Sect 14 6 Bernoullis Equation Bernoullis Principle qualitative

Sect. 14. 6: Bernoulli’s Equation

• 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. – NOT a 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 = (P 1 – P 2) V.

• Net work done on the segment: W = (P 1 – P 2) V. Part of this goes into changing kinetic energy & part to changing the gravitational potential energy. • Change in kinetic energy: ΔK = (½)mv 22 – (½)mv 12 – 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 = mgy 2 – mgy 1. Work also equals change in energy. Combining: (P 1 – P 2)V =½ mv 22 - ½ mv 12 + mgy 2 – mgy 1

Bernolli’s Equation • Rearranging and expressing in terms of density: P 1 + ½ rv 12 + rgy 1 = P 2 + ½ rv 22 + rgy 2 • This is Bernoulli’s Equation. Often expressed as P + ½ rv 2 + rgy = constant • When fluid is at rest, this is P 1 – P 2 = 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 P 1 + (½)ρ(v 1)2 + ρgy 1 = P 2 + (½)ρ(v 2)2 + ρgy 2 (1) Fluid flowing out of spigot at bottom. Point 1 spigot Point 2 top of fluid v 2 0 (v 2 << v 1) P 2 P 1 (1) becomes: (½)ρ(v 1)2 + ρgy 1 = ρgy 2 Or, speed coming out of spigot: v 1 = [2 g(y 2 - y 1)]½ “Torricelli’s Theorem”

Flow on the level P 1 + (½)ρ(v 1)2 + ρgy 1 = P 2 + (½)ρ(v 2)2 + ρgy 2 (1) • Flow on the level y 1 = y 2 (1) becomes: P 1 + (½)ρ(v 1)2 = P 2 + (½)ρ(v 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 P 1 + (½)ρ(v 1)2 = P 2 + (½)ρ(v 2)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!) P 1 + (½)ρ(v 1)2 = P 2 + (½)ρ(v 2)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 P 1 + (½)ρ(v 1)2 = P 2 + (½)ρ(v 2)2 (2) “Where v is high, P is low, where v is low, P is high. ” PTOP < PBOT LIFT! A 1 Area of wing top, A 2 Area of wing bottom FTOP = PTOP A 1 FBOT = PBOT A 2 Plane will fly if ∑F = FBOT - FTOP - Mg > 0 !

Sailboat sailing against the wind! P 1 + (½)ρ(v 1)2 = P 2 + (½)ρ(v 2)2 (2) “Where v is high, P is low, where v is low, P is high. ”

“Venturi” tubes P 1 + (½)ρ(v 1)2 = P 2 + (½)ρ(v 2)2 (2) “Where v is high, P is low, where v is low, P is high. ” Auto carburetor

Application #2 e) “Venturi” tubes P 1 + (½)ρ(v 1)2 = P 2 + (½)ρ(v 2)2 (2) “Where v is high, P is low, where v is low, P is high. ” Venturi meter: A 1 v 1 = A 2 v 2 With (2) this P 2 < P 1 (Continuity)

Ventilation in “Prairie Dog Town” & in chimneys etc. P 1 + (½)ρ(v 1)2 = P 2 + (½)ρ(v 2)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 P 1 + (½)ρ(v 1)2 = P 2 + (½)ρ(v 2)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: y 1 = 0 v 1 = 0. 6 m/s, P 1 = 3. 8 atm Diameter d 1 = 5. 0 cm (r 1 = 2. 5 cm). A 1 = π(r 1)2 18 m up: y 2 = 18 m, d 2 = 2. 6 cm (r 2 = 1. 3 cm). A 2 = π(r 2)2 v 2 = ? P 2 = ? Continuity: A 1 v 1 = A 2 v 2 = (A 1 v 1)/(A 2) = 2. 22 m/s Bernoulli: P 1+ (½)ρ(v 1)2 + ρgy 1 = P 2+ (½)ρ(v 2)2 + ρgy 2 P 2 = 2. 0 atm