MFGT 242: Flow Analysis
Chapter 3: Stress and Strain in Fluid
Mechanics
Professor Joe Greene
CSU, CHICO
1
Types of Polymers
•
•
•
•
Stress in Fluids
Rate of Strain Tensor
Compressible and Incompressible Fluids
Newtonian and Non-Newtonian Fluids
2
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
3
Stresses, Pressure, Velocity, and Basic Laws
• Stresses: force per unit area
– Normal Stress: Acts perpendicularly to the surface: F/A
• Extension
• Compression
Cross Sectional A
Area A
A
F
F
– Shear Stress,  : Acts tangentially to the surface: F/A
• Very important when studying viscous fluids
• 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

 = µd /dt
Deformed Shape
4
F
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
• Pressure inside a balloon from a gas what is usually normal to surface
• Fig 3.13
zz 
zy
zx
5
Some Greek Letters
•
Nu: 
• gamma: 
•
rho: 
•
delta: 
•
tau: 
•
epsilon: 
•
eta: 
•
mu: 
•
Alpha: 
6
Pressure
• The stress in a fluid is called hydrostatic pressure and force per unit area acts
normal to the element.
   pI  
– Stress tensor can be written
• where p is the pressure, I is the unit tensor, and Tau is the stress tensor
• 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 D0
• 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, D0, will
return
7
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
• General motion of a fluid involves translation, deformation,
and rotation.
v
– 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
  v  (v)T
deform according to the following:
– where the dagger is the transposed matirx
– For injection molding the velocity gradient = shear rate in each cell
dt dx x2  x1 
    2 1
d dv v  v 
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Compressible and Incompressible Fluids
• Principle of mass conservation
t
 (  v)

– where  is the fluid density and v is the velocity
• For injection molding, the density is constant
(incompressible fluid density is constant)
v  0
9
Velocity
• 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.
• In microscopic treatment of fluids, it is essential to consider the
variations with position.
• Three fluxes that are based upon velocity and area, A
– Volumetric flow rate, Q = u A
– Mass flow rate, m = Q =  u A
– Momentum, (velocity times mass flow rate) M = m u =  u2 A
10
Equations and Assumptions
• Mass
• Momentum
• Energy
v  0
 Dt 
    P      g
 Dv 
Force
=
Pressure
Force
Viscous
Force
Gravity
Force
 Dt 
C p      q   p  v    v
 DT 
Energy
volume
= Conduction
Energy
Compression Viscous
Energy
Dissipation
11
Basic Laws of Fluid Mechanics
• Apply to conservation of Mass, Momentum, and Energy
• In - Out = accumulation in a boundary or space
Xin - Xout = X system
• Applies to only a very selective properties of X
– Energy
– Momentum
– Mass
• Does not apply to some extensive properties
– Volume
– Temperature
– Velocity
12
Physical Properties
• Density
– Liquids are dependent upon the temperature and pressure
• Density of a fluid is defined as
mass
M

 3
volume L
– mass per unit volume, and
– indicates the inertia or resistance to an accelerating force.
• Liquid
– Dependent upon nature of liquid molecules, less on T
– Degrees °A.P.I. (American Petroleum Institute) are related to
specific gravity, s, per:
141.5
 A.P.I . 
s
 131.5
– Water °A.P.I. = 10 with higher values for liquids that are less
dense.
– Crude oil °A.P.I. = 35, when density = 0.851
13
Density
• 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 T0
  0 1   T  T0 
• 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)

s
 SC
14
Viscosity
• Viscosity is defined as a fluid’s resistance to flow under an applied
shear stress
• Liquids are strongly dependent upon temperature
Moving, u=V
Y= h
V
y
x
Stationary, u=0
Y= 0
• 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
• Velocity is v = (y/h)V
• Shear stress is tangential Force per unit area,
 = F/A
15
Viscosity
• Newtonian and Non-Newtonian Fluids
–
–
–
–
–
Need relationship for the stress tensor and the rate of strain tensor
Need constitutive equation to relate stress and strain rate
For injection molding it is the rate of strain tensor is shear rate
  
For injection molding use power law model
For Newtonian liquid use constant viscosity     (m n 1 )
16
Viscosity
• For Newtonian fluids, Shear stress is proportional to velocity gradient.
du
V
 

dy
h
• The proportional constant, , is called viscosity of the fluid and has
dimensions
M
 
LT
• Viscosity has units of Pa-s or poise (lbm/ft hr) or cP
• 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.
17
Viscosity Models
• Models are needed to predict the viscosity over a range of shear rates.
• Power Law Models (Moldflow First order)
where m and n are constants.
If m =  , and n = 1, for a Newtonian fluid,
  m
n 1
you get the Newtonian viscosity, .
• For polymer melts n is between 0 and 1 and is the slope of the
viscosity shear rate curve.
• Power Law is the most common and basic form to represent the way
in which viscosity changes with shear rate.
• Power Law does a good job for shear rates in linear region of curve.
• Power Law is limited at low shear and high shear rates
18
Viscosity
• Kinematic viscosity, , is the ratio of viscosity and density
• Viscosities of many liquids vary exponentially with temperature and
are independent of pressure
• where, T is absolute T, a and b
• units are in centipoise, cP
e
T=200
a b lnT
Ln

T=300
T=400
0.01
0.1
1
Ln shear rate,
10

100
19