Transcript Chapter 15

Chapter 15
Fluid Mechanics
Fluids
• Fluids (Ch. 5) – substances that can flow (gases,
liquids)
• Fluids conform with the boundaries of any container
in which they are placed
• Fluids lack orderly long-range arrangement of
atoms and molecules they consist of
• Fluids can be compressible and incompressible
Density and pressure
• Density
m
  lim
V  0  V
• SI unit of density: kg/m3
• Pressure (cf. Ch. 14)
P  lim
A0
F
A
• SI unit of pressure: N/m2 = Pa (pascal)
Blaise Pascal
(1623 - 1662)
• Pressure is a scalar – at a given point in a fluid the
measured force is the same in all directions
• For a uniform force on a flat area
F
P
A
Atmospheric pressure
• Atmospheric pressure:
• P0 = 1.00 atm = 1.013 x 105 Pa
Fluids at rest
• For a fluid at rest (static equilibrium) the pressure is
called hydrostatic
• For a horizontal-base cylindrical water sample in a
container
F2  F1  mg
P2 A  P1 A  A( y1  y2 ) g
P2  P1   ( y1  y2 ) g
P  P0  hg
Fluids at rest
• The hydrostatic pressure at a point in a fluid
depends on the depth of that point but not on any
horizontal dimension of the fluid or its container
• Difference between an absolute pressure and an
atmospheric pressure is called the gauge pressure
Pg  P  P0  hg
P  P0  hg
Chapter 15
Problem 28
Barometric pressure in the eye of a hurricane is 0.91 atm (27.2 in. of mercury).
How does the level of the ocean surface under the eye compare with the level
under a distant fair-weather region where the pressure is 1.0 atm?
Measuring pressure
• Mercury barometer
P2  P1   ( y1  y2 ) g
y1  0; P1  P0
y2  h; P2  0
P0  hg
• Open-tube manometer
P2  P1   ( y1  y2 ) g
y1  0; P1  P0
y2  h; P2  P
Pg  P  P0  hg
Pascal’s principle
• Pascal’s principle: A change in the pressure applied
to an enclosed incompressible fluid is transmitted
undiminished to every portion of the fluid and to the
walls of its container
• Hydraulic lever
F1 F2
P 

V  A1x1  A2x2
A1 A2
A2
A1
x1  x2
F1  F2
A1
W  F1x1  F2 x2
A2
• With a hydraulic lever, a given force applied over a
given distance can be transformed to a greater force
applied over a smaller distance
Archimedes’ principle
• Buoyant force:
For imaginary void in a fluid
p at the bottom > p at the top
B  mf g
Archimedes
of Syracuse
(287-212 BCE)
• Archimedes’ principle: when a body is submerged in
a fluid, a buoyant force from the surrounding fluid
acts on the body. The force is directed upward and
has a magnitude equal to the weight of the fluid that
has been displaced by the body
Archimedes’ principle
• Sinking:
mg  B
• Floating:
mg  B
• Apparent weight:
weight apparent  mg  B
• If the object is floating at the surface of a fluid, the
magnitude of the buoyant force (equal to the weight
of the fluid displaced by the body) is equal to the
magnitude of the gravitational force on the body
Chapter 15
Problem 29
On land, the most massive concrete block you can carry is 25 kg. Given
concrete’s 2200-kg/m3 density, how massive a block could you carry
underwater?
Motion of ideal fluids
Flow of an ideal fluid:
• Steady (laminar) – the velocity of the moving fluid at
any fixed point does not change with time (either in
magnitude or direction)
• Incompressible – density is constant and uniform
• Nonviscous – the fluid experiences no drag force
• Irrotational – in this flow the test body will not rotate
about its center of mass
Equation of continuity
• For a steady flow of an ideal fluid through a tube
with varying cross-section
V  Ax  Avt  A1v1t  A2v2t
A1v1  A2v2
Av  const
Equation of continuity
Bernoulli’s equation
• For a steady flow of an ideal fluid:
Etot  K  U g  Eint
• Kinetic energy
mv 2 Vv 2
K

2
2
• Gravitational potential energy
U g  mgy  Vgy
• Internal (“pressure”) energy
Eint  VP
Daniel Bernoulli
(1700 - 1782)
Bernoulli’s equation
• Total energy
Etot  K  U g  Eint
Vv

 Vgy  VP
2
2
Etot v

 gy  P  const
V
2
2
v1
2
2
 gy1  P1 
v2
2
2
 gy2  P2
Chapter 15
Problem 33
Water flows through a 2.5-cm-diameter pipe at 1.8 m/s. If the pipe narrows to
2.0-cm diameter, what’s the flow speed in the constriction?
Questions?
Answers to the even-numbered problems
Chapter 15
Problem 16
10-14
Answers to the even-numbered problems
Chapter 15
Problem 24
5.0 km
Answers to the even-numbered problems
Chapter 15
Problem 36
(a) 1.9 m/s
(b) 31 m/s