Download fluids - School of Physics

VISUAL PHYSICS
School of Physics
University of SydneyAustralia
gold
m1 V1
r V
m=rV
rgold = m1 / V1 = m2 / V2
gold
m2 V2
r m
V=m/r
pressure !!!
F
A
Gauge and absolute pressures
Pressure gauges measure the pressure above and below
atmospheric (or barometric) pressure.
Patm = P0 = 1 atm = 101.3 kPa = 1013 hPa = 1013 millibars =
760 torr = 760 mmHg
Gauge pressure Pg
Absolute pressure P
P = Pg + Patm
200
100
0
300
400
200
100
0
300
400
Impact of a molecule on the wall of the
container exerts a force on the wall and
the wall exerts a force on the molecule.
Many impacts occur each second and
the total average force per unit area is
called the pressure.
The pressure in a fluid can be defined
as the ratio of the force exerted by the
fluid to the area over which it is
exerted. To get the pressure at a point
you need to take the limit as this area
approaches zero. Because of the weak
cohesive forces between the molecules
of the fluid, the only force that can be
applied by the fluid on a submerged
object is one that tends to compress it.
This means the force of the fluid acts
perpendicular to the surface of the
object at any point.
p0
pressure acting at on surface
Weight of
column
of liquid
F
h
A
Liquid – uniform density r
ph
ph
p0’
p
p
0
0
(0,0)
h
(0,0)
h
Linear relationship between pressure and depth.
If the pressure at the surface increases then the pressure at
a depth h also increases by the same amount.
h
The pressure exerted by a static fluid depends only upon the depth of
the fluid, the density of the fluid, and the acceleration of gravity
ph = p0 + r g h
Static pressure does not depend upon mass or surface area of liquid
and the shape of container due to pressure exerted by walls.
convergence
divergence
HIGH - more uniform
conditions - inhibits cloud
formation
sunshine
sunshine
divergence
convergence
LOW - less uniform
conditions - encourages cloud
formation
Cloudy / rain
?
D
h
A
B
C
A
h
patm
patm
B
C
r
F2
F1
h1
oil
h2
A1
A2
A sharp blow to the front of an eyeball will produce a higher pressure which is
transmitted to the opposite side
Another example is the pressure exerted by a growing tumour. This
increased pressure is transmitted down the spinal column via the
cerebrospinal fluid, and may be detected lower in the spinal cavity
which is less invasive than trying to detect it in the brain itself.
tumor
Increased
pressure
transmitted down
spinal cord
Partially submerged
floating
Floating: partially submerged
Weight of object < weight of fluid
that can be displaced by object
Volume of displaced water <
volume of object
Weight of liquid displaced by
partially submerged object =
weight of object
Water
displaced
Floating: fully submerged
Weight of object = weight of fluid
displaced by object
Water
displaced
Volume of displaced water =
volume of object
Static equilibrium
Some fish can remain at a fixed depth
without moving by storing gas in their
bladder.
Submarines take on or discharge
water into their ballast tanks to
rise or dive
Sinks
Weight of object > weight of fluid
displaced by object
Volume of displaced water =
volume of object
Water
displaced
A steel ship can encompass a great deal of empty
space and so have a large volume and a relatively
small density.
Volume of water displaced
Weight of ship = weight of water displaced
The buoyant force is equal to the weight of the water displaced, not
the water actually present. The missing water that would have filled
the volume of the ship below the waterline is the displaced fluid.
Volume of water displaced. This
volume is not necessarily the
volume present.
Weight of ship = weight of water displaced
FLOATING: weight of
object = buoyant force
+
F
B
F
Object partially submerged
G
Object fully submerged
top
bottom
h
rF
A
bottom
top
ro
A
h
w
ro
rF
oil
water
?
Flift + FB
m
a=0
FG
Flift + FB = FG
Cohesion: attractive forces between “like” molecules
Surface of any liquid
behaves as though it is
covered by a stretched
membrane
F
Net force on molecule
at surface is into bulk of the liquid
T
SF = 0
SF
pull up on surface
push down on surface
restoring forces
Which shape corresponds to a soap bubble?
Surface of a liquid acts like an elastic skin 
minimum surface potential energy 
minimum surface area for given volume
FLOATING NEEDLE
Not a buoyancy phenomena
FT = 2 T L
Length of needle, L
Coefficient of
surface tension T
FT
Equilibrium
FT = FG
F
G
Surface tension acts along
length of needle on both sides
k = 0.70 N.m-1
x = 3410-3 m
radius of ring
R = 2010-3 m
Fspring = Fe = k x
ring
FT + FG
mass of ring
m = 7.0 10-4 kg
FLOATING NEEDLE
Not a buoyancy phenomena
FT = 2 T L
Length of needle, L
Coefficient of
surface tension, T
FT
Equilibrium
FT = FG
F
G
Surface tension acts along
length of needle on both sides
Why can an insect walk on water?
q
FT
FT cosq
Surface tension force acts
around the surface of the leg
FG
q
FT = T L = 2 p R T
For one leg
FG = mg / 6
Flow of a viscous fluid
plate moving with speed v
vz = v
high speed
Z
X
linear velocity
gradient
L
d
vz = (v / L) d
low speed
stationary wall
vz = (d / L) v
vz = 0
Flow of a viscous newtonain fluid through a pipe
Velocity Profile
Cohesive forces
between molecules 
layers of fluid slide past
each other generating
frictional forces 
energy dissipated (like
rubbing hands together)
Parabolic velocity
profile
Adhesive forces between fluid and surface  fluid
stationary at surface
Poiseuille’s Law: laminar flow of a newtonian fluid through a pipe
Q = dV = Dp p R4
8hL
dt
p1 > p2 pressure
drop along pipe 
energy dissipated
(thermal) by friction
between streamlines
moving past each
other
volume flow rate Q = dV/dt
parabolic
velocity profile
Dp = p1 - p2
2R
p1
h
p2
Q = dV/dt
L
streamlines
Streamlines for
fluid passing an
obstacle
v
Velocity of particle
- tangent to streamline
Velocity profile for the laminar flow of
a non viscous liquid
A1
A2
r
r
v2
v1
A1
A1
A2
v1
Low speed
Low KE
High pressure
v2
high speed
high KE
low pressure
v1
Low speed
Low KE
High pressure
Dx2
Y
p2
m
v2
X
time 2
r
p1
Dx1
y2
A1
m
y1
v1
time 1
A2
force
high speed
low pressure
force
high velocity flow
high
pressure
(patm)
low pressure
velocity increased
pressure decreased
slow flow
(streamlines
further apart)
high pressure
5
1
Same speed
and pressure
across river
faster flow
(streamlines
closer together)
low pressure
p large
p large
p small
v small
v large
v small
artery
Flow speeds up at
constriction
Pressure is lower
Internal force acting on
artery wall is reduced
External forces causes
artery to collapse
(1)
Point on surface of liquid
y1
v2 = ? m.s1
y2
(2) Point just outside hole
(1)
(2)
rF
v1 =
?
h
rm
C
yC
A
yA
B
yB
D
Ideal fluid
Real fluid
arm
head
arm
lung
lung
heart
trunk
leg
leg
Floating ball
Resultant
FR
Lift FL
C
B
A
drag
FD
D
Drag force due
to pressure difference
low pressure region
rotational KE of eddies 
heating effect  increase in
internal energy 
temperature increases
motion of air
high pressure region
motion of object
Drag force due
to pressure difference
low pressure region
rotational KE of eddies 
heating effect  increase in
internal energy 
temperature increases
NO CURVE
high pressure region
Drag force is
opposte to the
direction of motion
Tear drop shape for streamlining
v
v
vT
vT
t
t
Object falling from rest
Object thrown
down with initial
speed v0 > vT
Drag force due
to pressure difference
flow speed (high) vair + v
 reduced pressure
v
vair (vball)
MAGNUS EFFECT
flow speed (low) vair - v
 increased pressure
v
high pressure region
low pressure region
Boundary layer – air
sticks to ball (viscosity)
– air dragged around
with ball
The trajectory of a
golf ball is not
parabolic
Golf ball with backspin (rotating CW) with air stream going from
left to right. Note that the air stream is deflected downward with a
downward force. The reaction force on the ball is upward. This
gives the longer hang time and hence distance carried.
lift
Direction plane is moving w.r.t. the air
Direction air is moving w.r.t. plane
low
pressure
lift
q
low pressure drag
attack angle
momentum transfer
high
pressure
downwash
huge vortices