ChBE421
Problem Set #5
Fall 2009
1.
Evaluate the stress tensor
σ
ij
for each of the following flows (
α
, R: constants).
a.
p
= 0,
u
= 0.6
x
,
v
= 0.3
y
, w = z
b.
p
= 0,
u
= 0,
v
= 
α
x
2
, w = 0
c.
p
= R
x
,
𝑢𝑢
=
𝑅𝑅
2
𝜇𝜇
(
−
2
ℎ𝑦𝑦
+
𝑦𝑦
2
)
,
v
= 0, w = 0
•
Sketch the velocity profile for (b) and (c).
•
For part (c), find the surface stress
f
as a function of
y
, for a surface with normal vector
n
= (0, 1,
0).
2. Answer briefly:
a.
Consider the boundary layer on a flat plate in uniform flow. Shear stress:
τ
= μ ∂u/∂y. If
we double the viscosity, by how much will the shear stress change?
b.
For laminar flow in a pipe, we stated that pressure was uniform across parallel
streamlines, and found that the highest velocity was in the center of the pipe. On the other
hand, the Bernoulli equation tells us that high velocity goes with low pressure, so the
pressure is lower in the center of the pipe. How do you explain this disagreement?
c.
The Bernoulli equation was derived under the assumption of inviscid, irrotational flow;
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 Spring '10
 1`
 Fluid Dynamics, Force, Bernoulli equation

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