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Padnos School of Engineering
Grand Valley State University
EGR – 365
Fluid Mechanics
Dr. Blekhman
Laboratory Report
Viscosity
Matt Reimink
Grading Rubric
Theory
T r W r W
s s
s
3
side
h
μR ω2πLL
T
b
4
bottom
2h
μR ωπL
T
t
2Ih h
μRπL4Lh Rh
b s
3
s s b sb
b s
3
1 e
μRπL4Lh Rh
2rWh h
ω t
2 πLπLL ω
r Wh
μ
3
s s
Present Major Results
Since the values of the torques are small to begin with, it was reasonable to neglect the
value of the bottom torque.
The distance needed to reach 90% of the terminal was determined to be 0.0195m or
19.5mm.
Mass
(kg)
Distance
(m)
Time
(s)
Velocity
(m/s)
Angular Velocity
(rad/s)
Viscosity (Ns/*
m
2
Average Viscosity 0.
Average Corrected
Viscosity 0.
Error Discussion
Measurement Value Tolerance
Weights, W 0.15 ± 0.0kg
Timing, t 14.68 ± 0.19s
Liquid Length, L 0.8863 ± 0.0005m
Thickness on Side hs 0.00437 ± 0.0005m
Spool Radius, rs 0.03 ± 0.0005m
Cylinder Radius, R 0.05771 ± 0.0005m
Falling Mass Length, d 0.9144 ± 0.02m
Mass of Cylinder 1.125 ± 0.00kg
Uncertainty
Viscosity 0.
Conclusions
1.) Glycerin is a newtonian fluid.
2.) The viscosity of glycerin is 0.8754 Ns/m
2
3.) The distance required before timing is 19.5mm.
4.) There was 0.2233% uncertainty in our viscosity value.
and the torque on the bottom is:
b
4
bottom
2h
μR ωπL
T
Derivations of both equations can be seen in Appendix A. Dimensions for the test apparatus are
contained in Table 1.
Table 1 : Apparatus Dimensions
Lengths (m)
Liquid Height, L 0.
Large Cylinder ID 0.
Small Cylinder OD 0.
Small Cylinder Radius, R 0.
Side Fluid, hs 0.
Bottom Fluid, hb 0.
Spool Radius, rs 0.
Mass (kg)
Cylinder Assembly 1.
Theoretically, neglecting air resistance, the torques cancel each other out, and the following force
diagram can be setup.
Figure 2 : FBD of Cylinder
Figure 2 showed that there needs to be a bottom torque and a side torque acting against the
torque created by the falling weight. Laying out the force balance equations yields:
ΣAppliedApplied Torques rW T T 0
s side bottom
Inserting Equations (1) and (2) into Equation (3) allowed us to solve for viscosity. Equation (4)
is the formula for viscosity.
2 πLπLL ω
r Wh
μ
3
s s
If a ratio of the two torques is set up, it can be easily noticed that the bottom torque can be
neglected. The derivation of this assumption is shown in Appendix A. Setting up our definition
of applied torques yields:
dt
dω
ΣAppliedApplied Torques rW T T I
s side bottom
Inserting Equations (1) and (2) into Equation (5) yields the first order differential equation:
t
2Ih h
μRπL4Lh Rh
b s
3
s s b sb
b s
3
1 e
μR πL4Lh Rh
2rWh h
ω t
Equation (6) yields the angular velocity of the cylinder in terms of time. However, we will be
measuring the velocity of the falling weight. Equation (7) links angular and linear velocity
together.
v t r ω
The acceleration of the falling weight was not to be included; therefore, the weight had to fall far
enough to reach its terminal velocity. Theoretically, it need only reach roughly 90% of its
terminal velocity in order for acceleration not to be an issue. The time and length it takes to
reach the set amount are shown below:
3
s
terminal
2 LR
ln(0.1)h I
TIME 0.90ω
2
3
2
s
2
s
90%
2 πLμL R
ln(0.1)r Wh I
LENGTH L
Our assumption is to let the weight fall 20 inches in order to eliminate the acceleration of the
mass.
Discussion of Results
apparatus. With such a small weight, the data could easily have been skewered by the friction in
the pulley or the friction in the bearings of the cylinder or the estimating in our moment of inertia
equation. The rest of the data indicates that glycerin has a viscosity of roughly 0.8754 Ns/m
2
and
is a newtonian fluid. This was determined because the data in Table 2 shows that viscosity does
not vary with velocity, therefore making glycerin a newtonian fluid.
Looking up published values of the viscosity of glycerin, it appears that our value is relatively
close. The viscosity chart in the back of our Fluid Mechanics book lists glycerin at room
temperature at roughly 0.83 Ns/m
2
Error Discussion
In every engineering application, error persists and this laboratory exercise is no different. For
every measurement taken, a certain level of error was present and all of that error needs to be
accounted for. The error that will be present in our measurements of viscosity is error
propagation. The way to deal with error propagation is to determine the uncertainty in every
variable used when solving for another variable. If f was a function of x
1
and x
2
, f (x1, x2) the
uncertainty of f would be:
2
x
2
2
x
2
2
u
x
f
u
x
f
f
% uncertaintyin f
Using Equation (11) it will be possible to determine the propagated error in all our
measurements. Table 3 tabulates all of the measurements taken along with a reasonable
assessment of the corresponding error and then all of the propagated errors. Derivations for the
error equation can be seen in Appendix A.
Table 3 : Estimate Error Values
Measurement Value Tolerance
Weights, W 0.15 ± 0.0kg
Timing, t 14.68 ± 0.19s
Liquid Length, L 0.8863 ± 0.0005m
Thickness on Side hs 0.00437 ± 0.0005m
Spool Radius, rs 0.03 ± 0.0005m
Cylinder Radius, R 0.05771 ± 0.0005m
Falling Mass Length, d 0.9144 ± 0.02m
Uncertainty
Viscosity 0.2233%
There was a 0.2233% of uncertainty in our viscosity value. This is reasonable because our errors
for measured values were relatively small compared to the actual measurement, and our viscosity
is close to the publish value.
Conclusions
1.) Glycerin is a newtonian fluid.
2.) The viscosity of glycerin is 0.8754 Ns/m
2
3.) The distance required before timing is 19.5mm.
4.) There was 0.2233% uncertainty in our viscosity value.