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7/27/2019 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
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
stress and velocity gradient
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