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A wide flat belt moves vertically upward at constant speed,
U
, through a large bath of viscous liquid as
shown in the figure.
The belt carries with it a layer of liquid of constant thickness,
h
.
The motion is steady
and fully-developed after a small distance above the liquid surface level.
The external pressure is
atmospheric (constant) everywhere.
a.
Simplify the governing equations to a form applicable for this particular problem.
b.
State the appropriate boundary conditions
c.
Determine the velocity profile in the liquid.
d.
Determine the volumetric flow rate per unit depth.
SOLUTION:
Make the following assumptions.
1.
steady flow
⇒
(
)
0
t
∂
=
∂
"
2.
planar flow
⇒
(
)
0
z
u
z
∂
=
=
∂
"
3.
fully-developed flow in the
y
-direction
⇒
0
y
x
u
u
y
y
∂
∂
=
=
∂
∂
4.
gravity acts only in the –
y
-direction
⇒
0,
x
z
y
g
g
g
g
=
=
= −
Consider the continuity equation.
(
)
N
0 #3
0
y
x
u
u
x
y
=
∂
∂
+
=
∂
∂
⇒
0
x
u
x
∂
=
∂
⇒
constant
x
u
=
(Note that
u
x
does not vary with either
y
or
z
either.)
Since there is no flow through the belt,
0
x
u
=
(Call this condition #5.)
(1)
Consider the Navier-Stokes equation in the
x
-direction.
(
)
N
(
)
(
)
N
(
)
N
(
)
N
(
)
N
2
2
2
2
0 #4
0 #1,#5
0 #5
0 #5
0 #3,#5
0 #3,#5
x
x
x
x
x
x
y
x
u
u
u
u
u
p
u
u
g
t
x
y
x
x
y
ρ
µ
ρ
=
=
=
=
=
=
⎛
⎞
⎡
⎤
⎜
⎟
⎢
⎥
∂
∂
∂
∂
∂
∂
⎜
⎟
⎢
⎥
+
+
= −
+
+
+
⎜
⎟
∂
∂
∂
∂
⎢
⎥
∂
∂
⎜
⎟
⎢
⎥
⎣
⎦
⎝
⎠
±²³
0
p
x
∂
∴
=
∂
(2)
liquid
atmosphere
belt
U
gravity
h
x
y

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