Pertemuan 01 - 02Introduction

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INTRODUCTION TO FLUID MECHANICS

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Definition of a FluidA fluid is a substance that flows under the action of shearing forces. If a fluid is at rest, we know that the forces on it are in balance.

A gas is a fluid that is easily compressed. It fills any vessel in which it is contained.

A liquid is a fluid which is hard to compress. A given mass of liquid will occupy a fixed volume, irrespective of the size of the container.

A free surface is formed as a boundary between a liquid and a gas above it.

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Density• Regardless of form (solid, liquid, gas) we can

define how much mass is squeezed into a particular space

density mass

volume

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Pressure• A measure of the amount of force exerted on a

surface area

pressure forcearea

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Pressure in a Fluid• The pressure is just the weight of all the fluid

above you• Atmospheric pressure is just the weight of all the

air above on area on the surface of the earth• In a swimming pool the pressure on your body

surface is just the weight of the water above you (plus the air pressure above the water)

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Pressure in a Fluid• So, the only thing that counts in fluid pressure is

the gravitational force acting on the mass ABOVE you

• The deeper you go, the more weight above you and the more pressure

• Go to a mountaintop and the air pressure is lower

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Pressure in a Fluid

Pressure acts perpendicular to the surface and increases at greater depth.

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Pressure in a Fluid

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BuoyancyNet upward force is called the buoyant force!!!

Easier to lift a rock in water!!

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Displacement of Water

The amount of water displaced is equal to the volume of the rock.

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Archimedes’ Principle• An immersed body is buoyed up by a force equal

to the weight of the fluid it displaces.• If the buoyant force on an object is greater than

the force of gravity acting on the object, the object will float

• The apparent weight of an object in a liquid is gravitational force (weight) minus the buoyant force

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Flotation• A floating object displaces a weight of fluid equal

to its own weight.

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Flotation

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Fluids:Statics vs Dynamics

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DensityThe density of a fluid is defined as its mass per unit volume. It is denoted by the Greek symbol, .

=V m3kgm-3

If the density is constant (most liquids), the flow is incompressible.If the density varies significantly (eg some gas flows), the flow is compressible.(Although gases are easy to compress, the flow may be treated as incompressible if there are no large pressure fluctuations)

water= 998 kgm-3

air =1.2kgm-3

kgm

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Density• Regardless of form (solid, liquid, gas) we can

define how much mass is squeezed into a particular space

density mass

volume

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PressurePressure is the force per unit area, where the force is perpendicular to the area.

p=A m2

Nm-2

(Pa)

NF

This is the Absolute pressure, the pressure compared to a vacuum.

pa= 105 Nm-2

1psi =6895Pa

The pressure measured in your tyres is the gauge pressure, p-pa.

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Pressure• A measure of the amount of force exerted on a

surface area

pressure forcearea

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PressurePressure in a fluid acts equally in all directions

Pressure in a static liquid increases linearly with depth

p=increase in depth (m)

pressure increase

g h

The pressure at a given depth in a continuous, static body of liquid is constant.

p1 p2

p3 p1 = p2 = p3

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Measuring pressure (1)Manometers

h

p1 p2=pa

liquiddensity

x y

z

p1 = px

px = py

pz= p2 = pa

(negligible pressure change in a gas)

(since they are at the same height)

py - pz = gh

p1 - pa = gh

So a manometer measures gauge pressure.

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Measuring Pressure (2)Barometers

A barometer is used to measure the pressure of the atmosphere. The simplest type of barometer consists of a column of fluid.

p1 = 0vacuum

h

p2 = pa

p2 - p1 = gh

pa = gh

exampleswater: h = pa/g =105/(103*9.8) ~10m

mercury: h = pa/g =105/(13.4*103*9.8) ~800mm

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Atmospheric PressurePressure = Force per Unit AreaAtmospheric Pressure is the weight of the column of air above a unit area. For example, the atmospheric pressure felt by a man is the weight of the column of air above his body divided by the area the air is resting on

P = (Weight of column)/(Area of base)

Standard Atmospheric Pressure:

1 atmosphere (atm) 14.7 lbs/in2 (psi) 760 Torr (mm Hg) 1013.25 millibars = 101.3 kPascals

1kPa = 1Nt/m2

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Fluid StaticsFluid StaticsBasic Principles:

Fluid is at rest : no shear forces

Pressure is the only force acting

What are the forces acting on the block?

Air pressure on the surface - neglect

Weight of the water above the block

Pressure only a function of depth

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Units

SI - International SystemLength MeterTime SecMass KgTemp 0K = 0C + 273.15Force Newton = Nt = 1 kg m / s2

Gravity 9.81 m/s2

Work = Fxd Joule = Nt-m

Power = F/t Watt = Joule/sec

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Units

English Length in Ft

Time in Sec

lbm (slug) - 1 slug = 32.2 lbm

Force - lb

Gravity - 32.2 ft/sec2

Work = slug-ft/s2

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Properties of FluidsDensity = (decreases with rise in T) mass per unit volume ( lbs/ft3 or kg/m3 )

for water density = 1.94 slugs/ft3 or 1000 kg/m3

Specific Weight = (Heaviness of fluid) weight per unit volume = g

for water spec wt = 62.4 lbs/ft3 or 9.81 kN/m3

Specific Gravity = SG Ratio of the density of a fluid to the density of water

SG = f / w SG of Hg = 13.55

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Ideal Gas Law relates pressure to Temp for a gas

P = RT

T in 0K units

R = 287 Joule / Kg-0K

PressureForce per unit area:

lbs/in2 (psi), N/m2, mm Hg, mbar or atm

1 Nt/m2 = Pascal = Pa

Std Atm P = 14.7 psi = 101.33 kPa = 1013 mb

Viscosity fluid deforms when acted on by shear stress

= 1.12 x 10-3 N-s/m2

Surface tension - forces between 2 liquids or gas and liquid - droplets on a windshield.

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Section 1: PressurePressure at any point in a static fluid not fcn of x,y,or z

Pressure in vertical only depends on of the fluid

P = h + Po

Gage pressure: relative to atmospheric pressure: P = hThus for h = 10 ft, P = 10(62.4) = 624 psf

This becomes 624/144 = 4.33 psi

P = 14.7 psi corresponds to 34 ft

10 ft

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What is the pressure at point A? At point B? G = 42.43 lbs/ft3

SG = 0.68 W = 62.4 lbs/ft3 At point A: PA = G x hG + PO

= 42.43 x 10 + PO

424.3 lbs/ft2 gage

At point B: PB = PA + W x hW

= 424.3 + 62.4 x 3

611.5 lbs/ft2 gage

Converting PB to psi:

(611.5 lbs / ft2) / (144 in2/ft2)

= 4.25 psi

Pressure in a Tank Filled with Gasoline and Water

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Measurement of Pressure

Barometer (Hg) - Toricelli 1644

Piezometer Tube

U-Tube Manometer - between two points

Aneroid barometer - based on spring deformation

Pressure transducer - most advancedQuickTime™ and aTIFF (LZW) decompressor

are needed to see this picture.

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Manometers - measure PRules of thumb:

When evaluating, start from the known    pressure end and work towards the    unknown end At equal elevations, pressure is    constant in the SAME fluid When moving down a monometer,    pressure increases When moving up a monometer,    pressure decreases Only include atmospheric pressure on    open ends

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Manometers

Find the pressure at point A in this open u-tube monometer with an atmospheric pressure Po

PD = W x hE-D + Po

Pc = PD

PB = PC - Hg x hC-B

PA = PB

Simple Example:

P = x h + PO

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Archimedes’ Principle• An immersed body is buoyed up by a force equal

to the weight of the fluid it displaces.• If the buoyant force on an object is greater than

the force of gravity acting on the object, the object will float

• The apparent weight of an object in a liquid is gravitational force (weight) minus the buoyant force

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Flotation• A floating object displaces a weight of fluid equal

to its own weight.

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Flotation

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Gases• The primary difference between a liquid and a

gas is the distance between the molecules• In a gas, the molecules are so widely separated,

that there is little interaction between the individual moledules

• IDEAL GAS• Independent of what the molecules are

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Boyle’s Law

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Boyle’s Law• Pressure depends on density of the gas• Pressure is just the force per unit area exerted by

the molecules as they collide with the walls of the container

• Double the density, double the number of collisions with the wall and this doubles the pressure

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Boyle’s LawDensity is mass divided by volume.

Halve the volume and you double the density and thus the pressure.

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Boyle’s Law• At a given temperature for a given quantity of

gas, the product of the pressure and the volume is a constant

P1V1 P2V2

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Atmospheric Pressure• Just the weight of the air above you• Unlike water, the density of the air decreases with

altitude since air is compressible and liquids are only very slightly compressible

• Air pressure at sea level is about 105 newtons/meter2

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Barometers

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Buoyancy in a Gas• An object surrounded by air is buoyed up by a

force equal to the weight of the air displace.• Exactly the same concept as buoyancy in water.

Just substitute air for water in the statement• If the buoyant force is greater than the weight of

the object, it will rise in the air

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Buoyancy in a Gas

Since air gets less dense with altitude, the buoyant force decreases with altitude. So helium balloons don’t rise forever!!!

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Bernoulli’s Principle

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Bernoulli’s Principle• Flow is faster when the pipe is narrower• Put your thumb over the end of a garden hose• Energy conservation requires that the pressure

be lower in a gas that is moving faster• Has to do with the work necessary to compress a

gas (PV is energy, more later)

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Bernoulli’s Principle• When the speed of a fluid increases, internal

pressure in the fluid decreases.

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Bernoulli’s Principle

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Bernoulli’s Principle

Why the streamlines are compressed is quite complicated and relates to the air boundary layer, friction and turbulence.

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Bernoulli’s Principle

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REVIEW

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Fluid Mechanics• Pressure• Pascal’s Law• Archimedes’ Principle• Fluid Dynamics• Bernoulli’s Equation

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Pressure

Fluids apply a compressive force to submerged objects from all sides. This means that the force is spread out over a surface area.

AFP Pressure:

Force per unit area

(1 Pa = 1 N/m2)

PdAdF If pressure varies over the area:

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Example – Q14.2Both dams have the same height and width.

Which needs to be stronger?

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(a)What is the force experienced by your finger; what is the force experienced by your thumb?

(b)Your thumb holds the pointy end. What is the pressure on the thumb; what is the pressure on your finger?

Example

You hold a thumb tack between your index finger and thumb with a force of 10 N. The needle has a point that is 0.1mm in radius whereas the flat end has a radius of 5 mm.

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Variation of Pressure with Depth

ghPP 0

Pressure exerted by a liquid increases with depth.

Pressure at sea level is taken to be 1 atmospheres (atm)

Pa 10013.1atm 1 5

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Example – 14.4

F = ?

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Pascal’s Law• A change in the pressure applied to a fluid is transmitted to

every point of the fluid and to the walls of the container.

2

2

1

1

21

AF

AF

PP

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Example 14.2

d1 = 5.00 cmd2 = 15.0 cmmgcar = 13300 NF1 = ?P = ?

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Buoyant Forces – Archimedes’ Principle

Archimedes’ Principle:The magnitude of the buoyant force on an object equals the weight of the fluid displaced by the object.

gVgmB fff

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Example – 14.5

Weight in air = 7.84 NWeight in water = 6.84 N

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Totally Submerged Objects

gVgVB objfff

objf VV

gVgVF objobjobjf a is upward if f > obj

a is downward if f < obj

aMgMBF objobj

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Floating Objects

gVgM

BF

ffobj

g

gVgV ffobjobj

f

obj

obj

f

VV

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ExampleConsider an object that floats in water but sinks in oil. When the object floats in water, half of it is submerged. If we slowly pour oil on top of the water so it completely covers the object, the object

1. moves up.2. stays in the same place.3. moves down.

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Fluid Dynamics• We now put the fluid in motion (flow).• Here are several assumptions about the fluid and

its flow:– The flow is to be laminar (steady) not turbulent.– The fluid is non-viscous (negligible internal friction).

Think water, not honey.– The fluid in incompressible.– The flow irrotational (no angular momentum).

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Equation of Continuity

constant2211 vAvA

The product of the velocity of flow and the area of the pipe remains constant.

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Example

A blood platelet drifts along with the flow of blood through an artery that is partially blocked by deposits. As the platelet moves from the narrow region to the wider region, its speed

1. increases.2. remains the same.3. decreases.

What about the pressure?

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Bernoulli’s Equation

constant21 2 gyvP

Using conservation of energy:

WE

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Example – 14.9

d1 = 5 cmd2 = 3 cmv2 = 15 m/s

Vout in 10 min = ?v1 = ?P1 - P2 = ?

outV

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ReviewAFP ghPP 0

constant2211 vAvA

constant21 2 gyvP

gmB f

Pascal’s Law: A change in the pressure applied to a fluid is transmitted to every point of the fluid and to the walls of the container.

Archimedes' Law: The magnitude of the buoyant force on an object equals the weight of the fluid displaced by the object.

Equation of Continuity:

Bernoulli’s Equation:

Pressure:

1221 AFAF

Fluid Dynamics