Lecture 5
Solve Laplace’s equation for velocity above and below the boundary condition.
Assume Fourier series expansion
() (
)
)
,0
,,
c
o
s c
o
s
,,,
, c
o
o
s
ii
m
m
im
m
xyt
A
t
x
m
y
xyzt
F
zt
x
my
η
φ
∞
=
∞
=
=
=
∑
∑
A
A
A
A
A
A
Conservation of mass is guaranteed by LaPlace’s equation.
Plugging
into LaPlace’s equation on each side of interface:
()
2
22
2
0
mm
Fm
F
z
∂
−+
=
∂
AA
A
has solution (
upper layer)
1,
i
=
)
11
,
m
kz
m
Fz
tCt
e
−
=
A
A
in the upper fluid layer,
, the positive root.
22 2
0, with
, using
0
zk
m
k
>=
+
A
>
Which satisfies the condition at
.
z
→+∞
In the lower fluid layer,
( )
2,
,
kz
iF
z
t
C
t
e
+
==
since
0
z
<
and this satisfies the condition
as
.
z
→−∞
Now drop the summation
∑
for all components of the Fourier series, and work with a single
particular mode.
Whatever is found for one mode is valid for all modes.
m
A
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View Full DocumentThe Kinematic B.C. gives
12
1,2
c
o
sc
o
s
c
o
o
s
c
o
o
s
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 Fall '08
 Staff
 Fourier Series, Laplace, Cos

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