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Hydro Mechanics of
Offshore Structures
Hydro Mechanics of
Offshore Structures
Lecture (4)
MM771 M.A. Kotb
11/26/2014 1
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Hydro Mechanics of
Offshore Structures MM771 M.A. Kotb
11/26/2014 2
In 1755, Leonard Euler applied Newtons second law (force = mass
x acceleration) to non-viscous and incompressible fluids.
Euler Equations
Leonhard Euler
(1707-83)
Isaac Newton
(1642-1 727)
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Hydro Mechanics of
Offshore Structures
2
x
x
pp
MM771 M.A. Kotb
11/26/2014 3
C v
w
u
x
z
y
CV
y
z
x
Dt
DudmFx
rgxyz
2
x
x
pp
Fluid Element
in the x-direction for a fluid mass element dm
-
x
y
2
y
y
pp
2
y
y
pp
2
x
x
pp
2
x
x
pp
p
zy zy
zx
zx
C (xc , yc , zc)
Consider the flow in the x-y plane
Hydro Mechanics of
Offshore Structures
MM771 M.A. Kotb
11/26/2014 4
Dt
DudmFx
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Hydro Mechanics of
Offshore Structures
z
uw
y
uv
x
uu
t
uzyxzyx
x
pypzyp
z
u
t
z
y
u
t
y
x
u
t
x
t
uzyxzy
x
x
ppzy
x
x
pp
r
r22
MM771 M.A. Kotb
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2
x
x
pp
2
x
x
pp
zy zyDt
DudmFx
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Hydro Mechanics of
Offshore Structures
z
uw
y
uv
x
uu
t
u
x
p
z
uw
y
uv
x
uu
t
u
x
p
z
uw
y
uv
x
uu
t
uzyxzyx
x
p
r
r
r
1
MM771 M.A. Kotb
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z
uw
y
uv
x
uu
t
u
x
p
r
1
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Hydro Mechanics of
Offshore Structures
MM771 M.A. Kotb
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z
vw
y
vv
x
vu
t
v
y
p
r
1
direction-Similarly in y
and in z-direction
z
ww
y
wv
x
wu
t
wg
z
p
r
1
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Hydro Mechanics of
Offshore Structures
MM771 M.A. Kotb
11/26/2014 8
Thus, the Euler equations for a non-viscous and incompressible
flow are:
in x-direction
z
uw
y
uv
x
uu
t
u
x
p
r
1
z
vw
y
vv
x
vu
t
v
y
p
r
1in y-direction
z
ww
y
wv
x
wu
t
wg
z
p
r
1in z-direction
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Hydro Mechanics of
Offshore Structures MM771 M.A. Kotb
11/26/2014 9
Bernoulli Equation
Swiss mathematician, son of Johann Bernoulli, who showed that as
the velocity of a fluid increases, the pressure decreases, a
statement known as the Bernoulli principle. He won the annual prize
of the French Academy ten times for work on vibrating strings,
ocean tides, and the kinetic theory of gases. For one of these
victories, he was ejected from his jealous father's house, as his
father had also submitted an entry for the prize. Daniel
Bernoulli
(1700-1782)
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Hydro Mechanics of
Offshore Structures
MM771 M.A. Kotb
11/26/2014 10
fSubstitute in terms of in x-direction
zxzyxyxxxxtx
p
xzzxyyxxxxtx
p
z
uw
y
uv
x
uu
t
u
x
p
fffffff
r
fffffff
r
r
1
1
1
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Hydro Mechanics of
Offshore Structures
MM771 M.A. Kotb
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tzyConstzyxt
p
zyxt
p
x
zyxtx
p
x
zxyxxxtx
p
x
,,2
1
2
1
2
1
2
1
2
1
2
10
2
1
2
1
2
1
2
1
2
1
2
1
222
222
222
222
ffff
r
ffff
r
ffff
r
ffff
r
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Hydro Mechanics of
Offshore Structures
MM771 M.A. Kotb
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Similarly in y-direction
tzConstzyxt
p
zyxt
p
y
,2
1
2
1
2
1
2
1
2
1
2
10
222
222
ffff
r
ffff
r
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Hydro Mechanics of
Offshore Structures
MM771 M.A. Kotb
11/26/2014 13
Similarly in z-direction
tConstgzzyxt
p
zyxt
p
zg
222
222
2
1
2
1
2
1
2
1
2
1
2
1
ffff
r
ffff
r
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Hydro Mechanics of
Offshore Structures
MM771 M.A. Kotb
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tConstV
gzt
p
tConstVgzt
p
tConstwvugzt
p
tConstzyx
gzt
p
2
2
1
2
1
2
1
2
2
222
222
f
r
f
r
f
r
ffff
r
Hence
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Hydro Mechanics of
Offshore Structures
Stationary or Steady Flow
tConstV
gzt
p
2
2f
r
In steady flow ,the derivative of the potential, df/dt, becomes
zero and the function of time, C (t), becomes a constant, C; all
time functions are now absent. In this case, the Bernoulli equation
becomes:
Recall
ConstV
gzp
2
2
r
MM771 M.A. Kotb
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Hydro Mechanics of
Offshore Structures
Constg
Vz
g
p
2
2
r
Often, the Bernoulli equation for a stationary flow is written
in other forms:
ConstVgzp 2
2
1rr
Or
MM771 M.A. Kotb
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Hydro Mechanics of
Offshore Structures MM771 M.A. Kotb
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Isaac Newton (1 642-1 727)
English mathematician, physicist
and astronomer; studied at the
University of Cambridge. His three
big discoveries of the spectral
analysis of light, universal
gravitation and differential and
integral calculus are only too well
known. There are so many
scientific terms named after
Newton (Newton's rings and
Newton's law of
otion/viscosity/resistance) that he
can be regarded as the greatest
contributor to the establishment of
modern natural science.
Isaac Newton (1 642-1 727)
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Hydro Mechanics of
Offshore Structures MM771 M.A. Kotb
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Leonhard Euler (1707-83)
Mathematician born near Basle in
Switzerland. A pupil of Johann Bernoulli
and a close friend of Daniel Bernoulli.
Contributed enormously to the
mathematical development of Newtonian
mechanics, while formulating the equations
of motion of a perfect fluid and solid. Lost
his sight in one eye and then both eyes, as
a result of a disease, but still continued
his research.
Leonhard Euler (1707-83)
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Hydro Mechanics of
Offshore Structures MM771 M.A. Kotb
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Daniel Bernoulli (1700-82)
Mathematician born in Groningen in the
Netherlands. A good friend of Euler. Made
efforts to popularise the law of fluid motion,
while tackling various novel problems in
fluid statics and dynamics. Originated the
Latin word hydrodynamics, meaning fluid
dynamics.
Daniel Bernoulli (1700-82)
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Hydro Mechanics of
Offshore Structures MM771 M.A. Kotb
11/26/2014 20
Thank you .!!!!!
