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Mechanics

Oct 29, 2013 (4 years and 5 months ago)

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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

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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

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

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Some Greek Letters

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

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)

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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

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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

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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

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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

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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.

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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

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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