Download 16-6 The Equation of Continuity

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Physics for Scientists and
Engineers, 3rd edition
Fishbane
Gasiorowicz
Thornton
© 2005 Pearson Prentice Hall
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Chapter 16
Properties of Fluids
Main Points of Chapter 16
• States of matter: solid, liquid, gas
• Density and pressure
• Pressure in a fluid at rest
• Buoyancy and Archimedes’ principle
• Fluids in motion
• Equation of continuity
• Bernoulli’s equation and applications
• Real fluids: viscosity, turbulence
16-1 States of Matter
Matter has three states: solid, liquid, and gas
In a gas, the molecules are far apart and the
forces between them are very small
16-1 States of Matter
Matter has three states: solid, liquid, and gas
In a solid, the molecules are very close
together, and the form of the solid depends
on the details of the forces between them;
that form is often a lattice. Solids resist
changes in shape.
16-1 States of Matter
Matter has three states: solid, liquid, and gas
In a liquid, the molecules are also close
together and resist changes in density, but
not in shape.
16-1 States of Matter
Qualitative behavior of the
force between a pair of
molecules as a function of
their separation, r:
A substance with no resistance to shear is called
a fluid (it flows). This includes gases and liquids.
16-2 Density and Pressure
Definition of density is mass per unit volume:
(16-1)
Specific gravity of a substance is the
density of that substance divided by the
density of water.
Definition of pressure is force per unit area:
(16-2)
16-2 Density and Pressure
SI units of pressure: pascals
(16-4)
Other units used: pounds per square inch
(psi), atmospheres (atm), bar, and torr.
The pressure in a static fluid at any point
must be the same in all directions.
16-3 Pressure in a Fluid at Rest
Pressure increases with depth, due to the
force of gravity:
Here, the upper surface of
the imaginary cylinder has
the pressure from the rest of
the fluid on it; the lower
surface has that also, but in
addition has the weight of
the imaginary cylinder.
From this, we find how the pressure varies with
depth:
(16-6)
16-3 Pressure in a Fluid at Rest
Since the pressure
varies only with depth
(assuming an
incompressible fluid),
the connected tubes at
right all have fluid in
them up to the same
level.
16-4 Buoyancy and Archimedes’ Principle
Assume block below is in equilibrium. Then
upward forces must equal downward forces.
Upward force: pressure from fluid
Downward force: atmospheric pressure plus weight
Therefore,
(16-8)
16-4 Buoyancy and Archimedes’ Principle
In this case, the object is less
dense than the fluid, and it
floats
Here, the object has the same
density as the fluid, and is in
neutral equilibrium
Now, the object is more dense
than the fluid, and it sinks
16-4 Buoyancy and Archimedes’ Principle
In all cases, the upward force on the object is
called the buoyancy force:
(16-11)
Again, we have the three cases of object density:
16-4 Buoyancy and Archimedes’ Principle
Archimedes’ principle: The buoyant force on an
immersed object equals the weight of displaced
fluid.
The picture below shows an object in the air,
partially submerged, and completely submerged.
16-5 Fluids in Motion
Assumptions for Sections 16-6 and 16-7:
1. The fluid is incompressible.
2. The temperature does not vary.
3. The flow is steady, so that the velocity and
pressure do not depend on time.
4. The flow is laminar rather than turbulent.
5. The flow is irrotational, so there is no circulation.
6. There is no viscosity in the fluid.
Example of fluids where there is circulation:
16-6 The Equation of Continuity
The equation of continuity says that if a flow is
contained within streamlines, whatever goes in one
end has to come out the other end without piling up
anywhere.
16-6 The Equation of Continuity
(16-14)
Therefore:
For an incompressible fluid, where the density is
constant, we have the conservation of flow or flux:
(16-15)
The definition of flux:
(16-16)
16-7 Bernoulli’s Equation
If a fluid is incompressible and has no
viscosity, energy is conserved. Therefore,
the work done on a “piece” of fluid is equal
to the change in its kinetic energy.
Along a streamline:
(16-22)
This is Bernoulli’s equation.
16-8 Applications of Bernoulli’s Equation
For flow at constant height:
(16-24)
Therefore, higher speed means lower pressure. This
is evident when blowing across a light sheet of
paper (below), and is also responsible for air flow in
chimneys and in prairie dog burrows.
16-8 Applications of Bernoulli’s Equation
For flow from holes in a tank, the pressure just
outside the hole is the same as the pressure on the
upper surface. The only variables are the speed of
the flow (assumed zero at the top) and the height of
the fluid.
Therefore:
(16-26)
16-8 Applications of Bernoulli’s Equation
Lift is a result of the Bernoulli effect – where
the air travels faster, the pressure is lower,
resulting in a net upward force on the wing:
(16-28)
Here, L is the lift, w the width of the wing, S the
wingspan, ρ the air density, K a constant depending
on the geometry of the wing, and v the air speed.
16-9 Real Fluids
Viscosity acts like a drag or frictional force; it
slows flow speed relative to a surface.
Top: an ideal fluid without
viscosity; this is called a velocity
profile.
Bottom: the same, with viscosity.
16-9 Real Fluids
The viscosity of a fluid is described by its
coefficient of viscosity; this varies widely
among fluids, and also with temperature.
16-9 Real Fluids
Turbulence is a complex and chaotic
movement of a fluid. Streamlines can no
longer be defined. Turbulent flow is much
more difficult to analyze than laminar flow.
Summary of Chapter 16
• Liquids and gases are fluids – they deform in
response to external forces, and flow.
• Liquids and solids both have closely-spaced
molecules, but in solids they form a lattice.
• If a fluid is incompressible, its density is
constant.
• Pressure is defined as force per unit area.
• Pressure increases with depth in a static fluid.
Summary of Chapter 16, cont.
• An upward buoyant force acts on an object
immersed in a fluid:
(16-11)
• The equation of continuity is a statement of
the conservation of mass:
(16-14)
• A fluid moving under the influence of
gravity follows Bernoulli’s equation:
(16-22)