# Lectures 1-2 Introduction Fluid Mechanics1

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Lectures 1 & 2

INTRODUCTION: CONCEPTS AND DEFINITIONS(Fluid Mechanics; 4th edition; Frank M. White)

Dr. Vinh Q. Tang

[email protected]

Office: ME2186

MAAE 2300 - Fluid Mechanics

Department of Mechanical and Aerospace Engineering

Carleton University

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Contents

1. Basic Concepts & Definitions

2. Dimensions and Units

3. Properties of the Velocity Field4. Thermodynamic Properties

5. Transport Properties

6. Flow Patterns

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1. Basic Concepts & Definitions

Fluid Mechanics - Study of fluids at rest, in

motion, and the effects of fluids on

boundaries

Fluid - A substance which moves and deforms

continuously as a result of an applied shear

stress

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Continuum Concept

Fluids consist of discrete particles, the molecules.But for engineering purpose, we only needaverage effects due to many molecules.

We can assume that fluids (and solids) arecontinua, i.e., continuous distributions ofmatter

This is satisfactory if the mean free path, , of theparticle is much less than the significant length of our

problem. Example for hydrogen at 150C and 1 atm: =1.8x10-7 m, so

for many practical problems continuum assumption is ok.

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Pressure and Velocity

Two important properties in the study of fluid mechanics:

Pressure: The normal stress on any plane through a

fluid element at rest The direction of pressure forces will always be

perpendicular to the surface of interest.

Velocity: The rate of change of position at a point in aflow field.

It is used to specify flow field characteristics; flow rate;momentum; and viscous effects for a fluid in motion

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2. Dimensions and Units

Primary Dimensions in SI and BG Systems SI: International System of units

BG: British gravitational units

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Secondary Dimensions in

Fluid Mechanics

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Other Units

S.I.: 1 Newton (N) = 1 kg m/s2

British Gravitational System: 1 lbf = 1 slug ft/s2

(1 slug = 32.174 lbm)

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Unit Consistency

Units in an equation must be consistent Example 1Mechanical Energy = Kinetic Energy + Potential Energy

2

2

1mV mgzME

)550)(81.9)(1.7()23)(1.7(21 2

2 cmsmkg

hrkmkgME

Substituting given values of parameters

As shown the units are not consistent.

Must convert

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)100

1)(550)(81.9)(1.7()1000()sec3600

()23)(1.7(21

2222

cmmcm

smkg

kmmhr

hrkmkgME

)550)(81.9)(1.7()23)(1.7(2

12

2

cms

mkghr

kmkgME

Recall:

Converting units:

2222 /.383/.9.144 smkgsmkg

)/.1)(/.9.527(2

22

smkg

N

smkg

mN.9.527

N.m1J1where9.527 J

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Other Notes on Units

Weight is a force (but mass is not) Force is defined by the acceleration that it

produces on a standard mass:

On earth 1kg weighs:

amF .)1).(1(1

2s

mkgN

)81.9)(1()).(1( 2s

mkggkg earth

N81.9

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English system:

On earth 1lbm weighs:

To give 1 slug an acceleration of 1 ft/sec2 requires a force of 1 lbf

)sec

2.32).(1(12

ftlblb

mf

)sec2.32).(1()).(1( 2ft

lbglb mearthm

flb1

definitionby2.321m

lbslug

mlbslug 2.321

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2

sec

.2.321ft

lblbmf

2sec.11ft

sluglbf

From previous slide:

mlbslug 2.321 And :

Thus :

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Example 2

Given: Following pump power requirements

Determine: The power required in kW

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Recall

Substitute values and introduce conversions, we

get:

(Note: We used 1 lbf = 1 slug . ft/s2

)

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Tips

At the start of the problem, convert all

parameters with units to the base units being

used in the problem, e.g.:

For S.I. problems, convert all parameters to kg,

m, and s

For BG problems, convert all parameters to slug,

ft, and s Then convert the final answer to the desired

final units.

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3. Properties of the velocity Field (2)

Mass flow rate

Note: Vn is measured relative to the

moving boundary

Vn : the component of the

velocity normal to the

surface area across which

the fluid flows

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4. Thermodynamic Properties

The usual thermodynamic properties are also

important in fluid mechanics

P : Pressure (kPa, psi)

T : Temperature (0C, 0F)

: Density (kg/m3, slug/ft3)

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State Relations for Gases

All gases at high temperatures and low pressures

(relative to their critical point) are in good agreement

with theperfect-gas law

Each gas has its own constant R, equal to a universal constant

divided by the molecular weight

where = 49,700 ft2/(s2.R) = 8314 m2/(s2.K).

Most problems in this class are for air, with M = 28.97

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Standard Air Density

Recall:

Substitute values for and M(for air), we get:

Density can be determined from

Where, standard atmospheric pressure is 2116 lbf/ft2, and

standard temperature is 60F or 520R. Thus standard airdensity is

P= RT

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Specific heats

As a first approximation in airflow analysis we

commonly take cp, cv, and k to be constant The ratio of specific heats of a perfect gas : ; kair=1.4

Actually, for all gases, cp and cvincrease gradually withtemperature, and k decreases gradually

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State Relations for Liquids

Liquids are nearly incompressible and have a

single reasonably constant specific heat

Thus an idealized state relation for a liquid is

Water is normally taken to have a density of 1.94slugs/ft3 and a specific heat cp = 25,200 ft

2/(s2..R).

The steam tables may be used if more accuracyis

required.

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The density of a liquid usually decreases slightly with

temperature and increases moderately with

pressure.

If we neglect the temperature effect, an empirical

pressure-density relation for a liquid is:

where B and n are dimensionless parameters which vary

slightly with temperature and pa anda are standard

atmospheric values. Water can be fitted approximately to

the values B 3000 and n 7.

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Alternative for Density

: Specific Weight (weight per unit volume)(N/m3, lbf/ft3)

For H2O: = 9790 N/m3 = 62.4 lbf/ft3

For Air: = 11.8 N/m3 = 0.0752 lbf/ft3

S.G. : Specific Gravity = / (ref)

(ref)= (water at 1 atm, 20C) for liquids = 998 kg/m3

= (air at 1 atm, 20C) for gases = 1.205 kg/m3

= g

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Example 3

Determine the static pressure difference indicated by

an 18 cm column of fluid (liquid) with a specific

gravity of 0.85.

P = g h= (SG) . . h

= (0.85) (9790 N/m3 ) (0.18 m)

= 1498 N/m2

= 1.5 kPa

Note:

See note

below

g

gSG

liquid

water

liquid

.)(SG).(

.water

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5. Transport Properties

e.g.:

coefficient of viscosity (dynamic viscosity)

{M / L t }

kinematic viscosity ( / ) { L2 / t }

They relate to the diffusion of momentum due to

shear stresses, as seen in following slides

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Stress and Pressure

AF

AreaForceStress :

Area

ForceTangentialstress, Shear

Area

ForceNormalStress, Normal

A

ve)(compressiFStresseCompressiv:Pr essure

Fluid cannot

sustain tensile

stress

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A solid can resist a shear stress by a static

deformation; a fluid cannot

Any shear stress applied to a fluid, no matter

how small, will result in motion of that fluid

The fluid moves and deforms continuously as

long as the shear stress is applied.

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Newtonian fluid

A fluid which has a linear relationship between shear

Note: The linearity coefficient in

the equation is the coefficient

of viscosity

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Classic Problem: Viscous flow induced by

relative motion between two parallel plates

Note: No slip at either plate

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No-Slip Condition

At solid boundaries, the solid and fluid

molecules interlock, and there is no relative

velocity between the fluid and the solid (i.e.,

no slip) Its based on empirical fact, no matter how

smooth the solid surface is. Exceptions are for

conditions at very low pressures, such as inouterspace.

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Surface Tension

Molecules deep within the liquid repel eachother because of their close packing

Molecules at the surface are less dense and

attract each other. Since half of theirneighbors are missing, the mechanical effect isthat the surface is in tension

We can account adequately for surface effectsin fluid mechanics with the concept of surfacetension

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Surface tension acts in the plane of the liquid

surface, and its magnitude measured in N/m (see

Example 1.9) The two most common interfaces are water-air and

mercury-air

For a clean surface at 20C 68F, the measuredsurface tension is:

0.0050 lbf/ft = 0.073 N/m air-water

0.033 lbf/ft = 0.48 N/m air-mercury

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Pressure change across a curved interface due to

surface tension

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Flow regions

Flows constrained by solid surfaces can typically bedivided into two regimes:

A) Flow near a bounding surface with1. significant velocity gradients

2. significant shear stresses This flow region is referred to as a "boundary layer

B) Flows far from bounding surface with

1. negligible velocity gradients2. negligible shear stresses

3. significant inertia effects

This flow region is referred to as "free stream" or "inviscidflow region"

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Reynolds number (Re)

An important parameter in identifying the characteristics ofthe flow regions

This physically represents the ratio of inertia forces in theflow to viscous forces

For most flows of engineering significance, both thecharacteristics of the flow and the important effects due tothe flow, e.g., drag, pressure drop, aerodynamic loads, etc.,are dependent on this parameter

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Laminar Flow & Turbulent Flow

Very low Re indicates viscous creeping motion

where inertia effects are negligible

Moderate Re implies a smoothly varying

laminarflow

High Re probably spells turbulent flow

which is slowly varying in the time-mean but has

superimposed strong random high-frequency

fluctuations

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Low Re : laminar flow

High Re : turbulent flow

Moderate Re : transition flow

Flow issuing at constant speed from a pipe:

(a) high viscosity, low-Reynolds-number, laminar flow;

(b) low-viscosity, high-Reynolds-number, turbulent flow.

(a) (b)

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Formation of a turbulent puff in pipe flow:(a)and (b) nearthe entrance;

(c) somewhat downstream;

(d) far downstream

(a)

(b)

(c)

(d)

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6. Flow Patterns

Fluid flow is generally three-dimensional

Pressures and velocities change in all directions

But, one or two-dimensional flow analysis can

be useful for many practical applications

Where the changes are most significant in one or

two directions only

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One-Dimensional Flow

Conditions vary only in the direction of flow notacross the cross-section

In certain pipe flow application we can assume 1-D flow

(Average or mean velocity over the cross section is used)

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Streamlines & Streamtube

The most common method of flow-pattern presentation:(a)Streamlines are everywhere tangent to the local velocity

vector

(b)Streamtube is formedby a closed collection of streamlines

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Streamline over an aerofoil an example

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Examples

Example 1.1

Example 1.3

Example 1.7

Example 1.9