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Chapter 3. Elementary Fluid
Dynamics-The Bernoulli
Equation
©2014 Ming-Tsung Sun Ph. D.
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Learning Objectives
• Discuss the application of Newton’s second
law to fluid flows.
• Explain the development, uses, and limitations
of the Bernoulli equation.
• Use the Bernoulli equation to solve simple
flow problems.
• Apply the concepts of static, stagnation,
dynamic, and total pressures.
• Calculate various flow properties using the
energy and hydraulic grade lines.
©2014 Ming-Tsung Sun Ph. D.
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Newton’s Second Law
F = ma
Consider Inviscid Fluids only:
(Net pressure force on particle) + (net gravity force on particle) =
(particle mass)× (particle acceleration )
V2
ds
dV ∂V ds
∂V
=
=V
, an =
V = , as =
dt
dt
∂s dt
∂s
ℜ
V
dV = Vdθ n
dθ
V+dV
dθ = ds ℜ
©2014 Ming-Tsung Sun Ph. D.
dV V ds
=
dt ℜ dt
V2
3
=
ℜ
an =
F = ma Along a Streamline
∑ δF
s
= δmas = δmV
∂V
∂V
= ρδVV
∂s
∂s
δWs = −δW sin θ = −γδV sin θ
δFps = ( p − δps )δnδy − ( p + δps )δnδy
= −2δpsδnδy = −
∑ δF
s
∂p
∂p
δsδnδy = − δV
∂s
∂s
∂p 

=δWs + δFps =  − γ sin θ − δV
∂s 

©2014 Ming-Tsung Sun Ph. D.
δps ≈
− γ sin θ −
∂p ds
∂s 2
∂p
∂V
= ρV
∂s
∂s
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Ex. 3.1 Pressure Variation Along a
Streamline
 a3 
V = V0 1 + 3 
 x 
θ =0
⇒
 a 3  3V a 3 
∂p
∂V
= − ρV
= ρV0 1 + 3  04 
∂x
∂x
 x  x 
 a 3 1  a  6 
p = − ρV02   +   
 x  2  x  
©2014 Ming-Tsung Sun Ph. D.
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Fluids in the News
Incorrect Raindrop Shape
• Teardrop shaped raindrop can be found everywhere.
• This happens only they run down a windowpane.
• D < 0.5 mm: sphere with small pressure ρV02/2.
• 0.5 mm < D < 2 mm: sphere with a flattened bottom.
• D = 2 mm: shaped like hamburger buns.
• D > 4 mm: shaped like inverted bag with an annular ring
around its base.
• The ring finally breaks up into smaller drops.
©2014 Ming-Tsung Sun Ph. D.
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Pressure Variation Along a
Streamline
sin θ =
− γ sin θ −
dz
ds
V
( )
dV 1 d V 2
=
ds 2 ds
( )
∂p
∂V
dz dp 1 d V 2
= ρV
⇒ −γ
−
= ρ
∂s
∂s
ds ds 2
ds
( )
dp +
1
ρd V 2 + γdz = 0 (along a streamline)
2
p+
1
ρV 2 + γ z = constant along a streamline
2
The celebrated Bernoulli equation
©2014 Ming-Tsung Sun Ph. D.
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Ex. 3.2 The Bernoulli Equation
• Determine the difference between p1 & p2
1
1
ρV12 + γ z1 = p2 + ρV22 + γ z 2
2
2
z1 = z 2 , V1 = V0 , V2 = 0
p1 +
⇒ p2 − p1 =
©2014 Ming-Tsung Sun Ph. D.
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1
ρV12 = ρV02
2
2
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2015/11/11
F = ma Normal to a Streamline
∑ δFn =
δmV 2
ℜ
=
ρδVV 2
ℜ
∂p 

= δWn + δFpn =  − γ cos θ − δV
∂n 

dz
cos θ =
dn
dz ∂p ρV 2
=
−γ
−
dn ∂n
ℜ
p + ρ∫
V2
dn + γ z = constant across the streamline
ℜ
©2014 Ming-Tsung Sun Ph. D.
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Ex. 3.3 Pressure Variation Normal to a
Streamline
−γ
dz ∂p ρV 2
−
=
dn ∂n
ℜ
dz
= 0, r = − n
dn
∂p ρV 2
⇒
=
∂r
r
∂p
2
= ρ (V0 r0 ) r
∂r
2

ρV02  r 
  − 1
p − p0 =
2  r0 


∂p ρ (V0 r0 )
=
∂r
r3
2
p − p0 =
ρV02 
r 
1 −  0 
2   r 
2



©2014 Ming-Tsung Sun Ph. D.
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Physical Interpretation of Bernoulli’s
Equation
p+
×1 γ
1
ρV 2 + γ z = constant along a streamline
2
V2
⇒ +
+z = constant along a streamline
γ 2g
p
Elevation head
Velocity head
Pressure head
©2014 Ming-Tsung Sun Ph. D.
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Ex. 3.4 Kinetic, Potential, and Pressure
Energy
• Discuss the energy of the fluid
at points (1), (2), & (3) using
the Bernoulli equation.
p+
1
ρV 2 + γ z = constant along a streamline
2
Energy Type
Point
Kinetic
Potential
Pressue
ρV2/2
γz
p
(1)
(2)
(3)
©2014 Ming-Tsung Sun Ph. D.
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2015/11/11
Fluids in the News
Armed with a Water Jet for Hunting
• A archerfish can shoot down insects like a water pistol.
• The high-speed water jet ejected from its snout has a
pressure head of 2 to 3 m, meaning the water jet can
reach the height but hit its prey only within 1 m range.
• Within 0.1 second, the fish has extracted all the
information needed to predict the point where the prey
will hit the water.
• http://youtu.be/fhBZ40jIo4Q
©2014 Ming-Tsung Sun Ph. D.
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Ex. 3.5 Pressure Variation in a Flowing
Stream
• Water flows in a curved, undulating waterslide.
• Describe the pressure variation between (a) points (1) and (2);
(b) points (3) and (4).
(a) Section A to B : p + γ z = constant
(b) Section C to D :
©2014 Ming-Tsung Sun Ph. D.
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Static, Stagnation, Dynamic, and Total
Pressure
• In the Bernoulli equation:
p+
1
ρV 2 + γ z = constant along a streamline
2
– p, (thermodynamic pressure) static pressure (measurement at
static relative to the fow)
– γz, hydrostatic pressure (pressure change due to elevation
change)
– ρV2/2, dynamic pressure
– p + ρV2/2, stagnation pressure
p2 = p1 +
1
ρV12
2
– Point (2): stagnation point
©2014 Ming-Tsung Sun Ph. D.
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Fluids in the News
Pressurized Eyes
• Our eyes maintain a 10 to 20 mmHg pressure to function
properly through balancing the in and out fluid.
• Optic nerve at the exit of eye damages under high
pressure – loss of sight at the visual field termed glaucoma.
• Noninvasive type of eye-pressure measurement uses a
calibrated “puff” of air blown against the eye.
• The stagnation pressure causes the eyeball to deform
whose magnitude is correlated with the eye pressure.
©2014 Ming-Tsung Sun Ph. D.
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Pitot-static Tube
• Total pressure, pT
p+
1
ρV 2 + γ z = pT = constant along a streamline
2
• Static pressure
©2014 Ming-Tsung Sun Ph. D.
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Ex. 3.6 Pitot-Static Tube
• An airplane flies 300 km/h at 3000 m.
• Determine (a) p1, (2) ∆p measured with the Pitot-static tube.
©2014 Ming-Tsung Sun Ph. D.
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Fluids in the News
Bugged and Plugged Pitot Tubes
• Many accidents have been caused by inaccurate Pitot
tube readings resulting from blocked holes.
• Reasons are the protective cover not being removed or
nest within the tube built by bugs.
• The most serious accident was the Boeing 757 took off
from Puerto Plata in the Dominican Republic.
• The incorrect airspeed data caused the autopilot to
increase AOA and engine power, and confused the crew.
• It stalled and plunged into the Caribbean Sea killing all.
©2014 Ming-Tsung Sun Ph. D.
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Example of Using Bernoulli’s Equation
Free Jets
1
1
p1 + ρV12 + γ z1 = p2 + ρV22 + γ z 2
2
2
• Along the streamline from point (1) to point (2)
• The jet speed is
• Further down to (5)
Well-contoured nozzle
©2014 Ming-Tsung Sun Ph. D.
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Example of Using Bernoulli’s Equation
Free Jets
• Vena Contracta
• Contraction Coefficient
©2014 Ming-Tsung Sun Ph. D.
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Example of Using Bernoulli’s Equation
Confined Flows
• Mass flowrate m& (kg/s)
• Volume flowrate Q (m3/s)
• Continuity equation
• Incompressible fluid
©2014 Ming-Tsung Sun Ph. D.
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Ex. 3.7 Flow from a Tank – Gravity Driven
• Determine the flowrate, Q.
©2014 Ming-Tsung Sun Ph. D.
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Ex. 3.8 Flow from a Tank – Pressure Driven
• Determine the flowrate Q and hose pressure p2.
p1 +
1
1
1
ρV12 + γ z1 = p2 + ρV22 + γ z2 = p3 + ρV32 + γ z3
2
2
2
©2014 Ming-Tsung Sun Ph. D.
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Fluids in the News
Hi-tech Inhaler
• Inhaler helps asthma or bronchitis treatment.
• More kinds of illness than respiratory ailments can be
treated with hi-tech inhaler, such as diabetes.
• The concept is to make micro-scaled spray droplets to reach
alveoli and enter blood stream.
• A laser-machined nozzle sprays the medicine solution
through an array of very fine holes.
• The patient breathes through the device to create a
pressure difference that activates an electrically actuated
piston, which drives the liquid from its reservoir through the
nozzle array and into the respiratory system
©2014 Ming-Tsung Sun Ph. D.
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Ex. 3.9 Flow in a Variable Area Pipe
• Determine the manometer reading, h.
p1 +
1
1
ρV12 + γ z1 = p2 + ρV22 + γ z2
2
2
©2014 Ming-Tsung Sun Ph. D.
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Cavitation
•
•
• In a Venturi channel:
•
•
©2014 Ming-Tsung Sun Ph. D.
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Ex. 3.10 Siphon and Cavitation
• Siphoned: z3 < z1; z2 moderate.
• Determine Hmax without cavitation.
©2014 Ming-Tsung Sun Ph. D.
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Flowrate Measurement
Q − Qactual
≈ 0.40
Qactual
Orifice
meter
• Assumptions: horizontal and
steady flow, inviscid and
incompressible fluid.
Q − Qactual
≈ 0.10
Qactual
Nozzle
meter
Q − Qactual
≈ 0.02
Qactual
Venturi
meter
©2014 Ming-Tsung Sun Ph. D.
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Ex. 3.11 Venturi Meter
• Determine the range in ∆p to
measure 0.005 < Q < 0.050 m3/s
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Sluice Gate Flow Meter
• Apply Bernoulli’s and continuity equations:
©2014 Ming-Tsung Sun Ph. D.
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Ex. 3.12 Sluice Gate
• Determine Q/b
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Energy Line & Hydraulic Grade Line
• Hydraulic grade line (HGL) & Energy line (EL)
p
γ
+
V2
+ z = constant along a streamline = H
2g
©2014 Ming-Tsung Sun Ph. D.
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Energy Line & Hydraulic Grade Line
• Hydraulic grade line (HGL) & Energy line (EL)
p
γ
+
V2
+ z = constant along a streamline = H
2g
©2014 Ming-Tsung Sun Ph. D.
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Ex. 3.13 EL and HGL
• Will the water leak at the small hole at (1)
©2014 Ming-Tsung Sun Ph. D.
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Restrictions on the Use of the BE
• Only suitable for incompressible fluid.
– Ma < 0.3 (~102 m/s or 367 km/hr)
• Steady flow.
• Inviscid flow.
• No mechanical devices (pumps or turbines) in
the system.
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