Transcript Part V

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 = (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
With (2) this  P2 < P1
(Continuity)
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