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ME 363 - Fluid Mechanics
Final Exam
May 15, 2008
Spring Semester 2008
Problem 1a (5 points)
A 6-mm diameter hole is punched near the bottom of a 32-oz drinking
cup full of cold water (ρ = 1000 kg/m3, μ = 0.0018 kg/m-s). Estimate the
velocity of the stream that issues through this hole.
Problem 1b (5 points)
A 6-mm-inside-diameter straw, 10 cm long, is held up
to the hole so that the water flows through it. Assume
the average velocity V in the straw is equal to your
answer to problem 1. Do you expect the flow through
this straw to be laminar or turbulent? (no matter what
you find, assume it is laminar for all calculations
below: this straw is short enough to prevent the flow
from completely developing turbulence)
Problem 1c (10 points)
The setup is exactly as in problem 1b. Use the Moody chart or similar information to
estimate the pressure gradient dP/dx [Pa/m] in the straw. You may ignore minor losses
and flow kinetic energy (the  V terms).
Problem 1d (10 points)
In fluid mechanics
textbooks, one can
read about extra
losses existing at
pipe entrances. For
example, Figure 6.6
from White, Fluid
Mechanics (6th
edition) highlights
the “entrance
pressure drop”:
Your answer to 1c corresponds to the slope of the curve at the label “1C” above. Below
we will estimate the (steeper) pressure gradient at the point labeled “1E” above. To do
this, we will consider only the first 1 cm of the straw:
Here, boundary layers grow from the straw’s interior surface inward. If we imagine
slicing the whole straw lengthwise and unrolling it to form a flat plate (18.8 mm by 10
cm), we can apply our understanding of flat plate boundary layers to approximately
understand this boundary layer region. Using flat plate theory, how thick is the boundary
layer 1 cm from the entrance to the straw [mm]?
Problem 1e (20 points)
Assume your answer to 1d is thin enough that (1) the flat plate analogy makes sense and
(2) the inviscid core flow does not accelerate significantly in the first 1 cm. By considering the
friction (drag or shear stress) in the first 1 cm, estimate the pressure gradient [Pa/m] in
the first 1 cm of the straw (your answer corresponds to the slope at label “1E” above).
Problem 2a (20 points)
A large horizontal culvert with severe scale build-up on its interior can be considered as a
2-m diameter pipe with a relative roughness of 0.05. Initially, this culvert is full of water
(ρ = 1000 kg/m3, μ = 0.001 kg/m-s) at rest. Then, as a valve is opened upstream, the
water velocity V increases linearly in time, reaching 20 m/s after 6 seconds. Ignoring
viscous friction, what is the horizontal reaction force required to hold a 1-m length of the
culvert stationary during this valve-opening event?
Problem 2b (20 points)
Now we will reconsider the inviscid assumption from problem 2a. Including viscosity, at
what average velocity V [m/s] will the total reaction force be double your answer in 2a?
Problem 3 (10 points)
A 1.24-mm-diameter rod, 4.4 mm long, is coated in glue. This rod is to be inserted
completely into a cylindrical hole with a diameter of 1.26 mm. The maximum force F
that can be used to insert the rod is 1 Newton. During insertion, the velocity of the rod
must always exceed 0.2 mm/s. What is the maximum allowable viscosity of the glue
note untis to answer above should be Pa/m