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1ME-203
Fluid Mechanics-I
Kannan Iyer
[email protected]
Department of Mechanical Engineering
Indian Institute of Technology, Bombay
Why Study Fluid Mechanics
Let us see some connections with technology
The world is full of fluid Water and air are indispensable for
human
existence No technology can survive without fluids It is one of
the most widely used subject in
Engineering
Fluid Mechanics and Technology
Not just academic requirement, but an engineering necessity
Civil - Dams, Rivers, Canals
Mechanical - Hydraulic machines, Automobiles Elect/CS -
Packaging for cooling
Metallurgy - Blast furnace, Casting
Chemical - Reactors, Distillation
Pharmacy - Blood, Drugs manufacture
Agriculture - Pumps, Irrigation networks
Power - Power plants, Gas handling
Aerospace - Aircrafts, Rockets
No. of Gate:4 Span: 2.0m 1, 20.6m1, 22.15m2
TYPICAL PIPING SYSTEM
TURBULENT JET
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2Dam construction in order to hold large body of water
Design of gates and tanks
Wright Brothers - 1903
A GENERAL PIPE NETWORK
GEAR PUMP
CENTRIFUGAL PUMP PELTON TURBINE
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3AIRCRAFT PROPULSION SYSTEM
What is a Fluid
Matter can be divided into Solid and Fluid
Resists Deformation
Strong Cohesive Force (can resist shear)Retains Shape
Deforms Easily
Weak Cohesive Force (cannot resist shear)Cannot retain Shape
Fluid is one that deforms continuously on application of shear
stress.
Solid Fluid
Liquid and Gas
Weak Cohesive Force
Retains volume
Very Weak Cohesive Force
Cannot retain volume
Liquid Gas
Fluids can be divided into Liquid and Gas
How to Study
Microscopic - Kinetic theory of gases Keeping track of a large
number of
molecules is difficult and impractical
Macroscopic- Continuum methodology Continuous representation
instead of
discreteness Average properties defined Empirical closure
defined This course will emphasize this aspect
What is continuum?
Concept of a fluid particle
Validity of Continuum
Molecular mean free path small
It can be expressed in terms of Knudsen Number
01.0Kn For continuum to be valid
Knudsen Number,L
Kn = Mean free path______________System Length Scale
For Ideal Gas
http://hyperphysics.phy-astr.gsu.edu/hbase/kinetic/menfre.html#c1
Refer
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4Consequences of Continuum
...
2x)z,y,x(
x
ux)z,y,x(
x
u)z,y,x(u)z,y,xx(u2
2
2
+
+
+=+
Most important are
1. All parameters vary continuously
2. So Taylor series can be applied
...
2x)z,y,x(
x
uvx)z,y,x(
x
uv)z,y,x(uv)z,y,xx(uv2
2
2
+
+
+=+
We shall use this concept extensively
Analysis by Continuum Can be very detailed
Called Differential analysis Results in Partial Differential
Equation Mostly needs computers except for simple cases This
becomes the basis for CFD
Can be overall Called Integral analysis Details of flow
distribution not important Only system analysis is resorted to
Results in Algebraic equations
Lagrangian vs Eulerian
Eulerian This is control volume analysis The equations are
generated for the
volume The volume may or may not move We shall follow this
approach
Lagrangian This is control mass analysis The equation are
generated for the
fixed mass When mass moves, then you move
with the mass
Dimensions and UnitsSystems of Dimensions [M], [L], [t], and [T]
[F], [L], [t], and [T] [F],[M], [L], [t], and [T]
Dimensions and UnitsSystems of Units MLtT
SI (kg, m, s, K) FLtT
British Gravitational (lbf, ft, s, oR) FMLtT
English Engineering (lbf, lbm, ft, s, oR)
Dimensions and UnitsPreferred Systems of Units SI (kg, m, s,
K)
British Gravitational (lbf, ft, s, oR)
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5Principle of Dimensional Homogenity
All additive terms in a physical equation must have same
dimensions
ttanconsgH2Vp
2
=++
Experimentalists sometimes derive empirical equations that are
not dimensionally consistent
21
32
SR49.1V =
V (ft/s)=Average velocity, R=Hydraulic Radius (ft), S=slope
Mannings Equation for open channel flow
Principle of Dimensional Homogenity
21
32
SR49.1V = [ ]49.1
S)ft(R)s/ft(V
21
32
=
Let us look at the dimensions of the constant 1.49
1.49 is not dimensionless but has units of (ft1/3/s)Therefore if
you measure in another system of Units say SI, the value of 1.49
would not be the same. If we convert into SI, 1.49 (ft1/3/s)
becomes 1.00(m1/3/s)
21
32
S)m(R00.1)s/m(V =
Scalar, Vector and Tensor - I
Certain propoerties like mass, temperature have only magnitude
and they are called scalars
Certain other characteristics, such as velocity, Force, etc.,
have directional effects. Hence we need to specify magnitude and
direction. The parameter can be decomposed into its components V =
Vx i + Vy j + Vz k
Scalar, Vector and Tensor - II
A tensor has magnitude and two or more directions associated
with it
Velocity Field - I
V = u i + v j + w kwhere i, j and k are the unit vectors in x, y
and z directions.
Velocity is a vector quantity
Since velocity can continuously vary it is called a velocity
field
In general V (x,y,z,t)
Velocity Field - II
In stationary Eulerian context, Steady state implies that local
velocity does not change with time or V (x,y,z)
Mathematically 0t
=
V
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6Dimensionality of a problem
The number of dimensions required to specify the variations of
properties of a system is the dimensionality
The dimensionality can be reduced by intelligently choosing a
coordinate system
Examples to be discussed
Time, Path, Stream and Streak Lines
Path line is the path traced by a fluid particle in a flow
field
Streak line is the locus of all the particles that have passed
through a given point
Stream line is line whose tangent at a point defines the
velocity direction at that point
Time line is the locus of all points that originated from a
line
Examples to be discussed
Property Review
Density( ) =
Velocity(V) =
Pressure(p) =
Momentum = Mass X Velocity
0dd
dM
0dtdtdS
AreaForce
For ideal gas where, T (K), R = Ru/MRTp
=
Equations for Ideal Gas
=v
p
c
c
For ideal gas where, T (K), R = Ru/MRTp
=
R = cp - cv
Ru = 8314 J/k-mol-K (Universal Gas Constant)
For Monoatomic gas = 1.67
For Diatomic Gas = 1.4
= 1.33For gases having more than two atoms
Concept of Distortion - I Concept of Distortion - II
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7Fluid Distortion
As the top plate is moved, the top layer moves with plate and
the time line looks like what has been shown
FLUID
F
tot1 t2
t2 > t1 > to
Strain Rate
baTan =
tUa =
btU
=
tLt0t
=
&
dydu
bU
==&
Newtonian Fluid
dydu=
SI Unit : Pa-s or N-s/m2
CGS Unit : dyne-s/cm2 (called Poise) = 0.1 Pa.s
&dydu
Many fluids follow a linear relationship between shear stress
and strain rate
or
Dynamic Viscosity
For one dimensional flow
Non-Newtonian Fluids
Non-Newtonian Fluids Special fluids (e.g., most biological
fluids,
toothpaste, some paints, etc.) Non-linear fluids
Behavior of Fluids Viscosity Decreases with
temperature for liquids This is because force of
cohesion decreases
Viscosity increases with temperature for gases
This is because force of molecular kinetic energy increases
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8Kinematic Viscosity
Kinematic viscosity =
=
SI Unit : m2/s
CGS Unit : cm2/s (called Stoke) = 0.0001 m2/s
Surface Tension -I
A B
Liquid
C DFree surface
Cohesion - force of attraction between molecules of a liquid
Adhesion force of attraction between two unlike molecules
In the interface surface tension arises due to
Surface Tension -II
Rppp ei
2========
ei pp >>>>
2pipipipi R = p pipipipiR2
Adhesion >> Cohesion Cohesion >> Adhesion
pi=pi cosR2ghR 2Force balance implies
gRcos2h
=
Contact Angle Gives indication of
Wettability (ability to stick to surfaces)
Fully-wetting Absolutely Non-
wetting
0=180=
Wetting < 90o
Non-wetting > 90o
Error AnalysismaF = )a,m(FF =
mmm = and aaa = What is uncertainty in F2/122
aa
Fm
m
FF
+
=
This procedure can be applied to any functional form
