Friction (and Shear)

1. Friction (and Shear) • Gas • Origin of Viscosity • Mix of gases • Liquid • Origin of Viscosity • Effect of foreign materials • Dilute vs Concentrated (sol-gel) • Non-newtonian Fluids • Concentrated • Effect of non-spherical dispersed materials • Presence of structure

2. Gas 3 2 • Gas • Kinetic Theory of gas • Non polar, low density V 1 Y • Mean Free Path is large X • Molecular movement between 1 and 2 (and 2 and 1, etc) • Momentum Transfer between planes • ==> viscosity • Increase Temp ==> Increase velocity, Viscosity • Rigid Spheres

3. Gas • Accounting for van der Waals attractive force • Lennard-Jones potential • Sigma- collision dia • omega- collision integral • M -molecular wt • Mix of gases

4. Liquids • Theory is not as well developed • Eyring’s Theory • Inter-molecular forces cause viscosity (NOT moving molecules) • Temp increase ==> more energy for molecule ==> less viscosity • Similar to reaction equilibrium

5. Force B C C’ A A’ Energy State Liquid Viscosity B • To go from A to C, the particle should have energy DEAct(Activation Energy) • Energy released is heat of reaction DERxn A Energy C State • For Liquid movement • EA and EC are same • Application of stress shifts A up and C down • ==> Movement from left to right

6. Dilute solutions • Assume • No interaction between particles • Spherical, uncharged • Liquid velocity on particle surface = particle surface velocity • Newtonian behavior • Emulsions will show lower viscosity • particles do not shear, emulsions will • surface contamination will increase emulsion viscosity

7. Non newtonian fluids • When one or more of the assumptions are violated • Usually heterogenous • Higher concentration (eg 40% of blood has red blood cells in plasma) ==> interaction between particles • Non spherical particles • Electrically charged (not discussed here)

8. Non newtonian fluids • High concentration (high is relative) • Interaction, structure formation • Structural viscosity • Application of shear stress • breaks structure over time ==> thixotropic • breaks structure quickly, more stress ==> more disintegration ==> pseudoplastic • alternate: cylinders, ellipses align better with flow under higher shear ==> pseudoplastic • thixotropic (60 sec) --> pseudoplastic L D Axis Ratio = L/D

9. Bingham Plastic Pseudo plastic Dilatant Stress Newtonian Strain Non newtonian fluids • Dilatant: Mostly solids with some fluid in between • Low stress ==> lubrication and less viscosity • higher stress ==> insufficient lubrication, more viscosity • Bingham Plastic • Minimum yield stress • Newtonian

10. Non newtonian Fluids: Models

11. Non Newtonian Fluids:Models • Viscoelastic: • usually coiled or connected structure • stretched (not broken) by stress • recoil after stress is released • normal stress on pipe != 0 • eg. Pull back after the applied force is removed • Non-newtonian != high viscosity • Many polymers added to reduce friction in water

12. Assumptions • Laminar flow • steady state • no-slip • incompressible Fluid flow in a pipe • Hagen-Poiseuille’s law r x r • Momentum balance • Pressure drop = friction

13. r x r Fluid flow in a pipe • Newtonian • Flow Rate • Average Velocity

14. Fluid flow in a pipe • Non newtonian: power law fluid • Flow Rate • Average Velocity • Double the pressure != double velocity

15. Fluid flow in a pipe • Non newtonian: Bingham Plastic • Flow Rate • Average Velocity • Double the pressure != double velocity

16. Sample 1 Sample 2 Flow between plates • Micro fluidics • Identification of DNA fragments (for example) • Flow rate depends on • Viscosity • Surface Tension • Sample movement rate depends on affinity

17. Flow between plates Y • Steady state • Incompressible • Laminar flow • no-slip 2b Z X • Element of width length DX, height DY and width (or depth) of 1 unit

18. Flow between plates • By symmetry, at the center, shear stress =0 • Newtonian • Flow rate • Average velocity

19. Flow between plates • Non newtonian : Power law fluids • Flow rate • Average velocity

20. Resistance Examples • Pipe flow • Fluid flow ~= Current flow • DP = Voltage, Vavg = current • Non-newtonian fluid: non-linear relation between DP and Vavg • Newtonian fluid: easier prediction of results of changing one or more parameters

21. Example • Non newtonian : Bingham Plastic H=10 m D=0.1m L=20m/5m

22. Example • Find the time taken to drain the tank H=10 m D=0.1m • V2 is a function of H L=20m/5m • Tank will not drain completely!

23. Example • Non newtonian : Power law fluid 1m/s 25 m Long, 1cm dia • If flow rate has to be doubled, pressure needed

24. Example L23 L12 • Pbm. 8.2 • Given, L12=22 km, L23=18 km, Q, DP known • Consider this as resistance model

25. Viscometers • Tube,Cone&Plate,Narrow gap cylinder, infinite gap cylinder

26. A Y • Goal: Shear Stress, Velocity Profile, Torque Viscometers • Cone and Plate Viscometer • Ref: BSL, pbm 2B.11 X • Fluid between two plates, linear profile

27. Shear Stress at cone: Viscometers • Shear stress vs Velocity: Spherical Co-ordinates • Force, Torque • Practical Y0 ~ 1o

28. Viscometers • Cylindrical viscometer • Vary W, obtain Torque and velocity gradients for plots

29. Continuity (Velocity Divergence) Differential momentum balance:Navier-Stokes Equation • Newtonian Fluids ONLY • Assumptions/applicability: • Isothermal conditions • both Compressible/Incompressible • both laminar/turbulent • Stokes assumption for bulk-viscosity (needed for compressible fluids)

30. Appendix 1 2 • Pbm. 8.6 • Given, L=8m,DP=207kPa, d=.635 cm, m,r • Find velocity for no friction vs friction • Frictional effects

31. Appendix: Blood Flow in Arteries • 40% red blood cells in plasma, non-newtonian • Pulsating motion, varying pressure • Re = 600, during exercise 6000 • Blood vessel dilation , short term, long term • Shear stress vs platelet activation (wound vs stenosis); ultrasonic detection • Tensile vs compressive stress; structure of blood vessel • Collapse of vessel during BP measurement • Collapse near stenosis and cardiac arrest • Mass & heat transport