Lecture 9.

Surface Tension

Bernoulli Principle

Fluid flow

Pressure

Fluid flow

L

A v

v is the average speed = L/t

A is the area

Volume V =AL

Flow rate Q is the volume flowing per

unit time (V/t) Q = (V/t)

Q = AL/t = A v

Flow rate Q is the area times the average speed

Relating:

Fluid flow rate to Average speed

Q = A v

Speed of a fluid in a pipe is not the same

as the flow rate

Depends on the radius of the pipe.

example:

Low speed

Large flow rate

Same low speed

Small flow rate

Fluid flow -- Pressure

Pressure in a moving fluid

with low viscosity and laminar flow

Bernoulli Principle

Relates the speed of the fluid to pressure

Speed of a fluid is high—pressure is low

Speed of a fluid is low—pressure is high

Daniel Bernoulli

(Swiss Scientist 1700-1782)

21

2P v gh constant

Bernoulli Equation

P =pressure at some chosen point

h= height of the point above some reference level

Bernoulli’s principle allows the combination of

pressure, speed, and height of a fluid at one point

to be compared to the same three properties at

a different point in the fluid

Fluid flow -- Pressure

2 2

1 1 1 2 2 2

1 1

2 2P v gh P v gh

Bernoulli Principle

Bernoulli Equation

Fluid flow -- Pressure

2 2

1 1 1 2 2 2

1 1

2 2P v gh P v gh

Bernoulli Principle

P1 P2

Fluid

2 2

1 1 2 2

1 1

2 2P v P v

1 2if h h

2 2

2 1 1 2

1 1

2 2v v P P

2 2if v is higher then P is lower

v1

v2

Fluid flow

P1 P2

P3

A1

A2

A3 v1 v3 v2

If fluid is incompressible, flow rate Q is the

same everywhere along tube

A1 = A2 v1 v2 v2 v1

A1

A2

=

> v2 v1 Since A2 < A1

Thus from Bernoulli’s principle P1 > P2

therefore

Venturi Effect

–constricted tube enhances the Bernoulli effect

Q = A v

Continuity of flow

Fluid flow

Explanation

Speed increases in smaller tube

P1 P2

Potential energy associated with pressure is

employed to increase kinetic energy.

Therefore pressure decreases.

Speed increases pressure decreases

High speed—low pressure

Therefore kinetic energy increases.

(Tube horizontal so no change in gravitational

potential energy)

Bernoulli’s principle:

Fluid

Fluid flow

Example

If the average speed of blood in a capillary of

diameter 4 x10-4cm is 3.5 x10-2cms-1,

calculate the flow rate in litres per second.

Q = A v

Q =flow rate A = area

v =average speed

A = pr2 = (2 x10-4cm)2 p

Q = [(2 x10-4cm)2 p](3.5 x10-2cms-1)

Q = 44 x 10-10 cm3s-1

Q = 44 x10-13 litres.s-1

Fluid flow

Plaque build-up on an artery wall reduces its effective

diameter from 1.1 cm to 0.75 cm. If the speed of the blood

Is 15 cms-1 before reaching the region of plaque build-up.

Find the speed of the blood within the plaque region?

Q =flow rate A = area v =average speed

Assume blood is incompressible,

flow rate Q is the same everywhere

along artery

A1 = A2 v1 v2

v2 v1

A1

A2

=

Q = A v

2

211 0.95

2

dA cmp

2

222 0.44

2

dA cmp

1 1

2

0.95 215 32

0.44

cmv cms cms

cms

Fluid flow

Bernoulli effect not limited to

fluid flow in tubes.

Airplane wing profile

Air moves faster over the

upper side of the wing

Pressure is lower,

resulting in lift

Shower curtain

Curtain is “sucked inwards” when water

is switched on

Increased water/air speed inside curtain results

in reduced pressure

Forces between Molecules

Otherwise forces are attractive

Molecules close together: forces repulsive

>>>Liquids and solids almost incompressible

Attractive forces

>>>> phenomena such as surface tension

Water droplet spherical shape

why?

Surface is subject to tension:

makes surface area as small

as possible

Liquid surface behaves like a rubber membrane

under tension

Intermolecular forces mainly attractive

Forces between Molecules

Surface Tension Molecule at B

surrounded on all sides by

other similar molecules.

Net attractive force is zero

since it is attracted equally

in all directions

A

B

Molecule at A

no liquid molecules above,

therefore net force exists which pulls it

towards the interior of the liquid

Net effect of the pull on all molecules at surface

Surface of liquid contracts

>>surface area becomes a minimum

Minimum surface area for a given volume is

when shape is a sphere

Reason why drops of water have

a spherical shape

Forces between Molecules

Surface Tension g = F/L

Measuring surface tension

Measure force (F) required

to stretch liquid film

F

Liquid g(Nm-1)

Blood 0.058

Ethyl

Alcohol

0.023

Mercury 0.44

Water

(0oC)

0.076

Water

(20oC)

0.072

Water

(100oC)

0.059

Soapy

Water

0.037

Surface Tension

SI unit of surface tension

Newton per metre Nm-1

2

L

Surface Tension Phenomena

Liquid surface behaves like a rubber membrane

under tension

Needle on surface of water

mg

Force due to surface tension

Density of steel s » w

But steel needle does not sink

Surface tension results in

Upwards force on needle

Surface Tension Phenomena

Insects can walk on water

Depression in water surface

(increases surface area)

Liquid surface behaves like a rubber membrane

under tension

Surface tension opposes this, which results

in an upwards force that tends to bring back

surface to original flat shape.

Surfactant (surface active) substances (soaps)

When added to liquid will lower its surface tension

Soapy water can penetrate the fine structure

of clothes or skin more easily than water

and hence clean better

Temperature of liquid increases:

Surface tension of liquids decreases

Surface Tension Phenomena

Uses

Molecules moving faster

–bound together less tightly

Surface Tension Phenomena

Lungs

Similar effect occurs

Surface tissue of air sacs (alveoli) has a liquid

with large surface tension that would result in

difficulty in lungs expanding during inhalation

the body

secretes a fluid (surfactant) into the tissue of the

air sacs that lowers the surface tension

of the liquid and allows easy inflation of the air

sacs

This surfactant produced late on in the

development of the child

Result-

premature infant suffer respiratory distress

Premature birth

Surface Tension

Capillary action

Forces between like molecules are

called cohesive forces

Forces between unlike molecules are

called adhesive forces

e.g. between water molecules

e.g. between water and glass

H2O

F F

Adhesive forces

(between water and glass)

greater than the cohesive forces

between waters molecules

Result: water rises in the capillary tube

until the weight of the water column supported

= the upward force

h qq

Capillary tube in water

meniscus

Surface Tension

Capillary action

H2O

Therefore small radius (r) → large h

h

h = 2g

gr Cosq

F F

qq

Surface Tension g = F/L

F = gL = g2pr

Vertical component of force Fv = FCosq

q F Fv

Fv = g2prCosq

This force must equal the weight w of the liquid

which rises to height h,

w = mg = Vg = (pr2h) g

Thereforepr2h g = g2prCosq

Surface Tension

Hg

F F

When adhesive forces between liquid and

glass are less than the cohesive forces

between liquid molecules

Capillary action

Cohesive forces are dominant.

Liquid in capillary tube is

depressed to a distance h

below the surface of the

surrounding liquid.

q

h = 2g

gr Cosq

h

Mercury

Surface Tension

Capillary action

Applications

•Used to draw samples of blood

•Plants: feed using capillary action

•Kitchen towels: absorb using capillary action

Surface Tension

Dental application: filling

Adhesion of material to tooth surface

Mechanical interlock

Chemical bond

Mechanical (amalgam) no bonding

•undercutting required: chamber that is

smaller at the surface and wider inside.

Adhesion:

interaction force between two materials at their

contact interface

Advantage

conserves tooth structure

Alternative

Surface Tension

Contact angles

Important characteristic is the way in which the

adhesive “wets” the surface

Effectiveness of adhesion

Wetting characterised by the way in which the

substance spreads out:

f>90o large surface tension

f<90o small surface tension

f

Water drop

f

Water drop

+wetting agent f

wetting agent reduces surface tension

Adhesion

•good intimate contact

•Large area

Enamel normally covered with thin layer of pellicle

(organic substance deposited from saliva)

Clean surface to achieve good adhesion

Low surface tension adhesive

desirable in promoting adhesion

Surface Tension

Dental application:

Chemical Adhesion

Bond strength depends on contact area

Rough surfaces>>>small contact area

Small force >>large stress at local points

>>result failure

Smooth surface –large contact area –lower stress

Use fluid that flows into irregularities to provide

intimate contact over larger surface area

Example- glass slides with water

Rough surfaces when

viewed on atomic scale

Surface Tension

Dental restoration

Bonding to tooth surfaces impaired by

contamination

-Etching debris and saliva

Viscosity

Adhesive should spread out (wet)

therefore a low viscosity adhesive is important

Wetting of enamel and dentine surfaces reduced

by application of aqueous fluoride solution

less plaque adheres to enamel surface treated

with fluoride

Fluid must flow easily (wetting) to achieve bonding

Surface Tension

Chemical Adhesion Dental restoration

Forces between Molecules

Like water droplets,

Bubbles are also spherical

Inward force due to surface tension

increases pressure of the gas inside

Excess pressure DP inside bubble given by

DP = 4g/r

Surface Tension

Example

Calculate the excess pressure in (a) SI units

and (b) in mm Hg inside a water bubble of

radius 0.25mm

DP = 4g/r

DP = 4 (0.072N/m)/(0.25 x10-3m)

(a)

DP = 1.152 x103 Nm-2

(b) P = gh

h = P/g

h = 1.152 x103 Pa

(13.6 x103 kgm-3)(9.8ms-2)

h = 8.6 x10-3m

h = 8.6 mm Hg

DP = 1.152 x103 Pa

Surface Tension

Calculate the pressures inside bubbles of water

and soapy water each of diameter 1.5cm.

Surface tension of water(gw) is 0.072Nm-1

Surface tension of soapy water (gsw) is 0.037Nm-1

Pressure inside a bubble is given by

P =(4g)/r

Water

Pressure (P) = (4 x 0.072Nm-1)/(0.75 x10-2m)

P = 38.4Nm-2

Soapy water

Pressure (P) = (4 x 0.037Nm-1)/(0.75 x10-2m)

P = 19.6Nm-2