buoyancy archimedes principle fluid flow viscosity · 2016-01-11 · buoyancy and archimedes...

Fluid flow

Viscosity

Lecture 8

Buoyancy

Archimedes Principle

Buoyancy and Archimedes Principle

Archimedes (287 BC – 212 BC)

Greek Physicist, Mathematician

Archimedes principle:

Any object completely or partially immersed

in a fluid experiences an upward or buoyant

force equal to the weight of the fluid it displaces.

Net upward force is the buoyant force

is equal to the weight of the displaced fluid.

Pressure is greater at

greater depth: P = rgh

therefore the upward force is

greater at the bottom of an

object than the downward

force at the top of the object

One’s body ,arms and legs etc, feel lighter under water.

Easy to lift someone if they are in a swimming pool.

Buoyant force easily explained

F1

F2

F2 > F1


• If an object floats then the buoyant force

must equal its weight.

Buoyant force Fb

Fb = P2A2- P1A1

Fb = weight of fluid

= weight of fluid

displaced

Fb = mfg =rfluidVfluidg

P1A1

P2A2 mg

Cube of same fluid

h2

h1

P1 = rgh1

P2 = rgh2

• If an object sinks then its weight must be

greater than the buoyant force.

Cube remains stationary

Fb = P2A2- P1A1= mfg

where mf =mass of fluid

Cube of different material (only mass of cube

changes) buoyant force remains unchanged

(Fb = mfg)


Apparent weight = net downward force

= wobject - Fb

0 0 wobject - Fb

wobject

Fluid

Fb = weight – apparent weight

Fb = weight of liquid displaced


Example

Calculate the volume and density of an

Irregular shaped object.

A person has a mass of 75kg in air and an

apparent mass of 2kg when submerged in water.

Calculate the volume and density of the person.

mass of water displaced = Mass – apparent mass

mass of water displaced = 75kg – 2kg = 73kg

Volume of water displaced =(mwater)/(rwater)

= 73kg/1000kgm-3 = 73x10-3m3

Therefore volume of person = 73x10-3m3

rperson =(mperson)/(volumeperson)

rperson =(75kg)/(73x10-3m3) = 1027kgm-3

Fb = weight – apparent weight

Viscosity and Fluid flow

One characteristic of fluids is that they flow

Fluid flow in tubes:

examples:

Flow rate Q defined as volume flowing

per unit time (V/t)

Flow rate depends on

•pressure difference

•other characteristics of the fluid and tube

Q = (V/t)

•IV tubes,

•garden hoses,

•circulatory system

SI units: m3 per second


Flow rate depends on pressure difference

P1 = P2 no flow

P1 > P2 flow direction

P1 < P2 flow direction

Resistance is all factors that impair flow; example,

• friction between fluid and tube,

•friction within the fluid*

Q = (V/t)

Flow rate 1 2P PQ

R

SI unit of viscosity N.s.m-2

L

P1 P2

where R is the resistance to flow.

* known viscosity h (Greek letter eta).

Flow can be characterised as

• laminar

• turbulent


Turbulent flow

Fluctuating flow patterns

Caused by constrictions or obstructions

Turbulence causes increased resistance to flow

Laminar flow


Smooth, streamlined, quiet

Resistance to laminar flow of an incompressible

fluid in a tube is a function of

• tube length (L),

• radius (r)

• viscosity h

4

8 LR

r

h

4

1R

r double radius; resistance

decreases by a factor of 16


4

8 LR

r

h

Poiseuille’s Law

French scientist Jean Poiseuille (1797-1869)

studied fluid flow in tubes; in particular blood flow

Flow rate 1 2P PQ

R

( 4

1 28

rQ P P

L

h


Fluid flow

dentinal tubules (microscopic channels)

radiate outward through the dentine from

the pulp to the dentine-enamel interface.

Sensitivity to cold, heat, air,

Concerned with fluid flow

within the tooth

If outer enamel develops a crack or cavity

when you eat cold food, for example

normal fluid flow within the dentine may be

disrupted – affecting the pulp (nerves , blood

vessels etc), resulting in pain.

Teeth

Fluid flow


To change flow rate; change radius of tube

•Blood flow rate in circulatory system changed

by constricting or dilating blood vessels

Examples

•Clamp on IV tubing

•Tap on garden hose

( 4

1 28

rQ P P

L

h

If effective radius of a vein or artery is reduced

by a constriction (deposits)

•blood circulation problems occur.

4Q r

Result: heart has to work considerably harder to

produce a higher blood pressure in order to

maintain the required flow rate.

Flow rate Q

Viscosity

Resistance to fluid flow

Restorative materials are manipulated in fluid

state to achieve desired result

Dentistry

Viscosity of cement should be low so that it

will flow over tooth surface to achieve good

retention

Materials

Prepared as fluid pastes

adapted to the required shape

subsequently solidify

Setting of such materials

Change of viscosity with time

Materials

Viscosity V

iscosity

Time

Vis

cosity

Time

Vis

cosity

Time

No well defined

working time or

setting time

Well defined long

working time and

sudden setting time

Long working time

reasonable

setting time

Initial low viscosity for dispensing and

moulding.

Followed by large increase in viscosity

during setting

Working time – time during which the material

can be easily manipulated (low viscosity)

Change of material viscosity with time

Restorative Dental Material

Setting time – time at which viscosity

becomes very high


Example

If effective radius of the artery is halved, by

what factor does the blood flow rate change?

( 4

0 1 28

rQ P P

L

h

(

4

1 2

2

8

r

Q P PL

h

4

0

1 1

2 16

Q

Q 0

16

QQ

Factor of 16


By what percentage would the radius of the

arteries have to decrease to reduce the blood

flow rate by a factor of 3.

( 4

0 1 28

rQ P P

L

h

(

4

01 2

3 8

nQ rP P

L

h

Example

( 4

01 2

3 8

nQ rP P

L

h

( 4

0 1 28

rQ P P

L

h

4

4

4

1

3

10.76

3

n

n

r

r

r

r

Dividing the two equations

0.76nr r

Reduction of (1-0.76)x100% = 24%


Example

Patient receiving blood transfusion through a needle

of length 2.3cm and radius 0.2mm. The reservoir

supplying the blood is 0.7m above the patients arm.

Determine the flow rate through the needle.

Density of blood is 1050kgm-3. Viscosity of blood is

2.7x10-3N.s.m-2

Pressure difference P1-P2 = rgh

( 4

1 28

rQ P P

L

h

P1 – P2 = (1050kgm-3)(9.8ms-2)(0.7m)

= 7.20 x103Pa.

( 4 4

3

-3 -2 2

(2 10 )7.20 10

8(2.7 10 Nsm ) 2.3 10

mQ Pa

m

Q = 7.28 x 10-8m3s-1


Flow rate and pressure drop 1 2P P

QR

1 2P P QR

P1 P2

Water supply

to homes

If P1 = P2 no flow

If many users draw water simultaneously

large flow large pressure drop

Hence P2 is smaller than when usage is light.

Solutions: increase P1 and/or

decrease R (increase diameter of water main)

Water main

Both occur during vigorous exercise

in the circulatory system

Blood pressure increases and arteries dilate


Objects that float

Object’s weight is equal to the buoyant force Fb

wobject = Fb

Applying Archimedes principle

wobject = wfluid (displaced)

Fb is equal to weight of fluid displaced

therefore

True only if object floats

Fb = mfg = rfluidVfluidg Fb = mfg = robjectVobjectg

rfluidVfluidg = robjectVobjectg

robject Vfluid

rfluid Vobject =

“The fraction of the object that is submerged is

equal to the ratio of the density of the object

to the density of the fluid.”

volume of object submerged

= volume of fluid displaced

Why do ships float

Average density is less than that of water


Hot air (or Helium) balloons

Hot air less dense that cold air resulting in a

net upward force on the balloons

Applications

Human brain is immersed in cerebrospinal fluid,

density ≈ 1007 kgm-3

average density of brain ≈ 1040 kgm-3

Most of brain’s weight is supported by

the buoyant force

Ship partially submerges until weight of ship

equals weight of water displaced.


Objects that sink (totally submerged)



Fb = mfluidg = (Vfluid)(rfluid)g

Downward gravitational force on object

mobjectg = (robject) (Vobject)g

Net force upwards on object = Fb - wobject

Net force upwards= (rfluid-robject)(Vobject)g

If rfluid > robject object will float

If rfluid < robject object will sink

Fb = (Vobject)(rfluid)g

But volume of fluid displaced = volume of object

therefore

= (Vobject)(rfluid)g -(robject) (Vobject)g


iceberg icebergwater water

iceberg water iceberg water

V m

V m

r r

r r

Calculate the fraction of an iceberg’s volume that is

submerged when it floats in water. Density of ice 917kgm-3

wiceberg = wwater

miceberg = mwater

3

3

9170.917

1000

iceberg water

water iceberg

V kgm

V kgm

r

r

Fb = rfluidVfluidg

Weight of object (w)

= robjectVobjectg

w = Fb

Vwater = volume of water displaced


Example

Determine the density of a liquid if an object of

known volume (50cm3) and mass (0.150kg)

has an apparent mass of 0.105kg in the liquid.

0.150kg - 0.105kg

=0.045kg is displaced by the object


Buoyant force is equal to weight of liquid displaced

Mass – apparent mass =mass of liquid displaced

Volume of object =50cm3 = 50x10-6m3

= Volume of liquid displaced

Therefore density of liquid =

(0.045kg) /(50x10-6m3)

= 900kgm-3

Weight – apparent weight =weight of liquid displaced


A 70 kg statue lies at the bottom of the sea

(Density 1.025x103 kg m-3). Its volume is

3.0x104cm3. How much force do you need

to lift it?

Weight - Apparent weight

= weight of fluid displaced

mg –Wapp = weight of fluid displaced

Wapp = mg –mfluid g

Wapp = mg –rfluidVg

Wapp=9.8ms-2(70kg –1.025x103 kg m-3 3.0x10-2m3)

Wapp = 9.8ms-2( 70kg –30.75kg)

384.65N

Wapp = g(m –rfluidV)


You immerse an object in water and measure

the apparent weight. You then repeat the

process in salt solution (density slightly

higher than water) would you expect the

new apparent weight to be

a) Higher

b) Same

c) Lower



Fb is equal to mg of fluid displaced

Fb is equal to rVg of fluid displaced


4m x 6m x d x (1000kgm-3) x (9.8ms-2) = 9.41kN

A small car ferry measures 4.00 metres wide by

6.00metres long. When several cars of total weight

9.41 x103N drive on to it, it sinks an additional

depth (d) into the water. Calculate the additional depth.

Density of water = 1000kgm-3

Additional volume (V) of ferry below water line

V = 4m x 6m x d

Additional mass of ferry = density x Volume

It floats, therefore:

object’s weight is equal to the buoyant force Fb

Wobject = Fb Applying Archimedes principle

wobject = wfluid


Additional weight of ferry = density x Volume x g

d = 4cm

buoyancy archimedes principle fluid flow viscosity · 2016-01-11 · buoyancy and archimedes...

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