CBE 320
September 3, 2010
Introductory Transport Phenomena
Problem Session I
Part B: Review of Mathematical Topics
Review Appendices A and C in
Transport Phenomena
before doing this assignment.
1.
Sketch the following functions; be sure to label the axes carefully.
a.
y
(
x
) =
cosh
bx
cosh
b
,
−∞
< x <
∞
,
b
= constant
b.
y
(
x
) =
e
bx
,
−∞
< x <
∞
,
b
= constant
c.
y
(
x
) =
tanh
x
x
,
−∞
< x <
∞
d.
v
z
(
r
) =
v
0
1
−
r
R
2
,
0
≤
r
≤
R,
v
0
, R
= constant
e.
v
z
(
r
) =
v
0
ln(
r/R
)
ln
κ
,
κR
≤
r
≤
R,
v
0
, R, κ
= constant
,
0
< κ <
1
2.
A particle is located at
x
= 4
,
y
= 2
,
z
= 1
in Cartesian (rectangular) coordinates.
a.
What are the cylindrical coordinates of the particle
(
r, θ, z
)
?
b.
What are the spherical coordinates of the particle
(
r, θ, φ
)
?
3.
A particle, located at
x
= 2
,
y
= 4
,
z
= 0
, is moving in a Northeasterly direction with speed 8 cm/sec (“North” = direction
of the
+
y
-axis; “East” = direction of the
+
x
-axis). By means of a carefully labelled sketch, show the vector
v
(assume
one unit in distance is numerically equal to one cm/sec). Then show the Cartesian components