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

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  = µd /dt  Deformed Shape F

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 zz  zy zx

6. Some Greek Letters • Nu:  • rho:  • tau:  • Alpha: • gamma:  • delta: • epsilon: • eta: • mu:

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 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

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 =  u2 A

11. Equations and Assumptions Force = Pressure Viscous Gravity Force Force Force Energy = Conduction Compression Viscous volume Energy Energy Dissipation • Mass • Momentum • Energy

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 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)

15. 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

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 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