normal blood flow in the human aorta is
laminar, but a small disturbance such as a
heart pathology can cause the flow to
become turbulent.
The turbulence makes noise, which is why
listening to blood flow with a stethoscope is
a useful diagnostic technique.

Q. Water near the bed of a deep river
is quiet while that near the surface
flows. Give reason?

If water in one flask and castor oil in
other are violently shaken and kept
on a table, which will come to rest
either?

Viscosity
It is the property of a liquid by which it
opposes the relative motion between its
different layers.
It is a measure of resistance of a fluid

Viscous force acts tangentially in a direction
opposite to the relative motion between the
different layers of the liquid.
Viscosity of a lubricating oil is one of the
factors which decides whether it is suitable
for use in the engine of a machine.

Reynold’s number (Osborne Reynold)
Velocity of flow of a liquid upto which the
flow is streamlined (above which, flow
becomes turbulent) is called critical velocity.
Critical velocity vc = Rη/ρD
R - Reynold’s number
η - coefficient of viscosity
D - diameter of tube, ρ - density of liquid

Reynold’s number (Osborne Reynold)
Flow is streamline if R< 2000
Flow is turbulent if R > 3000
Reynold’s number is dimensionless
Derivation - dimensional analysis

Reynold’s number
It is a dimensionless number.
It is a critical variable which determines
the process taking place inside a
cylindrical tube, when fluid flows
through it.
Law of similarity? And Reynold’s number

well I won't go any faster with fuel

Viscosity and density of some liquids
Fluid η Pl ρ SI
Hydrogen 8.4 *10-6 0.082 at 300K
Air 17.4 *10-6 1.161 at 300K
Water 0.8 * 10-3 1000
Mercury 1.526 * 10-3 13600
Blood at 370C (3 to 4) * 10-3 1060
Castor oil 0.985 956
Glycerol 1.49 1126
Honey 2 - 10 1420
ketchup 50 – 100
Molten glass 10 - 1000

Note : higher viscous, low dense liquid flowing
through narrow tube may be streamline (higher
critical velocity)
Low viscous, high dense liquid flow even through
wider tube may be turbulent
Above Vc, most of energy needed to drive liquid
is dissipated in setting up whirlpools, vortices
and eddies

Expression for Viscous Force (Newton’s equation)
Magnitude of viscous force F on a certain layer of
liquid is propotional to
i] area A ii] velocity gradient dv/dx
F = -η A dv/dx
η – constant depend upon the nature of the liquid and is
called coefficient of viscosity of the liquid
-ve sign indicates that viscous force acts in a direction
opposite to direction of flow of liquid

Coefficient or absolute or dynamic viscosity η
F = -η A dv/dx. η = F
A dv/dx
Coefficient of viscosity may be defined as the
tangential viscous force per unit area
required to maintain unit velocity gradient
normal to the direction of flow.

Unit of η
F = -η A dv/dx. η = F
A dv/dx
SI unit: Pa-s or Nsm-2 called poiseuille (Pl)
CGS uint : poise
Reyn is british unit for dynamic viscosity
1Pl = 10 poise or decapoise
Dimension : ML-1T-1

The velocity of water in a river is 18 km/hr
near the surface. If the river is 5 m deep,
find the shearing stress between the
horizontal layers of water. The coefficient of
viscosity of water = 10-2 poise.
dv/dx = (18 km/hr)/5m = 1.0 s-1.
Stress = F/A = η A (dv/dx)

What is the effect of temperature on coefficient of
viscosity of a liquid ?
η of liquid decreases with increase
in temperature.
η of gases increases with increase
in temperature.

Viscosity of water at different temperatures
Temperature oC η * 10-3 Pa-s
10 1.308
20 1.002
30 0.7978
40 0.6531
50 0.5471
60 0.4668
70 0.4044
80 0.3550
90 0.3150
100 0.2822
Decreaseswithincreaseintemperature

Q. Oils of different viscosities are used in
different seasons, for lubrication. Why ?
Q. Why do the machine parts get jammed in
winter ?

Can you stop it

Effect of pressure on viscosity
Viscosity of liquids and gases
increases with pressure.

Fluidity
Measure of ability to flow with ease
Reciprocal of coefficient of viscosity = 1/η
Unit : poise-1 some times called ‘rhe’
Fluidity is rarely used in engineering.

Kinematic viscosity
It is the ratio between coefficient of viscosity
and density = η / ρ
Viscosity index is a measure for the change of
kinematic viscosity with temperature.
It is used to characterise lubricating oil in
the automotive industry.
CGS unit : cm2s-1 or ‘stokes’
1 m2 / s = 10,000 stokes

youngest bodybuilder in the world.

George Gabriel Stokes 1819 - 1903

Sphere falling through a fluid : Stokes’ law
F = 6πηav
η – coefficient of viscosity of liquid
a – radius of the sphere
v – terminal velocity attained
Using dimensional method F = K ηx ay vz
From experiments k = 2π

Terminal velocity :
The constant velocity attained by a body
as it falls down through a fluid medium is
called the terminal velocity.
V = 2 a2 (ρ- σ) g
9 η
For a sphere falling through air, σ can be
neglected

Terminal velocity Derivation
Wt of the body = Vρg = 4/3 πa3 ρg
Buoyant force = Vσg = 4/3 πa3 σg
ρ – density of body σ – density of
liquid
Under dynamic equilibrium
Effective wt = Viscous force

Terminal velocity Derivation
Under dynamic equilibrium
Effective wt = Viscous force
4/3 πa3 (ρ- σ) g = 6πηav
V = 2 a2 (ρ- σ) g
9 η

Velocity time graph for a body moving in
viscous medium.
time
velocity
Vt

Note: In Biology terminal velocity is called
sedimentation velocity.
By performing experiments on
sedimentation, useful information
concerning very small particles maybe
obtained.

 Rain drops falling under gravity do not acquire
very high velocity. Why?
 Q. Find the terminal velocity of a rain drop of radius 0.01 mm.
Given ηair = 1.8 x 10-5 SI units and density 1.2 SI units. R. D of
water is 1. take g = 10 m/s2.
Hint:
since density of air << density of water – buoyancy neglected.

Viscosity vs Friction
i. Only when motion both at rest
as well as in motion
ii. Due to cohesion partly due to adhesion
iii. Viscous F α A independent of area
iv. F α dv/dx independent of relative
velocity of the surfaces
v. Depend on shape independent of shape

Frictional force between solids operates
even when they do not move with
respect to each other. Do we have
viscous force acting between two layers
even if there is no relative motion?

Variation of viscosity with temperature
Liquids:
Viscosity decreases with rise in temperature
For glycerine η = 46 poise at 0 oC and 3.5 poise
at 30 oC.
Gases:
Viscosity increases with rise in temperature.

Liquids?
Liquids – viscosity is due to attraction
among molecules and between molecules
and solid in contact.
As temperature increases, molecular
attraction is getting weakened, hence
viscosity decrease.

Gases?
Gases – molecules are farther apart and viscosity is
due to collision between fast moving molecules
with slow moving molecules. Fast molecules will
be impeded in collision.
As temperature increases, molecular activity
increases and this causes disorderly mixing of the
molecules. So viscosity increases.

A man jumping
without parachute, Vterminal = 120 km/h with
parachute, Vterminal = 14 km/h
Fog formation – tiny droplets and dust
particles have small terminal velocity, and
appear to suspend in air.
Hail storm does not cause much damage as
they come with terminal velocity rather
than acceleration.

Practical applications of viscosity
i. Selection of lubricant.
ii. Fountain pen ink – neither flows
down nor stuck up in the pen.
iii. Streamlining – shaping (aeroplane,
rocket)

An air bubble of 1cm radius is rising
at a steady rate of 0.5 cm/s through
a liquid of density of 0.8 g/cm3.
Calculate the co-efficient of viscosity
of the liquid. Neglect density of air.

Terminal velocity = - 0.5 cm/s
(σ - ρ) = - ρ = (-0.8)
-ve sign has been taken as bubble moves upward.
Η = 2 a2 ρ g = 2 * (-0.8) 981
9 V 9 ( -0.5)
= 348.8 poise

Eight rain drops of radius 1mm each
falling down with a terminal
velocity of 5cm/s coalesce to form a
bigger drop. Calculate the terminal
velocity of the bigger drop.

V’ = 2 R2 (ρ - σ) g for big drop
9 η
V = 2 r2 (ρ - σ) g for small drop
9 η
Volbig = 8 * Volsmall
4/3 πR3 = 8 * 4/3 πr3
R = 2r
V’/ V = 0.22/0.12 = 20 cm/s

What is the reason for floating of clouds in the sky ?
Water particles of clouds attain a terminal
velocity while moving through air. This
terminal velocity is very low and remain
suspended in the sky.
When cloud is getting denser, terminal
velocity increase and they fall freely.

A small and a big air bubbles rise up through a
liquid. Which rises faster ?
Terminal velocity is directly proportional to
radius2.
Hence big air bubble rises faster.
Direction of viscous force in this case is
downward or terminal velocity is –ve.

Poiseuille’s Equation
Volume of liquid flowing through a tube
per second [V] depends on
i. Coefficient of viscosity η
ii. radius of tube r
iii. Pressure gradient P/l

Poiseuille’s Equation
η r and P/l
Using dimensional analysis
M0L3T-1 = [ML-1T-1]a[L]b[ML-2T-2]c
On solving, V = πPr4
8lη

Essential conditions for Poiseuille’s
equation to hold good.
i. Capillary tube is horizontal ie. no
gravity effect
ii. Flow is stream line parallel to the
axis
iii. No radial flow of liquid
iv. Liquid is viscous

 The cylindrical layer in contact with the wall of the
capillary tube is at rest. The velocity of the liquid layers
goes on increasing as we move from the wall towared the
axis of the tube.
 acceleration of liquid at any point is zero .
 liquid can withstand small shearing stress.
These conditions are realized in actual
practice if the capillary tube is of fine bore
and the liquid flows with small velocity.

Calculate the mass of water flowing in
10 min. through a tube of radius 1 cm,
one meter in length and there is a
constant pressure head of 20 cm of
water. η = 0.009 cgs unit.

V = πPr4 P = hdg = 20 *1*980
8lη
= 3.14 * 20 *1*980 *14
8 * 100 * 0.009
= 8.56 * 103 cm3 per second
Volume collected in 600 seconds = 8.56 * 103*
600
Mass of water collected = Volume * density(1
for water)
= 5.136 * 106 g.Dr. Pius Augustine, SH College, Kochi

Application of viscosity
Knowledge of viscous drag is used to determine the molecular
mass using centrifuges in biological and medical labs.
High viscous liquids as buffers in trains.
For damping the motion of certain instruments
Proper shaping – streamlining (aeroplanes, ship hulls, rockets)
In the study of circulation of blood. – variation in η of blood
will talk about the efficiency of the person.
Production and transportation of oils
Quality of ink
Variation in η with temperature helps in identifying the best
lubricant for a particular machine.

Knowledge of viscous drag is used to determine the
molecular mass using centrifuges in biological
and medical labs.
High viscous liquids as buffers in trains.
For damping the motion of certain instruments
Production and transportation of oils

Proper shaping – streamlining (aeroplanes, ship hulls,
rockets)
In the study of circulation of blood. – variation in η of
blood will talk about the efficiency of the person.
Quality of ink
Variation in η with temperature helps in identifying
the best lubricant for a particular machine.

2T vs 4T oils
A 2T Oil is designed to lubricate the engine
components, mix-up with the fuel completely,
burn and go out in the exhaust. A 4T Oil is not
designed to do so. As a result, when a 4T Oil is
put into in a 2T Engine, it causes spark-plug
fouling, exhaust port blockage, smoke emissions
etc.

Q. In scooters more viscous mobile oil
is used in summer than in winter.
Why?
Q. Why do machine parts get jammed
in winter?

Parachute and Drogue
An umbrella like collapsible device.
Produce drag when pulled through fluid.
For slowing down while dropping a man or goods
through air.
If it is used under water or at high speeds
in air it is called drogue

Equation of continuity.
Fundamental equation of
fluid flow.
Extension of law of
conservation mass.
a1, v1 and a2 , v2 are area
and velocity respectively
at sections C and D
C
D
Since liquid is
incompressible
Mass of liquid crossing any
section per second is const.

Equation of continuity.
Since liquid is incompressible
Mass of liquid crossing any
section per second is const.
a1v1 ρ = a2v2 ρ
av = constant
If the fluid is compressible
a1v1 ρ1 = a2v2 ρ2 or avρ = constant
C
D

The product ‘av’ is the volume
flow rate dV/dt (rate at which
volume crosses a section of
the tube)
av = dV/dt

Q. Deep water runs slow. Comment
Q. Air in the atmosphere is nearly
incompressible. Use this fact to
explain why particularly fast moving
winds are found in mountain
passes.

A 20.0 litre bucket can be filled with water using
a water hose 3.00 cm in diameter in 2 minutes.
Calculate the speed with which the water leaves
the hose.
1 lit ~ 1m3.
Volume of water flowing/sec = 20/2x 60
Volume of water flowing/second = area x velocity.
velocity?

Q. As part of a lubricating system for heavy
machinery, oil of density 850 kg/m3 is
pumped through a cylindrical pipe of
diameter 8.0 cm at a rate of 9.5 litres per
second. i) What is the speed of the oil? Ii)
What is the mass flow rate? C) If the pipe
diameter is reduced to 4.0 cm, what are the
new values of the speed and volume flow
rate? Assume that oil is incompressible.

Why do the fire fighters attach brass jets
at the ends of water pipes?
av = constant.
As ‘a’ decreases, velocity
increases.
So water is able to reach the place
of fire.

When the water tap is closed
with our fingers jets of water
gush through the space between
fingers with high speed. Why?

Water is slowly coming out from a
vertical pipe. As the water
descends after coming out, its area
of cross – section reduces. Explain.

While watering a distant plant, a
gardener partially closes the exit
hole of the pipe by putting his
finger on it. Explain why this
results in the water stream going
to a larger distance.

Energy of fluid in steady flow - 3 Kinds
Kinetic energy = ½ mv2
Kinetic energy per unit mass = ½ v2
Kinetic energy / unit volume = ½ ρv2
1. Kinetic Energy

Potential energy = mgh
Potential energy / unit mass = gh
Potential energy / unit volume = ρgh
2. Potential Energy

Pressure energy = PV
Pressure energy /unit mass = P/ρ
Pressure energy / unit volume = P
3. Pressure Energy

i. kinetic energy = ½ mv2
Kinetic energy per unit mass = ½ v2
Kinetic energy / unit volume = ½ ρv2
ii. Potential energy = mgh
Potential energy / unit mass = gh
Potential energy / unit volume = ρgh
iii. Pressure energy = PV
Pressure energy /unit mass = P/ρ
Pressure energy / unit volume = P

Pressure energy
The energy possessed by a fluid by virtue of
its pressure is called its pressure energy.
A fluid under pressure can do work and
possess energy called pressure energy.

Pressure energy
Let A be the area of cross section of the piston.
Force acting on the piston = PA
Work done = PA x = PV
This is equivalent to the energy
contained in a volume V of the liquid
on account of its pressure called pressure energy.
x
P0 + P
P0

Daniel Bernoulli (1700 -1782)

Different forms of Bernoulli’s eqn. (practice derivation –
according to the figure you are plotting….)
Total energy = constant
PV + mgh + ½ mv2 = constant
Total energy of unit mass of fluid = constant
P/ρ + gh + ½ v2 = constant
Total energy of unit vol of fluid = constant
P + ρgh + ½ ρv2 = constant
static pressure + dynamic pressure = constant
Total pressure head = constant
P/ρg + h + v2 /2g = constant
Press head + gravitnal head +vel head= const.

Corrections to be applied?
Bernoulli’s equation is derived based on certain assumptions
i. Fluid is non-viscous?
ii. Velocity of the fluid particles at different points is the same
iii. No loss of energy
Assumptions are not absolutely correct.
Velocity of the fluid is maximum along the axis and
decreases towards the walls of the tube
Part of KE is converted into heat.

Q. Water flows through a tube of variable cross
section. Area of cross-section at A and B are 4
mm2 and 2 mm2 respectively. 1 cc of water enters
per second through A. Find i) the speed of water
at A, ii) the speed of water at B and iii) the
pressure difference PA-PB in the following three
cases.
Case 1. tube is horizontal
Case 2 tube is vertical (with A upwards) with
separation between A and B is 15/16 cm
Case 3 Tube is vertical (with B upwards and water
enters B at the rate of 1 cm2/s – note speed
decreases as the water falls down)

Ideal fluid?
A fluid that is incompressible
(density can not be changed)
and has no internal
friction(viscosity) is ideal fluid.

Applications of Bernoulli’s theorm
i. Venturimeter
ii. Pitot tube
iii.Atomiser
iv.Dynamic uplift
v. Carburettor
vi.Bunsen’s burner

When a sphere or cylinder moves in still air while
spinning about an axis perpendicular to the
direction of its motion, its curved path is more
curved .
Ball drags air forward.
Relative velocity of air w.r.t ball is different for the
two diametrically opposite points, which cause
difference in pressure.
Ball turns towards the region of low pressure.

A spinning cricket ball takes a
curved path. Comment

Why bullets are made cylindrical and not
spherical ?
Fired bullet spirals along the grooves in the
barrel and comes out spinning about its
axis.
Cylinder – direction of motion is parallel to
its spin axis. So the pressure on the sides
remains uniform throughout and will not
deviate from its straight path.

Air foil
• Any surface designed in such a way as to
obtain reacting force from the air through
which it moves is known as air foil
• Generally used for wings of aeroplane
• Upper surface is slightly convex and lower
slightly concave.
• Air splits at the leading edge and meets at
the trailing edge simultaneously.
• Upper side velocity of air is more, which
causes reduction in pressure.
• Resulting uplift is called dynamic uplift.

The snow accumulated on the wings
of an aeroplane may decrease the lift.
Why?

Venturimeter
B

Venturimeter
It is a gauge used for measuring the rate of steady
flow of a fluid, based on Bernoullis eqn.
(fixed horizontally, PE cancels )
P1 + ½ ρv1
2 = P2 + ½ ρv2
2
P1 – P2 = ½ ρ(v2
2 – v1
2 )
P1 – P2 = hρg manometer
hρg = ½ ρ(v2
2 – v1
2 )
In actual practice ,
manometer tube
contains mercury.
hρ'g = ½ ρ(v2
2 – v1
2 )
ρ' - density of mercury.

Venturimeter
hρg = ½ ρ(v2
2 – v1
2 )
From equation of continuity, a1v1 = a2v2 = V
volume of liquid crossing per second
v1 = V/a1 and v2= V/a2
Substitute and solve,
V = a1a2 √2 √g √h
(a1
2
- a2
2)1/2
V α √h Dr. Pius Augustine, SH College, Kochi

Atomiser or Scent sprayer
Air is blown past the
mouth of this tube at a
high speed by pressing,
creates a low pressure.

Velocity of Efflux:
The average flow rate of material emitted
into the atmosphere from a source such as
a smokestack. This is the average speed of
gas out of the top of a smokestack.

Torricelli’s theorm
Patm, V
=0
patm
h’

Torricelli’s theorm - law of efflux
The velocity of efflux of a liquid through an
orifice is equal to that which a body would
attain in falling freely from the free surface
of the liquid to the orifice.
P + ρg (h + h’ ) + 0 = P + ρg h’ + ½ ρv2
ρg h =½ ρv2
V = √ 2gh

Pitot Tube

Pitot Tube
It is a device used for measuring the velocity of
flow and hence the rate of flow at any depth in a
flowing liquid.
Apply Bernoulli’s theorem at openings a and b
PE cancels, and velocity at ‘a’ is zero as flow
stopped.
P1 + ½ ρ x 02 = P2 + ½ ρv2
P1 – P2 = ½ ρv2
hρg = ½ ρv2
Rate of flow = av = a √(2gh)
V = √(2gh)

Pitot-Static tubes, which are also
called Prandtl tubes, are used on aircraft as
speedometers. ...
The pitot-static tube is mounted on the
aircraft, or in a wind tunnel , so that the
center tube is always pointed in the
direction of the flow and the outside holes
are perpendicular to the center tube.

Ping pong ball in blown air

A light ping pong ball can be balanced on a continuous
stream of water or air coming out of a jet in vertically
upward direction. Explain
Fluid speed is same on all sides of ball.
So same pressure on all sides of ball.
If the ball is displaced towards left, fluid
will have greater speed on right and less
speed on left.
Bernoulli’s theorm, pressure will be
lesser on right side .
So ball will be displaced to right – eqbm.

Bunsen burner

Bunsen burner
Gas comes out of the nozzle with high
velocity causes reduction in pressure
in the stem.
Air from atmosphere rushes into the
burner.
Mixture of gas and air burns at the top.

A few other examples
Ping pong ball kept on a stream of
water
Blowing off roofs
Near a fast moving train
Two moving parallel ships etc.

Explain the working of
Carburettor – Bernoulli’s
principle

A gypsy car has got a canvas top. When the
car runs at high speed, the top bulges out.
Explain.

Q. To keep a piece of paper horizontal,
you should blow over, not under it.
Why?
When blown, velocity becomes high,
and pressure low above the paper.
High pressure in the lower side push
the paper to low pressure region.

Q. Why does flag flutter, when strong
winds are blowing on a certain day?
With the fluctuations in the velocity of
air on either side of the flag, pressure
will change, which makes it flutter.

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Dr. Pius Augustine, SH College, KochiDr. Pius Augustine, SH College, Kochi

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Dr. Pius Augustine, Dept of Physics, Sacred Heart College, Thevara
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Dr. Pius Augustine, Asst. Professor, Sacred Heart College, Thevara, Kochi.

19 pius augustine fluid flow and viscocity

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19 pius augustine fluid flow and viscocity