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Applying Bernoulli's Equa On 23/09/14 Lecture 5 Applying Bernoulli’s Equaon net work done on fluid element = ΔKE + ΔPE Example: The Venturi meter Bernoulli’s equation: 2 2 p1 + ρ g y1 + ½ ρ v1 = p1 + ρ g y1 + ½ ρ v1 The Venturi meter is used to measure flow speed in a pipe. Derive an expression for v1 Solving Bernoulli’s equation: • Identify points 1 and 2 along a streamline • Define your coordinate system: where y=0 • List your known and unknown variables • Solve for your unknowns, possibly using the continuity equation. Note: We can’t apply Bernoulli’s equation between e.g. point 1 and the liquid in either of Faster fluid speed leading to low pressure is the the vertical tubes. key to many common applications. Bernoulli’s equation only applies to points on the same streamline. • The chimney effect: partly hot air rises; wind across top of chimney ⇒ lower pressure at top ⇒ smoke forced up chimney 1 23/09/14 Rabbits use this effect to avoid suffocation in • Cyclones: strong wind generates a large negative their burrows. Different air flow across two pressure over the roof of a closed-up house holes produces a pressure difference, which forces a flow of air through the burrow. House destroyed by Cyclone Yasi in Queensland 2011 • Passing trucks: the high speed of air between two Aside: Bernoulli’s equation and gases trucks lowers the pressure between them, leading to a tendency for them to pull together. Gas flow can be considered incompressible so long as its velocity remains low (small compared with the speed of sound). Low pressure As long as v < 0.005vs, the change in volume of the gas ΔV < 0.005V and can be ignored. • Lift on an aeroplane: flow lines crowd Another way of looking at it: there is a net together above the wing downward change in momentum in air flowing ⇒ increased speed past ⇒ reduced pressure ⇒ reaction force is upward ⇒ lift 2 23/09/14 • Spin: a spinning ball creates a pressure The siphon difference on either side, resulting in a force C on the ball. The spinning ball slows the air on one side and speeds up the air on the other, so h3 there is a net force B h2 A h1 D v Quesons • p = 0 at point C Can you have negative • This analysis doesn’t deal with the most pressure? interesting aspect: that water is flowing uphill! If we have a cylinder full of • What does p = 0 at point C mean? liquid and we apply • Can you use a siphon in a vacuum? pressure to the top, we e.g. a mercury siphon on the Moon? increase the pressure in the liquid. • If so, why is the maximum height of a siphon What happens if we invert determined by p0? the cylinder? Negative pressure responsible for water transport in trees. Liquid under tension ⇒ negative pressure At top of 100m tree, c.f. tensile strength of solids. Due to cohesive require pressure forces in liquid. –10 bar. Tensile strength of water is somewhere in excess of 1000 atmospheres. 3 23/09/14 • So why is the maximum height of a siphon Next leCture determined by p ? 0 • A liquid under tension is in a metastable state, and susceptible to cavitation: the formation of bubbles which “break” the column of liquid. • So the answer seems to be that a siphon Real fluids without air pressure is possible in theory but – viscosity and turbulence difficult in practice. Further reading: See Physics Today ar:Cle, Negave Pressures and Cavitaon in Liquid Helium, hp://www.aip.org/pt/feb00/maris.htm 4 .
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