Chapter 9 solids and fluids
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Chapter 9: Solids and Fluids. 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.

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Chapter 9: Solids and Fluids

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Chapter 9 solids and fluids

Chapter 9: Solids and Fluids

  • 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


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 series

of 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 mole-

  • cules together in fixed position.

  • The molecules wander around in

  • random fashion.

States of Matter


States of matter1

  • 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 motion

and 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.

States of Matter

plasma : At high temperature, electrons of atoms are free from

nucleus. Such a collection of ionized atoms with equal amounts of

positive (nucleus) and negative charges (electrons) forms a state

called 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


Deformation of solids1

  • Young’s modulus: elasticity in length

  • 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.

  • Eventually the bar attains an

  • equilibrium in which:

  • (1) its length is greater than L0

  • (2) the external force is balanced

  • by internal forces.

Deformation of Solids

The bar is said to be stressed.

Young’s modulus

SI unit: dimensionless

tensile stress

tensile strain

SI unit: Pa = 1 N/m2


Deformation of solids2

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

  • Typical values

Deformation of Solids

  • Stress vs. strain


Deformation of solids3

  • Shear modulus: Elasticity of shape

  • 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.

  • The stress in this

  • situation is called

  • a shear stress.

Deformation of Solids

Shear modulus

SI unit: dimensionless

shear stress

shear strain

SI unit: Pa = 1 N/m2


Deformation of solids4

  • Bulk modulus: Volume elasticity

  • Suppose that the external forces acting on an object are all

  • perpendicular to the surface on which the force acts and are

  • distributed uniformly.

  • This situation occurs when a

  • object is immersed in a fluid.

Deformation of Solids

bulk modulus

SI unit: dimensionless

volume stress

volume strain

SI unit: Pa = 1 N/m2


Deformation of solids5

  • An example

  • Example 9.3 : Stressing a lead ball

A solid lead sphere of volume 0.50 m3, dropped in the ocean, sinks

to 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 in

volume of the sphere?

Deformation of Solids


Density and pressure

  • Density

  • The density r of an object is defined as:

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


Density and pressure1

  • Pressure

  • 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

  • If F is the magnitude of a force exerted

  • perpendicular to a given surface of area A,

  • then the pressure P is defined as:

F: force, A: area

SI unit: Pa = N/m2


Density and pressure2

  • 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.

Density and Pressure

  • Effect of atmospheric pressure:

P0 : atmospheric pressure, P: pressure at depth h


Density and pressure3

  • Examples

  • Example 9.5 : Oil and water

r=0.700 g/cm3

h1=8.00 m

r=1025 kg/m3

h2=5.00 m

Density and Pressure


Density and pressure4

  • 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

Density and Pressure

F2 > F1 if A2 > A1


Density and pressure5

  • Car lift

  • Example 9.7 : Car lift

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

weight=13,300 N

(b) Find air pressure.

Density and Pressure

circular x-sec

(c) Show the work done

by pistons is the same.

r1=5.00 cm

r2=15.0 cm


Pressure measurements

  • Absolute and gaugepressure

  • An open tube manometer (Fig.(a))

P=PA=PB

measures the gauge pressure P-P0

P : absolute pressure

P0 : atmospheric pressure

  • A mercury barometer (Fig.(b))

Pressure Measurements

measures the atmospheric pressure

vacuum

  • One atmospheric pressure

defined as the pressure equivalent of a

column of mercury that is exactly 0.76 m

in height.


Pressure measurements1

  • 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…

Pressure Measurements


Buoyant forces and archimedes s principle

  • Archimedes’s principle

Any object completely or partially submerged in a fluid is buoyed

up by a force with magnitude equal to the weight of the fluid

displaced by the object.

Upward force (buoyant force) :

Buoyant Forces and

Archimedes’s Principle

Downward force:


Buoyant forces and archimedes s principle1

  • Archimedes’s principle and a floating object

Upward force (buoyant force) :

Vobj

Downward force:

Vfluid

Buoyant Forces and

Archimedes’s Principle


Buoyant forces and archimedes s principle2

  • Examples

  • Example 9.8 : A fake or pure gold crown?

Is the crown made of pure

gold?

Tair =7.84 N

Twater =6.86 N

Buoyant Forces and

Archimedes’s Principle

rgold=19.3x103 kg/m3


Buoyant forces and archimedes s principle3

  • Examples

  • Example 9.9 : Floating down the river

What depth h is the bottom of

the raft submerged?

A=5.70 m2

rwood=6.00x102 kg/m3

Buoyant Forces and

Archimedes’s Principle


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 :

Fluid in Motion

  • The fluid is non-viscous :

  • There is no internal friction force between adjacent layers.

  • The fluid is incompressible :

  • Its density is constant.

  • The fluid motion is steady :

  • The velocity, density, and pressure at each point in the fluid do not

  • change with time.

  • The fluid moves without turbulence :

  • Each element of the fluid has zero angular velocity about its center.


Fluid in motion1

  • 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 Dt, the

fluid entering the bottom end of

the pipe moves a distance:

The mass contained in the bottom

blue region :

Fluid in Motion

From a similar argument :

Since DM1=DM2(flow is steady):

Equation of continuity


Fluid in motion2

  • An example

  • Example 9.12 : Water garden

Fluid in Motion


Fluid in motion3

  • Bernoulli’s equation

  • Consider an ideal fluid flowing through a pipe of non-uniform size.

Work done to the fluid at Point 1

during the time interval Dt:

Work done to the fluid at Point 2

during the time interval Dt:

Fluid in Motion

Work done to the fluid :


Fluid in motion4

  • Bernoulli’s equation (cont’d)

If m is the mass of the fluid passing

through the pipe in Dt , the change

in kinetic energy is:

The change in gravitational

potential energy in Dt is:

Fluid in Motion

From conservation of energy:


Fluid in motion5

  • Bernoulli’s equation (cont’d)

From conservation of energy:

Fluid in Motion

Bernoulli’s equation


Fluid in motion6

  • Venturi tube

Consider a water flow through

a horizontal constricted pipe.

Fluid in Motion


Fluid in motion7

  • Examples

h =0.500 m

y1 =3.00 m

  • Example 9.13 : A water tank

  • Consider a water tank with a hole.

  • Find the speed of the water

  • leaving through the hole.

y

x

Fluid in Motion

(b) Find where the stream hits the ground.


Fluid in motion8

  • Examples

  • Example 9.14 : Fluid flow in a pipe

A2=1.00 m2

A1=0.500 m2

h =5.00 m

Find the speed at Point 1.

Fluid in Motion


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.

Surface Tension, Capillary Action, and Viscous Fluid Flow

  • The surface tension is defined as :

where the surface tension force F

is divided by the length L along

which the force acts.

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


Surface tension capillary action and viscous fluid flow1

  • 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.

Surface Tension, Capillary Action, and Viscous Fluid Flow

  • 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 tension capillary action and viscous fluid flow2

  • Surface of liquid

Surface Tension, Capillary Action, and Viscous Fluid Flow

  • 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.


Surface tension capillary action and viscous fluid flow3

  • 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.

  • 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.

Surface Tension, Capillary Action, and Viscous Fluid Flow

ideal fluid, non-viscous

  • 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


Surface tension capillary action and viscous fluid flow4

  • 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 h is the coefficient of viscosity of

  • the fluid, and A is the area in contact with fluid.

Surface Tension, Capillary Action, and Viscous Fluid Flow

1 poise=10-1 N s/m2

1 cp (centipoise) = 10-2 poise

SI unit : N s/m2

cgs unit: dyne s/cm2= poise

h


Surface tension capillary action and viscous fluid flow5

  • 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:

Surface Tension, Capillary Action, and Viscous Fluid Flow


Surface tension capillary action and viscous fluid flow6

  • 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:

r : density of fluid

v : average speed of the fluid along the direction of flow

Surface Tension, Capillary Action, and Viscous Fluid Flow

d : diameter of tube

h : 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.


Surface tension capillary action and viscous fluid flow7

  • Examples

  • Example 9.18 : A blood transfusion

A patient receives a blood transfusion through a needle of radius 0.20

mm and length 2.0 cm. The density of blood is 1,050 kg/m3. The bottle

supplying the blood is 0.50 m above the patient’s arm. What is the rate

of the flow through the needle?

Surface Tension, Capillary Action, and Viscous Fluid Flow


Surface tension capillary action and viscous fluid flow8

  • Examples

  • Example 9.19: Turbulent flow of blood

Determine the speed at which blood flowing through an artery of

diameter 0.20 cm will become turbulent.

Surface Tension, Capillary Action, and Viscous Fluid Flow


Transport phenomena

  • 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.

  • Diffusion

  • In a diffusion process, molecules move from a region where their

  • concentration is high to a region where their concentration is lower.

  • Consider a container in which a

  • high concentration of molecules

  • has been introduced into the left

  • side (the dashed line is an

  • imaginary barrier).

Transport Phenomena

All the molecules move in random

direction. Since there are more

molecules on the left side, more

molecules migrate into the right

side than otherwise. Once a

concentration equilibrium is reached, there will be no net movement.


Transport phenomena1

  • Diffusion (cont’d)

  • Fick’s law

where D is a constant of proportion called

the diffusion coefficient (unit : m2/s), A is

the cross-sectional area, (…) is the change

in concentration per unit distance

(concentration gradient), and DM/Dt is the

mass transported per unit time. The

concentrations, C1 and C2 are measured

in unit of kg/m3.

Transport Phenomena


Transport phenomena2

  • 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 where

    its concentration is lower.

Transport Phenomena

  • A membrane that allows passage of some molecules but not others

    is called a selectively permeable membrane.

  • Osmosis is the diffusion of water across a selectively permeable

  • membrane from a high water concentration to a low water

  • concentration.


Transport phenomena3

  • 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 h with

  • speed v is given by:

resistive

frictional

force

Stokes’s law

  • Consider a small sphere of radius r falls

  • through a viscous medium.

Transport Phenomena

force of

gravity

buoyant

force


Transport phenomena4

  • 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.

  • As the sphere accelerates, its speed increases

  • and so does Fr.

resistive

frictional

force

  • When the net force goes to zero, the speed

  • of the sphere reaches the so-called terminal

  • speed vt.

Transport Phenomena

force of

gravity

buoyant

force


Transport phenomena5

  • 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.

Terminal speed condition

Transport Phenomena


Transport phenomena6

  • 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).

Transport Phenomena


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