Chapter 9: Solids and Fluids

Chapter 9: Solids and FluidsStates of Matter

Three states of matter • Normally matter is classified into one of three (four) states: solid, liquid, gas (, plasma).

solid : crystalline solid (salt etc.) amorphous solid (glass etc.)

ordered structure atoms arranged at almost at random

States of Matter Three (four) states of matter (cont’d)

• Normally matter is classified into one of three (four) states: solid, liquid, gas (, plasma).

liquid : A molecule in a liquid does random-walk through a seriesof interactions with other molecules.

- For any given substance, the liquid state exists at a higher temperature than the solid state.-The inter-molecular forces in a liquid are not strong enough to hold molecules together in fixed position.-The molecules wander around in random fashion.

States of Matter Three (four) states of matter (cont’d)

• Normally matter is classified into one of three (four) states: solid, liquid, gas (, plasma).

gas : In gaseous state, molecules are in constant random motionand exert only weak forces on each other.

-The average distance between the molecules of a gas is quite large compared with the size of molecules.- Occasionally the molecules collide with each other, but most of them move freely.- Unlike solids and liquids, gases can be easily compressed.

plasma : At high temperature, electrons of atoms are free fromnucleus. Such a collection of ionized atoms with equal amounts ofpositive (nucleus) and negative charges (electrons) forms a statecalled plasma.

Deformation of Solids Stress, strain and elastic modulus

• Until external force becomes strong enough to deform permanently or break a solid object, the effect of deformation by the external force goes back to zero when the force is removed – Elastic behavior.

• Stress : the force per unit area causing a deformation Strain : a measure of the amount of the deformation Elastic modulus : proportionality constant, similar to a spring constant

stress = elastic modulus x strain

Deformation of Solids

• Consider a long bar of cross-sectional area A and length L0, clamped at one end. When an external force F is applied along the bar, perpendicular to the cross section, internal forces in the bar resist the distortion that F tends to produce.

Young’s modulus: elasticity in length

• Eventually the bar attains an equilibrium in which: (1) its length is greater than L0

(2) the external force is balanced by internal forces.

The bar is said to be stressed.

0LLY

AF

tensile straintensile stress Young’s modulus SI unit: dimensionlessSI unit: Pa = 1 N/m2


• Typical values

Young’s modulus: elasticity in length (cont’d)

• Stress vs. strain


• Another type of deformation occurs when an object is subjected to a force F parallel to one of its faces while the opposite face is held fixed by a second force.

Shear modulus: Elasticity of shape

• The stress in this situation is called a shear stress.

hxS

AF

shear strainshear stress Shear modulus SI unit: dimensionlessSI unit: Pa = 1 N/m2


• Suppose that the external forces acting on an object are all perpendicular to the surface on which the force acts and are distributed uniformly.

Bulk modulus: Volume elasticity

• This situation occurs when a object is immersed in a fluid.

VVBP

volume strainvolume stressbulk modulus SI unit: dimensionlessSI unit: Pa = 1 N/m2


• Example 9.3 : Stressing a lead ball

An example

A solid lead sphere of volume 0.50 m3, dropped in the ocean, sinksto a depth of 2.0x103 m, where the pressure increases by 2.0x107 Pa.Lead has a bulk modulus of 4.2x1010 Pa. What is the change involume of the sphere?

VVPB/

3410

73

m 104.2Pa 102.4

Pa) 100.2)(m 50.0(

BPVV

Density and Pressure

• The density of an object is defined as:

Density

VM

M: mass, V: volume SI unit: kg/m3 (cgs unit: g/cm3 )

• The specific gravity of a substance is the ratio of its density to the density of water at 4oC, which is 1.0x103 kg/m3, and it is dimensionless.

Density and Pressure

• If F is the magnitude of a force exerted perpendicular to a given surface of area A, then the pressure P is defined as:

Pressure

AFP F: force, A: area SI unit: Pa = N/m2

• Fluids do not sustain shearing stresses, so the only stress that a fluid can exert on a submerged object is one that tends to compress it, which is a bulk stress.

• The force F exerted by the fluid on the object is always perpendicular to the surfaces of the object.

Density and Pressure Variation of pressure with depth

• When a fluid is at rest in a container, all portions of the fluid must be in static equilibrium – at rest with respect to the observer.• All points at the same depth must be at the same pressure. If this were not the case, fluid would flow from the higher pressure region to the lower pressure region.

• Consider an object at rest with area A and height h in a fluid.

012 MgAPAP

)( 21 yyAVM

2112 yygPP 0for 2121 yyPP• Effect of atmospheric pressure:

ghPP 0 P0 : atmospheric pressure, P: pressure at depth h

Density and Pressure Examples

• Example 9.5 : Oil and water

=0.700 g/cm3

=1025 kg/m3

h1=8.00 m

h2=5.00 mPa 101.56

m) 00.8()m/s 80.9)(kg/m 1000.7(

Pa 1001.1

5

232

5101

ghPP

Pa 1006.2 5

21

ghPPbot

Density and Pressure Pascal’s principle

• A change in pressure applied to an enclosed fluid is transmitted undiminished to every point of the fluid and to the walls of the container.

Hydraulic press

21 PP

2

2

1

1

AF

AF

1

212 AAFF

F2 > F1 if A2 > A1

Density and Pressure Car lift

• Example 9.7 : Car lift

N 1048.1 3

222

21

2

121

Frr

AAFF

r1=5.00 cm r2=15.0 cm

circular x-sec

weight=13,300 N

(a) Find necessary force by compressed air at piston 1.

(b) Find air pressure.

Pa 1088.1 5

1

1 AFP

(c) Show the work done by pistons is the same.

221121 xAxAVV

2112 // xxAA

2121 // AAFF 11

2

2

1

22

11

2

1

AA

AA

xFxF

WW

Pressure Measurements Absolute and gauge pressure

• An open tube manometer (Fig.(a))

ghPP 0P : absolute pressure

P=PA=PB

• A mercury barometer (Fig.(b))

ghPPP 00

P0 : atmospheric pressure

vacuummeasures the atmospheric pressure

measures the gauge pressure P-P0

ghPP 0

• One atmospheric pressuredefined as the pressure equivalent of acolumn of mercury that is exactly 0.76 min height.

atm 1 Pa 10013.1 50 ghP

Pressure Measurements Blood pressure measurement

• A specialized manometer (sphygmomanometer)

-A rubber bulb forces air into a cuff wrap.-A manometer is attached under cuff and is under pressure.-The pressure in the cuff is increased until the flow of blood through brachial artery is stopped.-Then a valve on the bulb is opened, and measurer listens with a stethoscope to the artery at a point just below the cuff.-When the pressure at the cuff and the artery is just below the max. value produced by heart (the systolic pressure), the artery opens momentarily on each beat.-At this point, the velocity of the blood is high, and the flow is noisy and can be heard…

Buoyant Forces and Archimedes’s Principle

Archimedes’s principleAny object completely or partially submerged in a fluid is buoyedup by a force with magnitude equal to the weight of the fluiddisplaced by the object.

Upward force (buoyant force) :

B

gV

Vg

hAgAPP

fluidfluid

fluid

fluid

)( 12

Downward force:

VgghAgM objobjobj


Archimedes’s principle and a floating objectUpward force (buoyant force) :

gVB fluidfluid

Downward force:

gVgM objobjobj Vfluid

obj

fluid

fluid

objobjobjfluidfluid V

VgVgV

Vobj


Examples• Example 9.8 : A fake or pure gold crown?

Is the crown made of puregold?

Tair =7.84 NTwater =6.86 N

0 mgTair

0 BmgTwater

0 BTT airwater

N 980.0 waterwaterwaterair gVTTB

34 m 1000.1 crownwater VV

kg 800.0/ gTm air33 kg/m 1000.8/ crowncrown Vm

gold=19.3x103 kg/m3


Examples• Example 9.9 : Floating down the river

What depth h is the bottom ofthe raft submerged? A=5.70 m2

wood=6.00x102 kg/m3

gmBgmB raftraft 0

gAhgVgmB waterwaterwaterwater )()(

gVgm raftraftraft )(

gVgAh raftwaterwater )()(

m 0632.0A

Vh

raft

raftraft

Fluid in Motion Some terminology

• When a fluid is in motion: (1) if every particle that passes a particular point moves along exactly the same smooth path followed by previous particles passing the point, this path is called streamline. If this happens, this flow is said to be streamline or laminar. (2) the flow of a fluid becomes irregular, or turbulent, above a certain velocity or under any conditions that can cause abrupt changes in velocity.• Ideal fluid :

1. The fluid is non-viscous : There is no internal friction force between adjacent layers.2. The fluid is incompressible : Its density is constant.2. The fluid motion is steady : The velocity, density, and pressure at each point in the fluid do not change with time.3. The fluid moves without turbulence : Each element of the fluid has zero angular velocity about its center.

Fluid in Motion Equation of continuity

• Consider a fluid flowing through a pipe of non-uniform size. The particles in the fluid move along the streamlines in steady-state flow. In a small time interval t, thefluid entering the bottom end ofthe pipe moves a distance:

tvx 11The mass contained in the bottomblue region :

tvAxAM 1111111 From a similar argument :

tvAxAM 2222222 Since M1=(flow is steady):

22211121 vAvAMM

ibleincompress if 2211 vAvA Equation of continuity

Fluid in Motion An example

• Example 9.12 : Water garden

Fluid in Motion Bernoulli’s equation

• Consider an ideal fluid flowing through a pipe of non-uniform size. Work done to the fluid at Point 1during the time interval t:

VPxAPxFW 1111111 Work done to the fluid at Point 2during the time interval t:

VPxAPW 22222

VPVPW fluid 21 Work done to the fluid :

Fluid in Motion Bernoulli’s equation (cont’d)

21

22 2

121 mvmvKE

The change in gravitationalpotential energy in t is:

If m is the mass of the fluid passingthrough the pipe in t , the changein kinetic energy is:

12 mgymgyPE

From conservation of energy:

PEKEW fluid

Fluid in Motion Bernoulli’s equation (cont’d)From conservation of energy:

PEKEW fluid

1221

22

21

21

21

)(

mgymgymvmv

VPP

Vm /

.21

21

22221

211 constgyvPgyvP

.21 2 constgyvP Bernoulli’s equation

Fluid in Motion Venturi tube

Consider a water flow througha horizontal constricted pipe.

222

211 2

121 vPvP

2211 vAvA

1221 vvAA

21 yy

Fluid in Motion Examples

Consider a water tank with a hole.(a)Find the speed of the water leaving through the hole.

• Example 9.13 : A water tank h =0.500 my1 =3.00 m

201210 2

1 gyPgyvP

m/s 13.32)(2 121

ghyygv

(b) Find where the stream hits the ground.

s 782.0210 0

21 ttvgtyy y

m 45.210 tvtvx x

x

y

Fluid in Motion Examples

Find the speed at Point 1.

• Example 9.14 : Fluid flow in a pipe A2=1.00 m2

A1=0.500 m2

h =5.00 m

22201

210 2

121 gyvPgyvP

2211 vAvA

12

12 v

AAv

2

2

12

101

210 2

121 gyv

AAPgyvP

m/s 4.11)/(1

22)(21

221

112

2

2

121

AA

ghvghyyg

AAv

Surface Tension, Capillary Action, and Viscous Fluid Flow

Surface tension• The net force on a molecule at A is zero because such a molecule is completely surrounded by other molecules.

• The net force on a molecule at B is downward because it is not completely surrounded by other molecules. There are no molecules above it to exert upward force. this asymmetry makes the surface of the liquid contract and the surface area as small as possible.• The surface tension is defined as :

LF

where the surface tension force Fis divided by the length L alongwhich the force acts.

SI unit : N/m=(N m)/m2=J/m2


Surface tension (cont’d)• The surface tension of liquids decreases with increasing temperature, because the faster moving molecules of a hot liquid are not bound together as strongly as are those in a cooler liquid.• Some ingredient called surfactants such as detergents and soaps decrease surface tension.

• The surface tissue of the air sacs in the lungs contain a fluid that has a surface tension of about 0.050 N/m. As the lungs expand during inhalation, the body secretes into the tissue a substance to reduce the surface tension and it drops down to 0.005 N/m.


Surface of liquid

• Forces between like-molecules such as between water molecules are called cohesive forces.• Forces between unlike-molecules such as those exerted by glass on water are called adhesive forces.

• Difference in strength between cohesive and adhesive forces creates the shape of a liquid at boundary with other materials.


Viscous fluid flow• Viscosity refers to the internal friction of a fluid. It is very difficult for layers of a viscous fluid to slide past one another.

ideal fluid, non-viscous

viscous fluid

• When an ideal non-viscous fluid flows through a pipe, the fluid layers slide past one another with no resistance.

• If the pipe has uniform cross-section each layer has the same velocity.

• The layers of a viscous fluid have different velocities. The fluid has the greatest velocity at the center of the pipe, whereas the layer next to the wall does not move because of adhesive forces between them.


Viscous fluid flow• Consider a layer of liquid between two solid surfaces. The lower surface is fixed in position, and the top surface moves to the right with a velocity v under the action of an external force F.• A portion of the liquid is distorted from its original shape, ABCD, at one instance to the shape AEFD a moment later. The force required F to move the upper plate at a fixed speed v is :

where is the coefficient of viscosity of the fluid, and A is the area in contact with fluid.

dAvF

SI unit : N s/m2

cgs unit: dyne s/cm2= poise

1 poise=10-1 N s/m2

1 cp (centipoise) = 10-2 poise


Poiseuille’s law• Consider a section of tube of length L and radius R containing a fluid under pressure P1 at the left end and a pressure P2 at the right.• Poiseuille’s law describes the flow rate of a viscous fluid under pressure difference:

LPPR

tV

8)( 21

4


Reynolds number• At sufficiently high velocities, fluid flow changes from simple streamline flow to turbulent flow, characterized by a highly irregular motion of the fluid. Experimentally the onset of the turbulence in a tube is determined by a dimensionless factor called Reynolds number, RN, given by:

vdRN

: density of fluidv : average speed of the fluid along the direction of flowd : diameter of tube : viscosity of fluid

•If RN is below about 2000, the flow of fluid through a tube is streamline.•If RN is above about 3000, the flow of fluid through a tube is turbulent.•If RN is between 2000 and 3000, the flow is unstable.


Examples• Example 9.18 : A blood transfusion

A patient receives a blood transfusion through a needle of radius 0.20mm and length 2.0 cm. The density of blood is 1,050 kg/m3. The bottlesupplying the blood is 0.50 m above the patient’s arm. What is the rateof the flow through the needle?

Pa 105.15m) 50.0)(m/s 80.9)(kg/m 1050( 32321 ghPP

sL

PPRtV /m 100.6

8)( 3821

4


Examples• Example 9.19: Turbulent flow of blood

Determine the speed at which blood flowing through an artery ofdiameter 0.20 cm will become turbulent.

m/s 9.3m) 1020.0)(kg/m 1005.1()1000.3)(s/m N 107.2()(

233

333

dRNv

Transport Phenomena

Diffusion• In a diffusion process, molecules move from a region where their concentration is high to a region where their concentration is lower.

• A fluid can move place to place as a result of difference in concentration between two points in the fluid. There are two processes in this category : diffusion and osmosis.

• Consider a container in which a high concentration of molecules has been introduced into the left side (the dashed line is an imaginary barrier).All the molecules move in randomdirection. Since there are moremolecules on the left side, moremolecules migrate into the rightside than otherwise. Once aconcentration equilibrium is reached, there will be no net movement.

Transport Phenomena Diffusion (cont’d)

• Fick’s law

LCCDA

tM 12

timemassrateDiffusion

where D is a constant of proportion calledthe diffusion coefficient (unit : m2/s), A isthe cross-sectional area, (…) is the changein concentration per unit distance (concentration gradient), and M/t is themass transported per unit time. Theconcentrations, C1 and C2 are measuredin unit of kg/m3.

Transport Phenomena Size of cells and osmosis

• Diffusion through cell membranes is vital in supplying oxygen to the cells of the body and in removing carbon dioxide and other waste products from them.

•A fresh supply of oxygen diffuses from the blood, where its concentration is high, into the cell, where its concentration is low.

•Likewise, carbon dioxide diffuses from the cell into the blood whereits concentration is lower.

•A membrane that allows passage of some molecules but not othersis called a selectively permeable membrane.•Osmosis is the diffusion of water across a selectively permeable membrane from a high water concentration to a low waterconcentration.

Transport Phenomena Motion through a viscous medium

• The magnitude of the resistive force on a very small spherical object of radius r moving slowly through a fluid of viscosity with speed v is given by:

rvFr 6 Stokes’s law

• Consider a small sphere of radius r falls through a viscous medium.

resistivefrictionalforce

buoyantforce

force ofgravity

3

34 rggVw

3

34 rggVB ff

Transport Phenomena Motion through a viscous medium (cont’d)

• At the instance the sphere begins to fall, the force of friction is zero because the velocity of the sphere is zero.

resistivefrictionalforce

buoyantforce

force ofgravity

• As the sphere accelerates, its speed increases and so does Fr.• When the net force goes to zero, the speed of the sphere reaches the so-called terminal speed vt.

wBFr

33

34

346 rgrgrv ft

)(9

2 2

ftgrv

Transport Phenomena Sedimentation and centrifugation

• If an object is not spherical, the previous argument can still be applied except for the use of Stokes’s law. In this case, we assume that the relation Fr=kv holds where k is a coefficient.

wBFr

)/(

mVmggVB ff

tr kvF

f

ttf

kmgvkvmgmg 1

Terminal speed condition

Transport Phenomena Sedimentation and centrifugation (cont’d)

• The terminal speed for particles in biological samples is usually quite small; the terminal speed for blood cells falling through plasma is about 5 cm/h in the gravitational field of Earth.

• The speed at which materials fall through a fluid is called sedimentation rate. The sedimentation rate in a fluid can be increased by increasing the effective acceleration g: for example by using radial acceleration due to rotation (centrifuge).

rrvac

22

f

tfc

t krmv

kmav 11

2

Chapter 9: Solids and Fluids

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Transcript of Chapter 9: Solids and Fluids