1 http://www.physics.usyd.edu.au/teach_res/jp/fluids/wfluids.htm

web notes: Fluidslect6.pdf

viscosity.pdf

Lecture 6 Dr Julia Bryant

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

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Why do some liquids splash more?

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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?

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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”

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stationary wall"

L! A viscous fluid"

plate!

X

Z

A useful model

Assumptions

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stationary wall"

L!

Viscous fluid will flow"

plate moves with speed v!

X

Z

A useful model

Q: Direction? Q: Highest speeds? Q: Lowest speeds?

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stationary wall"

low speed"

high speed"

plate moves with speed v!

X

Z

vx = 0"

vx = v!

A useful model: Newtonian fluids water, most gases

linear velocity gradient"

d

L! vx!

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stationary wall"

plate exerts force F!

velocity gradient"is proportional to shear "

stress"

(F/A) = η (v / L)

A useful model: Newtonian fluids over area A"

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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.

η = (F / A)(L / v)

(F/A) = η (v / L)

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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.

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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)

Viscosity is very temperature dependent. Viscosity of a liquid decreases with increasing temperature. Viscosity of a gas increases with increasing temperature.

DEMO

η = (F / A)(L / v)

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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?

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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.

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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."

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What happens to the velocity profile when a "Newtonian fluid flows through a pipe?"

Adhesive forces between fluid and surface "⇒ fluid stationary at surface"

Parabolic "velocity" profile"

Cohesive forces between molecules ⇒ layers of fluid slide past each other generating frictional forces ⇒ energy dissipated (like rubbing hands together) "

Flow rate ∝ radius R

DEMO

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A useful model: Poiseuilleʼs Law: "laminar flow of a Newtonian fluid through a pipe "

volume flow rate!

Q = dV/dt!η 2R!

L!

p1" p2"

Δp = p1 - p2"

parabolic velocity profile"

What causes a fluid to flow through a pipe?

Assumptions

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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

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A useful model: Poiseuilleʼs Law:"

Q = dV/dt!η 2R!

L!

Δp"

Q = dV = Δp π R4"

8 η L!dt!

p1 > p2⇒ pressure drop along pipe "⇒ energy dissipated (thermal) by friction

between streamlines moving past each other"

Volume Flow rate

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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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Q = dV = Δp π R4"

8 η L!dt!

Volume Flow rate

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."

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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."

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What happens to the flow as viscosity changes?"

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!

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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.

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Turbulence

• A transition from laminar flow to turbulent flow occurs very suddenly as the flow rate increases. Fast flow increases the chance of turbulence.

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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) .

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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 Re > ~ 2000 turbulent flow

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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.

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Re = ρ v L / η

ρ = 103 kg.m-3

v = 5 m.s-1

L = 10 m

η = 10-3 Pa.s

Re = (103)(5)(10) / (10-3)

Re = 5x107

Sydney Harbour Ferry

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Re = ρ v L / η Blood circulation

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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Re = ρ v L / η

Bacterium

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

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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

(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

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

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Flow of a viscous newtonain fluid through a pipeVelocity Profile

Adhesive forces between fluid and surface ⇒ fluid stationary at surface

Parabolic velocityprofile

Cohesive forces between molecules ⇒layers of fluid slide past each other generating frictional forces ⇒energy dissipated (like rubbing hands together)

(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)"