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MFGT 242: Flow Analysis
Chapter 3: Stress and Strain in Fluid
Mechanics
Professor Joe Greene
CSU, CHICO
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Types of Polymers
•
Stress in Fluids
•
Rate of Strain Tensor
•
Compressible and Incompressible Fluids
•
Newtonian and Non

Newtonian Fluids
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General Concepts
•
Fluid
–
A substance that will deform continuously when subjected to a
tangential or shear force.
•
Water skier skimming over the surface of a lake
•
Butter spread on a slice of bread
–
Various classes of fluids
•
Viscous liquids

resist movement by internal friction
–
Newtonian fluids: viscosity is constant, e.g., water, oil, vinegar
»
Viscosity is constant over a range of temperatures and stresses
–
Non

Newtonian fluids: viscosity is a function of temperature, shear rate,
stress, pressure
•
Invicid fluids

no viscous resistance, e.g., gases
–
Polymers are viscous Non

Netonian liquids in the melt state and
elastic solids in the solid state
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Stresses, Pressure, Velocity, and Basic Laws
•
Stresses: force per unit area
–
Normal Stress: Acts perpendicularly to the surface: F/A
•
Extension
•
Compression
–
Shear Stress,
: Acts tangentially to the surface: F/A
•
Very important when studying viscous fluids
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For a given rate of deformation, measured by the time derivative d
/dt of a small angle of deformation
, the shear stress is directly
proportional to the viscosity of the fluid
F
Cross Sectional
Area A
A
F
A
F
Deformed Shape
F
= µ
d
/dt
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Stress in Fluids
•
Flow of melt in injection molding involves deformation of
the material due to forces applied by
–
Injection molding machine and the mold
•
Concept of stress allows us to consider the effect of forces
on and within material
•
Stress is defined as force per unit area. Two types of forces
–
Body forces act on elements within the body (F/vol), e.g., gravity
–
Surface tractions act on the surface of the body (F/area), e.g., Press
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Pressure inside a balloon from a gas what is usually normal to surface
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Fig 3.13
zx
zy
zz
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Some Greek Letters
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Alpha:
•
gamma:
•
delta:
•
epsilon:
•
eta:
•
mu:
•
Nu:
•
rho:
•
tau:
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Pressure
•
The stress in a fluid is called hydrostatic pressure and force per unit area acts
normal to the element.
–
Stress tensor can be written
•
where p is the pressure, I is the unit tensor, and Tau is the stress tensor
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In all hydrostatic problems, those involving fluids at rest, the fluid molecules
are in a state of
compression.
–
Example,
•
Balloon on a surface of water will have a diameter D
0
•
Balloon on the bottom of a pool of water will have a smaller diameter
due to the downward gravitational weight of the water above it.
•
If the balloon is returned to the surface the original diameter, D
0
, will
return
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Pressure
•
For moving fluids, the normal stresses include both a pressure and
extra stresses caused by the motion of the fluid
–
Gauge pressure

amount a certain pressure exceeds the atmosphere
–
Absolute pressure is gauge pressure plus atmospheric pressure
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General motion of a fluid involves translation, deformation,
and rotation.
–
Translation is defined by velocity, v
–
Deformation and rotation depend upon the velocity gradient tensor
–
Velocity gradient measures the rate at which the material will
deform according to the following:
–
where the dagger is the transposed matirx
–
For injection molding the velocity gradient = shear rate in each cell
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Compressible and Incompressible Fluids
•
Principle of mass conservation
–
where
is the fluid density and v is the velocity
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For injection molding, the density is constant
(incompressible fluid density is constant)
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Velocity
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Velocity is the rate of change of the position of a fluid particle with
time
–
Having magnitude and direction
.
•
In macroscopic treatment of fluids, you can ignore the change in
velocity with position.
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In microscopic treatment of fluids, it is essential to consider the
variations with position.
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Three fluxes that are based upon velocity and area, A
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Volumetric flow rate, Q =
u
A
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Mass flow rate,
m
=
Q =
u
A
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Momentum,
(velocity times mass flow rate)
M =
m u
=
u
2
A
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Equations and Assumptions
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Mass
•
Momentum
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Energy
Force = Pressure Viscous Gravity
Force
Force Force
Energy
= Conduction Compression Viscous
volume
Energy Energy Dissipation
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Basic Laws of Fluid Mechanics
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Apply to conservation of Mass, Momentum, and Energy
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In

Out = accumulation in a boundary or space
Xin

Xout =
X system
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Applies to only a very selective properties of X
–
Energy
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Momentum
–
Mass
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Does not apply to some extensive properties
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Volume
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Temperature
–
Velocity
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Physical Properties
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Density
–
Liquids are dependent upon the temperature and pressure
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Density of a fluid is defined as
–
mass per unit volume, and
–
indicates the inertia or resistance to an accelerating force.
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Liquid
–
Dependent upon nature of liquid molecules, less on T
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Degrees
°
A.P.I. (American Petroleum Institute) are related to
specific gravity, s, per:
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Water
°
A.P.I. = 10 with higher values for liquids that are less
dense.
–
Crude oil
°
A.P.I. = 35, when density = 0.851
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Density
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For a given mass, density is inversely proportional to V
•
it follows that for moderate temperature ranges (
is constant) the
density of most liquids is a linear function of Temperature
•
0
is the density at reference T
0
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Specific gravity of a fluid is the ratio of the density to the density of a
reference fluid (water for liquids, air for gases) at standard conditions.
(Caution when using air)
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Viscosity
•
Viscosity is defined as a fluid’s resistance to flow under an applied
shear stress
•
Liquids are strongly dependent upon temperature
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The fluid is ideally confined in a small gap of thickness h between one
plate that is stationary and another that is moving at a velocity, V
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Velocity is v = (y/h)V
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Shear stress is tangential Force per unit area,
= F/A
Stationary, u=0
Moving, u=V
V
x
y
Y= 0
Y= h
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Viscosity
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Newtonian and Non

Newtonian Fluids
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Need relationship for the stress tensor and the rate of strain tensor
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Need constitutive equation to relate stress and strain rate
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For injection molding it is the rate of strain tensor is shear rate
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For injection molding use power law model
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For Newtonian liquid use constant viscosity
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Viscosity
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For Newtonian fluids, Shear stress is proportional to velocity gradient.
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The proportional constant,
,
is called viscosity of the fluid and has
dimensions
•
Viscosity has units of Pa

s or poise (lbm/ft hr) or cP
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Viscosity of a fluid may be determined by observing the pressure drop
of a fluid when it flows at a known rate in a tube.
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Viscosity Models
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Models are needed to predict the viscosity over a range of shear rates.
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Power Law Models (Moldflow First order)
where
m
and
n
are constants.
If m =
, and
n
= 1, for a Newtonian fluid,
you get the Newtonian viscosity,
.
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For polymer melts
n
is between 0 and 1 and is the slope of the
viscosity shear rate curve.
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Power Law is the most common and basic form to represent the way
in which viscosity changes with shear rate.
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Power Law does a good job for shear rates in linear region of curve.
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Power Law is limited at low shear and high shear rates
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Viscosity
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Kinematic viscosity,
, is the ratio of viscosity and density
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Viscosities of many liquids vary exponentially with temperature and
are independent of pressure
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where, T is absolute T, a and b
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units are in centipoise, cP
Ln shear rate,
Ln
0.01
0.1
1
10
100
T=400
T=300
T=200
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