u 0,t
cos
A=U
Thus, vel. distribution
where
u
y,t
co
k
2
s
=
ky
amplitude
phase
i
t
u y t
e
U
t
Ue
t
ky
ω
−
⇒
=
ω
→
←
ω
ν
=
ω −
±²²³²²´

(
)
velocity of fluid decreases exponentially as the distance from the plate
y
increases.
rate of decrease
k=
, will be faster for higher frequency and smaller viscosity.
2
fluid oscillates in time
•
ω
ν
•
(
)
(
)
-ky
with the same frequency as the input freq. in boundary.
amplitude of oscillation Ue
max. amp. at y=0
U
phase shift cos
t-ky
g
y
•
=
→
•
ω
specific time
instantaneous time
U
y

Can define thickness of oscillating shear layer,
δ
as again where velocity amplitude
drops to 1% of U.
∴
u/U=0.01.
4.6
4.6
0.01
=
k
6.5
2
ky
k
y
u
e
e
e
U
k
δ
δ
δ
δ
δ
−
−
−
=
=
=
=
=
⇒
ω
ν
=
⇒
≅
ν
ν
ω
∼
remember characteristic of laminar flows
(
)
w
0
0
:
For air at 20
, with a plate frequency of 1 Hz
=2
rd/sec
10
The wall shear stress at the oscillating plate
sin
4
lags the max. vel.
y
y
Ex
C
mm
u
U
t
y
shear stress
π
δ
π
τ
τ
µ
µ
=
=
ω
≈
⎛
⎞
∂
⎛
⎞
=
=
=
ρω
ω −
⎜
⎟
⎜
⎟
∂
⎝
⎠
⎝
⎠
D
by 135
2
4
π
π
⎛
⎞
+
⎜
⎟
⎝
⎠
D
Since governing eq. is linear, the method of superposition is applicable. Hence,
superposing several oscill. of diff. freq. & ampl. the sol. for arbitrary periodic motion
of plate can be
obtained.

UNIFORM FLOW OVER A POROUS WALL
(
)
non-linear inertia terms are not zero
V.
0
but linearized to permit a closed form solution
V
•
∇
≠
•
JG
JG
Example:
steady, fully-dev. flow over a plate with suction
y
fully dev.
u=u(y)
p=const.
δ
• suction on surface is sometimes used to prevent boundary layers from separating.

I. solution with primitive variables, u, v, p
u
x
∂
∂
2
2
0
v=const.
:
u
v
y
u
u
u
x
v
x
y
x
∂
+
=
⇒
∂
∂
∂
∂
+
= ν
∂
∂
∂
N
(
)
(
)
2
2
0
2
2
2
2
2
1
v=
V uniform
:
0
0
0
=0 &
V
y
u
p
y
x
p
y
y
du
d u
V
dy
dy
d u
V du
V
V
dy
d
A
y
u
y
Be
α
α
α
α
⎛
⎞
−
⎜
⎟
ν
⎝
⎠
⎛
⎞
∂
∂
+
−
⎜
⎟
⎜
⎟
∂
ρ ∂
⎝
⎠
−
∂
=
∂
−
= ν
+
=
+
=
⇒
= −
ν
+
ν
ν
=
(
)
(
)
. .
u y=0
0
u y
B C s
U
=
→ ∞ =

(
)
V
note if blowing instead of suction v=V
u y
y
u
y
A
Be
ν
⇒
=
+
→ ∞ →
•
→ ∞
not physically possible
u would be unbounded
at large y
( )
(
)
(
)
0
0
A+B=0
u y
B.L. thicknes
1
s
V
y
u
U
A
V
u
y
U
e
δ
−
ν
⎡
⎤
=
=
→
ν
⎛
⎞
→ ∞ =
=
−
⎢
⎥
⎣
⎜
⎟
⎝
⎠
⎦
∼
II.
solution using Vorticity transport eq
.
Vorticity Transport in 2-D
(
)
2
2
2
2
2
.
z
z
z
z
z
V
t
u
v
t
x
y
x
y
ω
ω
υ
ω
ω
ω
ω
ω
ω
υ
∂
+
∇
=
∇
∂
⎛
⎞
∂
∂
∂
∂
∂
+
+
=
+
⎜
⎟
∂
∂
∂
∂
∂
⎝
⎠
JG
JG
JG
JG

(
)
(
)
2
z
2
2
2
of
diffusion
vorticity toward
away fr
plate
0
&
0
fully-dev. flow
& v=
V=const.
from continuity
z
z
z
z
z
z
convection
viscous
steady
t
x
v
y
y
y
d
y
dy
d
d
V
dy
dy
ω
ω
υ
ω
ω
υ
∂ω
∂
→
=
⇒
∂
∂
∂
∂
=
⇒
ω = ω
∂
∂
∂
→
−
∂
−
=
±²³²´
om plate
2
2
1
z
1
Integrate once,
0
at
y
. .
0
0
at
y
z
z
z
z
z
d
d
V
dy
dy
d
V
c
dy
B C
c
d
dy
ω
ω
ω
ω
ω
−
=
ν
−
=
+
ν
⎫
ω =
→ ∞
⎪
⇒
=
⎬
=
→ ∞
⎪
⎭
±³´

z
2
2
z
z
2
yields,
0
at
y
,
0=c
c ??

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