Fluid dynamics - viscosity and
turbulent
flow
Fluid statics
• What is a fluid?
Density Pressure
• Fluid pressure and depth
Pascal’s principle
• Buoyancy
Archimedes’ principle
Fluid dynamics
• Reynolds number
• Equation of continuity
• Bernoulli’s principle
• Viscosity and turbulent flow
• Poiseuille’s equation
Lecture 6
Dr Julia Bryant
web notes:
Fluidslect6.pdf
viscosity.pdf
1
http://www.physics.usyd.edu.au/teach_res/jp/fluids/wfluids.htm
Why do some liquids
splash more?
2
Why do cars need different oils in hot
and cold countries?
Why do engines run more freely as it
heats up?
Have you noticed that skin lotions are
easier to pour in summer than winter?
Why is honey sticky?
3
When real fluids flow they have a certain
internal friction called viscosity. It exists in
both liquids and gases and is essentially a
frictional force between different layers of
fluid as they move past one another.
In liquids the viscosity is due to the cohesive
forces between the molecules whilst in
gases the viscosity is due to collisions
between the molecules.
“VISCOSITY IS DIFFERENT TO DENSITY”
4
A useful model
Z
plate!
Assumptions
X
L!
A viscous fluid"
stationary wall"
5
A useful model
Z
plate moves with speed v!
Viscous fluid will flow"
X
L!
Q: Direction?
Q: Highest speeds?
Q: Lowest speeds?
stationary wall"
6
A useful model:
Newtonian fluids
water, most gases
plate moves with speed v!
vx = v!
high speed"
Z
L!
X
linear
velocity
gradient"
vx!
d
low speed"
stationary wall"
vx = 0"
7
A useful model:
Newtonian fluids
shear "
stress"
plate exerts force F!
over area A"
is proportional to
velocity
gradient"
(F/A) = η (v / L)
stationary wall"
8
(F/A) = η (v / L)
η = (F / A)(L / v)
coefficient of viscosity η (Greek letter eta).
The greater the coefficient of viscosity η, the
greater the force required to move the plate at
a velocity v.
This relationship does not hold for all fluids.
Viscous fluids that obey this equation are
called Newtonian fluids and η = constant
independent of the speed of flow.
9
When η does depend upon the velocity of flow the fluids are called
non-Newtonian or rheological fluids.
Blood is an example of a non-newtonian mixture because it contains
corpuscles and other suspended particles. The corpuscles can
deform and become preferentially oriented so that the viscosity
decreases to maintain the flow rate.
Corn flour and water mixture is another non-Newtonian fluid.
Certain soils (more clay content) are non-Newtonian when moist to
wet.
10
Viscosity
SI unit is (N.m-2)(m).(m-1.s) ≡ Pa.s
A common unit is the poise P (1 Pa.s = 10 P)
Fluid
η (mPa.s)
water (0 °C)
1.8
water (20 °C)
1.0
water (100 °C)
0.3
Glycerine (20 °C)
1490
white blood (37 °C)
~4
blood plasma (37 °)
~1.5
engine oil (AE10)
~ 200
air
0.018
Beer
1.32 - 2.20 (av. 1.57)
η = (F / A)(L / v)
DEMO
Viscosity is very temperature dependent.
Viscosity of a liquid decreases with increasing
temperature.
Viscosity of a gas increases with increasing
11
temperature.
Why can't you simply remove dust just by
blowing across a surface?
Why does dust cling to a fast rotating fan?
How can a leaf stay on a car moving at high
speed?
12
Boundary layer
When a fluid moves over a surface, there is a thin layer of the
fluid near the surface which is nearly at rest. This thin layer is
called the boundary layer.
13
Flow through a pipe"
The rate of flow through a pipe is described by
Poiseuilleʼs law."
The derivation of this law is beyond this course
but we will discuss the motivation for the form
of the law."
14
What happens to the velocity profile when a "
Newtonian fluid flows through a pipe?"
Flow rate ∝ radius R
Parabolic "
velocity"
profile"
Adhesive forces between fluid and surface " DEMO
⇒ fluid stationary at surface"
Cohesive forces between molecules ⇒ layers
of fluid slide past each other generating
frictional forces ⇒ energy dissipated (like
rubbing hands together) "
15
What causes a fluid to flow through a pipe?
A useful model: Poiseuilleʼs Law: "
laminar flow of a Newtonian fluid through a pipe "
parabolic
velocity profile"
2R!
p1"
η
p2"
L!
Q = dV/dt!
volume flow rate!
Δp = p1 - p2"
Assumptions 16
We can guess that:
•  the bigger the pressure difference, the
higher the flow
flow rate ∝ Δp
•  the longer the pipe, the greater the
friction
flow rate ∝ 1/L
•  the more viscous the liquid, the lower
the flow
flow rate ∝ 1/η
17
A useful model: Poiseuilleʼs Law:"
Volume Flow rate Q = dV = Δp π R4"
8 η L!
dt!
2R!
η
Δp"
Q = dV/dt!
L!
p1 > p2⇒ pressure drop along pipe "
⇒ energy dissipated (thermal) by friction
between streamlines moving past each other"
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Poiseuilleʼs Law is only applicable to "
• laminar flow (not turbulent flow) in "
• Newtonian fluids (viscosity is
independent of the speed v, and the
force is proportional to the speed.)."
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Volume Flow rate
Q = dV = Δp π R4"
8 η L!
dt!
High viscosity ⇒ low flow rate"
Δp/L is the pressure gradient: the bigger the
pressure difference, the faster the flow"
The radius of the pipe makes a large
difference to the flow rate."
20
Applications of Poiseuilleʼs law:"
Irrigation pipes: Since Q ∝ Δp/L, it is uneconomical to
spray irrigation too far from the river (similar for air
conditioning, ducting, piping)."
Respiratory system: The resistance to flow is
determined primarily by the narrow tubes leading to the
alveoli. Any general constriction of the pipes, as occurs
in bronchospasm for instance, increases the resistance
to flow and makes breathing much more difficult.
Asthma. !
Circulatory system: Any constriction of the blood
vessels - like cholesterol build-up on the walls of the
arteries - increases the resistance and the heart has to
work harder to produce the same flow rate."
21
The heart is so responsive to the
changing needs of our body that cardiac
output can vary from as little as 5 to a
maximum of 35 litres of blood per
minute, a sevenfold change, over a very
short interval.
Q = dV = Δp π R4"
8 η L!
dt!
What happens to the flow as viscosity changes?"
22
Turbulence
Until now we have discussed laminar flow. When the motion
becomes too violent, eddies and vortices occur. We call this
turbulent motion.
The pattern of flow is no longer smooth and stable but
becomes irregular and chaotic. A complex flow pattern
changes continuously with time. The velocity of the particles at
each given point varies chaotically with time.
Eddies absorb a great deal of energy due their rotational
kinetic energy. "
Turbulence dissipates energy.
23
Turbulence
• A transition from laminar flow to
turbulent flow occurs very suddenly
as the flow rate increases. Fast flow
increases the chance of turbulence.
24
Turbulence
.
•  When
the flow becomes turbulent
there is a decrease in the volume flow
rate.
• When a fluid flows around an object
the shape of the object is a very
important parameter in determining the
type of flow.
• Thicker liquids like honey (or Kahlua
in milk) do not get turbulent as easily
as thin ones like water (or whiskey) .
25
Turbulence
• Turbulent flow occurs
when there are abrupt
changes in boundary
surfaces. The flow of blood
through a normal artery is
laminar. However, when
irregularities occur the flow
becomes turbulent. The
noise generated by the
turbulent flow can be heard
with a stethoscope.
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REYNOLDS NUMBER
Re = ρ v L / η
ρ
density of fluid
v
average flow velocity over the cross
section of the pipe
L
characteristic dimension
As a rule of thumb, for a flowing fluid
Re < ~ 2000 laminar flow
~ 2000 < Re < ~ 3000 unstable laminar
to turbulent flow
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Re > ~ 2000 turbulent flow
REYNOLDS NUMBER
• The Reynolds number is not a precise quantity.
• The quantities L and v are only typical values of size
and speed. It is often not possible even to say which
length you are talking about.
• For a body moving through a fluid it might be either
length or breadth or thickness - or any other
dimension you might think of.
• For a fluid flowing through a pipe, it turns out to be
the diameter of the pipe.
28
Sydney Harbour Ferry
Re = ρ v L / η
ρ = 103 kg.m-3
v = 5 m.s-1
Re = (103)(5)(10) / (10-3)
Re = 5x107
L = 10 m
η = 10-3 Pa.s
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Blood circulation
Re = ρ v L / η
ρ = 103 kg.m-3
v = 0.2 m.s-1
L = 10 mm for the aorta
ηblood ~ ηwater = 10-3 Pa.s
Re = (103)(0.2)(0.01) / (10-3)
Re = 2000
on the boundary of turbulent flow
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Bacterium
Re = ρ v L / η
ρ = 103 kg.m-3
v = 30x10-6 m.s-1
L = 1 µm
η = 10-3 Pa.s
Re = (103)(30x10-6)(1x10-6) / (10-3)
Re = 3x10-5
31
A large artery in a dog has an inner radius of 4.00×10-3 m. Blood flows
through the artery at the rate of 1.00×10-6 m3.s-1. The blood has a
viscosity of 2.084×10-3 Pa.s and a density of 1.06×103 kg.m-3.
Calculate:
(i) The average blood velocity in the artery.
(ii) The pressure drop in a 0.100 m segment of the artery.
(iii)  The Reynolds number for the blood flow.
Briefly discuss each of the following:
(iv) The velocity profile across the artery (diagram may be helpful).
(v) The pressure drop along the segment of the artery.
(vi) The significance of the value of the Reynolds number calculated in
part (iii).
2004 Exam question
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Solution
(i) The average blood velocity in the artery.
Equation of continuity: Q = A v
A = π R2 = π (4.00×10-3)2
= 5.03×10-5 m2
v = Q / A = 1.00×10-6 / 5.03×10-5 m.s-1
= 1.99×10-2 m.s-1
radius R = 4.00×10-3 m
Q = 1.00×10-6 m3.s-1
blood η = 2.084×10-3 Pa.s
blood ρ = 1.060×10-3 kg.m-3
(ii) The pressure drop in a 0.100m segment of the artery.
Poiseuille’s Equation Q = ΔP π R4 / (8 η L)
L = 0.100 m
ΔP = 8 η L Q / (π R4)
ΔP = (8)(2.084×10-3)(0.1)(1.00×10-6) / {(π)(4.00×10-3)4} Pa
ΔP = 2.07 Pa
(iii) The Reynolds number for the blood flow.
Re = ρ v L / η
where L = 2 R (diameter of artery)
Re = (1.060×103)(1.99×10-2)(2)(4.00×10-3) / (2.084×10-3)
Re = 81
use diameter not length
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(iv)
Discuss: The velocity profile across the artery (diagram may be helpful).
Parabolic velocity profile: velocity of blood zero at sides of artery
(v) Discuss: The pressure drop along the segment of the artery.
Viscosity ⇒ internal friction ⇒ energy dissipated as thermal energy ⇒"
pressure drop along artery"
(vi) Discuss: The significance of the value of the Reynolds number "
Re very small ⇒ laminar flow (Re < 2000)"
Flow of a viscous newtonain fluid through a pipe
Velocity Profile
Cohesive forces
between molecules ⇒
layers of fluid slide past
each other generating
frictional forces ⇒
energy dissipated (like
rubbing hands together)
Parabolic velocity
profile
Adhesive forces between fluid and surface ⇒ fluid
stationary at surface
34