Sect. 14.6: Bernoulli’s Equation
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Transcript of Sect. 14.6: Bernoulli’s Equation
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