Due:
In class, Friday
October 5, 2007
Name
For instructor or TA use only:
Problem
Score
Points
1
15
2
15
3
50
4
20
ME 521
Fall Semester, 2007
Homework Set # 5
Professor J. M. Cimbala
Total:
100
1
.
(15 pts)
Text problem 3.1
. For consistency, do three parts:
(
a
)
Show that
x
2
–
y
2
= constant along streamlines two ways: (i) Integrate Eq. (3.37). (ii) Integrate Eq. (3.36) to find an
equation for
ψ
as a function of
x
and
y
, then show that
x
2
–
y
2
= constant along lines of constant
ψ
.
(
b
)
Prove that this velocity field is indeed irrotational.
(
c
)
Sketch a few streamlines.
2
.
(15 pts)
In class, we used a surface
integral and Gauss’s law for a material
volume to show that the total rate of work
per unit volume done by surface stresses on a
fluid is equal to
(
1
1,1
1
u
u dx
+
3
3,1
1
u
u
dx
+
)
ji
i
j
u
x
τ
∂
∂
. Derive the same
expression by considering the work done on
each face of an infinitesimal rectangular
fluid element of dimensions
dx
1
by
dx
2
by
dx
3
. For consistency, use the sketch provided
here as a guide, where the velocities and
stresses on the two faces normal to the
x
1
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 Fall '07
 CIMBALA
 Fluid Dynamics, stress tensor, principal axes, Couette flow problem

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