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MATERIALS
1.0 FLUID
The physics of fluids, or commonly known as fluid mechanics is the basis of
hydraulic engineering, a denomination of engineering that is applied in various fields.
A medical physicist or engineer might study the fluid flow in the arteries of an
aging patient.
A nuclear physicist or engineer might be highly interested in the investigations of
the fluid flow in the hydraulic system of an aging nuclear reactor, while an environmental
engineer might be constructing a logistical plan to create efficient irrigation of
agricultural areas and resolve the drainage problems from waste sites.
An aeronautical engineer is meticulously working on improving and designing the
hydraulic systems controlling the wing flaps that allow a jet airplane to land. As high as
the sky as in the previous example, we now also have to look into the deep sea where we
peek into the role of naval engineer in supervising deep sea diving activities or escape
from malfunctioned submarine, which potentially pose great dangers to her crew
members.
Not surprisingly, the knowledge and applications of fluid mechanics are even
implemented in many Las Vegas and Broadway performance, where huge sets are
quickly put up and brought down by hydraulic systems.
Before we can even venture into the realms of the physics of fluid , we need to
face an issue that confronts us all, “What exactly is fluid?”
1.1 WHAT IS A FLUID?
Fluid is a term to describe a substance, such as liquid or gas, whose molecules
flow freely so that it has no fixed shape and little withstanding resistance to shearing
stress regardless of how small the applied stress is. In sharp contrast to a solid., fluid
readily yields or conforms to the boundaries of any container in which we place them.
Ever wonder why we collectively call liquids and gases as fluid? After all, you
might argue, isn’t liquid as distinct from gas as it is from solid? In reality, it is not. Solid,
a crystalline structure, has its constituent atoms arranged in a fairly rigid threedimensional array called a crystalline lattice. We won’t exactly witness such orderly
arrangement in any liquids and gases
Pascal’s Law
Pascal’s law, developed by French mathematician Blaise Pascal, states that the pressure on a fluid is equal
in all directions and in all parts of the container. As liquid flows into the large container at the bottom of
this illustration, pressure pushes the liquid equally up into the tubes above the container. The liquid rises to
the same level in all of the tubes, reguardless of the shape or angle of the tube.
1.2 DENSITY & PRESSURE
1.2.1 Density : Introduction
When we are dealing with fluids, what academically stimulates us more is the
idea of variation of properties in different points of the fluid, hence it’s imperative to
speak of density and pressure instead of mass and force, the two terminology widely
used in mechanics of solids.
Let us examine Petaling Jaya and Ipoh, Perak. The number of people living in
PJ per area is higher compared to the number of people living in Ipoh, Perak. Such
example gives us the idea of the ratio of the number of people to area. Density, however,
is a measure of the ratio of mass to volume.
Density = Mass / Volume or ρ = m/V
ρ is a Greek letter rh. Common units are g cm-3 or kg m-3
1 g cm-3 = 1000 kg m-3
A substance could be
(i) denser in large mass and smaller volume
(ii) less dense in lower mass and larger volume.
Density is not a vector quantity as it does not have direction. Take notice that the
density of gas changes considerably with pressure, where as the density of a liquid
doesn’t. This is due to the higher compressibility in gases than in liquids.
1.2.2 MEASURING DENSITY
Measure the masses in kilograms of a wide range of solid objects. Then measure
their voumes. Use a micrometer or vernier callipers to measure their dimensions if they
are regular, or use the method of displacement of a volume of water by a solid with
irregular shape. The density can be calculated by dividing the mass by volume.
As for finding the density of liquid, first calculate the mass of a measuring
cylinder, then add liquid into it, getting its total mass. The mass of the liquid alone is
obtained by subtracting the mass of the cylinder from the total mass. Again, get the ratio
of mass to volume of the liquid as its density.
Find the density of a cube of ice. Then observe its melting process in a measuring
cylinder and measure its density again. Perform the same experiment with a cube of wax.
How much, in general, do substances change in volume when they make a transition from
solids to liquids?
Measure the volume and mass of a small cube of solid carbon dioxide. Drop it
into a plastic bag and seal the bag immediately. Estimate the volume when all the carbon
dioxide has become gas (process of sublimation). Calculate the density of solid carbon
dioxide and gaseous carbon dioxide.
1.2.3 THE RANGE OF DENSITIES
Solids and liquids are having small difference of volume when one phase makes a
transition to the other phase. Hence, their densities do not differ much. However, there’s
a distinct increase in volume for substance in gaseous state, where the density becomes
very much smaller, compared to solids and liquids.
1.3 COMPRESSIBILITY
Let’s try to compress, one at a time, a plastic syringe containing air, water and a
piece of dowel. The compressibility is increasing in that order. A fairly small force,
which has no noticeable effect on a liquid or a solid, will cause a large fractional
reduction in the volume of a gas. Such observation gives us an oppurtunity to look into
the physical microscopic properties of solid, liquid and gas.
1.4 PRESSURE
Snowshoes prevent the person from sinking into the soft snow because the force on the snow is spread over
a larger area, reducing the pressure on the snow’s surface.
After an exciting but exhausting lecture, a physics professor stretches out for a nap on a bed of nails,
suffering no injury and only moderate discomfort. How is this possible?
Pressure is defined as force over an area in on which the force is acting upon.
In the above examples, the snowshoes provide a large area to contribute to less
pressure on the ground. The same can be said for the nail bed. High heels for example, is
a very treasured item for a woman who has very high level expectation of beauty and
class, despite the pain inflicted on the heel due to its small area. Can you name other
applications of pressure in our daily lives?
Example
A water bed is 2.00 m on a side (a square) and 30.0 cm deep.
(a) Find its weight.
(b) Find the pressure that the water bed exerts on the floor. Assume that the entire lower
surface of the bed makes contact with the floor.
Calculate the pressure exerted by the water bed on the floor if the bed rests on its
side.
At any point on the surface of a submerged object in the container that constrain them, the force exerted by
the fluid is perpendicular to the surface area of the object. The force exerted by the fluid on the walls of the
container is perpendicular to the walls at all points.
1.3.1 PRESSURE IN FLUIDS
When a fluid is at rest in a container, every part of the fluid must be in static
equilibrium. This means the pressure, which is defined as
is acting perpendicular at any point in the fluid from all direction. Besides, the
pressure is the same for all points at the same depth of fluid. If it’s different, there
would be a flow of fluid from higher pressure region to lower pressure region. Look at
the following example in diagram (a). If the pressure on the left side is larger than the
pressure on the right side, there would be a net force and the block would accelerate to
the right because F1 would be greater than F2.
However, as we inspect diagram (b), the depth is different. There are three forces
acting on the block, Mg (downward), P1A (downward) and P2A (upward). The net force
is zero as it is in equilibrium, therefore
P2A - P1A – Mg = 0
Since M = ρV = ρAh
Therefore
P2A - P1A – (ρAh)g = 0
Eliminating the A,
P2 - P1 – ρgh = 0
If the upper surface of the block is at the water surface, exposed to atmospheric pressure,
we will get the following equation
P2 = P1 + ρgh
Where
P1 = Atmospheric pressure above the liquid surface
P2 = Total pressure on a point in the liquid
ρgh = Pressure on a point due to depth of liquid
EXAMPLE
In a huge oil tanker, salt water has flooded an oil tank to a depth of 5.00 m. On top of the
water is a layer of oil 8.00 m deep, as in the cross-sectional view of the tank in figure
below. The oil has a density of 0.700 g/cm3. Find the pressure at the bottom of the tank.
(Take 1 025 kg/m3 as the density of salt water.)
Strategy
First, use it to calculate the pressure P1 at the bottom of the oil layer. Then use this
pressure in place of P0 and calculate the pressure Pbot at the bottom of the water layer.
Solution
Remark
The weight of the atmosphere results in P0 at the surface of the oil layer. Then the weight
of the oil and the weight of the water combine to create the pressure at the bottom.
EXERCISE
1. Calculate the pressure on the top lid of a chest buried under 4.00 meters of mud with
density 1.75 X 103 kg/m3 at the bottom of a 10.0-m-deep lake.
(2.68 X 105 Pa)
2. Estimate the net force exerted on your eardrum due to the water above when you are
swimming at the bottom of a pool that is 5.0 m deep. (Estimating the area of the eardrum
as 1 cm2.
(4.9 Pa)
3. An aeroplane launches off at sea level and elevate to a height of 425 m. Estimate the
net outward force on a passenger’s eardrum assuming the density of air is
approximately constant at 1.3 kg/m3 and that the inner ear pressure hasn’t been equalized.
(0.54 N)
4. An office window has dimensions 3.4 m by 2.1 m. As a result of the passage of a storm,
the outside air pressure drops to 0.96 atm, but inside the pressure is held at 1.0 atm. What
net force pushes out on the window?
5. Three liquids that will not mix are poured into a cylindrical container. The volumes
and densities of the liquids are 0.50 L, 2.6 g/cm3 ; 0.25 L, 1.0 g/cm3 and 0.40 L, 0.80
g/cm3. What is the force on the bottom of the container due to these liquids? One liter = 1
L = 1000 cm3. (Ignore the contribution due to the atmosphere.)
6. Find the pressure increase in the fluid in a syringe when a nurse applies a force of 42 N
to the syringe’s circular piston, which has a radius of 1.1 cm.
7. A partially evacuated airtight container has a tight-fitting lid of surface area 77 m2 and
negligible mass. If the force required to remove the lid is 480 N and the atmospheric
pressure is 1.0 x 10 Pa, what is the air pressure in the container before it is opened?
Pascal’s Principle
What important contribution to fluids did Blaise Pascal make in 1647?
Known as Pascal’s Principle, the law that Blaise Pascal develop states that any
force applied to an enclosed fluid is transmitted in all directions to the walls of the
container. The principle is extremely important in the field of hydrostatics and in the
development of hydraulics. For example, if a piston pushes against the liquid in a closed
cylinder, the force applied by the piston will translate into pressure on the walls of the
cylinder. This occurs because liquids cannot be compressed as gases can.
What is hydraulics?
Hydraulics is the use of a liquid that, when moved from one place to another,
accomplishes by its motion some type of function. The liquid used in hydraulic
mechanism us usually water or oil. Hydraulic engineers design such things as pumps, lifts,
faucets, cranes, shock absorbers, and many other devices.
How does a hydraulic lift work?
The basic theory behind a hydraulic lift is to multiply forces and give a device
mechanical advantage. An automobile lift, used in many automotive repair shops, allows
the operator to use very little force to lift an automobile off the ground, by pushing liquid
from a small-diameter cyclinder and piston through a thin tube that expands into a largerdiameter cyclinder and piston, which is located beneath the vehicle to be lifted. Since the
liquid cannot be compressed like air, the liquid from the small cyclinder is pushed into
the large cylinder, forcing the large piston to move upward. Although this is very
simplistic view of how hydraulic lift works, Pascal’s Principle states that if a small-area
piston pushes a large-area piston, the mechanical advantage can be quite great.
Example 1
In a car lift used in a service station, compressed air exerts a force on a small piston that
has a circular cross section and a radius of 5.00 cm. This pressure is transmitted by a
liquid to a piston that has a radius of 15.0 cm. What force must the compressed air
exert to lift a car weighing 13 300 N? What air pressure produces this force?
Because the pressure exerted by the compressed air is transmitted undiminished
throughout the liquid, we have
The air pressure that produces this force is
This pressure is approximately twice atmospheric pressure.
Eureka!
What major discovery did Archimedes make when he stepped into bath in the third
century B.C.?
To his astonishment, when Archimedes stepped into a tub of water, the water rose!
Of course, this wasn’t the first time water rose when Archimedes sat in the tub, but it was
the first time he would consider the reasons why. He proceeded to conduct an experiment
with gold and silver crowns that he immersed in the tub, measuring the water ……
Why does a small clump of steel sink, while a 50000-ton steel ship can float?
In order to remain afloat, a ship needs to displace an amount of fluid equal to its
own weight. Therefore, if a clump of steel is placed in water, it will sink because its size
wouldn’t allow it to displace an amount of water equal to its own weight. In this case,
there is no way that the water could apply enough upward force to keep it afloat. A
50000-ton steel ship can easily stay afloat as long as it can displace 50000 tons of water.
It can do this by widening the hull of the ship and increasing its volume.
Buoyant Force
Have you ever attempted to push a basketball into the water? It’s actually a
difficult thing to achieve as the upward force exerted by the water on the ball is extremely
high. This sort of force exerted by the liquid on any immersed object is known as
buoyant force.
When an object is immersed in a fluid (either a gas or a liquid), it experiences an
upward buoyancy force because the pressure at the bottom of the object is greater than
the pressure on the top. The great Greek scientist Archimedes (287-212 B.C.) made the
following craeful observation, now called Archimedes Principle.
Any object completely or partially immersed in a fluid is buoyed up by a force equal to
the weight of the displaced fluid.
In order to verify this, consider a small portion of water in a beaker of water. The
downward forces are weight and force P (due to pressure of fluid at the top of the object
multiply the area). The upward force is solely force Q (due to pressure of fluid at the
bottom of the object multiply the area). In equilibrium, the upward force balances the
downward forces.
FP + W = FQ
The net upward force due to the fluid is called the buoyancy force,
FB = FQ - FP
FB = FQ - FP = W
FB = W
This small portion of water is replaced by an object having similar shape and size.
The object will feel the same upward buoyant force. A fully immersed object indicates
that the object’s density is the same as the density of the fluid. The same cannot be said
of the object immersed partially, where it’s density is less than the water.
In this discussion about buoyant force, let’s remember two things:
#1 In a floating or submerged object, assuming ideal system, the weight downward, W
of an object is the same as the buoyant force upward, FB, giving us the following,
FB = WOBJ
#2 According to Archimedes’ Principle, the magnitude of the buoyant force always
equals the weight of the fluid displaced by the object, giving us the following,
FB =WFLUID
Therefore, we can establish that the weight of an object is equals to the weight of the
fluid being displaced.
WOBJ = WFLUID
Archimedes's Principle
An object is subject to an upward force when it is immersed in liquid. The force is equal to the weight of
the liquid displaced. The apparent weight of a block of aluminium (1) immersed in water is reduced by an
amount equal to the weight of water displaced. If a block of wood (2) is completely immersed in water, the
upward force is greater than the weight of the wood. (Wood is less dense than water, so the weight of the
block of wood is less than that of the same volume of water.) So the block rises and partly emerges to
displace less water until the upward force exactly equals the weight of the block.
Apparent Weight
When we attach an object to a spring balance, the weight reading is the actual
weight of the object. However, when we immerse the object in a pool of water, the
reading is less, why?
The upthrust force (buoyant force) is acting upward on the object, resulting in a
less-than-actual apparent weight.
Apparent Weight = Actual Weight - Upthrust
Let’s say, Gina has weight 500 N, when she goes into the swimming pool, the
upthrust, 100 N, causes her to feel the apparent weight of 400 N only.
Determination of Object’s Density
Archimedes’ principle also makes possible the determination of the density of an
object that is so irregular in shape that its volume cannot be measured directly. If the
object is weighed first in air (ACTUAL WEIGHT) and then in water (APPARENT
WEIGHT), the difference in weights will equal the weight of the volume of the water
displaced (equivalent to UPTHRUST), which is the same as the volume of the object.
Thus the density of the object (mass divided by volume) can readily be determined. In
very high precision weighing, both in air and in water, the displaced weight of both the
air and water has to be accounted for in arriving at the correct volume and density.
In the simplest language possible, an object 500 N in air, then immersed in water
where you get 400 N reading on the spring balance.
The 100 N is the difference between the two weights, equivalent to the buoyant
force, which is also equal to the weight of the water being displaced.
WH20 = ρVg
100 = 1000 (V )(10)
V = 0.01 m3
The volume of the water being displaced 1 m3 , then we shall conclude that 1 m3 is
also the volume for the irregular object.
Using the 50 kg as the mass of the irregular object (assuming g is 10 m s-2), we
use this value and divide by 0.01 m3, you will get a density of 5000 kg m-3 for the
irregular object.
Applications of Archimedes’ Principle
A submarine applies the Archimedes’ Principle to enable it to float and sink. The
ballast tanks are special compartments in a submarine. When the compressed air forces
the water out of the ballast tank, the submarine rises. This occur because the buoyant
force is greater than the weight of the submarine.
Meanwhile, when water is allowed to enter the ballast tank, the submarine sinks
because the buoyant force is less than the weight of the submarine.
In hot air balloon, the fire heats up the helium gas inside the balloon up to 100°C..
As the balloon expands, it displaces a large amount of the surrounding air, thus the
upthrust is very huge. The density inside is getting less and the weight is decreased. The
balloon starts to rise when the upthrust is larger than the weight of the balloon.
If the hot air ballon is neither going up nor down,
Fup = WBALLOON = WATMOSPHERIC AIR
Example
A helium-filled hot air balloon is rising.
The density of surrounding air is 1.3 kg m-3
The density and volume of helium gas is 0.18 kg m-3 and 0.06 m3 respectively.
A weight, W is attached to the balloon.
(a) What is the weight of the helium gas inside the balloon?
(b) Calculate the upthrust acting upward upon the balloon.
(c) How much W is needed to enable the balloon to be staticly floating in air?
The helium gas contained inside the balloon is considered contributing downward weight
to the balloon and MUST NEVER BE CONFUSED with the outside surrounding air
displaced by the balloon (equivalent to the upthrust).
Solution:
How Science Works The Plimsoll Line
What happenes to the buoyancy of the ship when cargo and passengers
are added?
Ship builders must always consider the floatation level of the ship when cargo and
passengers are added to it. This increases the weight of the ship. The ship will float as
long as the total weight of the ship plus contents is less then the weight of the water it
displaces. When the weight of the ship and contents exceed this value, the ship sinks.
The amount of ship lowers in the water as a result of cargo and passengers can be
critical for navigation and maneuverability. Large cargo and cruise ships have numbers
on the bow of the ship that indicate how far the ship is submerged. If the ship has a
twenty-foot draft and the water is only eighteen feet deep, cargo and passengers must be
unloaded to allow the ship to rise.
When a ship floats on the surface of the ocean, two forces are acting on the ship,
namely, the weight of the ship and the upthrust. The weight of the ship changes when the
total load in the ship varies. Upthrust is proportional with the density and the volume of
the displaced ocean water. Whenever the load increases, the weight of the ship increases.
The ship will submerge a little deeper into the ocean to displace more water. In this way,
the ship will gain more upthrust to support the increasing weight of the ship.
Ships that navigate to places throughout the world will sail through various
densities of ocean water. Plimsoll Line is drawn on the side of the ship to give indication
on the maximum load allowed for the ship. If the ocean level is directly on the line, the
ship has carried the maximum permissible load.
The International Load Line
(LR: Lloyd's Register of Shipping)
LTF
Lumber, Tropical, Fresh
TF
Tropical Fresh Water Mark
LF
Lumber, Fresh
F
Fresh Water Mark
LT
Lumber, Tropical
T
Tropical Load Line
LS
Lumber, Summer
S
Summer Load Line
LW
Lumber, Winter
W
Winter Load Line
LWNA Lumber, Winter, North Atlantic WNA Winter, North Atlantic
FLUID MECHANICS
The mechanics of fluid flow are complex and produce many unexpected effects, some
of which are still not fully understood. The onset of turbulence when smooth flow is
suddenly disrupted by complex motions like vortices and eddies is very difficult to
predict or model effectively. In some situations turbulence is desirable; for example, to
provide rapid mixing of fuel and oxygen inside a jet engine. In other situations it is
disastrous. Turbulent airflow over a wing destroys lift.
The motion of real fluids is very complicated and not yet fully understood. Instead,
the discussion of ideal fluid is proposed instead. The following are four assumptions
abpout ideal fluid, which are concerned with flow:

Steady or laminar flow: The velocity of the moving fluid any fixed point does not
change with time, either in magnitude or in direction. The gentle flow of water
near the center of a quiet stream is steady; the flow in a chain of rapids is not.
Assume flow is smooth (laminar) and not turbulent. Fluid flow is described using
streamlines. These are arrows that represent the velocity of the fluid at each point.
In steady flow they correspond to lines of motion which follow the paths of the
particles.

Incompressible flow: We assume that the ideal fluid is incompressible at rest. In
other words, it means the density has a uniform and constant value.

Nonviscous flow: Ignore fluid friction, that is, viscosity. Viscosity is the measure
of how resistive the fluid is to flow. Hgher viscosity means that the velocity
gradient is lower; lower viscosity allows the velocity to increase rapidly away
from the surface. For instance, compare water and raw petroleum. The organic
fluid is more resistive to flowing than the water, so the raw petroleum is
considered more viscous than water. Using the same analogy as the solid-friction
interaction, we can describe the “friction” of the fluids in terms of viscosity. Both
contribute to the changing of kinetic energy to thermal energy. If the fluid is in the
state of nonviscous flow, it would not experience viscous drag force. The British
scientist Lord Rayleigh noted that in an ideal fluid a ship’s propeller would not
work, however, on the other hand, in an ideal fluid a ship (once set into motion it
would not need a propeller!)

Irrotational flow: The particle of dust in an experiment in fluid may rotate in a
circular path, but it will not rotate at its own axis. Take the example of Ferris
wheel, the passenger doesn’t rotate, but their bodies rotate according to the axis of
the wheeel!
LAMINAR FLOW/STREAMLINE FLOW
The steady flow of a fluid around a cylinder, as revealed by a dye tracer that was injected into the fluid upstream of the
cylinder.
In laminar flow, sometimes known as streamline flow through a pipe, adjacent
layers of a liquid move parallel to one another, without disruption between the layers. It
is opposite of turbulent flow. In layman language, laminar flow is “smooth” while
turbulent flow is “rough”
Laminar and Turbulent Motion
At low velocities, fluids flow in a streamlined pattern called laminar motion. Laminar motion can be
described mathematically by equations derived by Claude Navier and Sir George Stokes in the mid 1800s.
At high velocities, fluids flow in a complex pattern called turbulent motion. For fluids flowing in pipes, the
transition from laminar to turbulent motion depends on the diameter of the pipe and the velocity, density,
and viscosity of the fluid. The larger the diameter of the pipe, the higher the velocity and density of the
fluid, and the lower its viscosity, the more likely the flow is to be turbulent.
Let’s look at the smoking man photo. We can observe that the smoke is rising up
off the cigarette in a still air surrounding. It will, initially rise up, increasing in its speed
in a vertical and smooth manner for some distance (laminar flow). After some time, the
smoke will start undulating into a turbulent, non laminar flow. The whole process is
changing from the steady to non-steady.
The Equation of Continuity
Imagine an ideal fluid flowing through a frictionless pipe which becomes
narrower and narrower. The fluid is incompressible, so as much fluid enters the pipe
each second as leaves it. If the cross-sectional areas and velocities at entry and exit are A1
and A2 and v1 and v2 respectively then:
volume entering per second = A1 v1 = volume leaving per second = A2v2
This is the equation of continuity and leads to:
The flow speed is greater in the thinner part of the pipe.
The flow within any tube of flows obeys the equation of continuity:
Rv = Av = constant
In which
Rv is the volume flow rate
A is the cross-sectional area of the tube of flow at any point, and
v is the speed of the fluid at that point.
How Science Works : Continuity
Why does a river’s current run faster when the river is narrow?
When water flows down a river, the current represents the amount of water that
passes by a section of the river in a unit of time. For example, if the current of a river is
2000 L/min, this means that, assuming the slope of the river is constant, every minute
2000 liters pass by every section of the river. If a section of the river narrows, the 2000
liters of water still must pass in one minute beacause the water from behind does not let
up in its desire to flow downriver. Since the river is narrower, the water needs to speed
up in order to accomplish this task. The principle behind this phenomenon is called
continuity.
Example 1
Water is flowing out from a water tap. The cross-sectional areas at the starting point is A0
= 1.2 cm2 and a point 45 mm below the starting point is A = 0.35 cm2. What is the initial
velocity of the flowing water? What is the flow rate Rv?
Example 2
A garden hose with an internal diameter of 1.9 cm is connected to a (stationary) lawn
sprinkler that consists merely of a container with 24 holes, each 0.13 cm in diameter. If
the water in the hose has a speed of 0.91 m s-1, at what speed does it leave the sprinkler
holes.
Solution 2
We use the equation of continuity. Let v1 be the speed of the water in the hose and v2
be its speed as it leaves one of the holes. A1 = πR2 is the cross-sectional area of the hose.
If there are N holes and A2 is the area of a single hole, then the equation of continuity
becomes
where R is the radius of the hose and r is the radius of a hole. Noting that R/r = D/d (the
ratio of diameters) we find
Example 3
The water flowing through a 1.9 cm (inside diameter) pipe flows out through three 1.3
cm diameter pipes.
(a) If the flow rates in the three smaller pipes are 26, 19 and 11 L/min, what is the
flow rate in the 1.9 cm diameter pipe?
(b) What is the ratio of the speed in the 1.9 cm diameter pipe to that in the pipe
carrying 26 L/min?
Solution 3
(a) Equation of continuity provides us the initial flow rate to be the same as final flow
rate. Giving us (26 + 19 + 11) L/min = 56 L/min in the main pipe (1.9 cm diameter)
(b) Using the equation of continuity, Using v = R/A and A = πd2/4,
Example 4
The merging of two streams, A and B, forms river C. The following are their respective
data:
A Width: 8.2 m Depth: 3.4 m Speed: 2.3 m s-1
B Width: 6.8 m Depth: 3.2 m Speed: 2.6 m s-1
C Width: 10.5m Depth: h Speed: 2.9 m s-1
What is the value of h?
What is fluid dynamics?
Fluid dynamics is the study of fluids in motion. There are several different types
of fluid motion: steady flow, where the liquid or gas moves in a constant and predictable
manner; unsteady flow, where the fluid makes turns and changes its velocity; and
turbulent flow, where the fluid motion is extremely difficult to predict.
What makes a fluid flow?
As in all of physics, objects move as a result of forces. Just as a basketball
dropped in the air falls to the ground because of gravitational force, a fluid flows because
there is an unbalanced force acting on the liquid – that is, a difference in pressure
between two points; fluid will flow in the direction of decreasing pressure.
Why does it always seem windier in the city?
The explanation behind this question is not meteorological, but physical. In major
cities, there ae skyscrapers and other tall buildings that obstruct the flow of wind. In order
to flow past these large obstacle, the wind speed increases in the corridors of the streets
and avenues. The same effect can also be found in tunnels and outdoor “breezeways”. It
is the continuity of the fluid speed rushing through the narrow corridors of streets and
avenues that makes the city such a windy place.
Drag Act
As we investigate the dynamics of objects in fluid dynamics, we soon encounter a
property called drag (also known as fluid resistance). Drag is the force that resist
movement
In fluid dynamics, drag (sometimes called fluid resistance) is the force that resists the
movement of a solid object through a fluid (a liquid or gas). The most familiar form of
drag is made up of friction forces, which act parallel to the object's surface, plus pressure
forces, which act in a direction perpendicular to the object's surface. For a solid object
moving through a fluid, the drag is the component of the net aerodynamic or
hydrodynamic force acting in the direction of the movement. The component
perpendicular to this direction is considered lift. Therefore drag acts to oppose the motion
of the object, and in a powered vehicle it is overcome by thrust.
Terminal Velocity
How Science Works : Stokes’ Law
Introduction
This investigation involves determining the viscosity and mass density of
an unknown fluid using Stokes’ Law. Viscosity is a fluid property that provides an
indication of the resistance to shear within a fluid. Specifically, you will be using a
fluid column as a viscometer. To obtain the viscometer readings you will use a
stopwatch to determine the rate of drop of various spheres within the fluid. You will
determine both density and viscosity.
2.Learning Outcomes
On completion of this laboratory investigation students will:
• Appreciate the engineering science of 'fluid mechanics.'
• Understand the concept of fluid 'viscosity.'
• Understand the concept of dimensionless parameters, and most specifically
the determination of Reynold's Number.
• Be able to predict the settling time of spheres in a quiescent fluid.
• Be able to calculate the viscosity of an unknown fluid using Stokes' Law and
the terminal velocity of a sphere in this fluid.
• Be able to correct for the diameter effects of fluid container on the
determination of fluid viscosity using a 'falling ball' viscomter
3. Definitions
Viscosity – a fluid property that relates the shear stress in a fluid to the angular rate of
deformation.
Fluid Mechanics – the study of fluid properties.
Reynold’s Number – dimensionless parameter that represents the ratio of viscous to
inertial forces in a fluid
Strength of Materials
The Physical Properties of Solids
Sugar Crystals
This electron microscope image of raw cane sugar reveals the shape of sugar crystals. The crystals form
after purified cane juice has been heated and some of the water in the juice has evaporated, leaving behind
a cane syrup. Seed crystals added to the syrup make the sugar molecules dissolved in the syrup separate
from the liquid to form larger, solid crystals around the seed crystals.
Stretching Materials
Stress, Strain and the Young Modulus
The strength of a wire has direct proportional relationship with its cross-sectional
area. Thicker wires are tougher than thinner wires of the similar material.
For a given substance, the tension needed to break the wire divided by the crosssectional area is constant. Stress, σ is defined as the average amount of tension (force), F
exerted on a given area, A The tension divided by cross-sectional area is called the
tensile stress, symbol σ (the small Greek letter sigma):
tensile stress = tension/cross-sectional area
Stress has the same units as pressure, N m-2 or the pascal, Pa.
The stress needed to break a material is called the breaking stress or ultimate
tensile stress. If you want to compare the strengths of different materials, you compare
the breaking stresses, since these do not depend on the cross-sectional area of the sample
that you are testing. A strong material like steel has a high ultimate tensile stress; a weak
material has a low ultimate tensile stress.
A steel wire, cross-sectional area 1.0 mm2 breaks under a tension of 250 N.
breaking stress = F/A = 250 N/(1.0 x 10-6 m2) = 250 MPa
The shape of a tension—extension graph depends on the dimensions of the
sample of wire you are testing.
If you have wires of the same length and the same material, the thicker wire
will need a larger tension for the same extension.
If the wires are of the same material and thickness, but different lengths, the
longer wire will have a larger extension for the same tension.
If you wish to compare the behaviour of different materials, you need to plot
quantities that do not depend on the size of the sample of material tested.
Strain is the deformation of materials caused by the action of tensile stress. The
difference in the initial displacement of two points, say, A & B and final displacement of
two points, A & B after strain is applied is known commonly as extension:
Strain, symbol ε (the small Greek letter epsilon), is the extension divided by the original
length
ε = strain
l = length of extension + original length
l0 = original length
Strain accounts for the fact that samples stretch in proportion to their lengths. Strain is a
unitless quantity. The ratio of extended length (meter) to original length (meter) cancels
the unit out, therefore it doesn’t have unit of measure.
A steel wire 2.0 m long, stretched to just below its breaking point, extends by 2.6 mm.
strain = (2.6 X 10-3 m)/(2.0 m) = 1.3 X 10-3
In solid mechanics, Young's modulus (E) is a physical property that describe the
stiffness of a material. It is defined as the ratio of stress over strain in the region in
which Hooke's Law is obeyed for the material.We can conduct an experiment to acquire
the value from the slope of a stress-strain graph drawn during tensile tests conducted on
a sample of the material.
Based on the graph above (left), the slope of the straight line is called the
"Young's Modulus", and has dimensions of force over area. The three different materials
have various numerical gradient values, indicating the different measure of “stiffness”,
namely the Young’s Modulus.
Smaller values (lowest gradient, polystyrene) indicate that less stress is required
for more strain. In other words, it experiences longer stretches for a small stretching force.
Likewise, larger values of Young's Modulus (largest gradient, steel) indicate that
more stress is required for a given small strain. Less stretching is achieved, even though
much stretching force ha been applied.
In general, the stress versus strain curve will differ for each material for each type
of similar stress.
The photo on above (right) shows steel cables used in lifting heavy objects.
These steel cables break under a tension of 300 kN.
Stiffness is important for engineering materials, since for most applications it is
important that the shape of a component changes very little when it is under stress. So,
most of the time, materials with higher Young’s Modulus such as steel and copper are
chosen for engineering projects. On the other hand, stiffness of consumer products is not
necessary since the significant changes in shape do not really matter for substance like
polystyrene.
How Science Works : Climbing Ropes
How Science Works : Uncertainties in Measurement
Characteristics of Solids
How Science Works : The Mohs Hardness Scale
In the Mohs scale, named for the German mineralogist Friedrich Mohs who
devised it, ten common minerals are arranged in order of increasing hardness and are
assigned numbers:
1. talc
2. gypsum
3. calcite
4. fluorite
5. apatite
6. orthoclase (feldspar)
7. quartz
8. topaz
9. corundum
10. diamond
The hardness of a mineral specimen is obtained by determining which mineral in the
Mohs scale will scratch the specimen. Thus, galena, which has a hardness of 2.5, can
scratch gypsum and can be scratched by calcite. The hardness of a mineral largely
determines its durability.
How Science Works : Materials Selection Charts
Materials In The Real World
How Science Works : Real World Materials