Download Pressure and Fluid Flow_ppt_RevW10

Physics 106 Lesson #7
Pressure and Fluid Flow:
Pascal and Bernoulli
Dr. Andrew Tomasch
2405 Randall Lab
[email protected]
Review: Density
• Density  (Greek letter rho):

mass
volume
The mass of uniform
object is its density
times its volume: m = V
water = 1000 kg/m3 (SI)
water = 1 g/cm3 (CGS)
The specific gravity is density expressed
in units of the density of water. The
specific gravity of x ≡ x /water
Review: Buoyancy
• The upward buoyant
force helps to balance F
B
the downward weight,
reducing the tension in
the string, which is the
apparent weight of the
block → apparent
weight is less than the
true weight
T +
W
Review: Archimedes’ Principle
FB  (  fluidVobject ) g
Magnitude of
Buoyant Force
Weight of
Displaced Fluid
•The buoyant force acting on
a body is equal to the weight
of the fluid it displaces
Review: Archimedes’ Principle Applied
• An object sinks if ρobject > ρfluid
• An object floats if ρobject < ρfluid
• An object is neutrally buoyant
if ρobject = ρfluid
FB
ρobject
Vobject
ρfluid
W
Completely immerse an
object of volume Vobject
with density ρobject in a
fluid of density ρfluid
Review: Floating Objects
• An object floats with some portion of
its volume protruding from the liquid
• When Floating in Equilibrium:
object
Vin

Vobject
liquid
FB
Vin
W
Review: Surface Tension
• Water molecules have an uneven
distribution of electrical charge,
slightly positive at one end, slightly
negative at the other.
• The molecules on the water’s surface
stick together because of electrical
forces and behave like a membrane.
• Small objects more dense than water
can be supported on the water’s
surface.
• Soaps and detergents disrupt this
surface tension.
Demo: soap
powered boat
Introduction to Fluids
• Atoms in fluids can change relative positions easily,
unlike those in solids
• Push on a fluid and it will move in any direction to
release the pressure, not just away from you
• Fluids have a definite volume but no definite shape
Pressure
Pressure is a scalar. Area is a vector. The direction of
an element of area is perpendicular to the surface.
• Pressure P is the force perpendicular to a
surface divided by the area of the surface:
P
force
area
A difference in
pressure across a
surface or object
exerts a net force
perpendicular to
the surface.

F
A
 F  PA
Units of pressure: N/m2 ≡ Pascals (Pa)
(also mm or inches of mercury and lbs/in2)
How Fluids Exert Pressure
• Due to their thermal motion, molecules
are continuously bombarding the walls of
a container even when there is no bulk
movement of the air (wind):
container
p
• Force =
for collisions
t
• Molecules colliding with
with the container walls
produce a pressure
(force/area)
air molecule
Fluids: Not Just Liquids
• We are talking about fluids, not just liquids
– air is a fluid
– fluids like air (a gas) and water (a liquid) differ
primarily in the separation between atoms
– in liquids, the atoms are not as far apart as in
gasses, making them difficult to compress
Fluids Include
Liquids &
Gases
incompressible
fluids
compressible
fluids
Atmospheric Pressure
•The weight of the overlying air produces an atmospheric
pressure at any depth in the Earth’s atmosphere
At sea level atmospheric pressure is:
Patm = 1.013 x 105 Pa ≡ 1 atm
gravity
atmosphere
1 Pa  1 N/m
2
1 atm  1.013  105 Pa
ground
1 mm Hg  133 Pa
Demo: Magdeburg Sphere
Equivalent to the weight of a 10,000 kg (22 ton)
mass distributed over 1 square meter (!)
Hydrostatic Pressure
• Hydrostatic Pressure
Water is incompressible 
mH2O  H2O (volume)
– for a fluid at rest the hydrostatic pressure at a depth h
is the weight of the fluid above an area A divided by
the area A
– the total pressure at depth h is the hydrostatic
pressure plus the atmospheric pressure Patm at the
top of the fluid:
WH2O  mH O g  (  H O )( Ah) g
Patm
2
2
volume
 PH2O 
h
Area = A
WH O
2
A
  H O gh
2
Ptotal  Patm   H O gh
AKA Pascal’s Law
2
Pascal’s Principle
• We have just seen that the pressure at a
depth h in a fluid is the sum of
atmospheric pressure applied above the
fluid and the hydrostatic pressure:
Ptotal  Patm   H O gh
2
• This is an example of Pascal’s Principle:
“Any change in the pressure applied to a
completely enclosed fluid is transmitted
undiminished to all parts of the fluid and
vessel walls”
• Atmospheric pressure is transmitted
throughout the fluid and to the vessel
walls by Pascal’s Principle
Pascal
Pascal’s Law
Caution
Quiz
Ahead
• At a depth h below the surface
of an incompressible fluid:
Pabsolute  Patm  fluid gh
Pascal
Gauge Pressure
≡ Difference From
Atmospheric
The pressure in a static fluid
is the same at all points that
have the same depth
regardless of the container’s
Demonstration
shape:
PA= PB= PC= PD
This is a mercury barometer
The pressure at the
bottom of the mercury
column (A) is the density
of mercury x g x h and is
the same as at the top of
the mercury (B) which is
at atmospheric pressure
and at the same height.
Concept Test #1
A static fluid in a container is
subject to both atmospheric
pressure at its surface and
Earth’s gravitation. The
pressure at A:
A. Depends on the cross
sectional area of the
container
B. Depends on the shape of
the container
C. Is equal to atmospheric
pressure
PA  0   gh
Water has 1/13 the density of
mercury, so a water barometer is
13 times taller or 32 feet tall!
PB  Patm
PA  PB   gh  Patm
760 mm Hg = 1 Atm
= 29.92 inches Hg
Practical Hydrostatics
• Why do your ears hurt when dive into deep
water?
If you dive to the bottom of the deep end of a
pool, 3 m or so, you’ll feel uncomfortable
pressure in your ears. This depth corresponds to
a gauge pressure of (1000 kg/m3)(9.8 m/s2)(3 m) =
2.9 x 104 Pa, about a 30% increase over
atmospheric pressure. This pressure increase is
enough to compress gas inside your eardrum so
that is bends inward in a painful way.
Practical Hydrostatics
• Why are snorkels always so short?
Ever wondered by no one markets 10 ft snorkels?
When you are swimming at a depth h, the pressure
outside you in the water is Patm+  gh. Inside your
lungs, which are directly connected to the air by the
snorkel, the pressure is Patm. So your lungs have to
breathe against a gauge pressure of  gh. You can
expand your chest against a pressure only a small
fraction above atmospheric pressure. At a depth of
3 m you would injure your lungs. At a depth of 10 m
your lungs would implode as the outside water
pressure crushed your chest. This is why scuba
divers breathe carefully regulated pressurized gas.
Practical Hydrostatics
• How does a straw work?
Do you really suck a drink up into a straw?
Actually, you remove the air from inside
the straw thereby lowering the pressure
inside to something close to zero.
Atmospheric pressure at the surface of the
drink acts unopposed to push the drink up
into the straw.
Applied Hydrostatics:
The Hydraulic Lift
• Pascal’s Principle: the pressure
induced by pushing down on a
fluid is transmitted equally
throughout the fluid
• This fact is the basis for many
useful devices including the
hydraulic brakes in your car and
the hydraulic jack shown here
• A small force applied to the small
area A1 generates a pressure that
in turn can apply a large force on
a large area A2
A2
A
B
F1 F2
PA  PB  
A1 A2
A2
 F2 
F1
A1
Continuity in Fluid Flow
• Mass flowing in = mass
flowing out per unit time
• For an incompressible
fluid mass is equivalent to
volume because density is
constant
What Flows In Must Flow Out
A1v1  A2v2
(volume in)/time = (volume out)/time
Units: m3/s
Conservation of Mass:
The Equation of Continuity
(mass in) / time
Incompressible Fluid
1 A1v1  2 A2v2
1  2
Caution
Quiz
Ahead
A1v1  A2v2
(mass out) / time
The product of the crosssectional area and flow speed
is everywhere the same.
Concept Test #2
A river is 10 m wide and 2 m deep at a
certain point where the speed of the
flow is 2 m/s. A little later on, it’s 15 m
wide and 1 m deep. What is the speed
of the flow there?
A.
B.
C.
D.
2.67 m/s
1.34 m/s
6.33 m/s
4.25 m/s
 A1v1  A2 v2
A1v1 (10 m  2 m)  2 m/s
 v2 

A2
(15 m  1 m)
 v2  2.67 m/s
Conservation of Energy:
Bernoulli's Equation
• Work-Energy
Theorem for a
drop of fluid:
Dividing
through by the
volume of the
drop replaces
the mass of
the drop with
the density of
the fluid.
KEi  PEi  Wnc  KEf  PEf
nc ≡ nonconservative
nc work/volume
Gravitational Potential
Energy per Volume
1 2
1 2
 v1   gh1  ( P1  P2 )   v2   gh2
2
2
Kinetic Energy per Volume
Bernoulli’s Equation
Bernoulli’s equation is the statement of energy conservation in a moving fluid
and is shown here for background only. Note that to apply a force to the fluid
(and do nonconservative work) requires a pressure difference across the pipe
Bernoulli’s Equation: A Surprising Result
• Flow in a horizontal pipe:
A1v1  A2v2
Continuity
Demos: Bernoulli Effects
and Venturi Tube
v1  v2
P1  P2
Bernoulli
As the speed of a fluid increases over
a surface, the pressure of the fluid
against the surface decreases.
Example: A Damaged Heart
Bernoulli’s equation dictates
that as the speed of the flowing
blood increases through a
narrowed area of the vessel to
satisfy continuity, the pressure
on the inside of the vessel wall
decreases, resulting in a
tendency to collapse the vessel
wall.
Example: Bernoulli in the Shower
• The Shower-Curtain Effect: The shower
curtain sometimes gets sucked in
towards the shower when the water flows
• The moving water induces moving
currents of air on the shower side of the
curtain, which lowers the air pressure
slightly on that side
Example: A Leak
• Suppose a tank open to the air
has a leak a distance h below
the surface.
• Q: How does the speed v2 of
the exiting water compare with
that of a ball dropped from a
height h ?
• A: It is exactly the same. An
exiting drop of water carries
away the gravitational potential
energy lost from the top of the
water as kinetic energy. From
a work-energy standpoint,
each exiting drop has simply
fallen a distance h under the
influence of gravity!
Patm
Point 1: v1 ≈ 0
h
Point 2
PE=0 Here
Patm v2
More Examples
If an airplane flies too
slowly and/or at too high an
angle of attack, turbulent
flow occurs over the top
of the wing and the wing stalls
resulting in a loss of lift and a
large increase in drag
Most of the lift a wing produces
actually comes from deflecting
the momentum of the moving air.