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D EPA RTM EN T OF C IV IL EN GIN EER IN G
BITS PILA N I , R A J A S THA N
BY
D R . S HIBA N I KHA N R A J HA
A U GU S T 2 0 1 3
Fluids at Rest:
Pressure and its Effect
Lectures 3, 4, 5, 6, 7
Course: CE F212 Transport Phenomena 3 0 3
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Course: CE F212 Transport Phenomena 3 0 3
Topics to be covered
Pressure at a Point
Basic Equation for Pressure
Field
Pressure Variation in a Fluid
at Rest
Incompressible fluid
Compressible fluid
Standard Atmosphere
Measurement of Pressure
Course: CE F212 Transport Phenomena 3 0 3
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Manometry
Piezometer Tube
U-Tube Manometer
Inclined Tube Manometer
Hydrostatic force on a Plane Surface
Buoyancy, Floatation and Stability
Archimedes Principle
Stability
Pressure Variation in a Fluid with Rigid-Body Motion
Linear Motion
Rigid-Body Rotation
Fluids at Rest - Pressure and its Effects
Course: CE F212 Transport Phenomena 3 0 3
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The study of fluids that are at rest or moving in such a manner that there is no relative motion between adjacent particles is what we are going to discuss in these lectures.
There will be no shearing stress in the fluid and the only forces that develop on the surfaces of the particles will be due to pressure.
Our first concern is to investigate pressure and its variation throughout a fluid and the effect of pressure on submerged surfaces.
Image of hot air balloon
An example1: use of fluid static
Course: CE F212 Transport Phenomena 3 0 3
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heated air, which is less
dense than the surrounding
air, is used in hot air balloon
to create an upward buoyant
force. According to
Archimedes Principle, the
buoyant force is equal to the
weight of the air displaced
by the balloon.
Image of hurricane Allen
An example2: use of fluid static
Course: CE F212 Transport Phenomena 3 0 3
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Although there is
considerable motion and
structure to a hurricane, the
pressure variation along
vertical plane is
approximated by pressure-
depth relationship for a
static fluid
Fluid Pressure
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A liquid or gas cannot sustain a
shearing stress - it is only restrained
by a boundary. Thus, it will exert a
force against and perpendicular to
that boundary.
The force F exerted by a fluid on the
walls of its container always acts
perpendicular to the walls.
Water flow
shows F
Fluid Pressure
Course: CE F212 Transport Phenomena 3 0 3
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Fluid exerts forces in many
directions. Try to submerge
a rubber ball in water to see
that an upward force acts on
the float.
F
Fluids exert pressure in all
directions.
Common units used for Pressure (ML-1T-2)
Course: CE F212 Transport Phenomena 3 0 3
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2
5
5
2
1 1 / [SI]
1 1.01325 10
1 10 1
1 760 760
1 14.7 / .
pascal N m
atmosphere Pa
bar Pa atm
atm mm of Hg torr
atm lb in
Pressure at a Point
The term pressure is used to indicate the normal force per unit area at a given point acting on a given plane within the fluid mass of interest.
(P=F/A)
There are no shearing stresses present in a fluid at rest
The pressure at a point in a fluid at rest or in motion is independent of direction as long as there are no shearing stresses present
This important result is known as Pascals law named in honor of Blaise Pascal (1623-1662), a French mathematician who made important contributions in the field of hydrostatics
Course: CE F212 Transport Phenomena 3 0 3
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Pressure at a Point (Pressure is a scalar quantity)
Force balance in the x-direction:
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How the pressure at a point varies with the orientation of the plane
passing through the point?
Forces on arbitrary wedge
shaped element of fluid
Force balance in the z-direction:
Vertical force on DA Vertical force on
lower boundary Total weight of wedge element
= specific weight
Course: CE F212 Transport Phenomena 3 0 3 11
From last slide:
Divide through by to get the following
Now reduce the element size to a point such that: which finally leads to
This can be done for any orientation , so that following mathematical statement can be given
Course: CE F212 Transport Phenomena 3 0 3 12
0sin2
1D lpp zn
This mathematical statement shows that the pressure is
independent of direction (or it is a scalar quantity)
Basic equation for pressure field
Consider a small rectangular element or differential fluid
element of size dxdydz
There are two types of forces acting on this element:
Surface forces due to the pressure
Body forces due to the weight of the element
Other possible body forces (like magnetic fields) are not
taken into consideration
Pressure may vary across a fluid particle
Course: CE F212 Transport Phenomena 3 0 3
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Basic equation for pressure field
Body Force can be given as
Where
g=acceleration due to gravity
dm=differential mass
=density of fluid
=differential volume of fluid element of size
Course: CE F212 Transport Phenomena 3 0 3
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Objective is.
How does the pressure in a fluid in which there are no shearing
stresses vary from point to point?
d dzdydx
Basic equation for pressure field
Surface Force can be given as
Force balance along y-direction
Force balance along y-direction
With
Substituting the above relations, we get the following
expression for force balance along y-direction
Course: CE F212 Transport Phenomena 3 0 3
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RRLLy dApdApFd
RL dAdxdzdA
dxdydzy
pjFd y
Basic equation for pressure field
Surface Force can be given as
Similarly for x and z directions
Total surface force=
Or finally we can write the following
Total force=body force +surface force
OR
Course: CE F212 Transport Phenomena 3 0 3
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dxdydzz
pkFd
dxdydzx
piFd
z
x
zyxS FdFdFdFd
Basic equation for pressure field
Now applying Newtons Second Law
(for a fluid at rest a=0)
(equation of motion for fluid at rest)
For fluid in motion with acceleration a, general equation of
motion can be written as below
Course: CE F212 Transport Phenomena 3 0 3
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ak
p
Pressure Variation in a Fluid at Rest
We begin with general equation of motion
Or in component form we can write
Independent of x-y plane; so pressure in x-y plane remains same; it only depends on z;
Thus the general motion of fluid now can be written as ordinary differential equation as:
The above equation give the Pressure height relation
Pressure only depends on z and decreases as we move upward in a fluid at rest
For liquids or gases at rest the pressure gradient in the vertical direction at any point in a fluid depends only on the specific weight of the fluid at that point
Course: CE F212 Transport Phenomena 3 0 3
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ak
p
,,0,0
z
p
y
p
x
p
Pressure Variation in a Fluid at Rest
Incompressible Fluid- CONSTANT DENSITY FLUID
Integration the above equation between two points 1 and 2
h= pressure head
Such distribution of pressure in space is called hydrostatic distribution
If reference pressure is pressure at free surface (atmospheric pressure p0)
Incompressible Fluid Manometers
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hpp 21
Pressure Variation in a Fluid at Rest
Incompressible Fluid- CONSTANT DENSITY FLUID
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The following figure helps us to understand how the pressure is same at all points along the line AB even though the container may
have a very irregular shape.
Thus, pressure only depends on depth h, surface pressure p0, specific weight of the liquid in the container,
independent of size or shape of the container
Course: CE F212 Transport Phenomena 3 0 3
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Pressure in = Pressure out
Ideal mechanical advantage=
Transmission of Fluid Pressure: Pascals Law
Fout Fin Aout Ain
in out
in out
F F
A A
Pascals Law: An external
pressure applied to an enclosed
fluid is transmitted uniformly
throughout the volume of the
liquid.
in
out
A
A
Transmission of Fluid Pressure: Pascals Law
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The transmission of pressure throughout a stationary fluid
is the principle upon which many hydraulic devices are
based
Operation of hydraulic jacks
Lifts and presses
Hydraulic controls on aircraft
Other types of heavy machinery
HYDRAULIC JACKS OR COMPRESSED AIR IS USED ON THE LIQUID
SURFACE DIRECTLY AS IS DONE IN HYDRAULIC LIFTS COMMONLY
FOUND IN SERVICE STATIONS
Pressure Variation in a Fluid at Rest
Compressible Fluid-VARYING DENSITY FLUID
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A compressible fluid is one in which the fluid density changes
accompanied by changes in pressure and temperature.
We mostly think of gases as being compressible; the specific
weights of gases are small so the pressure gradient in the vertical
direction is small and even over greater distances the pressure
remains constant.
This way, attention must be given to the variation in specific
weight.
In order to analyze compressible fluids we use the equation of
state for an ideal gas and combine it with the equation for
pressure variation in fluids at rest:
Pressure Variation in a Fluid at Rest
Compressible Fluid-VARYING DENSITY FLUID
We begin with equation of motion
Since, density is varying; we use ideal gas law
Using ideal gas law in equation of motion
Need additional information, e.g., T(z) for atmosphere
For isothermal condition (T constant over z1 to z2)
This equation provides the desired pressure elevation relationship for an isothermal layer
2
1
2
11
2ln
z
z
p
pT
dz
R
g
p
p
p
dp
0
1212 exp
RT
zzgpp
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Standard Atmosphere
Standard atmosphere is an idealized representation of mean conditions in the earths atmosphere
For measurement of pressure we need pressure versus altitude over the specific range for the specific conditions (temperature, reference pressure). However, this type of information is not available
Thus, a standard atmosphere has been determined that can be used in the design of aircraft, missiles and spacecraft and in comparing their performance under standard conditions
Currently accepted U.S Standard atmosphere is: Idealized representation of middle-latitude, year round mean conditions of the earths atmosphere
Course: CE F212 Transport Phenomena 3 0 3
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Standard Atmosphere
Several important properties of standard atmospheric
conditions at sea level are listed
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Standard Atmosphere
Temperature decreases with altitude in the region nearest the
earths surface (troposphere), then becomes essentially
constant in the next layer (stratosphere) and subsequently
starts to increase in the next layer
Course: CE F212 Transport Phenomena 3 0 3
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Measurement of Pressure
Pressure at a point in the fluid mass is designated as either absolute pressure or gage pressure
Absolute pressure is measured relative to a perfect vacuum (absolute zero pressure)
Gage pressure is measured relative to the local atmospheric pressure
Zero gage pressure corresponds to local atmospheric pressure
Absolute pressures are always positive
Gage pressure can be either positive or negative depending on whether the pressure is above atmospheric pressure (a positive value) or below atmospheric pressure (a negative value)
A negative gage pressure is also referred to as a suction or vacuum pressure
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Measurement of Pressure:
gage pressure and absolute pressure
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Measurement of Pressure:
how to measure pressure - mercury barometer
A mercury barometer is used to measure atmospheric
pressure
The column of mercury will come to an equilibrium position
where its weight plus the force due to the vapor pressure
(which develops in the space above the column) balances the
force due to the atmospheric pressure.
For most practical purposes the contribution of the vapor
pressure (0.000023 lb/in2)can be neglected since it is very
small
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31 2014/8/25 http://zimp.zju.edu.cn/~xinwan/
Mercury Barometer
E. Torricelli (1608-47)
How high will water rise?
No more than h = patm/g = 10.3 m
atmatm
PP gh h
g
If water was used in place of mercury, the
height of column would have to be
approximately 34 ft rather than 29.9 in
(760 mm Hg or 1 atm) mercury for an
atmospheric pressure of 14.7 psi
For mercury, h = 760 mm.
C atm
atm
P gh P
P gh
Manometry
The standard techniques are to use Manometers which are vertical or inclined liquid columns to measure pressure
The mercury barometer is an example of one type of manometer, but there are many other possible configurations
Three common types of manometers include: The Piezometer tube
The U-tube manometer
The inclined-tube manometer
The Mercury Barometric is mostly used to measure atmospheric pressure
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Piezometer Tube
To determine pressure from a manometer, simply use the fact that the pressure in the liquid columns will vary hydrostatically
The simplest manometer consists of a vertical tube, open at the top and attached to the container in which the pressure is desired
Since manometers involve columns of fluids at rest, the fundamental equation describing their use is as follows
This gives the pressure at any elevation within a homogeneous fluid in terms of a reference pressure and the vertical distance h between p and .
In a fluid at rest, pressure will increase as we move downward and will decrease as we move upward.
Manometry:
Piezometer or Simple Manometer
0php
0p0p
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Piezometer Tube - Limitations
It is only suitable, if the pressure in the container is greater
than atmospheric pressure, otherwise air would be sucked
into the system
The pressure to be measured must be relatively small so the
required height of the column is reasonable
Also, the fluid in the container in which the pressure is to be
measured must be a liquid rather than a gas
Manometry:
Piezometer or Simple Manometer
Course: CE F212 Transport Phenomena 3 0 3
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U-Tube Manometer
To overcome the difficulties aroused in the Piezometer, another type of manometer consisting of U shaped tube is widely used
The fluid in the manometer is called the gage fluid
Better for higher pressures.
Possible to measure pressure in gases.
Manometry:
U-Tube Manometer
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Manometry:
U-Tube Manometer
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Find pressure at center of pipe:
Can start either at open end or inside
pipe.
Here we start at open end:
If the fluid in the pipe is gas, the
contribution due to the gas column is
almost negligible and we get the
pressure at 4 as
p at open
end Change in p
from 1 to 2
Change in p
from 3 to 4 p in pipe
hp mp D
Manometry:
U-Tube Manometer
Course: CE F212 Transport Phenomena 3 0 3
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The U-tube manometer is also widely used to measure the difference in pressure between two containers or two points in a given system.
Consider a manometer connected between containers A and B as is shown in Fig.
The difference in pressure between A and B can be found by again starting at one end of the system and working around to the other end.
Differential U-tube manometer
Differential Manometer is
Used for measuring pressure differences between desired
points along a pipe
Manometry:
Differential U-Tube Manometer
Course: CE F212 Transport Phenomena 3 0 3
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Inclined-Tube Manometer
Inclined-Tube manometers can be used to measure small pressure variation accurately
or
Manometry:
Inclined-Tube Manometer
BA phlhp 332211 sin
113322 sin hhlpp BA
Course: CE F212 Transport Phenomena 3 0 3
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If the container A and B
contains the gas, then
or
sin22lpp BA
sin22
BA ppl
Fluid statics: Hydrostatic Force
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The only stress in fluid statics is normal stress Normal stress is due to pressure
Variation of pressure is only due to the weight of the fluid fluid statics is only relevant in presence of gravity fields.
Applications: Floating or submerged bodies, water dams and gates, liquid storage tanks, etc.
Example of elevation head z converted to
velocity head V2/2g.
We'll discuss this in
more detail in coming
lectures (22-25)
Significance of Hydrostatic Pressure in
Scuba Diving
Course: CE F212 Transport Phenomena 3 0 3
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100 ft
1
2
Pressure on diver at 100 ft?
Danger of emergency ascent?
,2 3 2
,2 ,2
1998 9.81 100
3.28
1298.5 2.95
101.325
2.95 1 3.95
gage
abs gage atm
kg m mP gz ft
m s ft
atmkPa atm
kPa
P P P atm atm atm
1 1 2 2
1 2
2 1
3.954
1
PV PV
V P atm
V P atm
Boyles law
If you hold your breath on ascent, your lung
volume would increase by a factor of 4, which
would result in embolism and/or death.
Hydrostatic Force on a Plane Surface
In the determination of the resultant force on an area, the effect of atmospheric pressure often cancels
For a submerged surface in a fluid, forces develop on the surface due to the fluid
Pressure and resultant hydrostatic force developed on the bottom of an open tank
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Hydrostatic Force on a Plane Surface
Determination of these forces is important in the design of
storage tanks, ships, dams and other hydraulic structures
For fluids at rest, the force must be perpendicular to the
surface since there are no shearing stresses present
Also pressure vary linearly with depth, if the fluid is
incompressible
Course: CE F212 Transport Phenomena 3 0 3
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Pressure and resultant hydrostatic force
developed on the bottom of an open tank
Hydrostatic Force on a Plane Surface-
Analysing forces on different surfaces
For a horizontal surface
such as the bottom of a liquid filled tank, the magnitude of the resultant force is simply
Where p is the uniform pressure on the bottom and A is the area of the bottom
For the open tank ,
If atmospheric pressure acts on both sides of the bottom, the resultant force on the bottom is simply due to the liquid in the tank
Since the pressure is constant and uniformly distributed over the bottom, the resultant force acts through the centroid of the area
Pressure on the sides of the tank is not uniformly distributed
pAFR
hp
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Pressure and resultant
hydrostatic force developed
on the bottom of an open
tank
Hydrostatic Force on a Plane Surface-
Analyzing forces on different surfaces
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Pressure on the sides of the
tank is not uniformly
distributed
CP
C
Pressure distribution on the sides
of an open tank
Hydrostatic Force on a Plane Surface
For inclined submerged plane surface:
We assume that the fluid surface is open to the atmosphere
Let the plane in which the surface lies intersect the free surface at 0 and make an angle with this surface
The x-y coordinate is defined so that 0 is the origin and y is directed along the surface
Course: CE F212 Transport Phenomena 3 0 3
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Hydrostatic Force on a Plane Surface
For inclined submerged plane surface:
The area can have an arbitrary shape
We need to determine the direction, location and magnitude of the resultant force acting on one side of this area due to the liquid in contact with the area
Course: CE F212 Transport Phenomena 3 0 3
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Hydrostatic Force on a Plane Surface
At any depth, h, the force acting on dA (the differential area) is
Thus, the magnitude of the resultant force can be found by summing
these differential forces over the entire surface as given below
Where
For constant and
The integral appearing in above equation, is the first moment of the
area with respect to the x-axis, so we can write
hdAdF
AA
R dAyhdAdFF sin
sinyh
A
R ydAF sin
AyydA cA
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Hydrostatic Force on a Plane Surface
Where is the y coordinate of the centroid of area A measured from the x-axis which passes through 0. Thus the resultant force can be written as
or more simply as
Where is the vertical distance from the fluid surface to the centroid of the area
Note:
The magnitude of the force is independent of angle of inclination and depends only on the specific weight of the fluid, the total area, and the depth of the centroid of the area below the surface
The magnitude of the resultant force is the pressure at the centroid multiplied by the total area
cy
sincR AyF
AhF cR
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ch
Hydrostatic Force on a Plane Surface
The y coordinate, , of the resultant force can be determined by
summation of moments around the x-axis. That is the moment of the
resultant force must equal the moment of the distributed pressure
force, or
And ,therefore
The integral in the numerator is the second moment of the area (moment of inertia), Ix, with respect to an axis formed by the
intersection of the plane containing the surface and the free surface (x-
axis). Thus, we can write
dAyydFyFAA
RR
2sin
Ry
sincR AyF
Ay
dAy
yc
AR
2
Ay
Iy
c
xR
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Hydrostatic Force on a Plane Surface
Use can now use the parallel axis theorem to express Ix as
Where , is the second moment of the area with respect to an axis passing through its centroid (C) and parallel to the x-axis. Thus,
The above relation shows that the resultant force does not pass through the centroid but is always below it, since .
The x coordinate, , for the resultant force can be determined in a similar manner by summing moments about the y-axis. Thus,
and, therefore, where is the product of inertia with respect to the x and y axes
2
cxcx AyII
xcI
c
c
xcR y
Ay
Iy
0/ AyI cxc
Rx
xydAxFA
RR sin
Ay
I
Ay
xydA
xc
xy
c
AR
xyI
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Hydrostatic Force on a Plane Surface
Using the parallel axis theorem, we can write
where is the product of inertia with respect to an orthogonal
coordinate system passing through the centroid of the area and formed
by a translation of the x-y coordinate system
The point through which the resultant force acts is called the center of
pressure (CP) (xR, yR)
c
c
xyc
R xAy
Ix
xycI
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Hydrostatic Force on a Plane Surface
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The following diagram shows the geometric properties of some common shapes:
Buoyancy, Floatation and Stability
Archimedes Principle
In the figure, we see that the
difference between the weight in AIR
and the weight in WATER is 3 lbs.
This is the buoyant force that acts
upward to cancel out part of the
force. If you were to weigh the water
displaced; it also would weigh 3 lbs.
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B fluid fluid
object
F Vg m g
mg Vg
Archimedes Principle
Buoyancy, Floatation and Stability
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Buoyant force: A resultant body force that is generated when a stationary body is completely or partially submerged in a fluid.
Archimedes principle: A body wholly or partly immersed in a fluid is buoyed up by a force equal to the weight of the fluid it displaces. The buoyant force can be considered to act vertically upward through the center of gravity of the displaced fluid.
FB = buoyant force = weight of displaced fluid
Buoyancy, Floatation and Stability
Archimedes Principle
A net upward vertical force results because pressure increases with depth and the pressure forces acting from below are larger than the pressure forces acting from above
Consider the forces F1, F2, F3 and F4 are simply the forces exerted on the plane surfaces of the parallelepiped, W is the weight of the shaded fluid volume, and FB is the force the body is exerting on the fluid.
The forces on the vertical surfaces, such as F3 and F4 are all equal and cancel, so the equilibrium equation of interest is in the z direction and can be expressed as
WFFFB 12Course: CE F212 Transport Phenomena 3 0 3
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Buoyancy, Floatation and Stability
Archimedes Principle
If the specific weight of the fluid is constant, then
Where A is the horizontal area of the upper (or lower) surface of the parallelepiped, can be written as
Simplifying, we arrive at the desired expression for the buoyant force
Where is the specific weight of the fluid and is the volume of the body.
The direction of the buoyant force is directed vertically upward and the magnitude is equal to the weight of the fluid displaced by the body
This result is commonly referred as Archimedes principle in honour of Archimedes (287-212 B.C.), a Greek mechanician and mathematician who first enunciated the basic ideas associated with hydrostatics
AhhFF 1212
AhhAhhFB 1212
BF
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Buoyancy, Floatation and Stability
Archimedes Principle
The location of the line of action of the buoyant force can be determined by summing moments of the forces shown on the free body diagram wrt some convenient axis
And on substitution for the various forces
Where is the total volume
The right hand side of above equation, is the first moment of the displaced volume w r t x-z plane so that is equal to the y coordinate of the centroid of the volume
Buoyant force passes through the centroid of the displaced volume. The point through which the buoyant force acts is called the center of buoyancy
21112 WyyFyFyF cB
T Ahh 12
cy
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21 yyy TTc
Buoyancy, Floatation and Stability
Stability
The stability of a body can be determined by considering what happens when it is displaced from its equilibrium position
A body is said to be in a stable equilibrium position if when displaced it returns to its equilibrium position
It is said to be in unstable equilibrium position if when displaced, even slightly, it moves to a new equilibrium position
Stability considerations are particularly important for submerged or floating bodies since the centers of buoyancy and gravity do not necessarily coincide.
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Buoyancy, Floatation and Stability
Stability
A small rotation can result in either a restoring or overturning couple
For the completely submerged body, which has a center of gravity below the center of buoyancy, a rotation from its equilibrium position will create a restoring couple formed by the weight W and the buoyant force FB, which causes the body to rotate back to its original position
If center of gravity falls below the center of buoyancy, the body is in stable equilibrium position
If center of gravity falls above the center of buoyancy, the body is in unstable equilibrium position
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Buoyancy, Floatation and Stability
Stability
For floating bodies the stability problem is more
complicated, since as the body rotates the location of the
center of buoyancy (which passes through the centroid of the
displaced volume) may change.
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Pressure Variation in a Fluid with Rigid-Body Motion
Even though a fluid may be in motion,
if it moves as a rigid body there will be
no shearing stresses present
The general equation of motion
(discussed earlier)
A general class of problems involving
fluid motion in which there are no
shearing stresses occurs when a mass
of fluid undergoes rigid body motion
ak p
z
y
x
az
p
ay
p
ax
p
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Pressure Variation in a Fluid with Rigid-Body Motion
Linear Motion
We first consider an open
container of a liquid that is
translating along a straight path
with a constant acceleration
Since , it follows that the
pressure gradient in the x-
direction is zero .
z
y
agz
p
ay
p
a
0xa
0/ xp
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Linear acceleration
of a liquid
With free surface
Pressure Variation in a Fluid with Rigid-Body Motion
Linear Motion: In the y and z directions
The change in pressure between two closely spaced points
located at y, z and y+dy, z+dz can be expressed as
or in terms of acceleration
Along a line of constant pressure, , and therefore
from the above equation it follows that the slope of this line
is given by the relationship
dzz
pdy
y
pdp
dzagdyadp zy
0dp
z
y
ag
a
dy
dz
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Pressure Variation in a Fluid with Rigid-Body Motion
Linear Motion
Along a free surface the pressure is constant, so that for the accelerating mass, the free surface will be inclined if
Additionally, all lines of constant pressure will be parallel to the free surface
For the special circumstances in which , which corresponds to the mass of fluid accelerating in the vertical direction
indicates that the fluid surface will be horizontal
The pressure distribution is not hydrostatic, but is given by the equation
zagdz
dp
z
y
ag
a
dy
dz
0ya
0,0 zy aa
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Pressure Variation in a Fluid with Rigid-Body Motion
Linear Motion
For fluids of constant density this equation shows that the
pressure will vary linearly with depth, but the variation is
due to the combined effects of gravity and the externally
induced acceleration
What is the pressure gradient for a freely falling fluid
mass???
Example: the pressure throughout a blob of a juice
floating in an orbiting space shuttle (a free fall) is ???
The only force holding the liquid together is ???
Course: CE F212 Transport Phenomena 3 0 3
66 zagdz
dp
Pressure Variation in a Fluid with Rigid-Body Motion
Rigid-Body Rotation of a liquid in a tank
A fluid contained in a tank that is rotating with a constant angular velocity about an axis will rotate as a rigid body
In terms of cylindrical coordinates the pressure gradient can be expressed as
Thus, in terms of this coordinate system
Course: CE F212 Transport Phenomena 3 0 3
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p
zrz
pp
rr
pp eee
1
0
0
2
z
rr r
a
a
ea
Pressure Variation in a Fluid with Rigid-Body Motion
Rigid-Body Rotation
Thus,
This type of rigid body rotation shows that the pressure is a
function of two variables r and z and therefore differential
pressure is
or
Course: CE F212 Transport Phenomena 3 0 3
68
z
p
p
rr
p
0
2
dzdrrdp 2
dzz
pdr
r
pdp
It is greater than ZERO. Thus, the pressure increases in the radial direction because of
centrifugal acceleration.
Pressure Variation in a Fluid with Rigid-Body Motion
Rigid-Body Rotation
Along a surface of constant pressure, such as the free surface, , so that we can write
The equation for surface of constant pressure is
Integrating the pressure equation, we get
or Surfaces of constant pressure are parabolic.
They are curved rather than flat.
This result shows that the pressure varies with the distance from the axis of rotation, but at a fixed radius, the pressure varies hydrostatically in the vertical direction
Course: CE F212 Transport Phenomena 3 0 3
69
0dp
tconsg
rz tan
2
22
dzrdrdp 2
tconszr
p tan2
22
g
r
dr
dz 2
Examples of Rigid Body Motion
Course: CE F212 Transport Phenomena 3 0 3
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Summary of the lectures 3-7
Pascals law
Surface force
Body force
Incompressible fluid
Hydrostatic pressure distribution
Pressure head
Compressible fluid
U.S standard atmosphere
Absolute pressure
Gage pressure
Course: CE F212 Transport Phenomena 3 0 3
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Vacuum pressure
Barometer
Manometer
Centre of pressure
Buoyant force
Archimedes principle
Centre of buoyancy
Rigid body motion