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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

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

- 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.

- 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

- 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

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

- Typical values

Deformation of Solids

- Stress vs. strain

- 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

- 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

- 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

- 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

- 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

- 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

- 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

- 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

- 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

- 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.

- 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

- 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:

- Archimedes’s principle and a floating object

Upward force (buoyant force) :

Vobj

Downward force:

Vfluid

Buoyant Forces and

Archimedes’s Principle

- 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

- 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

- 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.

- 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

- 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 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 :

- 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:

- Bernoulli’s equation (cont’d)

From conservation of energy:

Fluid in Motion

Bernoulli’s equation

- Venturi tube

Consider a water flow through

a horizontal constricted pipe.

Fluid in Motion

- 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.

- 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

- 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 (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 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.

- 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

- 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

- 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

- 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.

- 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

- 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

- 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.

- 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

- 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.

- 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

- 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

- 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

- 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