Problem 5.2
[2]
Given:
Velocity fields
Find:
Which are 3D incompressible
Solution:
Basic equation:
x
u
∂
∂
y
v
∂
∂
+
z
w
∂
∂
+
0
=
Assumption:
Incompressible flow
a)
uxy
,
z
,
t
,
()
y
2
2x
⋅
z
⋅
+
=
vxy
,
z
,
t
,
2
−
y
⋅
z
⋅
x
2
y
⋅
z
⋅
+
=
wxy
,
z
,
t
,
1
2
x
2
⋅
z
2
⋅
x
3
y
4
⋅
+
=
x
,
z
,
t
,
∂
∂
2z
⋅
→
y
,
z
,
t
,
∂
∂
x
2
z
⋅
⋅
−
→
z
,
z
,
t
,
∂
∂
x
2
z
⋅
→
Hence
x
u
∂
∂
y
v
∂
∂
+
z
w
∂
∂
+
0
=
INCOMPRESSIBLE
b)
,
z
,
t
,
x
y
⋅
z
⋅
t
⋅
=
,
z
,
t
,
x
−
y
⋅
z
⋅
t
2
⋅
=
,
z
,
t
,
z
2
2
xt
2
⋅
yt
⋅
−
⋅
=
x
,
z
,
t
,
∂
∂
ty
⋅
z
⋅
→
y
,
z
,
t
,
∂
∂
t
2
x
⋅
z
⋅
−
→
z
,
z
,
t
,
∂
∂
zt
2
x
⋅
⋅
−
⋅
→
Hence
x
u
∂
∂
y
v
∂
∂
+
z
w
∂
∂
+
0
=
INCOMPRESSIBLE
c)
,
z
This is the end of the preview. Sign up
to
access the rest of the document.
This note was uploaded on 08/25/2011 for the course AME 331 taught by Professor Zohar during the Fall '08 term at Arizona.
 Fall '08
 ZOHAR

Click to edit the document details