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

• 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.

Bernolli’s Equation

• 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)mv2

2 – (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 mv1

2 + mgy2 – mgy1

• Rearranging and expressing in terms of density:P1 + ½ v1

2 + gy1 = P2 + ½ v22 + gy2

• This is Bernoulli’s Equation. Often expressed as P + ½ v2 + gy = constant

• When fluid is at rest, this is P1 – P2 = gh consistent with pressure variation with depth found earlier for static fluids.

• This general behavior of pressure with speed is true even for gasesAs the speed increases, the pressure decreases

Bernolli’s Equation

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 ofspigot: 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 = 0v1 = 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