Lecture notes for week 7

7/1
Week 7
Topics:
• Surface tension (Study guide 13. Section 11.7.)
• Capillarity (Study guide 13. Section 11.8.)
• Non-viscous fluid flow (Study guide 14. Sections 12.1–12.3.)
Next deadline:
Friday October 31 — Quiz #8 (12–13); requires labs 12 and 13
7/2
Surface tension (Sec. 11.7)
The surface of a liquid (e.g. water-air interface) tends to act as a
skin: the surface is under tension.
An important physical property of liquids is then their
surface tension γ (gamma).
The surface tension is a force per unit length and its unit is the
N/m (Newton per meter).
7/3
Different liquids have different surface tensions, and for the same
liquid, the surface tension depends on temperature (see text Tables
11-3, 11-4).
Surface tension at 20◦ C
Surface tension of water
liquid
γ (N/m)
Temperature (◦ C)
γ (N/m)
water
72.8 × 10−3
0
75.6 × 10−3
blood plasma
50.0 × 10−3
20
72.8 × 10−3
benzene
28.9 × 10−3
60
66.2 × 10−3
mercury
464 × 10−3
100
58.9 × 10−3
Surface tension is at play in many phenomena:
water forming drops; insects walking on water; mechanics
of breathing (box 11.2); formation of soap bubbles; etc.
7/4
The action of surface tension can easily be demonstrated with
soapy water.
A soap film in a wire frame will support the weight of a string to
which a small mass is attached.
l
surface
surface
soap film
string
F
front view
side view
The film has 2 interfaces with air, and the weight F is supported
by these two surfaces, each of which has a length `.
7/5
The surface tension is defined by
weight supported by film
surface tension =
2(length of film)
or
F
γ=
2`
This means that for the same liquid (same γ), a film twice as wide
can support twice the weight.
(Changing the height of the frame, and therefore the area of the
film, does not change the result.)
7/6
Soap bubble:
The pressure of the air inside a soap bubble is higher than the
pressure of the air outside.
This pressure difference is supported by the bubble’s surface
tension.
Both surfaces — inside and outside — contribute.
air (high pressure)
r
air (low pressure)
7/7
The pressure difference depends on the surface tension γ and the
bubble’s radius r:
P (inside) − P (outside) = 2
4γ
2γ
=
r
r
This means that for the same liquid (same γ), a smaller bubble
(smaller r) can support a larger pressure difference.
In this equation, P (outside) is often (but not always) the
atmospheric pressure.
The extra factor of 2 accounts for the fact that the bubble has two
interfaces with air.
7/8
Air bubble in water:
The pressure inside an air bubble is higher than the pressure of
the surrounding water.
water
air
r
In this case,
2γ
P (air) − P (water) =
r
The extra factor of 2 is missing because the bubble has a single
interface.
7/9
Example #1:
A soap bubble with a radius of 3 cm is floating in air. What is the
pressure difference across the surface of the bubble? (The surface
tension of soapy water is 69 × 10−3 N/m.)
Example #2:
An air bubble is located in water 3.0 m below the surface. The
gauge pressure inside the bubble is 4.0 atm. What is the radius of
the bubble?
7/10
Because a force is associated with stretching a film, there is also
work required to do this.
For an air bubble in liquid,
W = γ ∆A
while for a soap bubble in air,
W = 2 γ ∆A
Here ∆A is the change in area of the film.
You will also need to know formulas for the area and volume of a
sphere.
7/11
Capillarity (Sec. 11.8)
When a narrow tube is lowered into a liquid, the liquid either rises
or depresses inside the tube.
This phenomenon is called capillary rise, or capillary
depression.
It plays an important role in the movement of water in plants, or
the flow of blood in small blood vessels (capillaries).
A liquid will rise if it tends to wet a surface.
A liquid will depress if it tends not to wet a surface.
This property of liquids (to wet or not to wet) is characterized by
its contact angle θ (theta) with the surface.
7/12
Contact angle:
liquids on a glass plate
θ
θ
water (wets)
mercury (doesn’t wet)
liquids in a glass tube
air
air
meniscus
meniscus
θ
water
θ
mercury
A liquid wets if θ < 90◦ . A liquid doesn’t wet if θ > 90◦ .
7/13
Capillary rise or depression:
air
r
y
liquid
−y
air
liquid
r
rise (liquid wets)
depression (liquid doesn’t wet)
The rise (or depression) y is related to the liquid’s surface tension
γ, its density ρ, its contact angle θ, and the tube’s radius r:
y=
2γ cos θ
ρgr
This is positive if θ < 90◦ , and negative if θ > 90◦ .
For the same liquid (same γ, ρ, and θ), the rise (or depression)
increases if the tube is made smaller.
7/14
Example #1:
A tube of diameter 2.0 mm is inserted into water. By how much
does the water rise in the tube? (Water has a contact angle of 30◦
with the tube.) guide
Example #2:
The same tube is inserted into mercury, and the liquid is seen to
depress by 0.55 cm. What is the contact angle of mercury?
7/15
What you should know for quizzes about the lab for this study
guide:
Understand the blowing bubbles method. You do not need
to know molar masses or how to calculate relative
numbers of molecules.
7/16
Another example
During an experiment you are asked to determine the surface
tension of a blood sample using capillarity.
You know that the density of blood is 5% higher than the density
of water. You also know that blood and water have the same
contact angle with glass.
While you make your measurements you find that water rises in the
tube a factor 1.69 higher than blood. What is the surface tension of
the blood sample?
7/17
Study guide 14
Fluid dynamics
So far our description of fluids (gases or liquids) has been restricted
to static situations — nothing moved.
Now we shall study the dynamics of fluids — how they move.
The motion of fluids is important in the description of:
blood circulation; motion of airplane wings in air; motion
of fish in water; etc.
We will study the following topics:
continuity equation (conservation of flow rate); Bernoulli’s
equation (motion of nonviscous fluids); viscosity in
fluids; Poiseuille’s law (motion of viscous fluids).
7/18
Fluid flow (Sec. 12.2)
A fluid flowing through a pipe moves at an average speed v.
The speed changes when the cross-sectional area A of the pipe
changes: v increases if A decreases.
A1 v1
A2 v2
A3 v3
The law that describes this is called the equation of continuity.
It says
A1 v 1 = A 2 v 2 = A 3 v 3
7/19
Defining the flow rate Q = Av (whose unit is m3 /s), we have
Q = Av = constant
This relation holds over the entire length of the pipe.
(This result follows from the fact that equal quantities of fluid must
go through all parts of the pipe at all times.)
If a pipe branches off into two (or more) pipes (as happens in
arteries), then the statement of continuity generalizes to
A1 v 1 = A 2 v 2 + A 3 v 3 + · · ·
A1
A2
v2
A3
v3
v1
7/20
Example:
Suppose that a cylindrical pipe of radius r1 branches off into two
identical pipes of radius r2 = 12 r1 . By how much does the speed of
the fluid increase?
7/21
Bernoulli’s equation (Sec. 12.3)
We have seen that when a fluid moves in a horizontal pipe of
varying cross section, its speed changes, and it undergoes
acceleration.
v1 (low)
P1 (high)
v 2 (high)
P2 (low)
This acceleration must be produced by a net force acting on the
fluid.
This force, in turn, is produced by a pressure difference within
the fluid: where the speed is low there is a high pressure, and
vice-versa.
7/22
The law that expresses this is
1
1
2
P1 + ρv1 = P2 + ρv2 2
2
2
where ρ is the fluid’s density.
This follows from energy conservation:
Consider a piece of fluid of volume V . Its mass is ρV and
1
1
(ρV )v2 2 − (ρV )v1 2 = change of fluid’s kinetic energy
2
2
This change is produced by the work done by the pressure
difference:
work
=
(pressure difference)(area)(distance)
=
(P1 − P2 )V
Equating both sides and dividing by V gives back the result.
7/23
When the pipe is not horizontal, but changes in elevation, the law
must be modified to account for gravitational potential energy.
y2
y1
It becomes
1
1
P1 + ρv1 2 + ρgy1 = P2 + ρv2 2 + ρgy2
2
2
where y is the pipe’s elevation.
This result, known as Bernoulli’s equation, can also be
expressed as
1
P + ρv 2 + ρgy = constant
2
7/24
We have accounted for two types of forces acting on the fluid:
pressure differences, and gravity.
Are there other forces that might be relevant?
The answer is yes: viscosity.
This is an internal force that tends to slow the fluid down.
Bernoulli’s equation is valid for fluids whose viscosity is too low to
be significant.
It works well for water (and blood), but not for honey.
(Viscosity will be considered in the next section.)
7/25
Example #1:
Blood pressure is measured to drop by 40.0 Pa in an obstructed
artery. Supposing that the artery is horizontal, that its
unobstructed diameter is 3.00 mm, and knowing that the blood’s
flow rate is 1.00 × 10−6 m3 /s, what is the artery’s diameter in the
obstruction?
Example #2:
A pipe of constant cross section takes water from the ground to the
top of a 10.0 m tower. If the water’s pressure on the ground is
1.00 × 105 Pa, what is its pressure at the top of the tower?

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