制作张昆实 yangtze university

制作张昆实制作张昆实

Yangtze UniversityYangtze University

制作张昆实制作张昆实

Yangtze UniversityYangtze University

BilingualBilingual MechanicsMechanics

BilingualBilingual MechanicsMechanics Chapter 10

FluidsFluids

Chapter 10

FluidsFluids

Chapter 10 Chapter 10 FluidsFluids Chapter 10 Chapter 10 FluidsFluids

10-1 Fluids and the World Around Us10-2 What Is a Fluid? 10-3 Density and Pressure 10-4 Fluids at Rest 10-5 Measuring Pressure10-6 Pascal's Principle 10-7 Archimedes' Principle 10-8 Ideal Fluids in Motion10-9 The Equation of Continuity10-10 Bernoulli's Equation

10-1 Fluids and the World Around Us10-2 What Is a Fluid? 10-3 Density and Pressure 10-4 Fluids at Rest 10-5 Measuring Pressure10-6 Pascal's Principle 10-7 Archimedes' Principle 10-8 Ideal Fluids in Motion10-9 The Equation of Continuity10-10 Bernoulli's Equation

10-1 Fluids and the World Around Us 10-2 What Is a Fluid? 10-1 Fluids and the World Around Us 10-2 What Is a Fluid?

FluidsFluids, which include both liquidsliquids and gasesgases, play a central role in our daily lives. We breath and drink them, and a rather vital vital fluidfluid circulates in the human cardiovascularcardiovascular systemsystem. There are the fluid oceanthe fluid ocean and the the fluid atmospherefluid atmosphere.

A fluidA fluid is a substancesubstance that can flowcan flow. Fluids conformconform to the boundaries of any container

in which we put them.

With fluidsfluids, we are more interested in the extended substance, and in propertiesin properties that can varycan vary from point to pointfrom point to point in that substance. It is more usefulmore useful to speak of densitydensity and pre pressuressure than of mass mass and forceforce.

10-3 Density and Pressure 10-3 Density and Pressure

Density: To find the densitydensity of a fluid at any pointat any point, we isolate a small volume elementvolume element around that point and measure the massthe mass of the fluid contained within that element. The densitydensity is then

Vm

m

V

(10-1)

In theory, the density at any point in a fluid is the limit of the limit of this ratiothis ratio as the volume element at that point is made smaller and smaller. In practiceIn practice, for a “smoothsmooth” (with uniform densityuniform density) fluid, its densitydensity can be written as

m

V

V

(10-2)( uniform density )( uniform density )

10-3 Density and Pressure 10-3 Density and Pressure

Pressure: To find the PressurePressure at any point at any point in a fluid , we isolate a small area elementarea element around that point and measure the magnitude the magnitude of the force that acts normal to that element. The PressurePressure is then

AFF

PA

(10-3)

In theory, the PressurePressure at any point in a fluid is the limit of the limit of this ratiothis ratio as the area element at that point is made smaller and smaller. However, if the force is uniform over a flat area , the PressurePressure can be written as

FP

A

V

(10-4)( Pressure of uniform ( Pressure of uniform force on flat area )force on flat area )

A

The SI unit of pressure:51 1.01 10atm Pa Atmosphere (at sea lever)Atmosphere (at sea lever)

Millimeter of mercuryMillimeter of mercury (mmHg)

PascalPascal 1Pa=1N/m2

10-4 Fluids at Rest 10-4 Fluids at Rest

Three forces act on the columnThree forces act on the column:

The pressurepressure increasesincreases with depthdepth in waterin water. The pressurepressure decreasesdecreases with altitudealtitude inin atmosphereatmosphere..

Set up a vertical axis in a tank of water with its originorigin at the surfacethe surface. Consider an imaginary columnan imaginary column of water. and are the depthsthe depths below the surface of the upperthe upper and lower column lower column fasesfases, respectively.

y

2y1y

acts at the topthe top of the column;1F

The gravitational forcegravitational force of the column

mg

acts at the bottemat the bottem of the column;2F

★Pressure in a liquid

1 1 ,F p A 2 2F p A (10-6)

The columnThe column is in static equilibriumin static equilibrium, these three forces balanced.

2 ,y h 2p p1 0p p1 0,y and

2 1F F mg (10-5)

level 1: surfacesurface; level 2: h below ith below it

1 2( )m V A y y

2 1 1 2( )p A p A Ag y y

2 1 1 2( )p p g y y or (10-7)

0p p gh Eq. 10-7Eq. 10-7 : (10-8)



2 ,y d 2p p1 0p p1 1,y p and

level 1: surfacesurface; level 2: d above itd above it

2 1 1 2( )p p g y y (10-7)

★★Pressure in atmospherePressure in atmosphere

0 airp p gd Eq. 10-7Eq. 10-7 :

(Atmospheric density is uniform)(Atmospheric density is uniform)

d

Level 2

This case is different from the example in 漆安慎力学漆安慎力学 P387P387 。。

There the atmospheric densitythe atmospheric density is proportional toproportional to the pressurepressure!There the atmospheric densitythe atmospheric density is proportional toproportional to the pressurepressure!

(10-11)extp p gh

The load put apressure on the The load put apressure on the piston and thus on the liquid. The piston and thus on the liquid. The pressure at any point P in the liquipressure at any point P in the liquid is thend is then

extp

10-6 Pascal's Principle 10-6 Pascal's Principle

and

Pascal's PrinciplePascal's PrincipleA change in the pressureA change in the pressure applied to an enclosed incompressible an enclosed incompressible fluidfluid is transmitted undiminishedundiminished to every portion of the fluidto every portion of the fluid and to the wall of its container.

extp p

Add more shot to increase Add more shot to increase by , the and by , the and unchanged so the pressure unchanged so the pressure change at Pchange at P (10-12)

extpextp

h, ,g

Hydraulic LeverHydraulic LeverPistoni :

1d1,A1,F and

Pistono :

2d2 ,A2 ,F and


output forces

(10-13)i o

i o

F F

A A

The pressures on both sides are equal

oo i

i

AF F

A

,o iA A o iF F

i i o oV Ad A d

(10-11)io i

o

Ad d

A

The same volume of incompressible liquid is displaced at both pistons

,o iA A

o id d

Hydraulic LeverHydraulic LeverPistoni :

1d1,A1,F and

Pistono :

2d2 ,A2 ,F and


output forces

(10-13)

The output work

o io o i i i i

i o

A AW F d F d Fd

A A

o iW W

With a hydraulic leverhydraulic lever, a given force applied over a given distaover a given distancence can be transformed to a grea greater forceater force applied over a smaller a smaller distancedistance.

10-7 Archimedes' Principle 10-7 Archimedes' Principle

app bweight weight F (10-19)

(apparent weight)(apparent weight)

Archimedes' PrincipleArchimedes' PrincipleArchimedes' PrincipleArchimedes' Principle

When a body is partially or whollypartially or wholly immersedimmersed in a fluid, a buoyant forcea buoyant force from the surrounding fluid acts on the bodyacts on the body. The force is directed directed upwardupward and has a magnitude equal to the equal to the weight of the fluidweight of the fluid that has been displaced by displaced by the bodythe body.

Apparent Weight in a FluidApparent Weight in a FluidApparent Weight in a FluidApparent Weight in a Fluid

10-8 Ideal Fluids in Motion 10-8 Ideal Fluids in Motion

Ideal fluidIdeal fluid. There are . There are four assumptionsfour assumptions about about ideal fluidideal fluid::Ideal fluidIdeal fluid. There are . There are four assumptionsfour assumptions about about ideal fluidideal fluid::

1. Steady Flow1. Steady Flow. In steady flow the velocity of the moving fluid at any given point does not change as time goes on.

1. Steady Flow1. Steady Flow. In steady flow the velocity of the moving fluid at any given point does not change as time goes on.

2. Incompressible Flow2. Incompressible Flow. The ideal fluid is incompressible means its density has a constant value.

2. Incompressible Flow2. Incompressible Flow. The ideal fluid is incompressible means its density has a constant value.3. Nonviscous Flow3. Nonviscous Flow. An object moving through a nonviscous fluid would experience no viscous drag force.

3. Nonviscous Flow3. Nonviscous Flow. An object moving through a nonviscous fluid would experience no viscous drag force.

4. Irrotational Flow4. Irrotational Flow. In irrotational flow a test body will not rotate about an axis through its own center of mass.

4. Irrotational Flow4. Irrotational Flow. In irrotational flow a test body will not rotate about an axis through its own center of mass.

10-8 Ideal Fluids in Motion10-8 Ideal Fluids in Motion

Figure 10-12 shows strFigure 10-12 shows streamlines traced out by injeeamlines traced out by injecting cting dyedye into the moving fl into the moving fluid. uid. A streamlineA streamline isis the pat the path h traced out by a tiny fluid traced out by a tiny fluid elementelement which we may call which we may call a fluid “particle”.a fluid “particle”.

Figure 10-12 shows strFigure 10-12 shows streamlines traced out by injeeamlines traced out by injecting cting dyedye into the moving fl into the moving fluid. uid. A streamlineA streamline isis the pat the path h traced out by a tiny fluid traced out by a tiny fluid elementelement which we may call which we may call a fluid “particle”.a fluid “particle”.

Fig.10-12 streamlinesFig.10-12 streamlines

StreamlinesStreamlinesStreamlinesStreamlines

As As the fluid particlethe fluid particle moves, moves, its velocityits velocity may change, may change, both both in magnitudein magnitude and and in in directiondirection. . The velocity The velocity vectorvector at any point at any point will will alwaysalways be tangent tobe tangent to the the streamlinestreamline at that point. at that point.

As As the fluid particlethe fluid particle moves, moves, its velocityits velocity may change, may change, both both in magnitudein magnitude and and in in directiondirection. . The velocity The velocity vectorvector at any point at any point will will alwaysalways be tangent tobe tangent to the the streamlinestreamline at that point. at that point.

streamlinestreamline

fluid elementfluid element

v

10-8 Ideal Fluids in Motion 10-8 Ideal Fluids in Motion

StreamlinesStreamlinesStreamlinesStreamlinesStreamlines Streamlines never crossnever cross be because, if they did, a fluid pacause, if they did, a fluid particle arriving rticle arriving at the interseat the intersectionction would have to assume would have to assume two different velocities simtwo different velocities simultaneouslyultaneously, , an impossibilitan impossibilityy..

Streamlines Streamlines never crossnever cross be because, if they did, a fluid pacause, if they did, a fluid particle arriving rticle arriving at the interseat the intersectionction would have to assume would have to assume two different velocities simtwo different velocities simultaneouslyultaneously, , an impossibilitan impossibilityy..

streamlinestreamline

fluid elementfluid element

v

tube of flowtube of flow tube of flowtube of flow a tube of flowa tube of flowWe can build up We can build up a tube of flowa tube of flow

whose whose boundaryboundary is made up is made up of of streamlinesstreamlines. Such a tube . Such a tube acts acts like a pipelike a pipe because any because any fluid particle that enters itfluid particle that enters it

We can build up We can build up a tube of flowa tube of flow whose whose boundaryboundary is made up is made up of of streamlinesstreamlines. Such a tube . Such a tube acts acts like a pipelike a pipe because any because any fluid particle that enters itfluid particle that enters it

cannot escape through its cannot escape through its wallswalls; if it did, we would ; if it did, we would have a case of have a case of streamlines streamlines crossing each other.crossing each other.

cannot escape through its cannot escape through its wallswalls; if it did, we would ; if it did, we would have a case of have a case of streamlines streamlines crossing each other.crossing each other.

In a time intervala time interval a volumea volume a of fluid entersenters the tube at its left at its left endend. Then because the fluid is incompressibleincompressible, an identical identical volumevolume must emerge emerge from the right endfrom the right end.

Vt

V

10-9 The Equation of Continuity 10-9 The Equation of Continuity

Consider a tube segment (L) through which an idea fluid flows toward the right.

Consider a tube segment (L) through which an idea fluid flows toward the right.

Left end Right end Left end Right endCross-sec-t

ional areaCross-sec-tional area

Fluid speedFluid speed

1A

2v2A

1v

1 1 2 2V Av t A v t

1 1 2 2Av A v (10-23)( equation of continuity )

( equation of continuity )

A vFor an idea fluid, whenFor an idea fluid, when

VR Av (10-24)a constant(volume flow rate, equation of continuity)(volume flow rate, equation of continuity)

10-9 The Equation of Continuity10-9 The Equation of Continuity

1 1 2 2Av A v (10-23)( Equation of continuity )( Equation of continuity )

greatest speed

lower speed

For an idea fluidan idea fluid, when

A v Closer streamlines

VR 3( / )m s is the volume flow ratevolume flow rate ( volume per unit timevolume per unit time )is the volume flow ratevolume flow rate ( volume per unit timevolume per unit time )

If the density of the fluid is uniform, multiply Eq.10-24 by that density to get

If the density of the fluid is uniform, multiply Eq.10-24 by that density to get

m VR R Av (10-25)

a constant

( Mass flow rateMass flow rate )( Mass flow rateMass flow rate )

mR ( / )kg s is the mass flow ratemass flow rate

( mass per unit timemass per unit time )is the mass flow ratemass flow rate ( mass per unit timemass per unit time )

By applying the principle of conserva-the principle of conserva-tion of energytion of energy to the fluid, these quantthese quantities are related byities are related by

10-10 Bernoulli's Equation 10-10 Bernoulli's Equation

An idea fluidAn idea fluid is flowing through a tube a tube segmentsegment with a steady ratea steady rate. In a time intervala time interval , a volume of a volume of fluidfluid enters the tube at the leftenters the tube at the left end and an identical volumean identical volume emerges emerges at the right endat the right end because the fluid is incompressibleincompressible.

tV

Left end Right end Left end Right endelevationelevation

speedspeed1y

2v2y

1vpressurepressure 2p1p

2 21 11 1 1 2 2 22 2p v gy p v gy (10-28)

If the fluid doesn’t change its eleva-tieleva-tionon as it flows in a horizontal tube, take , Bernoulli's Equation is now in the following formin the following form

0y

10-10 Bernoulli's Equation10-10 Bernoulli's EquationIdeal fluidIdeal fluid. There are . There are four four assumptionsassumptions about about ideal ideal fluidfluid::

Ideal fluidIdeal fluid. There are . There are four four assumptionsassumptions about about ideal ideal fluidfluid::

2v

2y

2p

If If the speedthe speed of fluid element of fluid element increasesincreases as it travels along a horizontal stream-as it travels along a horizontal stream-line, line, the pressurethe pressure of the fluid must of the fluid must decreasedecrease, and , and converselyconversely..

If If the speedthe speed of fluid element of fluid element increasesincreases as it travels along a horizontal stream-as it travels along a horizontal stream-line, line, the pressurethe pressure of the fluid must of the fluid must decreasedecrease, and , and converselyconversely..

Bernoulli's Equation (only for ideal fluid ) Bernoulli's Equation (only for ideal fluid )

Eq.10-28 can be written as21

2p v gy a constanta constant (10-29)

2 21 11 1 1 2 2 22 2p v gy p v gy

(10-28)

y

2 21 11 1 2 22 2p v p v (10-30)

x

y

2v 2p1v 1p

We need be concerned only with chan-ges that take place at the input and out-put ends.

We need be concerned only with chan-ges that take place at the input and out-put ends.

10-10 Bernoulli's Equation10-10 Bernoulli's Equation

2v

2y

Take the entire volume of the fluid as our system; Apply the principle of con-sthe principle of con-servation of energyervation of energy to this system as it moves from initial state (Fig.(a))initial state (Fig.(a)) to the ffinal state (Fig.(b)).inal state (Fig.(b)).

Proof of Bernoulli's EquationProof of Bernoulli's Equation

ApplyApply energy conservationenergy conservation in the form in the form of of the work-kinetic energy theoremthe work-kinetic energy theoremApplyApply energy conservationenergy conservation in the form in the form of of the work-kinetic energy theoremthe work-kinetic energy theorem

W K (10-31)

2 21 12 12 2K mv mv

(10-32)2 212 12 ( )V v v


Proof of Bernoulli's EquationProof of Bernoulli's Equation

W K (10-31)

(10-32)2 212 12 ( )K V v v

The work done by the gravitational force on the fluid from the input level to the output level is

The work done by the gravitational force on the fluid from the input level to the output level is

2 1( )gW mg y y (10-33)2 1( )g V y y

Work must also be doneWork must also be done at the input at the input endend to push the entering fluid into theto push the entering fluid into the tubetube andand by the system by the system at the output at the output endend to push forward the fluid ahead of to push forward the fluid ahead of the emerging fluidthe emerging fluid..

Work must also be doneWork must also be done at the input at the input endend to push the entering fluid into theto push the entering fluid into the tubetube andand by the system by the system at the output at the output endend to push forward the fluid ahead of to push forward the fluid ahead of the emerging fluidthe emerging fluid..

2v

2y

2p

The work done at the input end is The work done at the input end is The work done at the input end is The work done at the input end is 1p VThe work done at the output end from The work done at the output end from the system is the system is The work done at the output end from The work done at the output end from the system is the system is

2p V


Proof of Bernoulli's EquationProof of Bernoulli's EquationW K (10-31)

(10-33)

(10-32)2 212 12 ( )K V v v

2 1( )gW g V y y

2v

2y

2p

Generally, the work done by a force F on an area A through , isGenerally, the work done by a force F on an area A through , isx

( ) ( )F x pA x p A x p V

2 1pW p V p V (10-34)2 1( )p p V

The work-kinetic energy theoremThe work-kinetic energy theorem now now becomesbecomesThe work-kinetic energy theoremThe work-kinetic energy theorem now now becomesbecomes

2v

2y

2p


Proof of Bernoulli's EquationProof of Bernoulli's EquationW K (10-31)

(10-33)

(10-32)2 212 12 ( )K V v v

2 1( )gW g V y y (10-34)

2 1( )pW p p V

2 1 2 1( ) ( )g V y y V p p 2 212 12 ( )V v v

Substituting from Substituting from (10-32), (10-33) and (10-34)

yieldsSubstituting from Substituting from (10-32), (10-33) and (10-34)

yields

g pW W W K

2 21 11 1 1 2 2 22 2p v gy p v gy (10-28)

Bernoulli's EquationBernoulli's Equation

制作 张昆实 yangtze university

Documents

制作张昆实 yangtze university