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R. Field 10/30/2012University of Florida
PHY 2053
Page 1
Ideal Fluids in Motion
2
(^22) 1 2 2 1
(^21) 1 2 1
gy
v
P
gy
v
P
ρ
ρ
ρ
ρ^
(^
1 2
(^21) (^22)
1 2
2 1
y
y
Mg
v
v
M
M
P
P^
ρ
-^
Bernoulli’s Equation:
-^
The Equation of Continuity:
Steady Flow, Incompressible Flow, Non viscous Flow, Irrotational Flow. If a fluid is incompressible, its density
ρρρρ^
is constant throughout.
Thus the volume of fluid entering a tube at one end per unit of timemust be equal to the volume of fluid leaving the other end per unittime. In the time
t we have A
v 1
t = A
v 2
t.
-^
Ideal Fluid:
A^1
v^1
= A
v 2
(continuity equation) 2
RV
= Av = volume flow rate = constant
Rm
ρρρρ
RV
ρρρρ Av
mass flow rate = constant
Application of W =
KE +
U:
∆∆∆∆ V = v
A 1 ∆∆∆∆ t 1
∆∆∆∆ V = v
A 2 ∆∆∆∆ t 2
F^1
= P
A 1
M^ ρ 1 P P V P P x A P x
AP
W
(^
2 1 2 1 2 2 2 1 1 1 − = ∆ − = ∆ − ∆
=^
F^2
= P
A 2 2
M
M ∆∆∆∆ x
= v 2
∆∆∆∆ 2 t
∆∆ x ∆∆
= v 1
∆∆∆∆ 1 t
R. Field 10/30/2012University of Florida
PHY 2053
Page 2
Bernoulli’s Equation:
Applications
2
(^22) 1 2 2 1
(^21) 1 2 1
gy
v
P
gy
v
P
ρ
ρ
ρ
ρ^
gh
A ghA
v^
A A^
2
)) / ( (^1) (
2
1 2 (^21) 2
2
→
− =^
<<
-^
Bernoulli’s Equation:
-^
Constant Height (y
= y 1
If the speed of a fluid element increases as the element travelsalong a horizontal streamline, the pressure of the fluid mustdecrease, and conversely.
P + ½
ρρρρ v
ρρρρ gy = constant (conservation of energy for a fluid)
v^1
P^1 Air Foil
v^1
>> v
2 v^2
P^2
P^1
<< P
2
v^1
= v
A 2
/A 2
1
gh
v
v^
(^21) 2 2
P^1
= P
= P 2
atm
(^22) 1 2 2 (^21) 1 2 1
v
P
v
P
ρ
ρ^
P + ½
ρρρρ v
2 = constant
-^
Example (velocity of efflux):
y-axis
y = 0
h
v^2
ρρρρ
v^1
Area A
1
Area A
2
We can use Bernoulli’s equation to calculate the speed ofefflux, v
, from a horizontal orifice (and area A 2
) located a 2
depth h below the water level of a large talk (with area A
2
(^22) 1 2 2 1
(^21) 1 2 1
gy
v
P
gy
v
P
ρ^
=
(Torricelli’s Law)
)
(^
(^22) (^21) 1 2 1 2
v v
P P P^
−
= − = ∆
R. Field 10/30/2012University of Florida
PHY 2053
Page 4
Bernoulli’s Equation:
Application
)
( 2
h^2 d g
v^
=
-^
Siphon:
The figure shows a siphon, which is a device forremoving liquid from a container. Tube ABC mustinitially be filled, but once this is done, liquid willflow until the liquid surface of the container islevel with the tube opening A. With what speeddoes the liquid emerge from the tube at C? Whatis the greatest possible height h
that a siphon can 1
lift water?
gd
v
P
V
P^
A
atm
ρ^
−
=
+^
2 1 2
2 1 2
A va V
va
VA
/
=^ gd
v
P
gd
a A
v
P
P^
A A a
A
atm
ρ^
−
→
−
^
−
=^
<<
2 1 2
2 2
2 1 2
1
y=
V
v
v
A = Area of containera = area of tube
)
(^
2
2 1 2
2 1 2
h d g
v
P
gd
v
P^
atm
A^
)
(^
2
2 1 2
h d g
v^
=
ρ^
)
(^
2
2 1 2
1
2 1 2
h d g
v
P
gh
v
P^
atm
B^
)
(
) (^
2
max 1
h d
g P
h^
atm
−
=
ρ
)
( )
(^
2
1
h d
g P
P
h^
B
atm
−
−
=
