Due:
In class, Friday
November 9, 2007
Name
For instructor or TA use only:
Problem
Score
Points
1
15
2
15
3
20
4
20
5
15
6
15
ME 521
Fall Semester, 2007
Homework Set # 9
Professor J. M. Cimbala
Total:
100
1
. (15 pts)
Text problem 3.8
.
Note
: Draw a sketch of a 3D fluid element of dimensions (
dr
,
rd
θ
,
dz
), then analyze the three
components individually, one face at a time. Check your answers with those given in Appendix B1.
2
.
(15 pts)
Consider the incompressible NavierStokes equation in planar (2D) polar coordinates, as given in Appendix B2.
Notice that the viscous term,
, contains some “extra” terms, involving
2
u
ν
∇
G
2
r
u
r
−
and
2
2
r
u
r
∂
∂
. Have you ever wondered
where these extra terms come from? Now you will find out.
(
a
)
Show that unit vectors
and
e
r
e
JJG
are functions of
, and that in fact
/
r
ee
∂ ∂=
JJ
GJ
G
J
and
/
r
∂∂
=
−
G
J
.
(
b
)
Expand the viscous term carefully, using
rr
uu
e u
e
=+
G
J
G
, and show that the equations in Appendix B2 are indeed
correct, including the “extra” terms.
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 Fall '07
 CIMBALA
 Fluid Dynamics, potential flow, stream function, Professor J. M. Cimbala, 2D potential flow

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