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. Types of Polymers. Stress in Fluids Rate of Strain Tensor Compressible and Incompressible Fluids Newtonian and Non-Newtonian Fluids. General Concepts. Fluid

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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

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

Professor Joe Greene

CSU, CHICO


Types of polymers

Types of Polymers

  • Stress in Fluids

  • Rate of Strain Tensor

  • Compressible and Incompressible Fluids

  • Newtonian and Non-Newtonian Fluids


General concepts

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


Stresses pressure velocity and basic laws

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

 = µd /dt

Deformed Shape

F


Stress in fluids

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


Some greek letters

Some Greek Letters

  • Nu: 

  • rho: 

  • tau: 

  • Alpha:

  • gamma: 

  • delta:

  • epsilon:

  • eta:

  • mu:


Pressure

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 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


Pressure1

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


Compressible and incompressible fluids

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)


Velocity

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


Equations and assumptions

Equations and Assumptions

Force = Pressure Viscous Gravity

Force Force Force

Energy = Conduction Compression Viscous

volume Energy Energy Dissipation

  • Mass

  • Momentum

  • Energy


Basic laws of fluid mechanics

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


Physical properties

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


Density

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

  • 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)


  • Viscosity

    Viscosity

    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


    Viscosity1

    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


    Viscosity2

    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.


    Viscosity models

    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


    Viscosity3

    Viscosity

    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


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