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Transcript
Hydrostatics:
Fluids at Rest
Fluid Mechanics
• applying Newtonian
principles to fluids
• hydrostatics—the study of
stationary fluids in which
all forces are in equilibrium
Fluid Mechanics
• hydrodynamics—the study
of fluids in motion
Density
•
•
•
•
abbreviation: ρ
mass per unit volume
g/cm³ is commonly used
SI unit: kg/m³
Density
• specific gravity: density
relative to water
• dimensionless number
• numerically equal to the
density of the substance in
g/cm³
Units of Pressure
• Pressure is defined as the
force exerted perpendicular
to a unit area.
• When a fluid is at rest, the
pressure is uniform
throughout the fluid in all
directions.
Units of Pressure
• At the boundaries of a
fluid, the container exerts a
pressure on the fluid
identical to the pressure
the fluid exerts on the
container.
Units of Pressure
• SI unit: Pascal (Pa)
• Earliest: atmosphere (atm)
• 1 atm = 1.013 × 105 Pa
• torr
• bars and millibars (mb)
• 1 atm = 1.013 bar = 1013 mb
Units of Pressure
• gauge pressure (Pg) often
used with piping systems
• absolute pressure (P)
Incompressible Fluids
• pressure changes with
depth
• density is usually assumed
to be constant throughout
depth
• y = d2 = d1 + Δd
• ΣF = 0 N
Incompressible Fluids
• ΣFy = Fd1 + Fd2 + Fw = 0 N
• to calculate the pressure at
any depth d:
Pd = Pref + ρgd
Incompressible Fluids
Pd = Pref + ρgd
• d is expressed as a negative
scalar distance
• g = -9.81 m/s²
• Pref is atmospheric pressure
if the liquid’s container is
open to the atmosphere
Compressible Fluids
• usually referring to gases,
since their density is not
constant with height/depth
-
P = Pref e
ρref
Pref
|g|h
Compressible Fluids
• must remember that
temperature also affects the
pressure of a gas
Hydraulic Devices
• Pascal’s principle: the
external pressure applied
to a completely enclosed
incompressible fluid is
distributed in all directions
throughout the fluid
Hydraulic Devices
• machines that transmit
forces via enclosed liquids
• small input forces can
generate large output
forces
Hydraulic Devices
• note the
crosssectional
areas of
each
• Fout = nFin
Hydraulic Devices
• note the
distance
each piston
travels
Pressure Indicators
• manometer
• barometer
• first instrument to
accurately measure
atmospheric pressure
• used mercury
Buoyancy
• famous problem:
Archimedes and the crown
• What happens when an
object is placed in a fluid?
Buoyancy
• for object in fluid:
• Fw-o: gravitational force
on object in fluid
• Fb: buoyant force on
object
• Fb = ρ|g|V
Buoyancy
• Fb = ρ|g|V
• ρ is the density of the
displaced fluid
Buoyancy
• Archimedes’ principle: any
system that is submerged
or floats in a fluid is acted
on by an upward buoyant
force equal in magnitude to
the weight of the fluid it
displaces
Buoyancy
• If the buoyant force is
equal to the system’s
weight, the forces are
balanced and no
acceleration occurs.
• requires object and fluid to
have equal density
Buoyancy
• If the weight of a system is
greater than that of the
displaced fluid, its density
is greater than the fluid’s.
• Since weight exceeds the
buoyant force, the object
will sink.
Buoyancy
• If the weight of a system is
less than that of the
displaced fluid, its density
is less than the fluid’s.
• Since buoyant force is
greater than weight, the
object will accelerate up.
Buoyancy
• When the object rises to
the surface of the liquid, its
volume remaining beneath
the surface changes the
buoyant force until they are
in equilibrium.
Buoyancy
• This is also true with
gases.
• The density of a gas
changes with altitude and
temperature.
• The object may respond to
a change in pressure.
Center of Buoyancy
• Every object submerged in
a fluid has both a center of
mass and a center of
buoyancy.
• These are the same for
objects of uniform density
that are completely
submerged.
Center of Buoyancy
• defined: the center of mass
of the fluid that would
occupy the submerged
space that the object
occupies
Center of Buoyancy
• If the center of mass and
center of buoyancy are not
the same, the object will
experience a torque and
rotate.
• The center of buoyancy will
be directly above the center
of gravity.
Hydrometer
• instrument used to
measure density
• has many uses
Hydrodynamics:
Fluids in Motion
Ideal Fluids
• assumptions:
• the fluid flows smoothly
• the velocity of the fluid
does not change with
time at a fixed location
in the fluid path
Ideal Fluids
• assumptions:
• the density of the fluid is
constant (incompressible)
• friction has no effect on
fluid flow
Ideal Fluids
• Streamlines
• not a physical reality
• laminar
• turbulent
• flow tube
Ideal Fluids
• The rate of volume and
mass flow into a segment
of a flow tube equals the
rate of volume and mass
flow out of the flow tube
segment.
Flow Continuity
• equation of flow continuity:
A1v1 = A2v2
• requires tubes with smaller
cross-sectional areas to
have higher fluid velocities
Bernoulli’s Principle
• background equations:
ΔK = ½ρΔVv22 – ½ρΔVv12
Equation 17.12
ΔU = ρΔV|g|h2 – ρΔV|g|h1
Equation 17.13
Bernoulli’s Principle
• background equations:
Wncf = ΔK + ΔU
Equation 17.14
Wncf = P1ΔV – P2ΔV
Equation 17.15
Bernoulli’s Principle
• Bernoulli’s Equation:
P1 + ½ρv12 + ρ|g|h1 =
P2 + ½ρv22 + ρ|g|h2
Bernoulli’s Principle
• if the velocity does not
change: v1 = v2
P1 + ½ρv12 + ρ|g|h1 =
P1 + ρ|g|h1 = P2 + ρ|g|h2
P2 + ½ρv22 + ρ|g|h2
Bernoulli’s Principle
• if the elevation of the fluid
does not change: h1 = h2
P1 + ½ρv12 2+ ρ|g|h1 = 2
P1 + ½ρv1 = P2 + ½ρv2
P2 + ½ρv22 + ρ|g|h2
Bernoulli’s Principle
• A faster-flowing fluid will
have streamlines that are
closer together.
• A lower-pressure fluid will
have streamlines that are
closer together.
Lift
• airfoil: any device that
generates lift as air flows
along its surface
• hydrofoil: object that
creates lift in liquid
Theories of Lift
• Bernoulli principle
• Conadă effect
Real Fluids
• viscosity: a measure of the
resistance of fluid to a flow
• caused by cohesive forces
between particles of a fluid
• a type of internal friction
• coefficient of viscosity (η)
Real Fluids
• lower coefficients of
viscosity indicate that the
fluids flow more easily
• viscosity is sometimes
referred to as the
“thickness” of a fluid
Real Fluids
• particles closest to the
walls move more slowly
than those farther from the
walls