1 Fluids Mechanics (1)

Mechanics Fluids

M.C. Mariana Olivares Avalos

MECHANICS

FLUIDS MECHANICS

FLUIDS STATICS(Hidrostatic)

FLUIDS DYNAMICS(Hidrodynamics)

General properties of fluids

Pressure and Density

Principles

Ideal Fluids

Conservation principles

Applications

Why is important to study Fluids?

Electronic Components

Human Body

Home

Transport

Biomedicine

Building trade

FLUIDS

Chemical Reaction

s

States of Matter A gas is a system in which each atom or molecule moves through space as a free particle.

A gas is compressible and fills its container.

A liquid is nearly incompressible. When a liquid is placed in a container, it fills only the volume corresponding to its initial volume.

A solid does not require a container but defines its own shape.

A solid can be compressed or deformed slightly.

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Is there anyother state of matter?

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So, What is a fluid?

Inability to sustain the Shear

Fluid Properties:

Any feature of a system is known as a property.

Extensive Properties

Intensive Properties

mVTPr

½ m ½ m

½ V ½ V

T T

P P

r r

… Fluid Properties

Density and Relative Density ( ; ) Specific weight () Energy (microscopic and macroscopic) Compressibility factor () Viscosity

Surface Tension

Capillarity Pressure

What kind of property are they?

Viscosity Weight VelocityThermal conductivity

Boiling Point Density

Temperature Electrical Conductivity

Volume

Mass Melting Point Heat

What’s going on here?https://www.youtube.com/watch?v=MzsORE0ae10

Density

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r (rho) Each substance has its own density. It is the mass per unit of volume, Modifying factors: Temperature and air pressure.

Relative density (dimensionless)

Aluminium (2.7)

Osmium (22.5)

Exercises: You have a rectangular metal piece (5 x 15 x 30 mm) and it has a mass of 0.0158 kg. The seller says that it is gold. How can we know if it is a fraud?The density of gold is

A 200 ml jar is full of water (4°C). If the temperature is rises to 80°C, 6 g of water are spilled. Give the water density at 80°C (expansion of the bottle is negligible and )

Resp.

What is pressure?

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http://www.youtube.com/watch?v=vo2iE94iAoA

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Pressure Pressure is defined as force per unit area:

The SI unit of pressure is N/m2, which has been given the name pascal, abbreviated Pa:

The average pressure of the Earth’s atmosphere, 1 atm, is:

Atmospheric pressure is often quoted in non-SI units:

Weighing Earth’s Atmosphere The Earth’s atmosphere is composed of 78.08% nitrogen, 20.95% oxygen, 0.93% argon, 0.25% water vapor, and traces of other gases, including carbon dioxide (CO2).

The CO2 content of the atmosphere currently is0.039%

POBLEM: What is the mass of Earth’s atmosphere and what is the mass of the CO2 in Earth’s atmosphere?

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Pressure-Depth Relationship The vertical forces on the cubemust sum to zero:

Where

So, if we substitute those terms

𝑝2= 𝑝1+𝜌 ( 𝑦2− 𝑦1 ) 𝑔

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Pressure-Depth Relationship A common problem involves the pressure as a function of depth below the surface of a liquid.

We define the pressure at the surface of a liquid (y1 = 0) to be p0.

We define the pressure at depthh (y2= h ) to be p.

We get the expression for thepressure at a given depth of aliquid:

This equation holds for any shapeof the vessel containing the liquid.

Submarine A U.S. Navy submarine of theLos Angeles class is 110 m longand has a hull diameter of 10. m.

Assume that submarine has a flattop with an area of A = 1100 m2.

The density of seawater is1024 kg/m3.PROBLEM:

What is the total force pushing down in the top of this submarine at a depth of 250. m?

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Gauge Pressure and Barometers

The pressure p in the pressure depth relationshipp = p0 + ρgh is the absolute pressure.

The difference between an absolute pressure and the atmospheric air pressure is called a gauge pressure.

A barometer is a device that is used to measure atmospheric pressure.

We will discuss two types of barometers: Mercury barometer Water barometer

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Manometer

An open tube manometer measures the gauge pressure of a gas.

The closed end is connected to a containerof gas whose gauge pressure is to bemeasured.

The other end is open andexperiences atmospheric pressure.

The gauge pressure of the gas in thecontainer is:

The gauge pressure can be positive or negative.11/16/22 Chapter 13 20

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Barometric Altitude Relation for Gases

In the derivation of the depth pressure relationship, we have made use of the incompressibility of liquids.

However, if our fluid is a gas, we cannot make this assumption.

For compressible fluids we find that the density is proportional to the pressure:

Combining the 2 equations, we get:

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Air Pressure on Mount EverestPROBLEM:

What is the air pressure on the top of Mount Everest?

h=8850𝑚

Exercise

A barrel containing a layer of oil ( has a thickness of 0.120 m, below it, there is a layer of water with a thickness of 0.250m.

Find the gauge pressure at the water / oil interface.

Find the gauge and absolute pressure at the bottom of the tank.

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Pascal’s Principle If we exert pressure on any part of an incompressible fluid, this pressure will be transferred to the rest of the fluid with no losses.

This concept is called Pascal’s Principle and can be stated:

“When a change in pressure occurs at any point in a confined fluid, an equal change in pressure occurs at every point in the fluid.”

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Pascal’s Principle Consider a cylinder partially filled with water.

Place a massless piston on the surfaceof the water.

Place a weight on the piston that exertsa pressure pt on the top of the water.

The pressure p at a depth h is then:

If we add a second weight, the change inpressure at depth h can only be due to the Δpt caused by the addition of the second weight:

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Now let’s consider two connected pistons partially filled with an incompressible fluid.

One piston has area Ain. The second piston has area Aout so that

Ain < Aout. We exert a force Fin on the first piston producing a change in the pressure in the fluid.

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This change will be transmitted throughout the fluid, including areas near the second piston, so we can write:

We can see that we have magnified the force by a factor equal to the ratio of the areas of the pistons.

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Archimedes’ Principle Let’s imagine a cube of water in avolume of water.

The weight of this cube of water issupported by the buoyant force FBresulting from the pressure differentialbetween the top and bottom of the cube:

For our imaginary cube of water, the buoyant force equals the weight:

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Now let’s replace the cube of waterwith a cube of steel.

The steel cube weighs more than thecube of water, so now there is a net

force in the y-direction given by:

This force is downward so the metal cube will sink.

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Let’s replace our steel cube with a wooden cube.

Now the weight of the wood is less than the weight of the water that the wood replaced, so the net force is upward.

The wooden block would rise toward the surface.

If we place an object less dense than water inwater, the object will float.

The object will sink into the water until theweight of the object equals the weight of the water displaced:

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Archimedes’ Principle A floating object displaces its own weight of fluid.

This is true independent the amount of fluid present.

Let’s look at a ship in a lock. In both positions, the ship floats with thesame fraction below the water level.

What matters is the amount of waterdisplaced, not the total amount of waterin the lock.

If an object with a higher density than wateris submerged, it will experience an upwardbuoyant force that is less than its weight:

PROBLEM: What fraction of an iceberg floating in seawater is visible above the surface?

Answer f=10.4%

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Floating Iceberg

Hot-Air Balloon A hot-air balloon has a volume of 2200. m3.

The density of air at 20. °C is 1.205 kg/m3.

The density of hot air inside the balloon at a temperature of 100. °C is 0.946 kg/m3.PROBLEM:

How much weight can the hot-air balloon lift (including the balloon itself)?

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http://phet.colorado.edu/sims/density-and-buoyancy/density_en.html

http://www.youtube.com/watch?v=vCJxDxSWFpo

ACTIVITY IN PAIRS

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Finding the Density of an Object Archimedes was charged by his king, Hiero of Syracuse, to determine if a new crown was pure gold or partly gold and partly silver.

Archimedes was able to determine that the crown was a combination of gold and silver by determining the density.

We now measure the density of a metal ball just by weighing, as follows: Weigh a beaker containing water. Submerge the metal ball without touching the bottom of the beaker.

Determine the mass of the new ball/beaker system. Let the ball rest on the bottom of the beaker and determine the mass.

What is the density of the metal ball?11/16/22 Chapter 13 35

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Finding the Density of an Object

a) Mass of water + glass = 437 g

b) Dip metal ball in water, but don’t let it touch bottom

c) New weight = 458 g

d) Now let ball rest on bottom of glass; new weight = 596 g

a) b)

c) d)

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Finding the Density of an Object

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In part (a) we measure the combined mass: m0 = 437 g

In parts (b) and (c) we measure in addition the mass of the water displaced by the volume of the ball: m1 = m0 + ρwVb = 458 g

In part (d) we measure the combined mass: m2 = mb = m0 + ρbVb = 596 g

Combining these two equations we get:

The density of the ball is then:

Finding the Density of an Object

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Floating in Two Different Liquids

Now pour paint thinner (density 80% of that of water) on top.

Paint thinner does not mix with water.

What will happen?

Pour water into a container.

Put a swimmer in it.

The swimmer is 90% submerged so its average density of 90% that of water.

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Floating in Two Different Liquids

Answer:Swimmer RISES!

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Ideal Fluid Motion The motion of real-life fluids is complicated. Here we will make some simplifying assumptions that will allow us to reach some relevant conclusions concerning the motion of fluids.

We assume that all fluids exhibit: Laminar flow

The velocity of the fluid at a given point in space does not change with time.

Incompressible flow The density of the fluid does not change as the fluid flows.

Non-viscous flow The fluid flows freely, no friction or losses.

Irrotational flow No part of the fluid rotates about its center of mass; no turbulence.

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Examples of Flow

Laminar Flow

Laminar to Turbulent Flow

Turbulent Flow

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Concept Check We blow air through a straw between two

empty soda cans sitting on straws as shown in below.

The soda cans willA. move apart.B. move together.C. not move.

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Equation of Continuity Consider the motion of an ideal fluid flowing with speed v in a container with cross sectional area A:

We can write the volume of fluid passing a given point in the pipe per unit time as:

Now consider a fluid flowing through a container that changes cross sectional area.

The fluid is initially flowing with speed v1 in a container with area A1 and then enters a section of the container where the fluid flows with speed v2 and cross sectional area A2.

The volume of fluid entering this section of the container must equal the volume of fluid leaving the container.

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Equation of Continuity

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We can the volume per unit time flowing in the first part of the container as:

And the volume per unit time flowing in the second part of the container as:

Using the fact that the volume per unit time passing any point must be the same we get:

This equation is called the equation of continuity.

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We can express a constant volume flow rateRV (volume per time):

We can also express a constant mass flow rateRm (mass per time):

Now let’s consider the case where an incompressible fluid (constant density ρ) is flowing through a pipe at a steady rate.

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p1 - pressurev1 - speedy1 - height

p2 - pressurev2 - speedy2 - height

Bernoulli’s Equation

Bernoulli’s Equation

This equation is Bernoulli’s Equation. One major consequence of Bernoulli’s Equation becomes clear if y = 0:

We can see that if the velocity of a moving fluid is increased, its pressure must decrease.

The decrease in pressure transverse to the fluid flow has many practical applications.

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Spray Bottle We squeeze the handle of a spray bottle, causing airto flow horizontally across the opening of of a tube that extends down into the liquid in the bottle.

If the air is moving at 50.0 m/s, what is the pressure difference at the top of the tube assuming that the density of air is 1.20 kg/m3?

Viscosity Water flows faster in the middle of a river than along its banks.

Water is not an ideal fluid. Water has some degree of “stickiness” called viscosity.

The viscosity of water is relatively low while the viscosity of honey is relatively high.

Viscosity causes the fluid streamlines in viscous flow of water in a river to partially stick to the banks and the neighboring streamlines to partially stick to each other.

Let’s look at the velocity profile of water flowing in a tube with and without viscosity.

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Viscosity

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Nonviscous idealfluid flow

Viscous flow

Viscosity Viscosity is measured by using two parallel plates of area A separated by a gap h filled with the fluid.

One of the plates is dragged and the force F required to do so is measured.

The viscosity is defined as the ratio of the force per unit area divided by the velocity difference between the top plate and the bottom plate over the distance between the plates.

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Viscosity The viscosity of a fluid depends on temperature.

The table shows some typical viscosities.

The viscosity of a fluid isimportant in determining howmuch fluid can flow through apipe of given radius and length.

The volume of fluid that can flowper unit time is:

This is often called theHagen-Poiseuille law.11/16/22 Chapter 13 54

Hypodermic NeedlePROBLEM:

If 2.0 cm3 of water is to be pushed out of a 1.0 cm diameter syringe through a 3.0 cm long need (interior diameter = 1.37 mm) in 0.40 s, what force must be applied to the plunger of the syringe?

With what speed does the water emerge from the needle?

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ACTIVITY

You will see some demostrations, working in teams explain why this is

going on.

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