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MFGT 242: Flow Analysis Chapter 3: Stress and Strain in Fluid Mechanics

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MFGT 242: Flow Analysis Chapter 3: Stress and Strain in Fluid Mechanics

Professor Joe Greene

CSU, CHICO

- Stress in Fluids
- Rate of Strain Tensor
- Compressible and Incompressible Fluids
- Newtonian and Non-Newtonian Fluids

- 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

- Newtonian fluids: viscosity is constant, e.g., water, oil, vinegar
- Invicid fluids- no viscous resistance, e.g., gases

- Viscous liquids- resist movement by internal friction
- Polymers are viscous Non-Netonian liquids in the melt state and elastic solids in the solid state

- A substance that will deform continuously when subjected to a tangential or shear force.

- 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

- Normal Stress: Acts perpendicularly to the surface: F/A

F

Cross Sectional

Area A

A

F

A

F

= µd /dt

Deformed Shape

F

- 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

- Nu:
- rho:
- tau:

- Alpha:
- gamma:
- delta:
- epsilon:
- eta:
- mu:

- 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

- Stress tensor can be written
- 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

- Example,

- 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

- 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)

- 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

Force = Pressure Viscous Gravity

Force Force Force

Energy = Conduction Compression Viscous

volume Energy Energy Dissipation

- Mass
- Momentum
- Energy

- 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

- 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

- 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

V

Moving, u=V

Y= h

y

Y= 0

x

Stationary, u=0

- 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

- 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

- 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.

- 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

T=200

T=300

Ln

T=400

0.01

0.1

1

10

100

Ln shear rate,

- 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