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1

MFGT 242: Flow Analysis

Chapter 3: Stress and Strain in Fluid
Mechanics

Professor Joe Greene

CSU, CHICO

2

Types of Polymers


Stress in Fluids


Rate of Strain Tensor


Compressible and Incompressible Fluids


Newtonian and Non
-
Newtonian Fluids

3

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


4

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


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


5

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


zx


zy


zz



6

Some Greek Letters


Alpha:




gamma:



delta:




epsilon:




eta:




mu:




Nu:



rho:



tau:


7

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



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

8

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.


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

9

Compressible and Incompressible Fluids


Principle of mass conservation


where


is the fluid density and v is the velocity


For injection molding, the density is constant
(incompressible fluid density is constant)



10

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

=


u
2

A

11

Equations and Assumptions


Mass



Momentum




Energy



Force = Pressure Viscous Gravity



Force


Force Force


Energy

= Conduction Compression Viscous

volume


Energy Energy Dissipation


12

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


13

Physical Properties


Density


Liquids are dependent upon the temperature and pressure


Density of a fluid is defined as


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:



Water
°
A.P.I. = 10 with higher values for liquids that are less
dense.


Crude oil
°
A.P.I. = 35, when density = 0.851




14

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 T
0



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)

15

Viscosity


Viscosity is defined as a fluid’s resistance to flow under an applied
shear stress


Liquids are strongly dependent upon temperature






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

Stationary, u=0

Moving, u=V

V

x

y

Y= 0

Y= h

16

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


17

Viscosity


For Newtonian fluids, Shear stress is proportional to velocity gradient.




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



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.


18

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,


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

19

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


Ln shear rate,

Ln

0.01

0.1

1

10

100

T=400

T=300

T=200