‫الكلية التقنية كركوك‬
Kirkuk - IRAQ
Refrigeration & Air Conditioning
Department
Second Year ‫المرحلة الثانية‬
First Semester
Fluid Mechanics
2014 – 2015
You can download the file from the website:
www.aumid.com
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Introduction to Fluid Mechanics:
1. Fluid Characteristics
2. Mass and weight density
3. Specific weight
4. Compressibility
5. Vapor pressure
6. Dynamic Viscosity
7. Kinematic Viscosity
Fluid Mechanics is the study of behavior of fluids (liquids and gases) and
the applications where fluid systems are used.
Fluid Mechanics is divided into TWO CATAGORIES
 Fluid Statics
 Fluid Dynamics
Fluid Statics deals with fluid systems where the fluid is at Rest
Fluid Dynamics deals with flowing fluids
Dynamics is divided into: HYDRODYNAMIS & GAS DYNAMICS
Hydrodynamics deals with flow systems where the fluid density does not
change significantly. This is called INCOMPRESSIBLE FLOW. This
applies to the flow of liquids at any speed and the flow of gases at speeds
less than 224 mph.
Gas Dynamics deals with flow systems where the fluid density changes
significantly (greater than 4%). E.g. high speed airflow in jet engine exhaust
nozzles.
Another category of fluid mechanics is aerodynamics which deals with the
effect of airflow on immersed bodies such as airplanes and automobiles
regardless of whether it is low speed incompressible flow or high speed
compressible flow.
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Fluids Characteristics: Concept of Shear Stress
A fluid is a substance that cannot resist a shearing force and remain at rest.
The principal difference in the mechanical behavior of fluids compared to
solids is that when a shear stress is applied to a fluid it experiences a
continuing and permanent distortion.
Fluids offer no permanent resistance to shearing, and they have elastic
properties only under direct compression: in contrast to solids which have all
three elastic moduli, fluids possess a bulk modulus only.
Thus, a fluid can be defined unambiguously as a material that deforms
continuously and permanently under the application of a shearing
stress, no matter how small. This definition does not address the issue of
how fast the deformation occurs and as we shall see later this rate is
dependent on many factors including the properties of the fluid itself. The
inability of fluids to resist shearing stress gives them their
characteristic ability to change shape or to flow; their inability to support
tension stress is an engineering assumption, but it is a well-justified
assumption because such stresses, which depend on intermolecular
cohesion, are usually extremely small..... Because fluids cannot support
shearing stresses, it does not follow that such stresses are nonexistent in
fluids. During the flow of real fluids, the shearing stresses assume an
important role, and their prediction is a vital part of engineering work. Without
flow, however, shearing stresses cannot exist, and compression stress or
pressure is the only stress to be considered. So we see that the most
obvious property of fluids, their ability to flow and change their shape, is
precisely a result of their inability to support shearing stresses (The stress
produced by two tectonic ‫ بنائي‬plates sliding past each other horizontally). .
Flow is possible without a shear stress, since differences in pressure will
cause a fluid lump to experience a resultant force and produce acceleration,
but when a fluid is deforming its shape, shearing stresses must be present.
With this definition of a fluid, we can recognize that certain materials that look
like solids are actually fluids. Tar, for example, is sold in barrel-sized chunks
which appear at first sight to be the solid phase of the liquid which forms
when the tar is heated. However, cold tar is also a fluid. If a brick is placed on
top of an open barrel of tar, we will see it very slowly settle into the tar. It will
continue to settle as time goes by --- the tar continues to deform under the
applied load --- and eventually the brick will be engulfed by the tar. Even then
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it will continue to move downwards until it reaches the bottom of the barrel.
Glass is another substance that appears to be solid, but is actually a fluid.
The glass flows under the action of its own weight. If you measure the
thickness of a very old glass pane you would find it to be larger at the bottom
than at the top of the pane. This deformation happens very slowly because
the glass has a very high viscosity, and the results can take centuries to
become obvious.
Mass and weight density
The mass of an object is a fundamental property of the object; a numerical
measure of its inertia; a fundamental measure of the amount of matter in the
object. Definitions of mass often seem circular because it is such a
fundamental quantity that it is hard to define in terms of something else. All
mechanical quantities can be defined in terms of mass, length, and time. The
usual symbol for mass is m and its SI unit is the kilogram. While the mass is
normally considered to be an unchanging property of an object, at speeds
approaching the speed of light one must consider the increase in the
relativistic mass.The weight of an object is the force of gravity on the object
and may be defined as the mass
times the acceleration of gravity
w = mg
Since the weight is a force, its SI unit is the newton. Density is mass/volume.
The weight of an object is defined as the force of gravity on the object and
may be calculated as the mass times the acceleration of gravity, w = mg.
Since the weight is a force, its SI unit is the newton.
For an object in free fall, so that gravity is the only force acting on it, then the
expression for weight follows from Newton's second law.
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You might well ask, as many do, "Why do you multiply the mass times the
freefall acceleration of gravity when the mass is sitting at rest on the table?”
The value of g allows you to determine the net gravity force if it were in
freefall and that net gravity force is the weight. Another approach is to
consider "g" to be the measure of the intensity of the gravity field in Nektons /
kg at your location. You can view the weight as a measure of the mass in kg
times the intensity of the gravity field, 9.8 Newtons/kg under standard
conditions. Data can be entered into any of the boxes below. Then click
outside the box to update the other quantities.
W=mg
At the Earth's surface, where g = 9.8 m / s2
Specific Weight
The specific weight (also known as the unit weight) is the weight per unit
volume of a material. The symbol of specific weight is γ (the Greek letter
Gamma).A commonly used value is the specific weight of water on Earth at
5°C which is 62.43 lbf/ft3 or 9.807 kN/m3
The terms specific gravity, and less often specific weight, are also used for
relative density. In fluid mechanics, specific weight represents the force
exerted by gravity on a unit volume of a fluid. For this reason, units are
expressed as force per unit volume (e.g., lb/ft3 or N/m3). Specific weight can
be used as a characteristic property of a fluid.
Specific weight of water and Air Water
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Compressibility
In thermodynamics and fluid mechanics, compressibility is a measure of the
relative volume change of a fluid or solid as a response to a pressure (or
mean stress) change.
Where V is volume and p is pressure.
The specification above is incomplete, because for any object or system
the magnitude of the compressibility depends strongly on whether the
process is adiabatic or isothermal. Accordingly isothermal compressibility
is defined:
Where the subscript T indicates that the partial differential is to be taken at
constant temperature adiabatic compressibility is defined:
Where S is entropy. For a solid, the distinction between the two is usually
negligible. The inverse of the compressibility is called the bulk modulus, often
denoted K (sometimes B). The compressibility equation relates the
isothermal compressibility (and indirectly the pressure) to the structure of the
liquid.
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Vapor pressure
Vapor pressure or equilibrium vapor pressure is the pressure exerted by a
vapor in thermodynamic equilibrium with its condensed phases (solid or
liquid) at a given temperature in a closed system.
The equilibrium vapor pressure is an indication of a liquid's evaporation
rate. It relates to the tendency of particles to escape from the liquid (or a
solid). A substance with a high vapor pressure at normal temperatures is
often referred to as volatile.
The vapor pressure of any substance increases non-linearly with
temperature according to the Clausius–Clapeyron relation. The atmospheric
pressure boiling point of a liquid (also known as the normal boiling point) is
the temperature at which the vapor pressure equals the ambient atmospheric
pressure.
With any incremental increase in that temperature, the vapor pressure
becomes sufficient to overcome atmospheric pressure and lift the liquid to
form vapor bubbles inside the bulk of the substance. Bubble formation
deeper in the liquid requires a higher pressure, and therefore higher
temperature, because the fluid pressure increases above the atmospheric
pressure as the depth increases.
The vapor pressure that a single component in a mixture contributes to the
total pressure in the system is called partial pressure. For example, air at sea
level, and saturated with water vapor at 20 °C, has partial pressures of about
23 mbar of water, 780 mbar of nitrogen, 210 mbar of oxygen and 9 mbar of
argon.
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The vapor pressure of water
Is the pressure at which water vapour is saturated? At higher pressures
water would condense. The water vapour pressure is the partial pressure of
water vapour in any gas mixture saturated with water. As for other
substances, water vapour pressure is a function of temperature and can be
determined with Clausius–Clapeyron relation.
As the temperature of a liquid or solid increases its vapor pressure also
increases. Conversely, vapor pressure decreases as the temperature
decreases
Dynamic Viscosity and Kinematic Viscosity
The viscosity of a fluid is a measure of its resistance to gradual
deformation by shear stress or tensile stress. For liquids, it corresponds to
the informal notion of "thickness". For example, honey has a higher viscosity
than water.
Viscosity is due to friction between neighboring parcels of the fluid that are
moving at different velocities. When fluid is forced through a tube, the fluid
generally moves faster near the axis and very slowly near the walls, therefore
some stress (such as a pressure difference between the two ends of the
tube) is needed to overcome the friction between layers and keep the fluid
moving. For the same velocity pattern, the stress required is proportional to
the fluid's viscosity. A liquid's viscosity depends on the size and shape of its
particles and the attractions between the particles.
A fluid that has no resistance to shear stress is known as an ideal fluid or
inviscid fluid. Zero viscosity is observed only at very low temperatures, in
superfluids. Otherwise all fluids have positive viscosity. If the viscosity is very
high, for instance in pitch, the fluid will appear to be a solid in the short term.
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A liquid whose viscosity is less than that of water is sometimes known as a
mobile liquid, while a substance with a viscosity substantially greater than
water is called a viscous liquid.
The kinematic viscosity is the ratio of the dynamic viscosity μ divided by
the density of the fluid ρ. It is usually denoted by the Greek letter nu (ν). It
is a convenient concept when analyzing the Reynolds number that expresses
the ratio of the inertial forces to the viscous forces:
Further explanations
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Mathematical Exercises to Follow:
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New Lecture
1. Ideal Fluid
2. Real Fluid
3. Surface Tension
4. Capillary property
Introduction
In physics,
An ideal or perfect fluid is a fluid that can be completely characterized by its
rest frame mass density ρ and isotropic pressure p.
(Isotropy is uniformity in all orientations; it is derived from the Greek
isos (equal) and tropos (way).
Real fluids are "sticky" and contain (and conduct) heat.
Perfect fluids are idealized models in which these possibilities are
neglected. Specifically, perfect fluids have no shear stresses, viscosity,
or heat conduction.
Ideal Fluid
 Consider a hypothetical fluid having a zero viscosity ( μ = 0). Such a
fluid is called an ideal fluid and the resulting motion is called as ideal
or inviscid flow. In an ideal flow, there is no existence of shear
force because of vanishing viscosity.
All the fluids in reality have viscosity (μ > 0) and hence they are
termed as real fluid and their motion is known as viscous flow.
 Under certain situations of very high velocity flow of viscous
fluids, an accurate analysis of flow field away from a solid
surface can be made from the ideal flow theory.
Non-Newtonian Fluids
 There are certain fluids where the linear relationship
between the shear stress and the deformation rate (velocity
gradient in parallel flow) as expressed by the
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is not valid. For these fluids the viscosity varies with rate of
deformation.
 Due to the deviation from Newton's law of viscosity they are
commonly termed as non-Newtonian fluids. Figure 2.1
shows the class of fluid for which this relationship is
nonlinear.
Shear stress and deformation rate relationship of different fluids
The abscissa in Fig. 2.1 represents the behaviour of ideal fluids since for the
ideal fluids the resistance to shearing deformation rate is always zero, and
hence they exhibit zero shear stress under any condition of flow.
under any loading condition.
ding to the law that shear stress is
linearly proportional to velocity gradient or rate of shear strain. Thus for these
fluids, the plot of shear stress against velocity gradient is a straight line
through the origin. The slope of the line determines the viscosity.
-Newtonian fluids are further classified as pseudo-plastic, dilatant
and Bingham plastic.
Surface Tension of Liquids
 The phenomenon of surface tension arises due to the two kinds of
intermolecular forces
(i) Cohesion : The force of attraction between the molecules of a liquid by
virtue of which they are bound to each other to remain as one assemblage of
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particles is known as the force of cohesion. This property enables the liquid
to resist tensile stress.
(ii) Adhesion : The force of attraction between unlike molecules, i.e. between
the molecules of different liquids or between the molecules of a liquid and
those of a solid body when they are in contact with each other, is known as
the force of adhesion. This force enables two different liquids to adhere to
each other or a liquid to adhere to a solid body or surface.
Figure 2.3 The intermolecular cohesive force field in a bulk of liquid with a free
surface A and B experience equal force of cohesion in all directions, C
experiences a net force interior of the liquid The net force is maximum for D
since it is at surface
inward force. Thus mechanical work is performed in creating a free surface or
in increasing the area of the surface. Therefore, a surface requires mechanical
energy for its formation and the existence of a free surface implies the
presence of stored mechanical energy known as free surface energy. Any
system tries to attain the condition of stable equilibrium with its potential
energy as minimum. Thus a quantity of liquid will adjust its shape until its
surface area and consequently its free surface energy is a minimum.
The magnitude of surface tension is defined as the tensile force acting across
imaginary short and straight elemental line divided by the length of the line.
he dimensional formula is F/L or MT-2 . It is usually expressed in N/m in SI
units.
Surface tension is a binary property of the liquid and gas or two liquids
which are in contact with each other and defines the interface. It decreases
slightly with increasing temperature. The surface tension of water in contact
with air at 20°C is about 0.073 N/m.
results in a greater pressure at the concave side of the surface than that at its
convex side.
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Capillarity
phenomenon of capillarity. When a liquid is in contact with a solid, if the
forces of adhesion between the molecules of the liquid and the solid are
greater than the forces of cohesion among the liquid molecules themselves,
the liquid molecules crowd towards the solid surface. The area of contact
between the liquid and solid increases and the liquid thus wets the solid
surface.
lace when the force of cohesion is greater
than the force of adhesion. These adhesion and cohesion properties result in
the phenomenon of capillarity by which a liquid either rises or falls in a tube
dipped into the liquid depending upon whether the force of adhesion is more
than that of cohesion or not (Fig.2.4).
θ as shown in Fig. 2.4, is the area wetting contact angle made by
the interface with the solid surface.
Vapour pressure
All liquids have a tendency to evaporate when exposed to a gaseous atmosphere.
The rate of evaporation depends upon the molecular energy of the liquid which in
turn depends upon the type of liquid and its temperature. The vapour molecules
exert a partial pressure in the space above the liquid, known as vapour pressure. If
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the space above the liquid is confined (Fig. 2.5) and the liquid is maintained at
constant temperature, after sufficient time, the confined space above the liquid will
contain vapour molecules to the extent that some of them will be forced to enter
the liquid. Eventually an equilibrium condition will evolve when the rate at which
the number of vapour molecules striking back the liquid surface and condensing is
just equal to the rate at which they leave from the surface. The space above the
liquid then becomes saturated with vapour. The vapour pressure of a given liquid is
a function of temperature only and is equal to the saturation pressure for boiling
corresponding to that temperature. Hence, the vapour pressure increases with the
increase in temperature. Therefore the phenomenon of boiling of a liquid is closely
related to the vapour pressure. In fact, when the vapour pressure of a liquid
becomes equal to the total pressure impressed on its surface, the liquid starts
boiling. This concludes that boiling can be achieved either by raising the
temperature of the liquid, so that its vapour pressure is elevated to the ambient
pressure, or by lowering the pressure of the ambience (surrounding gas) to the
liquid's vapour pressure at the existing temperature.
Figure 2.5 To and fro movement of liquid molecules from an interface in a
confined space as a closed surrounding
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Fluid statics and dynamics
Imagine a container which has a fluid such as water or oil in it.
Take a point in the fluid and blow it up. Imagine when you dive in water, there will
be pressure from all sides on your body.
The water wants to push you in and you want to push the water out.
If the Force is F on the wall of Area A then the ratio is called Pressure P
The unit for the pressure is Newton / meter2 and we call it Pascal
Take another example as a piston filled with Gas
If the piston is massless then the gas pressure inside is equal to the atmospheric air
pressure.
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Pascal's law or the principle of transmission of fluid-pressure
is a principle in fluid mechanics that states that pressure exerted anywhere in a
confined incompressible fluid is transmitted equally in all directions throughout
the fluid such that the pressure variations (initial differences) remain the same.
This is the idea of a hydraulic jack
If you want to increase the pressure inside the piston you have to put some weight on
the piston and measure that pressure by dividing mg on the piston Area A
So P = P0 + mg / A
P is called the ABSOLUTE PRESSURE
P0 is called the atmospheric pressure
Mg / A is called the gauge pressure
Example is when your car tire is FLAT, the pressure inside the tire is equal to the
atmospheric pressure and that does not help to move the tire.
You need to add some pressure inside the tire to have the wheel roll.
So when you measure the tire pressure you are measuring the pressure inside the tire
in excess of the atmospheric pressure.
A pressure is a condition in the fluid.
Now if you take an imaginary cylindrical shape inside a fluid
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Is the pressure equal on both sides of the cylinder?
If NO then the cylinder will move if YES then the cylinder is at equilibrium.
Because the water is pushing in both sides to keep it in Static condition.
Pressure can NOT change at a certain depth
Now imagine that the cylinder is in Vertical shape
Remember this is NOT a cylinder it’s WATER.
We have isolated water and you can draw it as dotted linesNow this cylinder has an area A at height h
h1 is the height from the top to upper cylinder and h2 is to the
lower cylinder
Now the pressure at the Top and bottom should NOT be equal.
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Otherwise the cylinder will fall down because the two forces upwards and
downwards will cancel but what is keeping the cylinder of water from falling down
There has to be a weight to equal that cylinder..
The pressure at the bottom is higher than the top which is pushing upwards
Now we calculate what the pressure difference is
We have to balance the gravitational force of the system
Upwards force - downwards force = (A) x (delta h) x (rho) x (g)
P2A - P1A = (A) x (delta h) x (rho) x (g)
A cancels from both sides and we end up in
Then we end up
Now if we move P1 to the top of the surface of water
P the pressure at any depth will equal to P0 atmospheric + (rho) (g) (h)
So when you dive in Water the pressure increases by this amount.
Also you are limited when you dive to certain depth but FISHES breath water
that is why they do not have problem.
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Now what is the origin of atmospheric pressure?
We as humans are living in a pool of air
We live at the bottom of an Ocean of air
P at elevated heights above earth is zero as its vacuum
H is the height of the atmosphere
Rho is the density of air
So on the earth surface P0 is equal to 105 Pascal
If the earth was a swimming pool, how high can a human go down?
The maximum depth that a human can resist down in water Now we can
think of a way to measure the pressure BAROMETER (Atmospheric pressure
is barometric Pressure) So 105 Pascal is the atmospheric pressure that
FLCTUATES.
One way of doing it is by taking a can of water and evacuate a test tube and
put it upside down
It will raise h amount due to the atmospheric
pressure.
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0 pressure inside the test tube on h height
and where h is the same pressure on the surface of the water outside the
test tube.
But if we want to measure the atmospheric pressure using water we need 32
feet column and this is too high.
What do we use?
We need bigger rho and smaller h
If you use Mercury it will be 760 mm
Imagine now you want to drink water using a straw
You have to create a vacuum in your mouth to
be less than the atmospheric pressure.
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So you cannot suck water from a well if its more than 32 feet deep.
Now the U-Tube
if you fill it with two P0 Two different fluids oil and water
Where the line is the two pressures are equal but above the line is different
because rho is different
So if you know the density of one fluid you can find the density of the other
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Amplification of FORCE
Another application:
Air exerts a pressure of 1 Kg/cm2
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Another example
Distance of moving the pistons will be different to maintain the law of
conservation of energy.
The brake system of the car is another example.
You move your brake pedal several centimeters but the distance of the
discs on the drums move few mms
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Pressure Measurement:
Pressure Measurement Devices
Moderate pressure can be measured with sufficient accuracy by using
simple pressure gauges like U-tube manometers. But when very low or
high pressures are to be measured with high accuracy, advanced and
more complex gauges are required. Common advanced pressure gauges
are listed with some details.
Inclined
or Sloping U-tube Manometer: It is basically the same U-
tube manometer just the tube is inclined at certain angle this time. This results in
more deflection in the liquid level in the tube for the same change in pressure. This
enables the measurement of small pressure changes with increased accuracy.
 Differential Manometer:
It is used to measure the pressure difference between two points or between
two systems. It is again a U-tube manometer with the two ends of the U-tube
connected to the two systems between which pressure difference is to be
measured. Depending on the range of pressure difference to be measured, a
suitable liquid or combination of liquids can be filled in the two arms of the Utube. If large pressure differences are to be measured a heavy manometer
liquid is filled in the U-tube. And to measure very small pressure difference
U-tube with long arms is used and two light liquids are filled in the two arms
of the U tube.
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 Bourdon Gage
The Bourdon Gauge has a coiled tube whose one end is connected to the
system under consideration and other end is sealed. With the application of
the pressure in the tube it straightens up causing deflection of the sealed
end. The sealed end is connected to the indicating needle through a gear
and linkage mechanism. The deflection of the sealed end results in
movement of the needle which moves on a calibrated dial. Bourdon gauges
can be used to measure a wide range of pressures
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 Diaphragm Gauge:
Similar to the Bourdon Gauge, but has a Diaphragm which deflects on pressure
changes and the deflection is indicated on the calibrated scale
 Bellows Gauge:
In such gauges indicating needle is driven by the deflection of bellows
chamber. This gauge is suitable for measurement of very low pressures
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Pressure measurement devices, continued…..
The concept of absolute and gage pressure is
discussed in this section. Several common
pressure measurement devices and
techniques are also introduced.
Absolute and Gage Pressure
Gage Pressure
Vacuum Pressure
When pressure is measured relative to
absolute zero pressure, it is called absolute
pressure. On the other hand, pressure
measuring devices often use gage pressure,
which is measured relative to atmospheric
pressure. Standard atmospheric pressure at
sea level is 101.3 kPa or 14.7 psi. Gage
pressure is referred to as vacuum or suction
pressure when it is negative.
Gage Pressure:
pgage = pabsolute - patm
Vacuum Pressure: pvacuum = patm - pabsolute
For example, when the driver's manual for
your car suggests that you keep your tires at a
pressure of 30 psi, it is refering to the gage
pressure. This is equivalent to 44.7 psi
absolute pressure.
Barometer
Mercury Barometer
Atmospheric pressure can be measured
through a mercury barometer. A simple
barometer consists of an inverted glass tube
filled with mercury with its open end
submerged in a mercury container.
According to the hydrostatic pressure
distribution derived in the last section, the
atmospheric pressure is given by
patm = ρgh + pvapor
The vapor pressure in the glass tube is
negligibly small, hence the atmospheric
pressure is simply given by the height of the
mercury column:
patm = ρgh
For example, a column height of 760 mm Hg
corresponds to the standard atmospheric
pressure of 101.3 kPa (14.7 psi) at sea level.
If water is substituted for mercury, then a
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column height of 10.3 m H2O is needed for
atmospheric pressure.
Piezometer
A piezometer is the simplest form of a
pressure measuring device. It has a vertical
tube connected to the container in which the
pressure is needed. The pressure head of the
fluid column indicates the pressure of the
container:
pA = ρgh
where pA is the gage pressure at point A
within the container.
Piezometer
The disadvantages of piezometers are:
(1) Cannot measure vacuum pressure since
air would be sucked into the container through
the tube.
(2) The measured pressure should be
reasonably low, otherwise a very long vertical
tube is needed.
Manometer
Another pressure measuring device is the
manometer. It consists of a U-tube with one
end connected to the container with an
unknown pressure and the other end open to
the known atmospheric pressure. The fluid in
the U-tube manometer (gage fluid) can be
different from the fluid in the container.
The procedure for determining the pressure
inside the container is:
(1) Start from one end, and work from one
fluid level to another, up to the open end of the
manometer.
(2) Remember that pressure increases linearly
with depth for a fluid at rest.
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Consider the U-tube manometer shown. The
pressure at point A inside the tank is
calculated as:
pA + ρ1gh1 - ρ2gh2 = 0
which gives:
pA = ρ2gh2 - r1gh1
U-Tube Manometer
Click to view movie (73k)
Once again, gage pressure is used in the
above equation (i.e., the atmospheric pressure
at the open end is zero gage). If the fluid in the
tank is a gas, then the pressure between point
1 and 2 is negligible, hence
pA = ρ2gh2
The U-tube manometer also can be used to
determine the pressure difference between
two systems. This type of manometer is called
a differential U-tube manometer. Consider the
differential manometer connected between
tanks A and B, as shown in the figure. The
pressure will be determined by moving from
point A to point B:
Differential U-Tube Manometer
Click to view movie (84k)
Inclined-Tube Manometer
Click to view movie (100k)
pA + ρ1gh1 - ρ2gh2 - ρ3gh3 = pB
The pressure difference is given by
pA - pB = ρ2gh2 + ρ3gh3 - ρ1gh1
Another type of manometer is the inclinedtube manometer which is used to measure
small pressure differences between two
systems (say for gases). The advantage of the
inclined manometer is that the differential
reading scales along the tube can be made
large compared to a vertical manometer for a
given pressure difference, hence improving
the accuracy in reading the scale.
The pressure difference between point A and
B is given by
pA + ρ1gh1 - ρ2gL2sinθ - ρ3gh3 = pB
For cases where the columns h1 and h3 are
gas, the weights can be neglected, simplifying
the equation
pA - pB = ρ2gL2 sinθ
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HYDROSTATIC FORCES ON SURFACES
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Archimedes' Principle
Any object completely or partially submerged in a fluid is buoyed
up by a force whose magnitude is equal to the weight of the fluid
displaced by the object
Archimedes
287 – 212 BC
physicist, and engineer
Buoyant force
Inventor
Greek mathematician,
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Daniel Bernoulli
1700 – 1782 Swiss physicist and mathematician
Wrote Hydrodynamic Also did work that was the beginning of the
kinetic theory of gases
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Examples on HYDROSTATIC FORCES ON SURFACES
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Force on a submerged plane area
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Continuity for Fluids
When fluids move through a full pipe, the volume of fluid that enters
the pipe must equal the volume of fluid that leaves the pipe, even if
the diameter of the pipe changes. This is a restatement of the law of
conservation of mass for fluids.
The volume of fluid moving through the pipe at any point can be
quantified in terms of the volume flow rate, which is equal to the
diameter of the pipe at that point multiplied by the velocity of the fluid.
This volume flow rate must be constant throughout the pipe, therefore
you can write the equation of continuity for fluids (also known as the
fluid continuity equation) as:
This equation says that as the cross-section of the pipe gets smaller,
the velocity of the fluid increases, and as the cross-section gets
larger, the fluid velocity decreases. You may have applied this yourself
in watering the flowers with a garden hose. If you want increase the
velocity of the water coming from the end of the hose, you place your
thumb over part of the opening of the hose, effectively decreasing the
cross-sectional area of the hose’s end and increasing the velocity of
the exiting water!
Question:
Water runs through a water main of cross-sectional area 0.4 m2 with a
velocity of 6 m/s. Calculate the velocity of the water in the pipe when
the pipe tapers down to a cross-sectional area of 0.3 m2.
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Question:
Water enters a typical garden hose of diameter 1.6 cm with a velocity of
3 m/s. Calculate the exit velocity of water from the garden hose when a
nozzle of diameter 0.5 cm is attached to the end of the hose.
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