Chapter 10Chapter 10FluidsFluids

FluidsFluids

A A fluidfluid isis a gas or a liquid.a gas or a liquid.

A A gasgas expands to fill any container expands to fill any container

A A liquidliquid (at fixed pressure and (at fixed pressure and temperature), has a fixed volume, but temperature), has a fixed volume, but deforms to the shape of its container.deforms to the shape of its container.

The atoms in a liquid are closely packed while those in a gas are separated by much larger distances.

Gas have a density ~ 1/1000 x liquid density

Density and PressureDensity and Pressure

The density of a substance of uniform The density of a substance of uniform composition is defined as its mass per unit composition is defined as its mass per unit volume:volume:

Units are kg/mUnits are kg/m33 (SI) or g/cm (SI) or g/cm33 (cgs) (cgs) 1 g/cm1 g/cm33 = 1000 kg/m = 1000 kg/m33

mV

Density, cont.Density, cont.

The densities of most liquids and solids The densities of most liquids and solids vary slightly with changes in temperature vary slightly with changes in temperature and pressureand pressure

Densities of gases vary greatly with Densities of gases vary greatly with changes in temperature and pressurechanges in temperature and pressure

Density = Mass/VolumeDensity = Mass/Volume = M/V= M/V SI unit: [kg/mSI unit: [kg/m33 ]]

Densities of some common things Densities of some common things (kg/m(kg/m33)) WaterWater 10001000 iceice 917917 (floats on water)(floats on water) bloodblood 10601060 (sinks in water)(sinks in water) lead lead 11,30011,300 Copper Copper 88908890 Mercury Mercury 13,60013,600 Aluminum Aluminum 27002700 Wood Wood 550550 airair 1.291.29 Helium Helium 0.18 0.18

DensityDensity

The The specific gravityspecific gravity of a substance is the of a substance is the ratio of its density to the density of water at ratio of its density to the density of water at 4° C4° C The density of water at 4° C is 1000 kg/mThe density of water at 4° C is 1000 kg/m33

Specific gravity is a unitless ratioSpecific gravity is a unitless ratio

PressurePressure

Pressure Pressure PP is the amount is the amount of force of force FF per unit area per unit area AA::

AF

P

Pressure is the outward force per unit area that the fluid exerts on its container.

A2

F1

A1

F2

By the Action-Reaction principle, Pressure is the inward force per unit area that the container exerts on the fluid.

The force exerted The force exerted by a fluid on a by a fluid on a submerged object submerged object at any point if at any point if perpendicular to perpendicular to the surface of the the surface of the objectobject

2m

NP ain

A

FP

A woman’s high heels sink into the soft ground, but the larger shoes of the much bigger man do not.

Pressure = force/area

The pressure exerted on the piston extends uniformly throughout the fluid, causing it to push outward with equal force per unit area on the walls and bottom of the cylinder.

The spring is The spring is calibrated by a calibrated by a known forceknown force

The force the fluid The force the fluid exerts on the exerts on the piston is then piston is then measuredmeasured

You are walking out on You are walking out on

a frozen lake and you a frozen lake and you

begin to hear the ice begin to hear the ice

cracking beneath you. cracking beneath you.

What is your best What is your best

strategy for getting off strategy for getting off

the ice safely?the ice safely?

1) stand absolutely still and don’t move a muscle1) stand absolutely still and don’t move a muscle

2) jump up and down to lessen your contact time with 2) jump up and down to lessen your contact time with the icethe ice

3) try to leap in one bound to the bank of the lake3) try to leap in one bound to the bank of the lake

4) shuffle your feet (without lifting them) to move 4) shuffle your feet (without lifting them) to move towards shoretowards shore

5) lie down flat on the ice and crawl toward shore5) lie down flat on the ice and crawl toward shore

ConcepTest 10.3 ConcepTest 10.3 On a Frozen LakeOn a Frozen Lake

You are walking out You are walking out

on a frozen lake and on a frozen lake and

you begin to hear the you begin to hear the

ice cracking beneath ice cracking beneath

you. What is your you. What is your

best strategy for best strategy for

getting off the ice getting off the ice

safely?safely?

1) stand absolutely still and don’t move a muscle1) stand absolutely still and don’t move a muscle

2) jump up and down to lessen your contact time 2) jump up and down to lessen your contact time with the icewith the ice

3) try to leap in one bound to the bank of the lake3) try to leap in one bound to the bank of the lake

4) shuffle your feet (without lifting them) to move 4) shuffle your feet (without lifting them) to move towards shoretowards shore

5) lie down flat on the ice and crawl toward shore5) lie down flat on the ice and crawl toward shore

As long as you are on the ice, your weight is pushing down. What is As long as you are on the ice, your weight is pushing down. What is

important is not the net force on the ice, but the force exerted on a important is not the net force on the ice, but the force exerted on a

given small area of ice (i.e., the pressure!). By lying down flat, you given small area of ice (i.e., the pressure!). By lying down flat, you

distribute your weight over the widest possible area, thus reducing the distribute your weight over the widest possible area, thus reducing the

force per unit area.force per unit area.

ConcepTest 10.3 ConcepTest 10.3 On a Frozen LakeOn a Frozen Lake

Atmospheric PressureAtmospheric PressureAtmospheric pressure Atmospheric pressure comes from the weight of the comes from the weight of the

column of air above us. At sea level, atmospheric column of air above us. At sea level, atmospheric pressure is:pressure is:

PPatat = 1.01 = 1.01 10 1055 N/m N/m22

= 1.01 = 1.01 10 1055 PaPa 1 Pascal= 1 N/m1 Pascal= 1 N/m22

= 14.7 lb/in= 14.7 lb/in22 (psi) (psi)

= 1 bar (tire pressure gauges in Europe read 1, = 1 bar (tire pressure gauges in Europe read 1, 2,..bar)2,..bar)

Hurricane Rita 2005: P = 882 millibar = 0.882 barHurricane Rita 2005: P = 882 millibar = 0.882 bar

F=MgF=PA

Pressure examplesPressure examples

1.1. Estimate the force of the atmosphere on the top of your Estimate the force of the atmosphere on the top of your head.head.

• A = (10cm)(15cm)=0.015mA = (10cm)(15cm)=0.015m22

• F=PA = [1.01 F=PA = [1.01 10 1055 N/m N/m22 ][0.015 m ][0.015 m22] = 1.5 kN] = 1.5 kN• A = (4in)(6in)=24 inA = (4in)(6in)=24 in22

• F=PA = [15 lb/inF=PA = [15 lb/in22][24in][24in22] = 360 lb.] = 360 lb.

2.2. Is atmospheric pressure on top of a mountain greater or Is atmospheric pressure on top of a mountain greater or less than at sea level?less than at sea level?

• Less. At higher altitude, there is less mass above.Less. At higher altitude, there is less mass above.

PressurePressure

ExampleExample

If a fluid is at rest in a container, all If a fluid is at rest in a container, all portions of the fluid must be in static portions of the fluid must be in static equilibriumequilibrium

All points at the same depth must be at All points at the same depth must be at the same pressurethe same pressure Otherwise, the fluid would not be in Otherwise, the fluid would not be in

equilibriumequilibrium The fluid would flow from the higher The fluid would flow from the higher

pressure region to the lower pressure regionpressure region to the lower pressure region

Examine the darker Examine the darker region, assumed to be region, assumed to be a fluida fluid It has a cross-It has a cross-

sectional area Asectional area A Extends to a depth h Extends to a depth h

below the surfacebelow the surface Three external forces Three external forces

act on the regionact on the region

P = PP = Poo + + ρρghgh

PPoo is normal is normal atmospheric atmospheric pressurepressure 1.013 x 101.013 x 105 5 Pa = Pa =

14.7 lb/in14.7 lb/in2 2 = 1 atm= 1 atm The pressure does The pressure does

not depend upon not depend upon the shape of the the shape of the containercontainer

Pressure in a FluidPressure in a FluidPressure in a fluid depends only on the depth Pressure in a fluid depends only on the depth hh below the below the

surface.surface.

P = PP = Patat + + gh gh = density of fluid = density of fluid

Weight/Area of fluidWeight/Area of fluid

Weight/Area of atmosphere above fluidWeight/Area of atmosphere above fluid

IFIF the density of the fluid is constant and it has atmospheric the density of the fluid is constant and it has atmospheric pressure (pressure (PPatat) at its surface.) at its surface.

Mass of fluid above depth h is (density)(volume) = hAForce of gravity on fluid above depth h: W= ghA

Pressure under waterPressure under water

To what depth in water must you dive to double the To what depth in water must you dive to double the pressure exerted on your body? pressure exerted on your body?

P = PP = Patat + + ghgh

gh = Pgh = Patat , h= P , h= Patat / /gg

m

smmkg

mNh

3.10

]/81.9][/10[

]/1001.1[233

25

Start to feel strong pressure at 3m

Pressure variation in fluidPressure variation in fluid

The variation in pressure at two different depths is The variation in pressure at two different depths is given by:given by:

PP22 = P = P11 + + ghgh

Pressure and DepthPressure and DepthBarometer: a way to measure Barometer: a way to measure

atmospheric pressureatmospheric pressure

p2 = p1 + gh

patm = gh

Measure h, determine patm

example--Mercury

= 13,600 kg/m3

patm = 1.05 x 105 Pa

h = 0.757 m = 757 mm = 29.80” (for 1 atm)

hp2=pat

m

p1=0

Absolute vs. Gauge PressureAbsolute vs. Gauge Pressure The pressure P is called the The pressure P is called the absoluteabsolute

pressurepressure Remember, P = PRemember, P = Poo + + ghgh

P – PP – Poo = = gh is the gh is the gaugegauge pressure pressure

One end of the U-One end of the U-shaped tube is open shaped tube is open to the atmosphereto the atmosphere

The other end is The other end is connected to the connected to the pressure to be pressure to be measuredmeasured

Pressure at B is Pressure at B is PPoo+ρgh+ρgh

One atmosphere of pressure is One atmosphere of pressure is defined as the pressure equivalent to a defined as the pressure equivalent to a column of mercury exactly 0.76 m tall column of mercury exactly 0.76 m tall at 0at 0oo C where g = 9.806 65 m/s C where g = 9.806 65 m/s22

One atmosphere (1 atm) =One atmosphere (1 atm) = 76.0 cm of mercury (760mm = 1 torr)76.0 cm of mercury (760mm = 1 torr) 1.013 x 101.013 x 1055 Pa Pa 14.7 lb/in14.7 lb/in22

Example:Example:

A change in pressure applied to an A change in pressure applied to an enclosed fluid is transmitted undiminished enclosed fluid is transmitted undiminished to every point of the fluid and to the walls to every point of the fluid and to the walls of the container.of the container. First recognized by Blaise Pascal, a French First recognized by Blaise Pascal, a French

scientist (1623 – 1662)scientist (1623 – 1662)

The hydraulic press is an The hydraulic press is an important application of important application of Pascal’s PrinciplePascal’s Principle

Also used in hydraulic Also used in hydraulic brakes, forklifts, car lifts, brakes, forklifts, car lifts, etc.etc.

2

2

1

1

A

F

A

FP

A small force F1 applied to a piston with a small area produces a much larger force F2 on the larger

piston. This allows a hydraulic jack to lift heavy objects.

Pascal’s Principle, ForcePascal’s Principle, Force

A external pressure P A external pressure P applied to any area of a fluid applied to any area of a fluid is transmitted unchanged to is transmitted unchanged to all points in or on the fluid.all points in or on the fluid.

This is just an application of This is just an application of the Action-Reaction the Action-Reaction principle.principle.

Hydraulic LiftHydraulic Lift

A Force F1 is applied to area A1, displacing the fluid by a distance d1.The pressure increase in the fluid is P=F1/A1.The Pressure F1/A1 creates a force on the car F2= A2 (F1/A1) = F1 (A2 /A1).A small force acting on a small area creates a big force acting over a large area!

Archimedes’ Principle:

The buoyant force acting on an object fully or partially submerged in a fluid is equal to the weight of the fluid displaced by the object.

The weight of a column of water is proportional to the volume of the column. The volume V is equal

to the area A times the height h.

Equilibrium…

The upward force is The upward force is called the called the buoyant buoyant forceforce

The physical cause of The physical cause of the buoyant force is the buoyant force is the pressure the pressure difference between the difference between the top and the bottom of top and the bottom of the objectthe object

The magnitude of the buoyant force The magnitude of the buoyant force always equals the weight of the displaced always equals the weight of the displaced fluidfluid

The buoyant force is the same for a totally The buoyant force is the same for a totally submerged object of any size, shape, or submerged object of any size, shape, or densitydensity

fl u i d fl u i d fl u i dB V g w

The buoyant force is exerted by the fluidThe buoyant force is exerted by the fluid Whether an object sinks or floats depends Whether an object sinks or floats depends

on the relationship between the buoyant on the relationship between the buoyant force and the weightforce and the weight

The upward buoyant force is B=ρThe upward buoyant force is B=ρfluidfluidgVgVobjobj

The downward gravitational force is The downward gravitational force is w=mg=ρw=mg=ρobjobjgVgVobjobj

The net force is B-w=(ρThe net force is B-w=(ρfluidfluid-ρ-ρobjobj)gV)gVobjobj

The object is less The object is less dense than the dense than the fluidfluid

The object The object experiences a net experiences a net upward forceupward force

The object is more The object is more dense than the dense than the fluidfluid

The net force is The net force is downwarddownward

The object The object accelerates accelerates downwarddownward

Question: How do steel ships float if steel is roughly 6 times more dense than water?

The object is in static equilibriumThe object is in static equilibrium The upward buoyant force is balanced by The upward buoyant force is balanced by

the downward force of gravitythe downward force of gravity Volume of the fluid displaced corresponds Volume of the fluid displaced corresponds

to the volume of the object beneath the to the volume of the object beneath the fluid levelfluid level

The forces balanceThe forces balance

obj

fluid

fluid

obj

V

V

Suppose you float a large ice-cube in Suppose you float a large ice-cube in a glass of water, and that after you a glass of water, and that after you place the ice in the glass the level place the ice in the glass the level of the water is at the very brim. of the water is at the very brim. When the ice melts, the level of the When the ice melts, the level of the water in the glass will:water in the glass will:

1. Go up causing the water to spill. 1. Go up causing the water to spill. 2. Go down.2. Go down.3. Stay the same.3. Stay the same.

Archimedes’ Principle: The buoyant force on an object equals the weight of the fluid it displaces.

Weight of water displaced = Buoyant force = Weight of ice

When ice melts it will turn into water of same volume

Example 9.9Example 9.9 A raft is constructed of wood having a density of A raft is constructed of wood having a density of

6.00 x 106.00 x 1022 kg/m kg/m33. Its surface area is 5.70m. Its surface area is 5.70m22, and , and volume is 0.60mvolume is 0.60m33. When the raft is placed in . When the raft is placed in fresh water, what depth h is the bottom of the fresh water, what depth h is the bottom of the raft submerged? raft submerged?

Concept QuestionConcept QuestionWhich weighs more: Which weighs more:

1. A large bathtub filled to the brim with water. 1. A large bathtub filled to the brim with water.

2. A large bathtub filled to the brim with water 2. A large bathtub filled to the brim with water with a battle-ship floating in it.with a battle-ship floating in it.

3. They will weigh the same.3. They will weigh the same.

Tub of water

Tub of water + ship

Overflowed water

CORRECT

Weight of ship = Buoyant force =

Weight of displaced water

Streamline flow Streamline flow Every particle that passes a particular point Every particle that passes a particular point

moves exactly along the smooth path followed moves exactly along the smooth path followed by particles that passed the point earlierby particles that passed the point earlier

Also called laminar flowAlso called laminar flow Streamline is the pathStreamline is the path

Different streamlines cannot cross each otherDifferent streamlines cannot cross each other The streamline at any point coincides with the The streamline at any point coincides with the

direction of fluid velocity at that pointdirection of fluid velocity at that point

Streamline flow shown around an auto in a wind tunnel

The flow becomes irregularThe flow becomes irregular exceeds a certain velocityexceeds a certain velocity any condition that causes abrupt changes in any condition that causes abrupt changes in

velocityvelocity Eddy currents are a characteristic of Eddy currents are a characteristic of

turbulent flowturbulent flow

The rotating blade The rotating blade (dark area) forms a (dark area) forms a vortex in heated airvortex in heated air The wick of the The wick of the

burner is at the burner is at the bottombottom

Turbulent air flow Turbulent air flow occurs on both occurs on both sides of the bladesides of the blade

Viscosity is the degree of internal friction in Viscosity is the degree of internal friction in the fluidthe fluid Measure of a fluid's ability to resist gradual

deformation by shear or tensile stresses The internal friction is associated with the The internal friction is associated with the

resistance between two adjacent layers of resistance between two adjacent layers of the fluid moving relative to each otherthe fluid moving relative to each other

Viscous Liquid!Viscous Liquid!

The fluid is nonviscousThe fluid is nonviscous There is no internal friction between adjacent layersThere is no internal friction between adjacent layers

The fluid is incompressibleThe fluid is incompressible Its density is constantIts density is constant

The fluid motion is steadyThe fluid motion is steady Its velocity, density, and pressure do not change in timeIts velocity, density, and pressure do not change in time

The fluid moves without turbulenceThe fluid moves without turbulence No eddy currents are presentNo eddy currents are present The elements have zero angular velocity about its The elements have zero angular velocity about its

centercenter

Equation of ContinuityEquation of Continuity

Mass is conserved as the fluid flows.Mass is conserved as the fluid flows.If a certain mass of fluid enters a pipe at one end at a If a certain mass of fluid enters a pipe at one end at a certain rate, the same mass exits at the same ratecertain rate, the same mass exits at the same rateat the other end of the tube (if nothing gets lost inat the other end of the tube (if nothing gets lost inbetween through holes, for instance). between through holes, for instance).

Mass flow rate at position 1 = Mass flow rate at position Mass flow rate at position 1 = Mass flow rate at position 22

11 A A11 v v11 = = 2 2 AA22 v v22

A v = constant along a tube that has a single entry A v = constant along a tube that has a single entry and a single exit point for fluid flow.and a single exit point for fluid flow.

What goes in comes out!What goes in comes out! If density is constant:If density is constant:

AA11vv11 = A = A22vv22 The product of the cross-The product of the cross-

sectional area of a pipe sectional area of a pipe and the fluid speed is a and the fluid speed is a constantconstant Speed is high where the Speed is high where the

pipe is narrow and pipe is narrow and speed is low where the speed is low where the pipe has a large pipe has a large diameterdiameter

Av is called the Av is called the flow rateflow rate

The equation is a consequence of conservation The equation is a consequence of conservation of mass and a steady flowof mass and a steady flow

A v = constantA v = constant This is equivalent to the fact that the volume of fluid This is equivalent to the fact that the volume of fluid

that enters one end of the tube in a given time interval that enters one end of the tube in a given time interval equals the volume of fluid leaving the tube in the equals the volume of fluid leaving the tube in the same intervalsame interval

• Assumes the fluid is incompressible and there are no leaksAssumes the fluid is incompressible and there are no leaks

Bernoulli’s EquationBernoulli’s Equation Work-Energy Theorem : WWork-Energy Theorem : Wncnc = change of total mechanical energy = change of total mechanical energy

applied to fluid flow :applied to fluid flow :

Difference in pressure => net force is not zero => fluid acceleratesDifference in pressure => net force is not zero => fluid accelerates

Pressure is due to collisional forces which is a nonconservative force:Pressure is due to collisional forces which is a nonconservative force:

WWncnc = (P = (P22-P-P11) V) V

Consider a fluid moving from height hConsider a fluid moving from height h1 1 to hto h22. Its total mechanical. Its total mechanical

energy is given by the sum of kinetic and potential energy. Thus,energy is given by the sum of kinetic and potential energy. Thus,

WWncnc = E = Etot,1tot,1 –E –Etot,2tot,2 = ½ m v = ½ m v1122+m g h+m g h11 –( ½ m v –( ½ m v22

22+m g h+m g h22) )

States that the sum of the pressure, kinetic States that the sum of the pressure, kinetic energy per unit volume, and the potential energy per unit volume, and the potential energy per unit volume has the same energy per unit volume has the same value at all points along a streamlinevalue at all points along a streamline

constant g yv2

1P 2

Shows fluid flowing Shows fluid flowing through a horizontal through a horizontal constricted pipeconstricted pipe

Speed changes as Speed changes as diameter changesdiameter changes

Can be used to Can be used to measure the speed of measure the speed of the fluid flowthe fluid flow

Swiftly moving fluids Swiftly moving fluids exert less pressure exert less pressure than do slowly moving than do slowly moving fluidsfluids

Objects Moving Through a FluidObjects Moving Through a Fluid Many common phenomena can be explained by Many common phenomena can be explained by

Bernoulli’s equationBernoulli’s equation At least partiallyAt least partially

In general, an object moving through a fluid is In general, an object moving through a fluid is acted upon by a net upward force as the result acted upon by a net upward force as the result of any effect that causes the fluid to change its of any effect that causes the fluid to change its direction as it flows past the objectdirection as it flows past the object

The air speed above the The air speed above the wing is greater than the wing is greater than the speed belowspeed below

The air pressure above the The air pressure above the wing is less than the air wing is less than the air pressure belowpressure below

There is a net upward There is a net upward forceforce

Called Called liftlift Other factors are also Other factors are also

involvedinvolved Designed to produce liftDesigned to produce lift Racecars designed to Racecars designed to

produce faster airflow on the produce faster airflow on the bottombottom

High velocity implies low High velocity implies low pressure, IN the fluidpressure, IN the fluid

Example: Example: A jet of water A jet of water

squirts out squirts out horizontally from a horizontally from a hole near the hole near the bottom of the tank bottom of the tank with a velocity of with a velocity of 1.33 m/s. What is 1.33 m/s. What is the height of the the height of the water level in the water level in the tank?tank?