CH 6304 - FLUID MECHANICS
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OUTLINE
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Why fluid mechanics?
What is fluid mechanics?
Properties of fluids
Fluid as a continuum
Velocity and stress field
Newtonian and non-Newtonian fluids
Classification of fluid motion
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FLUIDS
• There are number of fluids that when burnt, produce lots
of heat, which can be used for various applications.
– Examples of these fluids includes petrol and diesel for
vehicles.
• There are some fluids like oil that have a tendency to
exert very high pressure or force. These fluids can be
used for lifting various heavy loads.
– The fluids used in hydraulic machines and hydraulic lifters
are an example.
• Some fluids have excellent flow properties which can be
used for the lubrication of various machines
• Fluids like water posses kinetic and potential energy,
which is used for generation of electricity as in
hydroelectric power plants
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COMMON APPLICATIONS
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Hydroelectric power plants
Hydraulic machines
Automobiles
Refrigerators and Air conditioners
Thermal power plants
Nuclear power plants
Renewable energy resource
Operating various instruments
Heat engines
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WHAT IS FLUID MECHANICS?
• Mechanics – Study of forces and motions
• Fluid mechanics – Statics and Dynamics of
fluids
– Fluids?
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STRESS
• Stress – measure of internal force experienced
by an object per unit area (force per unit area)
• Types:
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TYPES OF STRESS
a)A rope is holding up a weight. Weight exerts a
force which tends to pull the rope apart.
b)A steel column is holding up a weight. The
weight exerts a force which tends to crush the
column.
c)Some glue is holding up a weight. The weight
exerts a force that tends to pull the weight down
the walls.
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SOLIDS VS. FLUIDS
Solids
Permanently
resist very
large shear
force
Fluids
Cannot
resist shear
force
Intermediate
Permanently
resist small
shear
Cannot
permanently
resist large
one
Magnitude of shear stress matters and not the type
of material
• Plastic deformation – at very high shear stress,
solids can be made to flow like fluid
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LIQUIDS VS. GASES
Liquids
Gases
Incompressible
Easy to
compress
Occupies fixed
volume
Volume changes
with P, T
Free surface
formed
No free surface
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DENSITY AND SPECIFIC GRAVITY
• Density is defined as the mass (m) per unit volume
(V) ρ = m/V. Density has units of kg/m3
• Specific volume is defined as v = 1/ρ = V/m.
• For a gas, density depends on temperature and
pressure.
• Specific gravity, or relative density is defined as the
ratio of the density of a substance to the density of
some standard substance at a specified temperature
(usually water at 4°C), i.e., SG= ρ / ρ H20. SG is a
dimensionless quantity.
• The specific weight is defined as the weight per unit
volume, i.e., gs = ρg where g is the gravitational
acceleration. gs has units of N/m3.
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DENSITY OF IDEAL GASES
• Equation of State: equation for the relationship
between pressure, temperature, and density.
• The simplest and best-known equation of state is the
ideal-gas equation.
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P V= n R T
• Ideal-gas equation holds for most gases.
• However, dense gases such as water vapor and
refrigerant vapor should not be treated as ideal gases.
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• To obtain a relation for viscosity, Consider,
– Fluid contained between two parallel plates, area ‘A’,
distance ‘h’ apart
– Upper plate moved in one direction at a constant velocity,
V
• µ is the dynamic viscosity and has units of kg/m·s, Pa·s, or
poise.
• NEWTON’S LAW OF VISCOSITY –
– the viscous force, F, opposing motion at the
interface between two liquid layers, flowing with
velocity gradient du/dy, is given as
F = µA (du/dy)
– µ is the fluid viscosity or coefficient of viscosity
F/A = µ (du/dy)
– F/A is the force applied per unit area – shear
stress, τ
– du/dy is the shear rate or the velocity gradient
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SURFACE TENSION
• Liquid droplets behave like small
spherical balloons filled with liquid,
and the surface of the liquid acts like
a stretched elastic membrane under
tension.
• The pulling force that causes this is
– due to the attractive forces between
molecules
– called surface tension ss.
• Attractive force on surface molecule
is not symmetric.
• Repulsive forces from interior
molecules causes the liquid to
minimize its surface area and attain
a spherical shape.
CAPILLARY EFFECT
• Capillary effect is the rise or
fall of a liquid in a smalldiameter tube.
• The curved free surface in the
tube is call the meniscus.
• Water meniscus curves up
because water is a wetting
fluid.
• Mercury meniscus curves
down because mercury is a
nonwetting fluid.
• Force balance can describe
magnitude of capillary rise.
METHODS OF ANALYSIS &
DESCRIPTION
• Three basic ways to address a fluid flow
problem:
– Integral analysis or control volume
– Differential analysis or infinitesimal system
– Dimensional analysis or experimental study
• In integral and differential analyses– mathematical models are developed
– solved by computational methods
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• Flow must satisfy the following conditions
in all the three analyses:
– Conservation of mass (continuity)
– Linear momentum (Newton’s second law)
– First law of thermodynamics (conservation of
energy)
– A state relation, such as density, ρ = f(P,T)
– Appropriate boundary conditions at solid
surfaces, interfaces, inlets and outlets
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FLUID AS A CONTINUUM
• Continuum – assuming that the material is not
a discrete mass but a continuous mass
• Fluids – aggregations of molecules widely
spread for a gas, closely spaced for a liquid
• Distance between particles is much greater
than molecular diameter – lot of empty space
• No fixed lattice – move freely in nature
• Number of molecules occupying a given
volume continually changes
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• Properties such as density, or conditions such
as pressure and temperature, does not imply
such properties or conditions of individual
molecules, but those of “fluid” as a whole
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TYPES OF FLUID
Compressible
• Density changes are
significant with moderate
changes in T and P
• Gases
Incompressible
• Density changes are
negligible or small with
change in T and P
• Liquids
Real
Ideal
• Shear stress cannot be neglected
• Incompressible and has nonzero viscosity
• Influence of the wall is
significant
• Shear stress is negligible
• Incompressible and has zero
viscosity
• Influence of the wall is
insignificant
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• Restriction:
– Continuum approach is applicable only when the
mean free path of the fluid is smaller than the
characteristic length of the system
– Diameter of the tube, size of the container
• Mathematically, for continuum approach based
model to hold good, λ (mean free path), Lc
(characteristic length), λ/Lc << 1
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• The parameter λ/Lc is known as Knudsen
number
• When K n> 0.01, the concept of continuum
does not hold good.
• Beyond this critical range of Knudsen number,
the flows are known as
– slip flow (0.01 < K n < 0.1),
– transition flow (0.1 < K n < 10) and
– free-molecule flow (Kn > 10)
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VELOCITY FIELD
• Eulerian method:
– Field of flow
– Properties of fluid as a function of position and
time
• Lagrangian method:
– Applicable to solid mechanics
– Follows individual particle moving through flow
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• Velocity field:
– V(x, y, z, t)
– Vector function: direction along with magnitude
– Three components, one in each direction, u, v, w in
the x, y, z directions
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STRESS FIELD
• Each fluid particle undergoes
– Surface forces – pressure, friction
– Body forces – gravity, electromagnetic
• Gravitational body force acting on an element
of volume dV is given by ρgdv,
– where ρ is the density and g is the acceleration due
to gravity
• Surface forces lead to stress
– Forces acting on the boundary transmitted
throughout the medium
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NEWTONIAN & NON-NEWTONIAN
FLUIDS
Rheology relationship
between shear
stress and
shear rate
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• The general relationship between shear stress
and shear rate:
τ = K rn
– K is the coefficient of consistency
– n is the power law index or flow behavior index
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• Newtonian:
– It is the simplest behavior represented by a straight
line passing through origin
– In the equation, τ = K rn
– for a Newtonian fluid, n = 1, K is the viscosity
– Newtonian fluids have constant viscosity
regardless of shear
– All gases and most liquids are Newtonian
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• Non-Newtonian:
– All other curves are non-Newtonian
– Complex mixtures, slurries, pastes, polymer solutions
are non-Newtonian
– Time independent - Bingham plastic, Pseudo plastic,
Dilatent
– Time dependent – Thixotropic, Rheopectic
– Does not obey Newton’s law of viscosity -no constant
viscosity
– The value of n in the equation, τ = K rn deviates from
1
– Follows power law model
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TIME INDEPENDENT – VISCOSITY CONSTANT
WITH TIME
• Bingham plastic:
– Resist a small shear stress but flow easily under large shear
stresses
– Requires a finite yield stress before it begins to flow
– Toothpaste, mayonnaise, clay suspensions
• Similar to Newtonian fluid
– but a threshold shear stress (yield stress or yield value τ0)
is needed for the shear rate to increase
– slope is the coefficient of rigidity or plastic viscosity
–
τ = τ0 + ɳᴦ
– ɳ is the coefficient of rigidity
– ᴦ is the shear rate
– τ0 is the yield stress
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• Pseudo plastic fluids:
– Viscosity decreases with increasing velocity
gradient
– Polymer solutions, blood, most slurries
– the long chain molecules tend to align with each
other at high shear rates
– results in easy flow with decrease in apparent
viscosity
• Flow is described by τ = K rn
– n is less than unity
– smaller the value of n, greater is the deviation from
Newtonian behavior
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• Dilatant fluids:
– Viscosity increases with increasing velocity
gradient
– Uncommon, Starch suspensions
• The flow is described by τ = K rn
– n is greater than unity
– greater the value of n, greater is the deviation from
Newtonian behavior
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TIME DEPENDENT – VISCOSITY CHANGES
WITH TIME
• Thixotropic fluids:
– Viscosity decrease with time
• Rheopectic fluids:
– Viscosity increases with time
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CLASSIFICATION OF FLUID MOTION
• To classify types of flow two conditions are
examined:
– the uniformity of the flow within the stream
– the steadiness of the flow over time.
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• Steady flow:
– When the velocity at each location is constant
– Velocity field is invariable with time
• Unsteady:
– Velocity at each location varies with time
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• Uniform flow:
– When the magnitude and
direction of velocity do
not change from point to
point in the fluid
– Flow of fluids through
long pipelines of constant
diameter is uniform
• Non-uniform:
– When velocity, pressure,
etc., change from point to
point in the fluid
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• Steady, uniform flow:
– Conditions do not change with position or time
– Flow of liquid through a pipe of uniform diameter,
running at constant velocity
• Steady, non-uniform flow:
– Conditions change from point to point but not with time
– Flow of a liquid at constant flow rate through a tapering
pipe
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• Unsteady, uniform flow:
– When a pump starts up
• Unsteady, non-uniform flow:
– Conditions of a liquid when pipetting
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FLOW PATTERNS
• The patterns of flow can be visualized in
different ways
• Four basic types:
– Streamline – line tangent to velocity vector at a
given instant
– Path line – actual path traversed by a fluid particle
– Streak line – locus of particles that have earlier
passed through a prescribed point
– Timeline – set of fluid particles that form a line at
a given instant
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Streamline
Streak line
Path line
Timeline
Mathematical
Experimental
Experimental
Experimental
Instantaneous
Generated
along with
time
Generated
along with
time
Instantaneous
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• STREAMLINE:
– Lines drawn at a given instant tangent to the
direction of flow at every point in the flow field
– No flow across a stream line
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• Stream tube:
– Streamlines cannot cross paths, when they are in a
closed pattern stream tubes
– By definition the fluid is confined within a stream
tube but it has fluid boundaries
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Streamlines
NASCAR surface pressure contours and
streamlines
Airplane surface pressure contours,
volume streamlines, and surface
streamlines
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Streaklines
• A Streakline is the
locus of fluid particles
that have passed
sequentially through a
prescribed point in the
flow.
• Easy to generate in
experiments: dye in a
water flow, or smoke in
an airflow.
• STREAKLINE:
– A Streak line is the line made by a dye injected
into a fluid at one point, and thus, marks the
position of all the particles of fluid which have
passed that point.
– Focus on a fixed location in space and identify all
fluid particles passing through this point
– Using dye or smoke
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Streak line
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• TIMELINE
– Number of adjacent fluid particles are marked at a
given time
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Pathline
• A pathline is a line made by a single particle as
it moves during a period of time.
• Pathline refers to the path of a single particle
• Streamline refers to an instnataneous picture of
the velocity directions of a number of particles
• In a steady flow, stream lines, streaklines and
pathlines are the same
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• PATH LINE:
– Path or trajectory traced out by moving a fluid
particle
– Identify the fluid particle at a given instant by
injecting a dye or smoke
– Used in studying the trajectory of a contaminant
leaving a smoke stack
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• In a steady flow the velocity remains constant
with time
• Particle located on a given streamline will
always move along the same stream line
• Consecutive particles passing through a fixed
point in space will be on the same streamline
• Streamlines, streak lines and path lines are
identical in steady flow
• Problems
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RECAP
• Fluid – properties – density, specific gravity,
viscosity, surface tension, vapor pressure
• Methods of analysis – differential, integral and
dimensional
• Continuum concept of fluid
• Velocity field, stress field
• Types of fluid –Rheology
• Types of fluid flow – Steady – unsteady, uniform –
non-uniform
• Flow patterns – Streamline, streak line, path line,
time line
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COEFFICIENT OF COMPRESSIBILITY
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How does fluid volume change with P and T?
Fluids expand as T ↑ or P ↓
Fluids contract as T ↓ or P ↑
Need fluid properties that relate volume changes to changes
in P and T.
– Coefficient of compressibility
 P 
 P 



  
 v T
 T
  v 
– Coefficient of volume expansion
1  v 
1   
      
v  T  P
  T  P
• Combined effects of P and T can be written as
 v 
 v 
dv  
dT


  dP

T

P
 P T
STREAKLINE AND PATHLINE
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