Problem Set 4 (2.006, Spring 08)
Problem 1
Flow past an infinite plate can be modeled using the steady twodimensional form of the continuity and
NavierStokes equations.
2
2
2
2
2
2
2
2
0
1
1
u
x
y
u
u
P
u
u
u
x
y
x
x
y
P
u
x
y
y
x
y
υ
υ
ν
ρ
υ
υ
υ
υ
υ
ν
ρ
∂
∂
+
=
∂
∂
⎛
⎞
∂
∂
∂
∂
∂
+
= −
+
+
⎜
⎟
∂
∂
∂
∂
∂
⎝
⎠
⎛
⎞
∂
∂
∂
∂
∂
+
= −
+
+
⎜
⎟
∂
∂
∂
∂
∂
⎝
⎠
From the problem statement we are told that the xvelocity,
u
, does not depend on the position x along the
plate.
Therefore,
0
u
x
∂
=
∂
Using the continuity equation and the boundary condition for the yvelocity at the plate, we conclude that the
yvelocity,
υ
, is constant and equal to the suction velocity,
o
υ
.
)
(
0
x
f
y
=
⇒
=
∂
∂
υ
υ
Apply the boundary condition:
o
υ
υ
=
at all x.
Since the yvelocity is constant the NavierStokes equation in the ydirection reduces to
0
=
∂
∂
y
P
)
(
x
f
P
=
⇒
The pressure outside the boundary layer is uniform and equal to
P
atm
.
Thus, the pressure inside the boundary
is
P = P
atm
Plugging in the pressure, yvelocity and
0
u
x
∂
=
∂
, the x N.S. equation reduces to
2
2
y
u
y
u
o
∂
∂
=
∂
∂
ν
υ
0
2
2
=
∂
∂
−
∂
∂
⇒
y
u
y
u
o
ν
υ
The general solution to the above differential equation is
2
1
exp
)
(
C
y
C
y
u
o
+
⎥
⎦
⎤
⎢
⎣
⎡
=
ν
υ
The boundary conditions: 1) u(y
→∞
) = u
o
, and 2) u(y=0) = 0
allow us to determine C
1
and C
2
.
C
2
= u
o
and
C
1
=  u
o
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
⎥
⎦
⎤
⎢
⎣
⎡
−
=
⇒
y
u
y
u
o
o
ν
υ
exp
1
)
(
b) The displacement thickness is defined as
∫
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
−
=
δ
δ
0
*
)
(
1
dy
u
y
u
o
⎟
⎠
⎞
⎜
⎝
⎛
−
−
=
⎟
⎠
⎞
⎜
⎝
⎛
=
∫
ν
δ
υ
υ
ν
ν
υ
δ
δ
o
o
o
dy
y
exp
1
exp
0
*
The momentum thickness is defined as
∫
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
−
=
δ
θ
0
)
(
1
)
(
dy
u
y
u
u
y
u
o
o
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⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
⎟
⎠
⎞
⎜
⎝
⎛
−
−
−
−
=
⎥
⎦
⎤
⎢
⎣
⎡
⎟
⎠
⎞
⎜
⎝
⎛
−
⎟
⎠
⎞
⎜
⎝
⎛
=
∫
ν
δ
υ
ν
δ
υ
υ
ν
ν
υ
ν
υ
θ
δ
o
o
o
o
o
dy
y
y
2
exp
1
2
1
exp
1
exp
1
exp
0
We now have to find
δ
.
At
y =
δ
, u = u
o
. Therefore,
δ→∞
, and we get
o
υ
ν
δ
−
=
*
and
o
υ
ν
θ
2
−
=
It is also acceptable to use the following definition:
y =
δ
when
u =0.99 u
o
.
This gives us
99
.
0
exp
1
=
⎟
⎠
⎞
⎜
⎝
⎛
−
ν
δ
υ
o
, and therefore,
o
υ
ν
δ
99
.
0
*
−
=
and
o
υ
ν
θ
49
.
0
−
≈
c) The drag force on the plate is
=
−
=
⋅
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
∂
∂
=
=
=
bL
u
bL
y
u
bL
D
o
o
y
w
ν
υ
μ
μ
τ
0
bL
u
o
o
υ
ρ
−
(
τ
w
is constant over x and b is the width of the plate into the page)
You can also use the momentum equation to derive the drag force – it will give you the same answer.
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 Spring '08
 Blah
 Fluid Dynamics, Aerodynamics, Shear Stress, Drag force

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