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Chapter

Chapter 4 Flowing Fluids And Pressure Variation Bernoulli

Department
Civil Engineering
Course Code
CE 2200
Professor
All

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CE 2200 Fluid Mechanics
Willson February 18, 2015 1
Eulers Equation
( )
zp
dl
d
al
γρ
+=
assumes only gravity and pressure forces
Gives the pressure variation due to weight and
acceleration
Valid for
inviscid
along a streamline
C
r
zp =+
2
22
ρω
γ
For rotational flows
Bernoulli Equation
0=++ udugdh
dp
ρ
Eulers equation
0
1=++ udugdhdp
ρ
constant
2
2
=++ u
gh
p
ρ
constant
2
2
=++
g
u
h
p
γ
The Bernoulli Equation is a
statement of the conservation
of ____________________
constant
2
2
=++ v
gh
p
ρ
p.e. k.e.

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CE 2200 Fluid Mechanics
Willson February 18, 2015 2
Bernoulli Equation
Assumptions needed for Bernoulli Equation
Eliminate the constant in the Bernoulli equation?
_______________________________________
Bernoulli equation does not include
___________________________
___________________________
Bernoullis Equation
(for irrotational flow in two dimensions)
y
u
v
x
u
uax
+
=
By definition h = z + p/ϒ
Eulers equation for the x-
coordinate system and
incompressible fluid
x
h
g
ax
=
+
=
y
u
v
x
u
u
gx
h1
y-direction
+
=
y
v
v
x
v
u
gy
h1
x-direction

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CE 2200 Fluid Mechanics
Willson February 18, 2015 3
Bernoullis Equation
(What is irrotational flow?)
u, v, w - velocities in the x, y, and z directions
x
v
y
u
=
x
w
z
u
=
x
v
y
w
=
u
v
x
y
y
y
u
uΔ
+
x
x
v
vΔ
+
Irrotational flow exists only when the average
rates of rotation are zero (Section 4.6)
Bernoullis Equation
(for irrotational flow in two dimensions)
and
+
=
y
u
v
x
u
u
gx
h1
x
v
y
u
=
For irrotational flow
+
=
x
v
v
x
u
u
gx
h1
=
2
2
u
xx
u
u
=
2
2
v
xx
v
v
+
=
22
122 v
x
u
xgx
h