PHYSICS OF FLUIDS

1. A fluid is matter that has no definite shape and adjusts to the container that it is
placed in. Gases and liquids are both fluids. All fluids are made of molecules. Every
molecules attracts other molecules around it.
2. Liquids exhibit surface tension. A liquid has the property that its free surface
tends to contract to minimum possible area and is therefore in a state of tension.
The molecules of the liquid exerts attractive forces on each other, which is called
cohesive forces. Deep inside a liquid, a molecule is surrounded by other molecules
in all directions. Therefore there is no net force on it. At the surface, a molecule is
surrounded by only half as many molecules of the liquid, because there are no
molecules above the surface.
3. The force of contraction is at right angles to an imaginary line of
unit length, tangential to the surface of a liquid, is called its surface
tension: . Here is the force exerted by the "skin" of the
F F
L
γ =
liquid. The SI unit of the surface tension is N/m.
4. Quantitative measurement of surface tension: let be the weight of the sliding wire,
force with which you pull the wire downward. Obviously
w
T = , net
downward force. Since film has both front and back surfaces, the force acts along a
total length of 2 . The surface tension in the film is defined as, 2 .
2
Hence, th
T w F
F
L FFL
L
γ γ
+ = =
= ⇒ =
e surface tension is .
2
5. Let's ask how much work is done when we stretch the skin of a liquid. If we move the
sliding wire through a displacement , the work done is . Now is a conser
w T
L
x Fx F
γ
+
=
Δ Δ vative
force, so there is potential energy where is the length of the surface
layer change in area of the surface. Thus . So we see that surface
tension is the
U F x L x L
L x A U
A
γ
γ
Δ = Δ = Δ
Δ
Δ = Δ = =
Δ
surface potential energy per unit area
L1= v1t
2 2 L = v t
6.When liquids come into contact with a solid surface, the liquid's molecules are attracted
by the solid's molecules (called "adhesive" forces). If these adhesive forces are stronger
than the cohesive forces, the liquid's molecules are pulled towards the solid surface and
liquid surface becomes curved inward (e.g. water in a narrow tube). On the other hand,
if cohesive forces are stronger the surface becomes curved outwards (e.g. with mercury
instead). This also explains why certain liquids spreadwhen placed on the solid surface
and wet it (e.g., water on glass) while others do not spread but form globules (e.g.,
mercury on glass).
0 p
p
7. The surface tension causes a pressure difference between the
inside and outside of a soap bubble or a liquid drop. A soap bubble
consists of two spherical surface films with a thin layer of
0
liquid
between them. Let pressure exerted by the upper half, and
external pressure. force exerted due to surface tension is
2(2 ) (the "2" is for two surfaces). In equilibrium the ne
p
p
π rγ
=
= ∴
2 2
0
0
0
t forces
must be equal: 2(2 ) . So the excess pressure is
4 . For a liquid drop, the difference is that there is only
surface and so, excess pressure 2 .
r p r r p
p p
r
p p
r
π π γ π
γ
γ
= +
− =
= − =
1 2 1 2 1 1 1 1 1
2
8. From the fact that liquids are incompressible, equal volumes of liquid must flow in both
sections in time t, i.e. V / V / V V . But you can see that V v and
similarly that V
t t AL A t
A
= ⇒ = = =
= 2 2 2 2 1 1 2 2 v . Hence v v . This means that liquid will flow
faster when the tube is narrower, and slower where it is wider.
L =A t A =A
1 v
2 v
This means that the pressure is smaller where the fluid is flowing faster! This is exactly why
an aircraft flies: the wing shape is curved so that when the aircraft moves through the air,
the air moves faster on the top part of the wing than on the lower part. Thus, the air pressure
is lower on the top compared to the bottom and there is a net pressure upwards. This is called
lift.
2
1 1 1
the gravitational potential cancels,
1 v
2
p+ ρ +ρgy 2
2 2 2
1 v
2
=p+ ρ +ρgy
( )
2 2
1 1 2 2
1 2 2
1 1 2 2 2 1 2 1 2 1
2
1 v 1 v.
2 2
Now, since the liquid is incompressible, it flows faster in the narrower part:
v v v v 1 v v .
2
p p
A A A p p
A
ρ ρ
ρ
⇒ + = +
⎛ ⎞
= ⇒ =⎜ ⎟ ⇒ = − −
⎝ ⎠
2
9. When a fluid flows steadily, it obeys the equation:
1 v constant
2
This famous equation, due to Daniel Bernoulli abou
p+ ρ +ρgy=
Bernoulli's Equation :
t 300 years ago, tells you how fast
a fluid will flow when there is also a gravitational field acting upon. For a derivation,
see any of the suggested references. In the following, I will only explain the meaning of
the various terms in the formula and apply it to a couple of situations.
10. Let us apply this to water flowing in a pipe whose crossection decreases along its
length. 1 1 2 (A is area of the wide part, etc).There is no change in the height so y= y and
Summary of Lecture 19 – PHYSICS OF SOUND
1. Sound waves correspond to longitudinal oscillations of density. So if sound waves are
moving from left to right, as you look along this direction you will find the density of air
greater in
-12 3
0
some places and less in others. Sound waves carry energy. The minimum
energy that humans can hear is about 10 watts per cm (This is called , the threshold
of hearing.)
2. To measure the
I
intensity of sound, we use as the unit. Decibels (db) are a relative
measure to compare the intensity of different sounds with one another,
relative in
decibels
R ≡ 10
0
tensity of sound log (decibels)
Typically, on a street without traffic the sound level is about 30db, a pressure horn creates
about 90db, and serious ear damage happens around 120db.
3
I I
I
=
( ) ( )
( )
. A sound wave moving in the direction with speed v is described by
, sin 2 v
where , is the density of air at a point at time . Let u
m
x
x t x t
x t x t
π
ρ ρ
λ
ρ
= −
s understand various aspects
of this formula.
a) Suppose that as time increases, we move in such a way as to keep v constant. So
if at 0 the value of is 0.23 (say), then at 1
t xt
t x t
−
= =
( )
( )
the value of would be v+0.23, etc.
In other words, to keep the density , constant, we would have to move with the
speed of sound , i.e. v.
b) In the expression for , , repla
x
x t
x t
ρ
ρ
( ) ( )
ce by . What happens? Answer: nothing,
because sin2 v sin2 v . This is why we call the "wavelength",
meaning that length after which a wave repeats itself.
c) In the
x x
x t x t
λ
π π
λ λ
λ λ
+
+ − = −
expression for ( , ) , replace by where / v. What happens? Again
the answer: nothing. T is called the time period of the sound wave, meaning that time
after which it repeats itsel
ρx t t t+T T=λ
f. The frequency is the number of cycles per second and is
obviously related to T through 1/ .
d) It is also common to introduce the the and :
T
wavenumber k angular frequency
ν
ω
=
k 2 and 2 2
T
π π
ω πν
λ
= = =
2 r
1 r
1 r
λ vO
4. : The relative motion between source and observer causes the observer to
receive a frequency that is different from that emitted by the source. One must distinguish
between
Doppler Effect
two cases:
0
. If the observer was at rest, the number of waves she
would receive in time would be / (or v / ). But if she is moving towards the
source with speed v
t tT t λ
Moving observer, source at rest
0
(as in the above figure), the additional number of waves received is
obviously v / . By definition,
frequency actually heard number of waves received
unit time
t λ
ν ′= =
0
0 0 0
0
v v
v v v v v v
v / v
We finally conclude that the frequency actuall heard is 1 v . So as the
v
observer runs towards the source, she hears a higher frequency (
t t
t
ν λ λ ν
λ ν
ν ν
∴ ′= + = + = + = +
′= ⎛⎜ + ⎞⎟
⎝ ⎠
s
higher pitch).
As the source runs towards the observed, more
waves will have to packed together. Each wavelength is reduced by v . So the
wavelength seen by
ν
Moving source, observer at rest :
( ) ( )
s
s s
o
s
the observer is v v . From this, the frequency that she
hears is v v v .
v v / v v
v v . As you can easily see, the
v v
above two results are s
λ
ν ν
ν ν
λ ν
ν ν
′ = −
′= = =
− −
′ = +
−
Moving source and moving observer :
pecial cases of this