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)mv22 – (½)mv1

2 – 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 v12 + gy1 = P2 + ½ v2

2 + 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 gases

As 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 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 = 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