Review Session Problem 1) A Newtonian fluid flows
upwards in a pipe, then meets a solid cylinder concentric
with the pipe rotating with angular velocity,
Ω
. The surfaces
of the conical shell and cone are described by the equations
on the diagram.
a)
We are interested in the flow around the junction of
the pipe and cylinder, over 0 < z < L1+L2. (L1 and L2
are comparable to R1 and R2). Postulate the form of
the velocity field v
. That is, what components of the
velocity are non‐zero, and what variables does each
component depend on?
b)
Write the boundary conditions within the region 0 < z
< L1.
c)
Write the boundary conditions within the region L1 <
z < L2
d)
Simplify the equations below by crossing out any
terms you expect to be zero.
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View Full DocumentReview Session Problem 1 Solution:
a) So we’ll assume:
Newtonian, incompressible, steady state, axisymmetric (no
θ
dependence).
Postulate:
v
θ
=
v
(
r
,
z
)
v
r
=
v
r
(
r
,
z
)
v
z
=
v
z
(
r
,
z
)
P
=
P
(
r
,
z
)
b
)
v
z
=
finite
at
r
=
0
for
0
<
z
<
L
1
v
r
=
0
at
r
=
0
for
0
<
z
<
L
1
v
=
0
at
r
=
0
for
0
<
z
<
L
1
v
r
=
0
at
r
=
R
for
0
<
z
<
L
1
v
z
=
0
at
r
=
R
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 Spring '10
 Muller
 Human mitochondrial DNA haplogroup, Human mtDNA haplogroups, Vθ, Review Session Problem

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