Fluid
• Concept of Fluid
• Density
• Pressure: Pressure in a Fluid. Pascal´s principle
• Buoyancy. Archimede´s Principle.
• Forces on submerged surfaces
Fluid on motion
• Continuity equation.
• Bernoulli´s Equation
• Viscous Flow. Viscosity. Poiseuille´s Law
• Laminar and turbulent fluxes
Fluid. Introduction
• A fluid is defined as a substance that flows
A fluid is defined as a substance that continually deforms (flows)
under an applied shear stress regardless of how small the applied stress. All
liquids and all gases are fluids. Fluids are a subset of the phases of matter and
include liquids, gases, plasmas and, to some extent, plastic solids.
Liquids flow under gravity until they occupy the lowest possible regions of their
containers (they have defined volume but not a defined shape). Gases expand
to fill their container (they have no neither defined shape and volume
Liquids form a free surface (that is, a surface not created by the container)
while gases do not.
Fluid. Introduction
•
A fluid is defined as a substance that continually deforms (flows)
under an applied shear stress regardless of how small the applied stress
The distinction between solids and fluid is not entirely obvious.
The distinction is made by evaluating the viscosity of the substance.
Fluids display such properties as:
- not resisting deformation, or resisting it only lightly (viscosity), and
- the ability to flow (also described as the ability to take on the
shape of the container).
These properties are typically a function of their inability to support a shear
stress in static equilibrium.
FLUID. Distinction between solids and fluid. About shear stress
Tensile stress
F/A
Tensile Strain
∆L/L
A solid body has a defined shape. Rigid Body is a
idealized concept of a solid body. Rigid Body is
defined as a body which does not deform under
acting forces on it
If a real solid body is subjected to forces that tend
to stretch, compress or shear the object, its shape
changes, reaching a new status of equilibrium
between external and internal forces.
- If the object returns to its original shape
when the external forces are removed, it is said to
be elastic. Most objects are elastic for forces up a
certain maximun, called the elastic limit. If the
acting forces exceed of elastic limit the object
remains permanently deformed or it is fractured.
Shear Stress
Fs/A
Shear Strain
∆X/L
Solids can be subjected to shear
stresses, and to normal stresses - both
compressive and tensile. In contrast,
ideal fluids can only be subjected to
normal, compressive stress which is
called pressure. Real fluids display
viscosity and so are capable of being
subjected to low levels of shear stress.
FLUID. Fluid Statics, Hydrostatics; Fluid dynamics; hydraulics
Fluid statics (also called hydrostatics) is the science of fluids at rest, and is
a sub-field within fluid mechanics. The term usually refers to the mathematical
treatment of the subject. It embraces the study of the conditions under which
fluids are at rest in stable equilibrium. The use of fluid to do work is called
hydraulics, and the science of fluids in motion is fluid dynamics.
Hydraulics is a topic of science and engineering dealing with the mechanical
properties of liquids. Hydraulics is part of the more general discipline of fluid
power. Fluid mechanics provides the theoretical foundation for hydraulics,
which focuses on the engineering uses of fluid properties. Hydraulic topics
range through most science and engineering disciplines, and cover concepts
such as pipe flow, dam design, fluid control circuitry, pumps, turbines,
hydropower, computational fluid dynamics, flow measurement, river channel
behavior and erosion.
FLUID. Density
• Density. An important property of a substance is the ratio of
the mass to its volume, which is called density
Specific weight
SI Units : kg/m3
The density of water at 4ºC is
1000 kg/m3 [1 kg/l] [1 g/cm3]
dm g
 g
dV
The ratio of the density of a substance to that a
water is called the specific gravity
Density must take temperature into account,
since the densities of most materials vary
with temperature. In the case of water,
maximum density occurs at 4ºC
Most solids and liquid change only slightly
in volume when heated or under an
increase of external pressure. The density
of a gas depends strongly of the pressure
and temperature
Density. Advanced concept
The concept of density is referred to a
determined point of the body
dm
m

 lim V 0
dV
V
Specific volume is
the inverse of
density and it is
defined as the ratio
of the volume to its
mass
FLUID. Density
• Density.
dm

dV
The density of
water at 4ºC is
1000 kg/m3 [1
kg/l] [1 g/cm3]
The density
of air at 0ºC
and 1 atm of
pressure is
1.293 kg/m3
Density of water
versus temperature
Temp
(°C)
Density
(g/cm3)
30
0.9957
20
0.9982
10
0.9997
4
1.0000
0
0.9998
−10
0.9982
−20
0.9935
−30
0.9839
Build a table of
densities of Gold,
Mercury, Water,
Wood, Air, and
Helium. Include also
typical soil density,
FLUID. Pressure
• Pressure in a fluid
When a body is submerged in a fluid such as water, the fluid exert a
force perpendicular to the surface of the body at each point on the
surface. This is a “distributed force”.
Pressure is the ratio between the normal force, FN and the
elementary area, A, where it is applied.
FN
P
A
SI Units: The Pascal [Pa] is the pressure
exerted by a force of one Newton uniformly
distributed on one square meter.
Another common unit of pressure is the atmosphere (atm), which equals
approximately the air pressure at sea level
1 atm = 101.325 kPa
The pressure due to a fluid pressing in on an object tends to
compress the object. The ratio of the increase in pressure ΔP to the
fractional decrease in volume -(ΔV/V) is called the bulk modulus
FLUID. Pressure
• Different behavior of liquids and gases to an increase of
pressure. Bulk modulus and the compressibility modulus
The pressure due to a fluid pressing in on an object tends to compress
the object.
The ratio of the increase in pressure ΔP to the fractional decrease in
volume -(ΔV/V) is called the bulk modulus.
P
 
V
V
The compressibility modulus is the
reciprocal of bulk modulus (1/B)
Liquids and solids are relatively incompressible, they have large values
of B. On the other way, the density of liquid and solids is relatively
constant with pressure changes
Gases are easily compressed and the values of B are strongly
dependent on pressure changes. The density of gases depends strongly
of pressure changes, besides of changes in temperature.
FLUID. Pressure.
• Change of pressure with height in a static fluid in a gravitational field.
Hydrostatic equation
dz
g
z
We can detach
the volume of
fluid from the rest
of the fluid, as
shown in the
figure. This
volume will be at
equilibrium in a
static fluid under
a gravitational
field.
Free-body diagram
Static condition  Fext  O
on the volume
 Fz  0  P dS  ( P  dP)dS   g dV  0
(P + dP) dS
and as
dm g   dV g
P dS
The vertical forces acting on the
volume are those exerted by the
rest of the fluid and by the weight.
Horizontal forces are not showed in
the figure because they will be
balanced.
dV  dS dz
dP    g dz
In the case of a liquid, ρ constant,
P   g h
P  P0   g h
The pressure P at the bottom must
be greater than the pressure at the
top to balance the weight of the fluid
The pressure in a pool (lake or ocean) increases with depth. Similarly, the
pressure of the atmosphere decreases with altitude.
Po
h
P
FLUID. Pressure.
• Hydrostatic equation. Change of pressure with depth in a static fluid in a gravitational field.
Po
dP    g dz
In the case of a liquid, ρ constant,
P   g h
h
P  P0   g h
P
• The pressure increase linearly with the depth, independent of the shape of the container
•The pressure is the same at all point at the same depth
Pascal´s principle: A pressure change applied to an enclosed liquid is
transmitted undiminished to every point in the liquid and to the walls of the
container.
Hydrostatic paradox: The pressure depends only on the depth of the water not
on the shape of the container, so at the same depth the pressure is the same in all
parts of the container
Many applications are based on the hydrostatic equation: Hydraulic lift,
hydraulic system, horizontal lines (using water connected vessels)…
FLUID. Pressure.
P   g h
Po
P  P0   g h
Hydraulic lift
h
Derive the relationship for the
forces in the pistons of
hydraulic lift, considering the
hydrostatic equation (Pascal´s
Principle
P
Hydrostatic paradox
Explain why :
1.- the free surface of
liquid will adopt the
horizontal line.
2.- The pressure at
the bottom at a
different point will be
the same
FLUID. Pressure.
P   g h
Po
P  P0   g h
h
Determine the difference of
pressure between point 1 and 2.
P
FLUID. Pressure
Measuring pressure
We can use the fact that the pressure difference is proportional to the depth of a
liquid to measure unknown pressures, that is, the pressure in the container.
Measuring gauge (gage) pressure:
the open-tube manometer.
Measuring
Atmospheric Pressure.
The mercury barometer
Pat=ρHggh
ρHg density of
mercury
The top of the tube is open to the
atmosphere at pressure Pat. The
other end of the tube is at pressure P,
which is to be measured. The
difference P – Pat, called the gauge
pressure, will be
P – Pat = ρ g h
What is the height of
the mercury column
in a barometer if the
atmospheric
pressure is 1 atm
(101.325 kPa)?. The
density of mercury
at 0ºC is 13.595x103
kg/m3. The same if
the liquid in the
column were water
at 4 ºC
The absolute pressure in the container is
obtained from the gauge pressure by
adding atmospheric pressure to it:
P = Pat + ρgh
FLUID. Pressure
Measuring pressure. Common pressure units
The pressure you measure in your automobile tire is
gauge pressure. The absolute pressure in the tire can
be obtained from the gauge pressure by adding the
actual atmospheric pressure measured by the
barometer. Exercise: The recommended pressure in a
type of tire is 2.5 bar. What is the absolute pressure if
the local atmospheric pressure is 933 mbar?
101325 Pa [Pascal]
1 atm [atmosphere]
1.01325 bar
760 mmHg [millimeter of mercury]
10.33 mH2O [meter of water]
1.0332 kgf/cm2
bar = 100 kPa
mbar [milibar]
Kilogram of force per square
centimeter also called
technical atmosphere
What is the minimum value of the absolute pressure? What is the maximum value of
suction that can be exerted?
FLUID. Buoyancy.
Buoyancy. Archimede´s Principle.
A body wholly or partially submerged in a fluid is buoyed
by a force equal to the weight of the displaced fluid
Apparent weight of
the submerged body
Deriving Archimede´s principle
from Newton´s Laws
W =ρF V g
Buoyant
force
B
The submerged body is replaced by an equal volume of
fluid (dashed line). The detached volume is at equilibrium
weight
balancing its own weight by the buoyant force exerted on it
by the rest of the fluid.
The weight of a body at air is
154.4 N. The same body
submerged in water has an
apparent weight of 146.4 N. From
which material is the body made ?
Then, the value of the buoyant force on the submerged
body must be the weight of the displaced fluid. The line of
action of the buoyant force passes through the center of
mass of the volume.
This result does not depend on the shape of the
submerged object
FLUID. Buoyancy.
The apparent weight of the child, when he is
completely submerged in water, with the air
completely exhaled from his lungs, is 5% of
his weight. What percentage of his body is
fat?. The density of fat ~0.9x103 kg/m3, and
the density of lean tissue (everything except
fat) ~1.1x103 kg/m3.
A raft of area A, thickness h, and mass
m=600 kg floats in calm water with 7
cm submerged. When a man stands on
the raft, 8.4 cm are submerged. What
is the man´s mass?.
A large helium
weather balloon is
spherical in
shape, with a
radius of 2.5 m
and a total mass
of 15 kg. What is
the initial upward
acceleration of the
balloon when it is
released at sea
level.
FLUID. Forces exerted on submerged areas
y
P dS
F   P dS    g y dS