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1 http://www.physics.usyd.edu.au/teach_res/jp/fluids09 web notes: lect4.ppt

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2 What do some liquids splash more? Why do we need to change brake fluid? Why do cars need different oils in hot and cold countries? Why do engines run more freely as it heats up? Have you noticed that skin lotions are easier to pour in summer than winter? Why is honey sticky?

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3 When real fluids flow they have a certain internal friction called viscosity. It exists in both liquids and gases and is essentially a frictional force between different layers of fluid as they move past one another. In liquids the viscosity is due to the cohesive forces between the molecules whilst in gases the viscosity is due to collisions between the molecules. “VISCOSITY IS DIFFERENT TO DENSITY”

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4 stationary wall L low speed high speed plate moves with speed v X Z linear velocity gradient v x / d = v / L d v x = 0 v x = v vxvx A useful model: Newtonian fluids water, most gases

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5 stationary wall plate exerts force F velocity gradient is proportional to shear stress (F/A) = (v / L) over area A A useful model: Newtonian fluids water, most gases

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6 coefficient of viscosity (Greek letter eta). The greater the coefficient of viscosity , the greater the force required to move the plate at a velocity v. This relationship does not hold for all fluids. Viscous fluids that obey this equation are called Newtonian fluids and = constant independent of the speed of flow. = (F / A)(L / v) (F/A) = (v / L)

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7 toothpaste velocity gradient v/ L grease wet sand corn flour (F/A) = (v / L) Newtonian fluid shear stress F / A slope

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8 Non-Newtonian or rheological fluids – viscosity is a function of the flow velocity Examples of non-Newtonian fluids * Blood - it contains corpuscles and other suspended particles. The corpuscles can deform and become preferentially oriented so that the viscosity decreases to maintain the flow rate. * Corn flour and water mixture. * Certain soils (more clay content) are non-Newtonian when moist to wet.

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9 Viscosity SI unit is (N.m -2 )(m).(m -1.s) Pa.s A common unit is the poise P (1 Pa.s = 10 P) Fluid (mPa.s) water (0 °C) 1.8 water (20 °C) 1.0 water (100 °C) 0.3 white blood (37 °C) ~4 blood plasma (37 °C) ~1.5 engine oil (AE10) ~ 200 air 0.018 Viscosity is very temperature dependent. Viscosity of a liquid decreases with increasing temp. Viscosity of a gas increases with increasing temp. 1 mPa = 10 -3 Pa

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10 Why can't you get all the dust off your car by just squirting water from a hose onto it? Why can't you simply remove dust just be blowing across the surface? Why does dust cling to a fast rotating fan? How can a leaf stay on a car moving at high speed?

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11 Boundary layer When a fluid moves over a surface, there is a thin layer of the fluid near the surface which is nearly at rest. This thin layer is called the boundary layer.

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12 What happens to the velocity profile when a Newtonian fluid flows through a pipe? Adhesive forces between fluid and surface fluid stationary at surface Parabolic velocity profile Cohesive forces between molecules layers of fluid slide past each other generating frictional forces energy dissipated (like rubbing hands together) Linear velocity profile

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13 A useful model: Poiseuille’s Law: laminar flow of a Newtonian fluid through a pipe volume flow rate RR L p1p1 p2p2 p = p 1 - p 2 parabolic velocity profile What causes a fluid to flow through a pipe? assumptions ? Q = dV/dt

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14 A useful model: Poiseuille’s Law Q = dV/dt RR L pp Q = dV = p R 4 8 L dtdt p 1 > p 2 pressure drop along pipe energy dissipated (thermal) by friction between streamlines moving past each other

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15 Irrigation pipes Pipes from Warragamba Dam Respiratory system Circulatory system Air conditioning, ducting, piping Soils Water will rise quicker in large grain soils (Q R 4 ) but it will rise to greater height by capillary attraction on fine grain soils (h 1/R) APPLICATIONS

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16 What happens to the flow as viscosity changes ? what happens to the flow as the radius changes ? The heart is so responsive to the changing needs of our body that cardiac output can vary from as little as 5 to a maximum of 35 litres of blood per minute, a sevenfold change, over a very short interval. Q = dV = p R 4 8 L dtdt

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17 FLUID FLOW STREAMLINE – LAMINAR FLOW TURBULENT FLOW REYNOLDS NUMBER How do we apply conservation of energy in a flow system?

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18 Velocity of particle -tangent to streamline streamlines Streamlines for fluid passing an obstacle v Velocity profile for the laminar flow of a non viscous liquid LAMINAR FLOW

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19 REYNOLDS NUMBER R e A British scientist Osborne Reynolds (1842 – 1912) established that the nature of the flow depends upon a dimensionless quantity, which is now called the Reynolds number R e. R e = v L / density of fluid v average flow velocity over the cross section of the pipe L characteristic dimension

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20 R e = v L / [R e ] [kg.m -3 ] [m.s -1 ][m] [Pa.s] -1 [kg] [m -1 ][s -1 ][kg.m.s -2.m -2.s] -1 = [1] R e is a dimensionless number As a rule of thumb, for a fluid flowing through a tube R e < ~ 2000 laminar flow ~ 2000 < R e < ~ 3000 unstable laminar to turbulent flow R e > ~ 2000 turbulent flow

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21 R e = v L / Sydney Harbour Ferry

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22 R e = v L / = 10 3 kg.m -3 v = 5 m.s -1 L = 10 m = 10 -3 Pa.s R e = (10 3 )(5)(10) / (10 -3 ) R e = 5x10 7

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23 R e = v L / Spermatozoa swimming

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24 R e = v L / Spermatozoa swimming = 10 3 kg.m -3 v = 10 -5 m.s -1 L = 10 m = 10 -3 Pa.s R e = (10 3 )(10 -5 )(10x10 -6 ) / (10 -3 ) R e = 10 -4

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25 Household plumbing pipes Typical pipes are about 30 mm in diameter and water flows at about 10 m.s -1 R e ~ (10)(30 10 -3 )(10 3 ) / (10 -3 ) ~ 3 10 5 The circulatory system Speed of blood ~ 0.2 m.s -1 Diameter of aorta L ~ 10 mm Viscosity of blood say ~ 10 -3 Pa.s R e ~ (0.2)(10 10 -3 )(10 3 ) / 10 -3 ) ~ 2 10 3 Impact Method of swimming/propulsion Pump design Flow systems … Re = v L /

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