IIT-Madras, Momentum Transfer: July 2005-Dec 2005 Friction (and Shear) n Gas u Origin of Viscosity u Mix of gases n Liquid u Origin of Viscosity u Effect of foreign materials F Dilute vs Concentrated (sol-gel) u Non-newtonian Fluids F Concentrated F Effect of non-s pherical dispersed materials F Presence of structure

IIT-Madras, Momentum Transfer: July 2005-Dec 2005 Gas V Y X n Gas u Kinetic Theory of gas u Non polar, low density n Mean Free Path is large n Molecular movement between 1 and 2 (and 2 and 1, etc) n Momentum Transfer between planes n ==> viscosity n Increase Temp ==> Increase velocity, Viscosity n Rigid Spheres

IIT-Madras, Momentum Transfer: July 2005-Dec 2005 Gas n Accounting for van der Waals attractive force n Lennard-Jones potential n Sigma- collision dia n omega- collision integral n M -molecular wt n Mix of gases

IIT-Madras, Momentum Transfer: July 2005-Dec 2005 Liquids n Theory is not as well developed n Eyring’s Theory u Inter-molecular forces cause viscosity (NOT moving molecules) u Temp increase ==> more energy for molecule ==> less viscosity n Similar to reaction equilibrium

IIT-Madras, Momentum Transfer: July 2005-Dec 2005 Liquid Viscosity State Energy A B C To go from A to C, the particle should have energy  E Act (Activation Energy) Energy released is heat of reaction  E Rxn n For Liquid movement u E A and E C are same u Application of stress shifts A up and C down u ==> Movement from left to right State Energy A B C A’ C’ Force

IIT-Madras, Momentum Transfer: July 2005-Dec 2005 Dilute solutions n Assume u No interaction between particles u Spherical, uncharged u Liquid velocity on particle surface = particle surface velocity u Newtonian behavior u Emulsions will show lower viscosity F particles do not shear, emulsions will F surface contamination will increase emulsion viscosity

IIT-Madras, Momentum Transfer: July 2005-Dec 2005 Non newtonian fluids n When one or more of the assumptions are violated n Usually heterogenous n Higher concentration (eg 40% of blood has red blood cells in plasma) ==> interaction between particles n Non spherical particles n Electrically charged (not discussed here)

IIT-Madras, Momentum Transfer: July 2005-Dec 2005 Non newtonian fluids n High concentration (high is relative) n Interaction, structure formation u Structural viscosity n Application of shear stress u breaks structure over time ==> thixotropic u breaks structure quickly, more stress ==> more disintegration ==> pseudoplastic u alternate: cylinders, ellipses align better with flow under higher shear ==> pseudoplastic u thixotropic (60 sec) --> pseudoplastic L D Axis Ratio = L/D

IIT-Madras, Momentum Transfer: July 2005-Dec 2005 Non newtonian fluids n Dilatant: Mostly solids with some fluid in between u Low stress ==> lubrication and less viscosity u higher stress ==> insufficient lubrication, more viscosity Stress Strain Newtonian Dilatant Pseudo plastic Bingham Plastic n Bingham Plastic u Minimum yield stress u Newtonian

IIT-Madras, Momentum Transfer: July 2005-Dec 2005 Non newtonian Fluids: Models

IIT-Madras, Momentum Transfer: July 2005-Dec 2005 Non Newtonian Fluids:Models n Viscoelastic: u usually coiled or connected structure u stretched (not broken) by stress u recoil after stress is released u normal stress on pipe != 0 u eg. Pull back after the applied force is removed n Non-newtonian != high viscosity u Many polymers added to reduce friction in water

IIT-Madras, Momentum Transfer: July 2005-Dec 2005 Fluid flow in a pipe n Hagen-Poiseuille’s law n Momentum balance n Assumptions n Laminar flow n steady state n no-slip n incompressible x r r n Pressure drop = friction

IIT-Madras, Momentum Transfer: July 2005-Dec 2005 Fluid flow in a pipe x r r n Newtonian n Flow Rate n Average Velocity

IIT-Madras, Momentum Transfer: July 2005-Dec 2005 n Non newtonian: power law fluid Fluid flow in a pipe n Flow Rate n Average Velocity n Double the pressure != double velocity

IIT-Madras, Momentum Transfer: July 2005-Dec 2005 n Non newtonian: Bingham Plastic Fluid flow in a pipe n Flow Rate n Average Velocity n Double the pressure != double velocity

IIT-Madras, Momentum Transfer: July 2005-Dec 2005 Flow between plates n Micro fluidics n Identification of DNA fragments (for example) n Flow rate depends on u Viscosity u Surface Tension u Sample movement rate depends on affinity Sample 1 Sample 2

IIT-Madras, Momentum Transfer: July 2005-Dec 2005 Flow between plates X Y Z n Steady state n Incompressible n Laminar flow n no-slip Element of width length  X, height  Y and width (or depth) of 1 unit 2b

IIT-Madras, Momentum Transfer: July 2005-Dec 2005 Flow between plates n By symmetry, at the center, shear stress =0 n Newtonian n Flow rate n Average velocity

IIT-Madras, Momentum Transfer: July 2005-Dec 2005 Flow between plates n Non newtonian : Power law fluids n Flow rate n Average velocity

IIT-Madras, Momentum Transfer: July 2005-Dec 2005 Examples n Pipe flow n Fluid flow ~= Current flow  P = Voltage, V avg = current Resistance Non-newtonian fluid: non-linear relation between  P and V avg n Newtonian fluid: easier prediction of results of changing one or more parameters

IIT-Madras, Momentum Transfer: July 2005-Dec 2005 Example n Non newtonian : Bingham Plastic H=10 m L=20m/5m D=0.1m

IIT-Madras, Momentum Transfer: July 2005-Dec 2005 Example n Find the time taken to drain the tank H=10 m L=20m/5m D=0.1m n V 2 is a function of H n Tank will not drain completely!

IIT-Madras, Momentum Transfer: July 2005-Dec 2005 Example n Non newtonian : Power law fluid 1m/s 25 m Long, 1cm dia n If flow rate has to be doubled, pressure needed

IIT-Madras, Momentum Transfer: July 2005-Dec 2005 Example n Pbm. 8.2 Given, L12=22 km, L23=18 km, Q,  P known n Consider this as resistance model L12 L23

IIT-Madras, Momentum Transfer: July 2005-Dec 2005 Viscometers n Tube,Cone&Plate,Narrow gap cylinder, infinite gap cylinder

IIT-Madras, Momentum Transfer: July 2005-Dec 2005 Viscometers n Cone and Plate Viscometer n Ref: BSL, pbm 2B.11 X Y A n Goal: Shear Stress, Velocity Profile, Torque n Fluid between two plates, linear profile

IIT-Madras, Momentum Transfer: July 2005-Dec 2005 Viscometers n Shear stress vs Velocity: Spherical Co-ordinates n Shear Stress at cone: n Force, Torque Practical  0 ~ 1 o

IIT-Madras, Momentum Transfer: July 2005-Dec 2005 Viscometers n Cylindrical viscometer Vary , obtain Torque and velocity gradients for plots

IIT-Madras, Momentum Transfer: July 2005-Dec 2005 Differential momentum balance: Navier-Stokes Equation n Newtonian Fluids ONLY n Assumptions/applicability: u Isothermal conditions u both Compressible/Incompressible u both laminar/turbulent u Stokes assumption for bulk-viscosity (needed for compressible fluids) n Continuity (Velocity Divergence)

IIT-Madras, Momentum Transfer: July 2005-Dec 2005 Appendix n Pbm. 8.6 Given, L=8m,  P=207kPa, d=.635 cm,  n Find velocity for no friction vs friction n Frictional effects 1 2

IIT-Madras, Momentum Transfer: July 2005-Dec 2005 Appendix: Blood Flow in Arteries n 40% red blood cells in plasma, non-newtonian n Pulsating motion, varying pressure n Re = 600, during exercise 6000 n Blood vessel dilation, short term, long term n Shear stress vs platelet activation (wound vs stenosis); ultrasonic detection n Tensile vs compressive stress; structure of blood vessel n Collapse of vessel during BP measurement n Collapse near stenosis and cardiac arrest n Mass & heat transport