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## Sect. 14.6: Bernoulli’s Equation

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**Bernoulli’s Principle (qualitative):**“Where the fluid velocity is high, the pressure is low, and where the velocity is low, the pressure is high.” • Higher pressure slows fluid down. Lower pressure speeds it up! • Bernoulli’s Equation (quantitative). • We will now derive it. • NOTa new law. Simply conservation of KE + PE (or the Work-Energy Principle) rewritten in fluid language!**Daniel Bernoulli**• 1700 – 1782 • Swiss physicist • Published Hydrodynamica • Dealt with equilibrium, pressure and speeds in fluids • Also a beginning of the study of gasses with changing pressure and temperature**Bernolli’s Equation**• As a fluid moves through a region where its speed and/or elevation above Earth’s surface changes, the pressure in fluid varies with these changes. Relations between fluid speed, pressure and elevation was derived by Bernoulli. • Consider the two shaded segments • Volumes of both segments are equal. Using definition work & pressure in terms of force & area gives: Net work done on the segment: W = (P1 – P2) V.**Net work done on the segment: W = (P1 – P2) V.**Part of this goes into changing kinetic energy & part to changing the gravitational potential energy. • Change in kinetic energy: ΔK = (½)mv22 – (½)mv12 • No change in kinetic energy of the unshaded portion since we assume streamline flow. The masses are the same since volumes are the same • Change in gravitational potential energy: ΔU = mgy2 – mgy1. Work also equals change in energy. Combining: (P1 – P2)V =½ mv22 - ½ mv12 + mgy2 – mgy1**Bernolli’s Equation**• Rearranging and expressing in terms of density: P1 + ½ rv12 + rgy1 = P2 + ½ rv22 + rgy2 • This is Bernoulli’s Equation. Often expressed as P + ½ rv2 + rgy = constant • When fluid is at rest, this is P1 – P2 = rgh consistent with pressure variation with depth found earlier for static fluids. • This general behavior of pressure with speed is true even for gases As the speed increases, the pressure decreases**Applications of Fluid Dynamics**• Streamline flow around a moving airplane wing • Lift is the upward force on the wing from the air • Drag is the resistance • The lift depends on the speed of the airplane, the area of the wing, its curvature, and the angle between the wing and the horizontal**In general, an object moving through a fluid experiences**lift as a result of any effect that causes the fluid to change its direction as it flows past the object • Some factors that influence lift are: • The shape of the object • The object’s orientation with respect to the fluid flow • Any spinning of the object • The texture of the object’s surface**Golf Ball**• The ball is given a rapid backspin • The dimples increase friction • Increases lift • It travels farther than if it was not spinning**Atomizer**• A stream of air passes over one end of an open tube • The other end is immersed in a liquid • The moving air reduces the pressure above the tube • The fluid rises into the air stream • The liquid is dispersed into a fine spray of droplets**Water Storage Tank**P1 + (½)ρ(v1)2 + ρgy1 = P2 + (½)ρ(v2)2 + ρgy2 (1) Fluid flowing out of spigot at bottom. Point 1 spigot Point 2 top of fluid v2 0 (v2 << v1) P2 P1 (1) becomes: (½)ρ(v1)2 + ρgy1 = ρgy2 Or, speed coming out of spigot: v1 = [2g(y2 - y1)]½“Torricelli’s Theorem”**Flow on the level**P1 + (½)ρ(v1)2 + ρgy1 = P2 + (½)ρ(v2)2 + ρgy2 (1) • Flow on the level y1 = y2 (1) becomes: P1 + (½)ρ(v1)2= P2 + (½)ρ(v2)2 (2) (2) Explains many fluid phenomena & is a quantitative statement of Bernoulli’s Principle: “Where the fluid velocity is high, the pressure is low, and where the velocity is low, the pressure is high.”**Application #2 a)Perfume Atomizer**P1 + (½)ρ(v1)2= P2 + (½)ρ(v2)2 (2) “Where v is high, P is low, where v is low, P is high.” • High speed air (v) Low pressure (P) Perfume is “sucked” up!**Application #2 b)Ball on a jet of air(Demonstration!)**P1 + (½)ρ(v1)2= P2 + (½)ρ(v2)2(2) “Where v is high, P is low, where v is low, P is high.” • High pressure (P) outside air jet Low speed (v 0). Low pressure (P) inside air jet High speed (v)**Application #2 c)Lift on airplane wing**P1 + (½)ρ(v1)2= P2 + (½)ρ(v2)2(2) “Where v is high, P is low, where v is low, P is high.” PTOP < PBOT LIFT! A1 Area of wing top, A2 Area of wing bottom FTOP = PTOP A1 FBOT = PBOT A2 Plane will fly if ∑F = FBOT - FTOP - Mg > 0 !**Sailboat sailing against the wind!**P1 + (½)ρ(v1)2= P2 + (½)ρ(v2)2 (2) “Where v is high, P is low, where v is low, P is high.”**“Venturi” tubes**P1 + (½)ρ(v1)2= P2 + (½)ρ(v2)2 (2) “Where v is high, P is low, where v is low, P is high.” Auto carburetor**Application #2 e)“Venturi” tubes**P1 + (½)ρ(v1)2= P2 + (½)ρ(v2)2 (2) “Where v is high, P is low, where v is low, P is high.” Venturi meter: A1v1 = A2v2(Continuity) With (2) this P2 < P1**Ventilation in “Prairie Dog Town” & in chimneys etc.**P1 + (½)ρ(v1)2= P2 + (½)ρ(v2)2 (2) “Where v is high, P is low, where v is low, P is high.” Air is forced to circulate!**Blood flow in the body**P1 + (½)ρ(v1)2= P2 + (½)ρ(v2)2(2) “Where v is high, P is low, where v is low, P is high.” Blood flow is from right to left instead of up (to the brain)**Example: Pumping water up**Street level:y1 = 0 v1 = 0.6 m/s, P1 = 3.8 atm Diameter d1 = 5.0 cm (r1 = 2.5 cm). A1 = π(r1)2 18 m up: y2 = 18 m,d2 = 2.6 cm (r2 = 1.3 cm). A2 = π(r2)2 v2 = ? P2 = ? Continuity:A1v1 = A2v2 v2 = (A1v1)/(A2) = 2.22 m/s Bernoulli: P1+ (½)ρ(v1)2 + ρgy1= P2+ (½)ρ(v2)2 + ρgy2 P2 = 2.0 atm