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 Area A 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

• • • • • Alpha:  • gamma:  delta:  epsilon:  eta: mu:  

Some Greek Letters

• Nu:  • • rho:  tau:  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 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 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.

– Translation is defined by velocity, v 

v

– Deformation and rotation depend upon the velocity gradient tensor deform according to the following:    

v

 ( 

v

)

T

– where the dagger is the transposed matirx   

d dt

 

dx dv

  

x v

2 2  

v x

1 1   8

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  

v

 0 (incompressible fluid density is constant) 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

= 

u 2

A 10

• Mass

Equations and Assumptions

 

v

 0 • Momentum    

Dt Dv

    

P

  

g

• Energy 

C p

   Force = Pressure Viscous Gravity Force Force Force

Dt DT

     

p

   

v

Energy volume = Conduction Compression Viscous Energy 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 per unit volume, and  

mass volume



M L

3 – 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: 

A

.

P

.

I

.

 141 .

5  131 .

5

s

– 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 T 0    0  1   

T



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)

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 V Y= h 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 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 a



b

ln

T

Ln  T=200 T=300 T=400 0.01

0.1

1 Ln shear rate, 10   100 19