MAE 5130: VISCOUS FLOWS

MAE 5130: VISCOUS FLOWS Lecture 4: Viscosity August 26, 2010 Mechanical and Aerospace Engineering Department Florida Institute of Technology D. R. Kirk

WHAT IS VISCOSITY? • What does viscosity mean? • Often related to ‘time to flow’ especially for petroleum products • Viscosity is a measure of a fluid's resistance to flow. A fluid with low viscosity flows easily and is often called "thin." Water is an example of a fluid with a relatively low viscosity. A fluid with high viscosity is often described as "thick." Maple syrup is an example of a fluid with a relatively high viscosity. • Remember: Time to flow is not viscosity • How long does it take for 60 ml of oil at specified temperature (100 ºC or 210 ºF, approx. engine operating temp) to flow out of a 1.76 cm hole in bottom of a cup? • SAE 10 motor oil takes 10 seconds • SAE 30 motor oil takes 30 seconds • W rating indicates that oil has been tested at a colder temperature • 10-W30 motor oil performs like a SAE 10 motor oil at colder temperatures (engine start-up) but still has the SAE 30 viscosity at higher temperatures (engine operating conditions) • What does a viscosity index (VI) number mean? • Measure of relative change in viscosity of oil over a temperature range • HIGHER VI → SMALLER viscosity change over temperature • VI not related to actual viscosity or SAE viscosity, but is measure of rate of viscosity change • Generally, multigrade oils (0W-40, 10W-30, etc.) will have high viscosity indexes. Monograde oils (SAE 30, 40, etc.) will have lower viscosity indexes.

COEFFICIENT OF VISCOSITY, m • More fundamental approach to viscosity shows it is property of fluid which relates applied stress (t) to resulting strain rate (e) • Consider fluid sheared between two flat plates • Bottom plate is fixed • Top plate moving at constant velocity, V, in positive x-direction only • u = u(y) only • Geometry dictates that shear stress, txy, must be constant throughout fluid • Perform experiment → for all common fluids, applied shear is a unique function of strain rate • For given V, txy is constant, it follows that du/dy and exy are constant, so that resulting velocity profile is linear across plates • Newtonian fluids (air, water, oil): linear relationship between applied stress and strain • Coefficient of viscosity of a Newtonian fluid: m • Dimension: Ns/m2 or kg/ms • Thermodynamic property (related to molecular interactions) that varies with T&P

VISCOUS BEHAVIOR OF VARIOUS MATERIALS • All true fluids can not resist shear, so must pass through origin on plot of t vs. e • Yielding fluids show finite stress at zero strain rate (part solid and part fluid) • Often called a Bingham plastic • Toothpaste, grease, hand creams • Pseudoplastic: shear-thinning • Usually solutions of large, polymeric molecules in a solvent with smaller molecules • Ketchup: When at rest it is hard to pour, however it has lower viscosity when agitated • Hair gel: much harder to pour off fingers (a low shear application), but that it produces much less resistance when rubbed between the fingers (a high shear application) • Dilatant: shear-thickening • Uncooked mix of cornstarch and water: • Under high shear the water is squeezed out from between the starch molecules, which are able to interact more strongly

VISCOUS BEHAVIOR OF VARIOUS MATERIALS • Behavior of some non-Newtonian fluids may be time-dependent • If strain rate is held constant, shear stress may vary • Thixotropic: shear stress decreases • The longer the fluid undergoes shear, the lower its viscosity • Yogurt • Paint • Many clutch-type automatic transmissions use fluids with thixotropic properties, to engage the different clutch plates inside the transmission housing at specific pressures, which then changes the gearset • Clay-like ground can practically liquefy under the shaking of a tremor • Ketchup is frequently thixotropic • Rheopectic: shear stress increases • The longer the fluid undergoes shear, the higher its viscosity • Some lubricants, thicken or solidify when shaken • Gypsum paste

POWER-LAW APPROXIMATION FOR NONNEWTONIAN FLUIDS • K and n are material parameters which in general vary with T & P • K is the flow consistency index • n is the flow behavior index • If n < 1: pseudoplastic • If n = 1: Newtonian (K = m) • If n > 1: dilatant • Power law is only a good description of fluid behavior across range of shear rates to which coefficients were fitted

VISCOSITY AS A FUNCTION OF T &P • Non-dimensionalization performed relative to critical point • Tr=T/Tc • General Trends • Viscosity of liquids ↓ as T ↑ • Viscosity of low-pressure gases (or dilute mixtures) ↑ as T ↑ • Viscosity always ↑ as P ↑ • Poor accuracy near Pc, Tc • Usually Pc ~ 10 atm • Common in many problems to ignore P dependence and consider only T dependence

CORRELATIONS FOR m AND k

EQUATIONS OF MOTION: CARTESIAN

EQUATIONS OF MOTION: CYLINDRICAL (r, q, z)

EQUATIONS OF MOTION: SPHERICAL (r, q, f)

PROBLEM: 1-4 • Steady viscous flow enters tube from a reservoir • Wall friction causes a viscous layer, initially probably laminar, to begin at inlet and grow in thickness downstream, possibly becoming turbulent further inside tube • Internal flow constrained by solid walls, so viscous layers must coalesce at some distance, xL, at which point tube is completely filled with boundary layer • Downstream of coalescence, flow profile ceases to change with axial position and is called ‘fully-developed’ • If ReD > 2,000, flow will end up turbulent • See picture above • At lower ReD flow remains laminar • See pictures to right Poiseuille-Paraboloid Laminar Pipe Flow Formula

PROBLEM: 1-6 • Vorticity, wz, calculated from equations in Appendix B • Instantaneous velocity and vorticity profiles are shown on the right for C=1, n=1. • At t=0, the flow is a ‘line’ vortex’, irrotational everywhere except at the origin where wz=∞

PROBLEM: 1-8 • Inviscid flow past a [non-rotating] cylinder • 2 stagnation points (r,q) = (R,0) and (R,p)

MAE 5130: VISCOUS FLOWS