Lecture 9.
Fluid flow
Pressure
Bernoulli Principle
Surface Tension
Fluid flow
Speed of a fluid in a pipe is not the same
as the flow rate
Relating:
Fluid flow rate to Average speed
L
v is the average speed = L/t
A
v
Volume V =AL
A is the area
Flow rate Q is the volume flowing per
unit time (V/t)
Q = (V/t)
Q = Av
Q = AL/t = A v
Flow rate Q is the area times the average speed
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
Daniel Bernoulli
(Swiss Scientist 1700-1782)
Speed of a fluid is high—pressure is low
Speed of a fluid is low—pressure is high
Bernoulli Equation
1 2
P   v   gh  constant
2
P =pressure at some chosen point
h= height of the point above some reference level
Fluid flow -- Pressure
Bernoulli Principle
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
Bernoulli Equation
1 2
1 2
P1   v1   gh1  P2   v2   gh2
2
2
Fluid flow -- Pressure
Bernoulli Principle
P1
P2
Fluid
v1
v2
1 2
1 2
P1   v1   gh1  P2   v2   gh2
2
2
if h1  h2
1 2
1 2
P1   v1  P2   v2
2
2
1 2 1 2
 v2   v1  P1  P2
2
2
if v2 ishigherthenP2 islower
Fluid flow
Venturi Effect
–constricted tube enhances the Bernoulli effect
P1
v1
P3
P2
A1
v2
A3 v3
A2
If fluid is incompressible, flow rate Q is the
same everywhere along tube
Q = Av
therefore A1v1 = A2 v2
A1
v2 =
v
A2 1
Continuity of flow
Since A2 < A1
v2 > v1
Thus from Bernoulli’s principle P1 > P2
Fluid flow
Bernoulli’s principle:
P1
Explanation
P2
Fluid
Speed increases in smaller tube
Therefore kinetic energy increases.
(Tube horizontal so no change in gravitational
potential energy)
Potential energy associated with pressure is
employed to increase kinetic energy.
Therefore pressure decreases.
Speed increases
pressure decreases
High speed—low pressure
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 =flow rate
A = area
v =average speed
Q = Av
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
Q = Av
A1v1 = A2 v2
A1
v2 =
v
A2 1
v =average speed
Assume blood is incompressible,
flow rate Q is the same everywhere
along artery
2
 d1 
A1  p    0.95cm2
2
2
 d2 
A2  p    0.44cm2
 2
0.95cm2
v2 
15cms 1  32cms 1
0.44cms
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
Molecules close together: forces repulsive
>>>Liquids and solids almost incompressible
Otherwise forces are attractive
Intermolecular forces mainly attractive
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
Forces between Molecules
Surface Tension Molecule at B
 surrounded on all sides by
A
other similar molecules.
 Net attractive force is zero
since it is attracted equally
B
in all directions
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
Measuring surface tension
Measure force (F) required
to stretch liquid film
F
L
2
Surface Tension g = F/L
SI unit of surface tension
Newton per metre Nm-1
Surface Tension
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 Phenomena
Needle on surface of water
Force due to surface tension
Density of steel s » w
mg
But steel needle does not sink
Surface tension results in
Upwards force on needle
Liquid surface behaves like a rubber membrane
under tension
Surface Tension Phenomena
Insects can walk on water
Depression in water surface
(increases surface area)
Surface tension opposes this, which results
in an upwards force that tends to bring back
surface to original flat shape.
Liquid surface behaves like a rubber membrane
under tension
Surface Tension Phenomena
Temperature of liquid increases:
Surface tension of liquids decreases
Molecules moving faster
–bound together less tightly
Surfactant (surface active) substances (soaps)
When added to liquid will lower its surface tension
Uses
Soapy water can penetrate the fine structure
of clothes or skin more easily than water
and hence clean better
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
Premature birth
This surfactant produced late on in the
development of the child
Resultpremature infant suffer respiratory distress
Surface Tension
Capillary action
Forces between like molecules are
called cohesive forces
e.g. between water molecules
Forces between unlike molecules are
called adhesive forces
e.g. between water and glass
Capillary tube in water
F q
h
q
F
meniscus
H 2O
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
Surface Tension
Capillary action
F q
q
F
Fv
h
H 2O
q F
Surface Tension g = F/L
F = gL = g2pr
Vertical component of force Fv = FCosq
Fv = g2prCosq
This force must equal the weight w of the liquid
which rises to height h,
w = mg = Vg = (pr2h) g
Thereforepr2h g = g2prCosq
2g
h = gr Cosq
Therefore small radius (r) → large h
Surface Tension
Capillary action
When adhesive forces between liquid and
glass are less than the cohesive forces
between liquid molecules
Mercury
q
h
F
Hg
F
Cohesive forces are dominant.
Liquid in capillary tube is
depressed to a distance h
below the surface of the
surrounding liquid.
2g
h = gr Cosq
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:
interaction force between two materials at their
contact interface
Chemical bond
Adhesion of material to tooth surface
Advantage
conserves tooth structure
Alternative
Mechanical (amalgam) no bonding
•undercutting required: chamber that is
smaller at the surface and wider inside.
Mechanical interlock
Surface Tension
Effectiveness of adhesion
Important characteristic is the way in which the
adhesive “wets” the surface
Contact angles f
f
Water drop
+wetting agent
f
Water drop
wetting agent reduces surface tension
Wetting characterised by the way in which the
substance spreads out:
f>90o large surface tension
f<90o small surface tension
Surface Tension
Dental application:
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
Chemical Adhesion
Dental restoration
Bond strength depends on contact area
Rough surfaces when
viewed on atomic scale
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
Surface Tension
Chemical Adhesion
Dental restoration
Fluid must flow easily (wetting) to achieve bonding
Bonding to tooth surfaces impaired by
contamination
-Etching debris and saliva
Wetting of enamel and dentine surfaces reduced
by application of aqueous fluoride solution
less plaque adheres to enamel surface treated
with fluoride
Viscosity
Adhesive should spread out (wet)
therefore a low viscosity adhesive is important
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
(a)
DP = 4 (0.072N/m)/(0.25 x10-3m)
DP = 1.152 x103 Nm-2
DP = 1.152 x103 Pa
(b) P = gh
h = P/g
1.152 x103 Pa
h=
(13.6 x103 kgm-3)(9.8ms-2)
h = 8.6 x10-3m
h = 8.6 mm Hg
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