Fluids.

Fluids

Phases of Matter Density and Specific Gravity Pressure in Fluids Atmospheric Pressure and Gauge Pressure Pascal’s Principle Measurement of Pressure; Gauges and the Barometer Buoyancy and Archimedes’ Principle

Fluids in Motion; Flow Rate and the Equation of Continuity
Bernoulli’s Equation Applications of Bernoulli’s Principle: from Torricelli to Airplanes, Baseballs, and TIA Viscosity Flow in Tubes: Poiseuille’s Equation, Blood Flow Surface Tension and Capillarity Pumps, and the Heart

Phases Summary

Phases of Matter The three common phases of matter are solid, liquid, and gas. A solid has a definite shape and size. A liquid has a fixed volume but can be any shape. A gas can be any shape and also can be easily compressed. Liquids and gases both flow, and are called fluids.

Water Earth is the only planet in the solar system
with a surface that is primarily liquid. It is the distance between the Earth and the sun that and the Earth’s atmosphere that make the temperature of the planet right to sustain the water cycle. The cycle of evaporation, condensation, precipitation and collection. The water cycle is a key to sustaining life on the planet.

Density and Specific Gravity
The density ρ of an object is its mass per unit volume: The SI unit for density is kg/m3. Density is also sometimes given in g/cm3; to convert g/cm3 to kg/m3, multiply by 1000. Water at 4°C has a density of 1 g/cm3 = 1000 kg/m3. The specific gravity of a substance is the ratio of its density to that of water.

Liquid A liquid is a fluid that has the particles loose and can freely form a distinct surface at the boundaries of its bulk material. The surface is a free surface where the liquid is not constrained by a container.

Liquids Liquids have a definite volume, but they assume the shape of their container. Molecules in a liquid move freely past other molecules in the liquid. Liquids are incompressible, so the density does not depend on depth (temperature effects ignored)

Pressure in Fluids Pressure = Force/Area.
In a fluid pressure depend on the density of the fluid and the depth of the fluid. Pressure = Patm + ρgh Pressure is not dependent on volume

Fluid Pressure Fluid pressure is not dependent on volume, this is important for dam construction as it must withstand the pressure due to the depth of the lake, the volume of water, or area of the lake does not matter for the construction of a dam. Which dam must withstand a greater pressure at the base (bottom)?

Question Why are metal bands closer together at bottom of water tower than at the top? Pressure at any level depends on depth of the fluid, so the pressure is higher at the bottom of the tank.

Fluid Pressure The pressure P is the same on the bottom of each vessel.

Pressure in Fluids Pressure is defined as the force per unit area.
Pressure is a scalar; the units of pressure in the SI system are Pascal's: 1 Pa = 1 N/m2 Pressure is the same in every direction in a fluid at a given depth; if it were not, the fluid would flow.

Pressure in Fluids Also for a fluid at rest, there is no component of force parallel to any solid surface – once again, if there were the fluid would flow.

Pressure in Fluids The pressure at a depth h below the surface of the liquid is due to the weight of the liquid above it. We can quickly calculate: This relation is valid for any liquid whose density does not change with depth.

Atmosphere The Earth’s atmosphere is an ocean of air. We are at the bottom of the ocean. 99% of Earth’s atmosphere is below 30km. Atmospheric pressure is due to the weight of the air column above us. Just as the water pressure depends on the depth.

Atmospheric Pressure and Gauge Pressure
At sea level the atmospheric pressure is about ; this is called one atmosphere (atm). Another unit of pressure is the bar: Standard atmospheric pressure is just over 1 bar. This pressure does not crush us, as our cells maintain an internal pressure that balances it.

Atmospheric Pressure and Gauge Pressure
Most pressure gauges measure the pressure above the atmospheric pressure – this is called the gauge pressure. The absolute pressure is the sum of the atmospheric pressure and the gauge pressure.

Measurement of Pressure; Gauges and the Barometer
There are a number of different types of pressure gauges. This one is an open-tube manometer. The pressure in the open end is atmospheric pressure; the pressure being measured will cause the fluid to rise until the pressures on both sides at the same height are equal.

Here are two more devices for measuring pressure: the aneroid gauge and the tire pressure gauge.

This is a mercury barometer, developed by Torricelli to measure atmospheric pressure. The height of the column of mercury is such that the pressure in the tube at the surface level is 1 atm. Therefore, pressure is often quoted in millimeters (or inches) of mercury.

Aneroid Barometer Partially evacuated with flexible membrane that is attached to a spring scale that records changes in air pressure. Note air pressure varies with altitude…air pressure is higher at low altitudes and lower at high altitudes.

Mercury is 14 times as dense as water
Mercury is 14 times as dense as water. If we measured the air pressure with a water barometer instead of a mercury barometer, how high would the column of water be, compared to the height of the mercury column? the same height B) 14 times the height. C) 1/14 times the height

Gases Gases flow like liquids. In gases, the
molecules are further apart and are moving faster than in the liquids. Note the space of any container will be filled with the gas in the system. Even outer space has some gas, typically 1 molecule/cc

Gases in the Atmosphere

Atmosphere The Earth’s atmosphere is an ocean of air. We are at the bottom of the ocean. 99% of Earth’s atmosphere is below 30km. Atmospheric pressure is due to the weight of the air column above us. Just as the water pressure depends on the depth.

Atmospheric Pressure Air pressure is 101,300 N/m2 or 1.01 Pa (Pascal)
The density of air is ~1.21 kg/m3

Question How many kilograms of air are in a room that is 10mX8mx3m
240m3*1.21kg/m3=300kg

What Happens if Air Pressure Changes?
Why do my ears pop? If you've ever been to the top of a tall mountain, you may have noticed that your ears pop and you need to breathe more often than when you're at sea level. As the number of molecules of air around you decreases, the air pressure decreases. This causes your ears to pop in order to balance the pressure between the outside and inside of your ear. Since you are breathing fewer molecules of oxygen, you need to breathe faster to bring the few molecules there are into your lungs to make up for the deficit.

Example Estimate the total mass of the Earth’s atmosphere. Remember,
P = F/A = mg/A

Answer m = P*A/g m=1.013x105Pa*4π(6.38x106m)2/(9.8m/s2)
m = 5.29x1018 kg

Straw The action of a barometer is similar to that of a drinking straw. When you drink liquid through a straw you are creating a vacuum and it is the air pressure that forces liquid up the straw…

Tire Pressure More air molecules means that
more collisions with the tire wall occur each second

Example Determine the gauge pressure of water at the house fed by the water tower as shown. How high would the water shoot if it came vertically out of a broken pipe in front of the house?

Answer First determine the gauge pressure. P =ρgh
= 1000kg/m3* 9.8 m/s2 * (5m+110m*sin(58°)) = 9.6x105 Pa It would shoot vertically to the top of the water in the tank h = 5m+110m*sin(58°) = 98.3m

Pascal’s Principle If an external pressure is applied to a confined fluid, the pressure at every point within the fluid increases by that amount. This principle is used, for example, in hydraulic lifts and hydraulic brakes.

Pascal’s Principle

Buoyancy and Archimedes’ Principle
This is an object submerged in a fluid. There is a net force on the object because the pressures at the top and bottom of it are different. The buoyant force is found to be the upward force on the same volume of water:

Buoyancy Buoyant force is due to pressure difference between top and bottom of object. The pressure difference does not depend on depth.

Archimedes Principle King Hiero asked Archimedes to devise a test which could determine if his crown was pure gold. Gold was known to be very dense if the volume of the crown could be determined the crown could be weighed and by taking the ratio of mass to volume and comparing to the known value of gold it could be

The net force on the object is then the difference between the buoyant force and the gravitational force.

Archimedes Principle

Need to know the volume of the blocks
The Wood and Iron have equal volumes. The wood floats while the iron sinks in water. Which has the greater buoyant force on it? Wood Iron It is the same Need to know the volume of the blocks

The Aluminum and the Lead have equal masses
The Aluminum and the Lead have equal masses. Which has the greater buoyant force on it? Aluminum Lead It is the same Need to know the volumes

Two bricks are held under water in a bucket
Two bricks are held under water in a bucket. One of the bricks is lower in the bucket than the other. The upward buoyant force on the lower brick is.. A) Greater than B) smaller than C) the same as the buoyant force on the higher brick.

Float or Sink Objects denser than a fluid will sink when added to the fluid Objects less dense than a fluid will float when added to the fluid. Object the same density as a fluid will be immersed in the fluid, they do not sink or float.

Float or Sink If the weight of the water displaced is less than the weight of the object, the object will sink Otherwise the object will float, with the weight of the water displaced equal to the weight of the object. Archimedes' Principle explains why steel ships float

Question Relative to the density of water what is the density of a fish? Since Fish do not float or sink the fish must have a density that is the same as the water they are in.

If the object’s density is less than that of water, there will be an upward net force on it, and it will rise until it is partially out of the water.

For a floating object, the fraction that is submerged is given by the ratio of the object’s density to that of the fluid.

A solid piece of plastic of volume V, and density plastic would ordinarily float in water, but it is held under water by a string tied to the bottom of bucket as shown. The density of water is water.. What is the buoyant force on the plastic? Zero plastic V C) water V D) water V g E) plastic V g

A solid piece of plastic of volume V, and density plastic is floating in a cup of water. The density of water is water. What is the buoyant force on the plastic? Zero B) plastic V C) water V D) water V g E) plastic V g

Two identical cups are filled with water to the same level
Two identical cups are filled with water to the same level. One of the cups has a plastic ball floating in it. The plastic ball has a lower density than water. Which cup weighs more? A) The cup with the ball. B) The cup without the ball. C) The two cups weigh the same.

A block of balsa wood with a rock tied to it floats in water
A block of balsa wood with a rock tied to it floats in water. When the rock is on top as shown, exactly half the block is below the water line. When the block is turned over so the rock is underneath and submerged, the amount of block below the water line is a) less than half b) half c) more than half and the water level at the side of the container will d) rise e) fall f) remain unchanged

Question Does the water level rise, fall, or remain the same when a boat drops anchor?

Answer The water level should fall, because when the anchor is in the boat it displaces a volume of water that makes up the weight of the anchor. When the anchor is in the lake it only displaces its own volume

An astronaut on earth notes that in her soft drink an ice cube floats with 9/10 its volume submerged. If she were instead in a lunar module parked on the moon, the ice in the same soft drink would float with a) less than 9/10 submerged. b) 9/10 submerged. c) more than 9/10 submerged.

Consider an air-filled balloon weighted so that it is on the verge of sinking ... that is, its overall density just equals that of water. Now if you push it beneath the surface, it will a) sink b) return to the surface c) stay at the depth to which it is pushed.

Question T or F, Heavy objects sink and light objects float.
Answer, False, it is the density of the object that determines if it floats or sinks.

This principle also works in the air; this is why hot-air and helium balloons rise. Internet Archive: Details: Physics B Lesson 22: Buoyancy

Floating on Air An object surrounded by air is buoyed by a force equal to the weight of displaced air. Balloons that float must have a density that is less than air.

Question Two balloons of identical size, one filled with helium the other with air are released, Which balloon experiences a greater buoyant force? Why does the Helium balloon float?

Answer The buoyant forces are equal.
The helium balloon is less dense than air so it floats, the one filled with compressed air has sinks because it is more dense than air.

Question On a fall day you are in a hot air balloon that is hovering, neither ascending or descending. The total weight of the balloon and its load is 20,000N. What is the weight of the displaced air?

Answer The displaced air must be the same as the weight it is supporting, 20,000N.

In the presence of air, the small iron ball and large plastic ball balance each other. When air is evacuated from the container, the larger ball a) rises b) falls c) remains in place

a) half b) the same c) twice d) more than twice
The weight of the stand and suspended solid iron ball is equal to the weight of the container of water . When the ball is lowered into the water, the balance is upset. The amount of weight that must be added to the left side to restore balance, compared to the weight of water displaced by the ball, would be a) half b) the same c) twice d) more than twice

If the flow of a fluid is smooth, it is called streamline or laminar flow (a). Above a certain speed, the flow becomes turbulent (b). Turbulent flow has eddies; the viscosity of the fluid is much greater when eddies are present.

We will deal with laminar flow. The mass flow rate is the mass that passes a given point per unit time. The flow rates at any two points must be equal, as long as no fluid is being added or taken away. This gives us the equation of continuity:

If the density doesn’t change – typical for liquids – this simplifies to Where the pipe is wider, the flow is slower.

Bernoulli’s Equation A fluid can also change its height. By looking at the work done as it moves, we find: This is Bernoulli’s equation. One thing it tells us is that as the speed goes up, the pressure goes down.

Applications of Bernoulli’s Principle: from Torricelli to Airplanes, Baseballs, and TIA
Using Bernoulli’s principle, we find that the speed of fluid coming from a spigot on an open tank is: This is called Torricelli’s theorem.

Applications of Bernoulli’s Principle
The lift experienced by airplanes The curveball TIA Air pressure in tornadoes and hurricanes

A pipe has input diameter a. The speed of water flow at the input is v0. Output diameter is 3a. What is the speed of water flow at the pipe output. 3vo 1/3 vo 9vo 1/9 vo

Sailboats A sailboat can move against the wind, using the pressure differences on each side of the sail, and using the keel to keep from going sideways. The sail acts like the wing.

Collapsing Umbrellas Top of umbrella acts like an airplane wing or
a sail.

Baseball Curves A ball’s path will curve due to its spin, which results in the air speeds on the two sides of the ball not being equal.

Lift on an airplane wing is due to the different air speeds and pressures on the two surfaces of the wing.

Airfoil Lift

The circulation is zero and the flow is symmetrical with respect to the x-axis. Note how the fluid particles closer to the cylinder are delayed with respect to those passing well above (or below) the cylinder. This delay is not due to friction, but to the fact that near the stagnation points velocity tends to decelerate to zero speed.

Direction of flow is from right to left and the angle of attack is varied by rotating the airfoil.
Note that, as the airfoil rotates, a streamline always leaves the trailing edge. Far from the airfoil the streamlines tend to become horizontal, as expected for a uniform flow.

This animation shows again the flow field, now described in terms of the velocity vectors.

We now examine the forces acting on the airfoil in terms of elementary forces instead of pressure

The figure shows how different shaped objects are affected by airflow
The figure shows how different shaped objects are affected by airflow. While the aircraft is flying, high-pressure air below the wing tends to flow into the low-pressure area above the wing. The two pressures mix at the wing tip and create a vortex (whirlpool). The vortex creates a suction effect at the ends of the wing and causes induced drag that varies directly with the angle of attack. Surface drag is created by the entire aircraft, excluding induced drag. It is caused by protrusions such as landing gear, rough surfaces, and air striking on the aircraft's frontal surfaces.

Secondary to surface drag, there is another type of fluid resistance called form drag. To examine this phenomenon let us look at a spinless ball in flight with laminar flow. The laminar flow of air around the ball is shown below. As the ball moves through the air a partial vacuum is created behind the ball. As a result the air pressure is greater in front of the ball. The unbalanced pressures result in unbalanced forces that slow the ball down. From drag can be reduced by streamlining a body. This streamlining eliminates the partial vacuum behind the body. Airplane wings are streamlined in such a manner so as to keep form drag at a minimum. It is obvious that objects such as baseballs and golf balls cannot be streamlined. Form drag will have a large effect on their motion if it is not reduced. This is accomplished by the rotation of the balls. As a ball spins it drags air around its surface. This air will help to fill the partial vacuum behind the ball. Subsequently, the unbalanced pressures on the ball will be reduced. The dimples on a golf ball and threads on a baseball will increase the amount of air dragged around them, thus reducing the pressure differences even more.

Bernoulli’s equation can also be used to show how the design of an airplane wing results in an upward lift. The flow of air around an airplane wing is illustrated below. In this case you will notice that the air is traveling faster on the upper side of the wing than on the lower. As a result the pressure will be greater on the bottom of the wing, and the wing will be forced upward. Let us consider an airplane wing where the flow of air (density = 1.3 kg/m3) is 250 m/sec over the top of the wing and 220 m/sec over the bottom. A. Calculate the pressure difference (P1 – P2) between the bottom and the top wing B. If the area of the two wings of the airplane is 10 m2 what is the upward force?

Calculate the pressure difference (P1 – P2) between the bottom and the top wing. Begin by listing the information given and checking that the units are consistent. Air velocity on bottom = v1 = 220 m/sec Air velocity on top = v2 = 250 m/sec Air density = 1.3 kg/m3 Pressure difference P1 – P2 = ? Now use Bernoulli’s equation B. If the area of the two wings of the airplane is 10 m2 what is the upward force? The result of the first part of this example told us that the net upward pressure is 9165 N/m2. Using the definition of pressure, P = F/A, we can write F = PA = 9165 N/m2 x 10 m2 = 91,650 N If we convert this force to pounds we obtain an upward force of approximately 20,000 lbs. This means that this air foil in question is capable of supporting a 10 ton airplane in level flight.

A Blood Platelet drifts along with the flow of blood through an artery that is partially blocked by deposits. As the platelet moves from the narrow region to the wider region Its speed Increases b. decreases c. stays the same and the pressure Increases b. decreases c. stays the same

A person with constricted arteries will find that they may experience a temporary lack of blood to the brain (TIA) as blood speeds up to get past the constriction, thereby reducing the pressure.

A venturi meter can be used to measure fluid flow by measuring pressure differences.

Air flow across the top helps smoke go up a chimney, and air flow over multiple openings can provide the needed circulation in underground burrows.

Example Estimate the air pressure at the center of a Hurricane with sustained winds of 300km/hour.

Hurricane Pressure Patm = Phurr + ½ ρv2 Phurr = Patm – ½ ρv2
½ ρ v2 = ½ (1.25 kg/m3) * (300000/3600)2 ½ ρ v2 = 4300 N/m2 Phurr = =101,300 Pa – 4300 Pa =97,000 Pa

Question Why does the stream of water from a faucet become narrower as it falls? As the stream of water falls, its vertical speed is faster away from the faucet than close to it, due to the acceleration caused by gravity. Since the water is essentially incompressible a faster flow has a smaller cross-sectional area. Thus the faster moving water has a narrower stream

Viscosity Real fluids have some internal friction, called viscosity.
The viscosity can be measured; it is found from the relation where η is the coefficient of viscosity.

Flow in Tubes; Poiseuille’s Equation, Blood Flow
The rate of flow in a fluid in a round tube depends on the viscosity of the fluid, the pressure difference, and the dimensions of the tube. The volume flow rate is proportional to the pressure difference, inversely proportional to the length of the tube and to the pressure difference, and proportional to the fourth power of the radius of the tube.

Flow in Tubes; Poiseuille’s Equation, Blood Flow
This has consequences for blood flow – if the radius of the artery is half what it should be, the pressure has to increase by a factor of 16 to keep the same flow. Usually the heart cannot work that hard, but blood pressure goes up as it tries.

Surface Tension and Capillarity
The surface of a liquid at rest is not perfectly flat; it curves either up or down at the walls of the container. This is the result of surface tension, which makes the surface behave somewhat elastically.

Soap and detergents lower the surface tension of water. This allows the water to penetrate materials more easily. Water molecules are more strongly attracted to glass than they are to each other; just the opposite is true for mercury.

If a narrow tube is placed in a fluid, the fluid will exhibit capillarity.

Pumps, and the Heart This is a simple reciprocating pump. If it is to be used as a vacuum pump, the vessel is connected to the intake; if it is to be used as a pressure pump, the vessel is connected to the outlet.

Pumps, and the Heart (a) is a centrifugal pump; (b) a rotary oil-seal pump; (c) a diffusion pump

Pumps, and the Heart The heart of a human, or any other animal, also
operates as a pump.

Pumps, and the Heart In order to measure blood pressure, a cuff is inflated until blood flow stops. The cuff is then deflated slowly until blood begins to flow while the heart is pumping, and then deflated some more until the blood flows freely.

Summary Phases of matter: solid, liquid, gas.
Liquids and gases are called fluids. Density is mass per unit volume. Specific gravity is the ratio of the density of the material to that of water. Pressure is force per unit area. Pressure at a depth h is ρgh. External pressure applied to a confined fluid is transmitted throughout the fluid.

Summary Atmospheric pressure is measured with a barometer.
Gauge pressure is the total pressure minus the atmospheric pressure. An object submerged partly or wholly in a fluid is buoyed up by a force equal to the weight of the fluid it displaces. Fluid flow can be laminar or turbulent. The product of the cross-sectional area and the speed is constant for horizontal flow.

An icecube is floating in a glass of water
An icecube is floating in a glass of water. As the icecube melts, the level of the water.. A) Rises B) falls C) stays the same

Bernoulli’s Equation Wrap-up
Lesson 23: Fluid Flow Continuity Lesson 24: Bernoulli's Equation

Summary Lesson 21: Hydrostatic Pressure
Internet Archive: Details: Physics B Lesson 22: Buoyancy Lesson 23: Fluid Flow Continuity Lesson 24: Bernoulli's Equation

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