Laboratory Module #1
Theory and Measurement Moving Through Fluids
Background & Experimental Procedure
Fall 2022
Objective
This lab module will investigate the dierence in motion at two regimes (1) the
regime
Eulerian
in which inertial forces dominate motion (this is the one you are familiar with from
everyday life) and (2) the
Stokes regime characterized by a dominance of viscous forces where
inertia has a negligible impact on motion.
The Stokes regime will seem odd and counter-
intuitive at rst glance but this is the regime in which most microorganisms, such as bacteria,
live their life. During this lab module, microorganisms will not be utilized. Instead, an understanding of these two dierent regimes will be developed through the study of the motion of
various types of beads through dierent uids. Specically, the motion of steel beads in water
(Eulerian regime) will be compared with the motion of aluminum beads in glycerol (Stokes
regime).
1
Introduction Understanding Motion
Remark The movement of microscopic objects in uids, such as the swimming of bacteria and
the diusion of particles, will be investigated in later laboratories and detailed discussions on the
uid dynamic concepts behind motion within a uid will occur during lectures toward the end of
the semester. For now, a basic foundation will be developed by focusing on a series of simple
objects falling through a resistive medium.
Figure 1:
Free body diagram for a falling sphere where
buoyant force, and
Fd
Fg
is the gravitational force,
Fb
the
is the drag force.
Consider the free body diagram of a sphere falling through a uid (Figure 1). As the sphere
falls through the uid there is a downward gravitational force from gravity acting on the mass
1
of the sphere
Fg = mg
where
Fg
is the gravitational force,
m
(1)
is the mass of the sphere and
g
is the acceleration due
to gravity. Additionally, there is a net upward force acting on the sphere due to the interaction
with the uid.
buoyant force,
The two forces contributing to the net upward force on the sphere are the
Fb ,
and the drag force,
Fd .
Buoyant force is simply the force from the weight
of the uid displaced by the sphere. Drag force, on the other hand, is not as straight forward.
The drag force,
Fd ,
is dependent on the velocity,
interesting the relationship between
Fd
and
v
v,
of the sphere and to make things more
can either be linear or quadratic. The parameter
that distinguishes between the two possible relationships is the
is dened as the ratio of the inertial force,
FI ,
Re =
1.1
Reynolds number, Re, which
to the viscous force,
Fη .
FI
Fη
(2)
What is Reynolds Number?
A person's everyday experiences are typically with high Reynolds number environments where
inertial forces dominate. Excellent examples of high Reynolds environments and low Reynolds
environments can be found in a majority of households.
For example, try stirring a glass of
water with a spoon and then compare it to stirring a jar of honey. When stirring the glass of
water there is essentially no noticeable resistance to the spoon by the water. Additionally, if the
spoon is released it will continue spinning around the glass for a while due to the inertial force
caused by accelerating the water with the spoon. Honey, on the other hand, has an apparent
thickness compared to water. While stirring the jar of honey a noticeable resistance to the
spoon is felt. It is the higher viscosity of the honey compared to the lower viscosity of water
that gives the apparent thickness and resists deformation by the spoon. When the spoon is
released, it will stop moving fairly quickly due to the domination of the viscous force over the
inertial force.
The general equation for inertial forces,
FI ,
is familiar
FI = ma
where
m
is mass and
a
(3)
is acceleration, but what does the equation for viscous forces look like?
Imagine a uid between two plates (Fig. 2). Intuitively, as the viscosity of the uid increases
Figure 2:
The velocity prole of a uid between a xed and a moving plate.
it requires more force to slide the plates apart (think water versus honey). Now assume that
the bottom plate is xed while the top plate, at some distance
2
L,
is free to move parallel to the
xed plate. If a force,
F,
is applied to the top plate, it will move at some velocity
v,
forming
a velocity gradient between the top and bottom plates. As the viscosity of the uid increases
it will take a larger force to form the same velocity gradient. Therefore, viscosity,
η ,i
can be
dened by the proportionality between the force per area (also known as shear stress) and the
velocity per length (shear rate). Given the above relationship a general equation for a viscous
force,
Fη ,
can be written as
Fη = η
Combining the general equations for
FI
Fη (Eq. 4) with the
number, Re, is found to be
Re =
ρ
is the uid density,
that will be conducted,
l
l
concept of an object
ρlv
η
(5)
is the objects characteristic length (in the case of the experiments
is the diameter of the sphere),
the dynamic viscosity of the uid.
1.2
(4)
(Eq. 3) and
moving through a uid (Eq. 2), the Reynolds
where
Av
L
ii
v
is the velocity of the object, and
η
is
Equation of Motion
Refer back to the free body diagram of a sphere falling through a uid (Fig. 1). From Newton's
second law of motion, the equation of motion for the falling sphere is
Fsphere = m
where
M′
dv
= (m − m′ )g − Fd = M ′ g − Fd
dt
(6)
is the eective mass of the object corrected for buoyancy and
Fd
is the velocity-
dependent drag force.
Fd , results from the mass of the uid countering the movement of the object
and a generalized relationship is easy to derive. In a given time interval δt, an object with crosssectional area A, moving with velocity v , sweeps out a volume V , where V = Avδt. During this
time, the object collides with a uid mass mf , where mf = V ρ and ρ is the uid density. The
The drag force,
resulting force is
Fd = mf a =
mf v
Av
ρ
= mf v
= mf Av 2
= ρAv 2
δt
V
mf
(7)
In practice, the drag depends not only on the cross-sectional area but also other properties
such as the 3-dimensional shape of the object and skin friction; so, to compensate for this
becomes
where the drag coecient,
1
Fd = ρCd Av 2
2
Cd ,
(8)
is measured empirically and is dependent on
the years many studies have been done to investigate the dependency of
curve ts have been published, each optimized for a particular range of
Re ≤ 2 × 105 : 3
Cd =
i
ii
24
6
√
+
+ 0.4
Re 1 + Re
Another common symbol to denote viscosity is
ν,
Re
1
(Fig. 3 ). Over
Cd on Re. Numerous
Re. 2 For instance, for
(9)
µ.
Typically, when referring to viscosity, the dynamic (or absolute) viscosity,
on occasion the relative viscosity,
Fd
η,
is the unit of interest. However,
is referred to. The relative viscosity is simply the dynamic viscosity divided
by the density of the uid.
3
Figure 3:
sphere.
1
Dependence of drag coecient,
1.2.1 Motion at Low Reynolds
For low Reynolds number (Re << 1)
Cd ,
on Reynolds number,
Re,
for ow around a
the rst term in the drag coecient equation (Eq. 9)
dominates. Therefore, the drag force equation (Eq. 8) becomes
Fd =
Given a sphere with radius
of
Re
to
v
(Eq. 5),
Fd
r
12ρAv 2
Re
and cross-sectional area
(10)
A = πr2
combined with the relationship
simplies to
Fd = 6πηvr
This is known as the
Stokes drag.
When the sphere reaches terminal velocity (
dv
dt
(11)
= 0), the equation of motion (Eq.
6) becomes
Fd = M ′ g
In this case, from Stokes drag (Eq. 11) the terminal velocity of the sphere,
vterm =
M ′g
6πηr
(12)
vterm ,
at low
Re
is
(13)
It can be inferred that the terminal velocity of a falling sphere has a quadratic relationship to
its radius,
vterm ∝ r2 ,
at low
1.2.1.1 Reaching Low
Re
Re
(Prelim. Question #3a).
Terminal Velocity
During the low Reynolds number experiment it will be assumed that the sphere's velocity is
relatively constant throughout its fall. Is this a reasonable assumption? For an object falling at
low
Re
the equation of motion is (ignoring the buoyant force)
m
dv
= mg − 6πηrv
dt
(14)
The solution to this equation is
v(t) =
mg −t
1 − e /τ
6πηr
4
(15)
where the time constant
τ
is
τ=
m
6πηr
(16)
Qualitatively it can be seen that the solution has the correct initial and asymptotic behavior.
At
t = 0, v = 0.
At
t = ∞, v =
mg/6πηr, the expected terminal velocity. The time it takes the
system to reach terminal velocity depends on the time constant
τ.
For a 1/8 diameter aluminum
sphere falling through glycerol
τ ≈1
ms
(17)
τ , it can be concluded that the sphere reaches terminal velocity nearly instantaneously
Re which suggests the initial assumption is reasonable.
Based on
at low
1.2.1.2 Wall Aect at Low
At low
Re,
Re
recall Stoke's law is used to calculate the drag force,
Fd ,
which assumes the sphere
is moving in an unbound or innite uid (Sec. 1.2.1). However, the experiments are not carried
out in an innite uid.
The aect of the viscous forces on an object is more signicant the
closer the object is to a surface due to a
boundary layer.
A boundary layer is a thin layer
of liquid in contact with a surface that is subjected to shear forces. As a result, objects falling
near a surface (like the wall of a cylinder) will fall slower than an object farther away from the
surface. During the experiment, this can be observed by simultaneously dropping two identical
spheres at the same time one near the wall and one at the center of the cylinder (feel free to
try it during the experiment!). to account for the eect of the cylinder wall on the motion of
the falling sphere, a correction factor needs to be applied. The velocity correction for a sphere
falling in the center of a uid-lled cylinder was derived from experimental measurements
found to be
vcorr =
where
vmeas
vmeas
d
1 − 2.10 D
+ 2.09
vcorr
(18)
d 3
D
is the experimentally measured terminal velocity,
is the diameter of the cylinder, and
4 and
d is the diameter of the sphere, D
is the expected velocity of the sphere if it were falling
in an unbounded uid.
1.2.2 Motion at High Reynolds
For high Reynolds number the equation of motion for the bead falling in a uid is
m
At terminal velocity,
vterm ,
dv
1
= M ′ g − ρCd Av 2
dt
2
(19)
equation 19 simplies to
2
vterm
=
2M ′ g
ρCd A
(20)
It can then be inferred that the terminal velocity scales with the square root of the radius of a
sphere,
iii
vterm ∝ r1/2 ,
falling through a high
Re
uid (Prelim. Question #3c).
iii
For further discussion on drag forces and how it relates to velocity, refer to the section on drag forces in
chapter 1 of An Integrated, Quantitative Introduction to the Natural Sciences, Part 1: Dynamical Models
(pages 33-41).
5
1.2.2.1 Reaching High
Re
Terminal Velocity
For high Reynolds number the equation of motion (eq. 19) can be written in the general
form
where
dv
= g − av 2
dt
a=
(21)
ρCd A
2m . Through separation of variables the solution is found to be
r
v(t) =
where the time constant
τ
t
τ
(22)
2m
ρCd Ag
(23)
g
tanh
a
is
1
τ=√ =
ag
r
Again, the time it takes the system to reach terminal velocity depends on the time constant
τ.
For a 1/8 diameter steel sphere falling through water
τ ≈ 0.1sec
(24)
This time constant is 100 times greater than the time constant found at low Reynolds number
(Sec. 1.2.1.1) suggesting that the sphere
does not
reach terminal velocity instantaneously at
high Reynolds number.
1.3
A Final Thought
This lab module will focus on just one of the unique properties of motion that is inuenced
by Reynolds number. As you delve further into Reynolds number and viscosity you will learn
about the other unique aects Reynolds number has on motion. Referring back to the equation
of Reynolds number
Re =
ρlv
η
activities that may be a high Reynolds activity for one object may be a low Reynolds activity
for a dierent object. For instance, swimming can be a low Reynolds activity when the length
scale (l) of the swimmer is small.
Microorganisms t into this category.
As will be seen in
Prelim. Question #2b, bacterium such as E. coli, have a Reynolds number much less than one
The viscous forces dominate the inertial forces.
To us this is a very alien hydrodynamic
world. For a human to swim at an equivalent Reynolds number, they would need to swim in
something viscous, such as honey, at speeds of about a foot a day, while cycling their arms at
about 1 stroke per hour.
For microscopic organisms, like E.coli, inertia is totally irrelevant. So, how far would E.coli
still glide in water if it all of a sudden stopped beating its agella? The answer is about 0.1
angstrom, and it takes it about 0.6 micro seconds to slow down (essentially instantaneous!).
This means that at low Reynolds number, the past does not inuence what is being done at
that exact moment. This observation has an important implication Time has no consequence
at low Reynolds number, all that matters is conguration.
In other words, at low Reynolds
number if a human tried to swim by bending their body one way and then bending it back, even
if one of these conformational changes is executed fast and the other slow, they go nowhere. A
video has been placed on Canvas which illustrates this oddity in a beautiful demonstration with
glycerol.
Additionally, the Reynolds number is a very important number in uid dynamics as it is
used to characterize dierent ow regimes, such as laminar or turbulent ow.
6
Laminar ow
exists for small Reynolds number, where viscous forces are dominant, whereas turbulent ow
occurs at high Reynolds number, where inertial forces dominate. The transition between laminar
and turbulent ow is often gradual and happens around a critical Reynolds number. Fluid
dynamics and the role Reynolds numbers play may seem unimportant but they are essential for
many everyday objects, such as the aerodynamics of airplanes and cars.
2
Day #1 Low Reynolds Number
The velocity of the falling sphere will be measured by capturing images of the sphere's motion.
A multitude of programs exist for imaging, processing and analysis ranging from commercially
available programs (such as MetaMorph) to in-house built programs utilizing platforms such
as MATLAB or LabView.
Throughout this course, a combination of these programs will be
utilized. Specically, the following programs will be utilized (1) imaging will be performed
with Pylon Viewer which is a software package by Basler for operating Basler cameras, (2) image
analysis will be completed with Fiji (Fiji is Just ImageJ) which is a freely available image
processing program written in Java and (3) data analysis will be conducted with MATLAB
(Matrix Laboratory) which is a multi-paradigm numerical computing environment.
2.1
Measuring Low
Re Terminal Velocity
Remark There are a total of 4 aluminum bead diameters (1/16, 5/64, 3/32 and 1/8). Each
group will only measure 2 of the 4 dierent beads. All groups will compile their data in a shared
Excel document and analysis will be performed on the class data.
2.1.1 Experimental Set-Up
1. Locate the 100 mL cylinder lled with glycerol and measure the
inner diameter with the
calipers
ˆ
Why This will be used to determine if there is any wall aect.
2. Mount the camera to the ring stand with the optical post and the C-mounting clamp
ˆ
Hint An example of the camera set-up can be found on the back counter.
3. Plug in the Basler Ace camera
ˆ
CAUTION! Do not force the plug into place or else you can break the connection
on the camera.
4. Open Pylon Viewer
5. If needed, initiate and set-up the Pylon Viewer interface outlined in the user's guide on
Canvas (Pylon Viewer User's Guide Sec. 1 through Sec. 2)
ˆ
Why Each time a new user uses the Basler Ace camera on a specic computer station
for the rst time the camera must be initiated and the Pylon Viewer interface must
be set-up.
6. In Pylon Viewer, begin a live image
7. Position the cylinder in front of the camera such that approximately 75% of the length of
the cylinder is in the eld of view making sure that neither the top nor the bottom portion
of the cylinder is visible
7
ˆ
Hint During set-up, make sure to get a clear image of the ball for the entire descent.
If anything blocks the image of the bead the image analysis will be dicult.
8. Adjust the camera and settings such that a clear picture of the cylinder can be seen.
ˆ
Remember There are several ways to adjust the clarity of the live image (a) Focus can be adjusted by twisting the front of the lens on the camera, but be
careful not to twist the lens o the camera.
(b) Gain, brightness and exposure time can be adjusted in Pylon Viewer to optimize
the contrast of the image. These settings should be recorded in your lab notebook
as well as any changes made to these settings over the course of the experiment.
ˆ
Remark The exposure time used is not crucial for the data analysis of the Low Re
experiments but it should always be set to a value lower then the amount of time
between frames based on the chosen frame rate when acquiring a series of images
(step 10).
2.1.2 Imaging
9. Using the forceps, fully submerge and drop a bead (any size) to develop an intuitive sense
of how the bead descends through the uid
ˆ
Remark The instructors will hand out the beads once the lab starts to ensure that
the dierent sizes are evenly distributed.
10. Refer to the Pylon Viewer user's guide on Canvas and set up the recording options for
acquiring an image series.
Initially, record 100 frames at a rate of 1 frame per second
(Pylon Viewer User's Guide Sec. 3)
ˆ
Hint You will want to optimize these settings later.
ˆ
Remember Document all settings in your lab notebook.
11. Remove a bead from the tube containing the smallest sized aluminum bead
12. Measure the diameter of the bead with the calipers
ˆ
Why The diameter of the beads provided on the tube is the average diameter of the
beads, but each individual bead may vary making it important to measure the diameter
of each bead
13. Using the forceps, fully submerge the bead in the glycerol in the center of the cylinder
ˆ
Why The mass of the bead is not large enough to overcome the surface tension of
the glycerol. If the bead is not fully submerged it will remain on the surface of the
glycerol.
14. Release the bead and begin recording with the camera
15. Once complete, verify the images successfully recorded and saved
ˆ
Hint The recording can be ended once the bead leaves the eld of view. However, at
least 1 image without the bead in view is needed!
8
ˆ
Why Whenever acquiring a set of images, it is good practice to verify that the images
saved successfully.
It is not common, but there have been times where it is later
discovered that no images saved after hours of imaging.
16. Transfer the image series to a personal H: drive, Google drive or thumb drive
ˆ
Why Laboratory computers are not easily accessed outside of the lab or lab oce
hours.
ˆ
Remember All lenames along with the location and a description of the le should
be recorded in your lab notebook.
17. Capture and save one image of a ruler at the
falling bead (Pylon Viewer User's Guide Sec.
same distance
from the camera as the
3)
ˆ
Why This will serve as a conversion factor from pixels to units of length.
ˆ
Contemplate Why is it important that the ruler is in the same plane as the bead
with respect to the camera?
18. View the images of the falling bead and make any necessary adjustments to the set-up
and settings
ˆ
Why Taking time to make adjustments for image acquisition now can save time,
improve results and prevent frustration during the image analysis portion. Most signicant problems that arise during image analysis could have easily been xed during
the acquisition portion of the experiment. For example Image artifacts, such as air bubbles or markings on the cylinder that block the
view of the bead, can prevent the tracking program from successfully tracking the
bead.
Too much noise in the background which prohibits the isolation of the bead from
the background during the threshold process during image analysis.
Insucient number of frames acquired prohibit tracking the bead for the full descent.
ˆ
19.
Etc.
Remember Record any changes to the set-up and settings in your lab notebook.
DO NOT SKIP THIS STEP! Before moving onto step 20 for the remaining trials, analyze the images (step 21 through step 41) and make any adjustments to the experimental
set-up or image acquisition parameters.
ˆ
Why Performing this step will prevent you from having to re-run all trials.
ˆ
Remark This step can be skipped and done at a later time as long as it has successfully
been completed for 1 run.
20. When ready, obtain
at least 3 runs for all bead sizes.
Record the frame rate for each run
ˆ
Why In science, the more data the better!
ˆ
Remember Obtain the diameter of each bead with the calipers and to drop the bead
in the center on the cylinder.
9
2.1.3 Image Processing
21. Open the folder that contains the set of images to analyze
22. Select 1 image where the bead is not in the eld of view and the lighting is equivalent the
majority of the remaining images
ˆ
Why This will serve as a background image for image processing.
ˆ
Why If the lighting of the background image is not equivalent (i.e. there is a shadow
from a persons hand, etc.), image processing may be more dicult.
23. Rename the background image and move it outside the folder containing all other frames
ˆ
Hint A specic background image folder can help keep all the background images
organized.
24. Delete all the remaining images where the bead is not in the eld of view
where the bead is not
fully
and any images
in the frame.
25. Open Fiji located on the Desktop
ˆ
Hint Fiji is a free image processing program that can be downloaded onto your personal computer in order to do image processing outside of lab if needed.
26. Import the image sequence of the falling bead using the following drop-down menu command: File
→
Import
→
Image Sequence
27. Open the background image by selecting File
28. Select Process
→
→
Open
→
Background
Image Calculator
29. Set the image sequence as Image1 and the background image as Image2 and set Operation to Dierence (this is dierent than Subtract)
ˆ
Remark Taking the dierence between the image sequence and the background will
create a new window with the result; the bead will appear white against a black background.
ˆ
Hint Background noise due to changes in lighting, air bubbles, etc. may be present If the background noise is small then it will be ltered out later
If there appears to be a large amount of background noise then adjust your set-up
and capture new images
30. Scroll through the resulting image sequence to verify the bead is present in all frames
ˆ
Hint The shading of the bead will be inhomogeneous due to the lighting As long as the center of the region that represents the bead remains the same
through the entire sequence then this is not a problem
If the center of the region that represents the bead does not remain the same
through the entire sequence, make adjustments to the imaging set-up and acquire
new images
10
31. Select Image
→
Adjust
→
Threshold.
32. Select Dark Background, and adjust the threshold such that the bead looks red in front
of the black background
33. Scroll through the images to verify the center of the thresholded bead remains the same.
Adjust the threshold as needed.
ˆ
Contemplate Why is it essential that the center of the thresholded bead remains the
same for the entire sequence?
ˆ
Hint Most background noise can be ltered during this step by adjusting the threshold
such that most of bead is visible but the noise is gone It is not essential to see the entire bead as long as the center of the region representing the bead remains the same through the entire sequence
If background noise is still an issue or the center of region does not remain the
same, investigate dilation and crop functions or else make adjustments to the
imaging set-up and capture a new series of images
34. Select Apply.
35. Set Background to Dark, verify that Calculate threshold for each image is not selected, verify Black background (of binary masks) is selected then hit OK
ˆ
Remark The result is a binary image with a white bead on a black background.
36. Select Analyze
→
Set Measurements to set the properties to be measured
37. Select Area, Center of Mass, and Stack Position. Everything else should be unselected
ˆ
Why Area will be useful for setting an area lter to get rid of noise which could
not be eliminated by applying the threshold.
ˆ
Why If two or more particles are detected in a single frame, Stack Position will
provide information to determine which frame this occurred.
38. To track the bead, select Analyze
→
Analyze Particles
39. Set Show to Outlines then select Display Results and Clear Results. Hit OK and
select Yes to process all of the images
ˆ
Remark This will open A new image sequence with outlines of the bead, numbered as it was tracked.
A results window containing the number of particles tracked in the rst column
followed by the area, the center of mass coordinates and image number. If all
settings are adjusted correctly only the bead is tracked through all the frames and
no noise is tracked.
40. Verify that the number of particles tracked is correct. If the number of particles tracked
diers from what is expected (1 particle per frame), then adjust the tracking settings and
reanalyze the images (Steps 38 and 39)
11
ˆ
Hint There are 3 criteria that must be met to verify the number of particles is correct
(a) The 1
st column of the results is the total number of particles tracked and this
must equal the number of frames
th column of the results is the slice (same as the Stack Position or frame
(b) The 4
number) and each number should show up only once (i.e. only 1 particle found
per frame)
(c) The values 1
ˆ
st and 4th column of the results must match for all values!
Hint If more particles are tracked than expected, this is where the area can be used
to distinguish background noise from the bead by applying an area lter (a) Scroll through the results. Typically, the area of the bead is signicantly dierent
than the background noise if the thresholding was done correctly and the noise is
minimal.
(b) Note the approximate area of the bead across all the frames.
(c) Close the results window
(d) Begin the tracking process (Step #38)
(e) Before selecting OK, adjust Size (pixel^2) based on the approximate area of
the bead. Make the range as large as possible while still excluding noise (i.e.
if the bead is the largest particle present then leave the max set to Innity and
only adjust the minimum area, etc.). It is better to make small adjustments and
repeat the analysis process to hone in on the best area range, than to make large
adjustments and potentially eliminate legitimate data.
ˆ
Remember Final tracking settings should be recorded in your lab notebook.
41. Save the results as a text le (use the .txt extension)
ˆ
Remember Record the lename along with the le location and a description of the
le in your lab notebook.
42. Repeat steps 21 through 41 for all runs.
2.1.4 Data Analysis
43. Read the tracking results for a single bead into MATLAB with
readmatrix
44. Calculate the velocities between frames (a) Calculate the distances traveled between frames
ˆ
Hint Theoretically the bead traveled straight down the cylinder so only 1 dimension changes. However, in reality there may be small changes in both dimensions
for various reasons (i.e. camera not perfectly aligned or table is not level). So,
make sure to account for this when calculating the distance.
(b) Convert distance from pixels to centimeters
(c) Calculate the velocity (cm/s) between each frame based on the frame rate
45. Copy and input the calculated velocities into the shared excel document of class data
found on Canvas
12
46. Repeat step 43 through step 45 for all runs and bead sizes
Analyze and upload all velocities to the class data by the
end of day #1! Move on to the remaining analysis in this section and Sec. 2.2 only after velocities are uploaded. Remaining
analysis can be completed outside of lab before day #2 if needed.
47. For the class data ˆ
Remark The MATLAB code for the following analysis can still be written if the class
data has not been completed then nalized once all class data is obtained.
(a) Calculate the average bead velocity with standard deviation for each bead size
ˆ
Remark Many averages will be calculated over the course of the semester. Anytime an average is calculated a value that represents the deviation in the data
(i.e. standard deviation or standard error) must also be calculated.
(b) Use the CurveFit toolbox to t the average bead velocity as a function of bead radius
with the equation found in Prelim. Question #3a
ˆ
Contemplate Does this conrm the expected low Reynolds number model of the
2
movement of a falling sphere (vterm ∝ r )?
(c) On the same plot, plot the average bead velocity (cm/s) as a function of bead radius
(cm) using the standard deviation as error bars and plot the t found in step 47b
(d) Calculate the average Reynolds number and standard deviation for each bead size.
Refer to App. A for physical properties.
ˆ
Hint Make sure to use the average radius of each bead and to account for the
standard deviation of both the average velocities and radius by propagating the
uncertainty. Refer to App. B for commonly used uncertainty propagation formulas.
(e) Plot the average Reynolds number as a function of bead radius (cm) using the standard deviation as error bars
2.2
Accounting for the Wall Aect
Recall, the aect of the viscous forces on an object is more signicant the closer the object is
to a surface. To account for the aect of the cylinder wall on the motion of the falling bead a
correction factor needs to be applied (Eq. 18).
1. For the class data calculate the average velocity and standard deviation for all bead sizes
corrected for the wall aect
2. On a single graph, plot the average velocity of the beads as a function of radius in MATLAB
for the data that has been corrected and uncorrected using the standard deviation as error
bars.
ˆ
Contemplate Does the wall appear to have a signicant aect on the calculated
average velocities?
13
2.3
Day #1 Clean Up
1. Wipe up all glycerol.
2. Throw away any trash (i.e. dirty paper towels, paralm, etc.)
3. Unplug the camera and break down the set-up.
4. Leave extra aluminum beads and the cylinder of glycerol at the station.
3
Day #2 High Reynolds Number
In contrast to Day #1 measurements, the velocity of even the smallest steel bead while falling
is so fast that the camera will only be able to capture 1 - 3 frames during acquisition. Instead
of measuring the distance traveled between frames, measurements will be based on the distance
the bead travels within the time span of a single exposure (or single frame).
3.1
Measuring High
Re Terminal Velocity
Remark Similar to day #1, there are a total of 4 aluminum stainless steel bead diameters (1/16,
1/8, 3/16 and 1/4). Each group will only measure 2 of the 4 dierent beads. All groups will
compile their data in a shared Excel document and analysis will be performed on the class data.
3.1.1 Experimental Set-Up
1. Locate the 250 mL cylinder lled with H2 O and measure the
2. Set up the camera to image the
ˆ
bottom half
inner diameter
of the cylinder
Why Since the bead does not reach terminal velocity instantaneously at high
Re,
capturing images as the bead falls in the bottom half of the cylinder will insure the
terminal velocity is measured.
3. Plug in the Basler Ace camera
ˆ
CAUTION! Do not force the plug into place or else you can break the connection
on the camera.
4. Open Pylon Viewer
5. If needed, initiate and set-up the Pylon Viewer interface outlined in the user's guide on
Canvas (Pylon Viewer User's Guide Sec. 1 through Sec. 2)
ˆ
Why Each time a new user uses the Basler Ace camera on a specic computer station
for the rst time the camera must be initiated and the Pylon Viewer interface must
be set-up.
6. Adjust the camera and program settings as needed to optimize image quality
ˆ
Hint Make sure to use an exposure time that is less then the time between frames
based on the frame rate chosen in step 8.
ˆ
Remember Record all camera settings in your lab notebook.
14
3.1.2 Imaging
7. Using the forceps, fully submerge and drop a bead (any size) to develop an intuitive sense
of how the bead descends through the uid
ˆ
Remark The instructors will hand out the beads once the lab starts to ensure that
the dierent sizes are evenly distributed.
ˆ
Contemplate How does the descent of the stainless steel bead through water dier
from the descent of the aluminum bead through glycerol?
8. Refer to the Pylon Viewer user's guide on Canvas and set up the recording options for
acquiring an image series with a frame rate between 30 and 55 frames/sec and set the number
of frames to a value that ensures the falling bead in captured in some of the frames (Pylon
Viewer User's Guide Sec. 3)
ˆ
Hint You will want to optimize these settings after your rst couple runs.
ˆ
Remember Record all camera settings in your lab notebook.
9. Remove a bead from the tube containing the smallest sized stainless steel bead
10. Measure the diameter of the bead with the calipers
11. Using the forceps, fully submerge the bead in the H2 O in the center of the cylinder
ˆ
Why The force exerted on the bead at the air-water interface may aect your measurements.
The potential aect is eliminated by fully submerging the bead in the
H2 O.
12. Begin acquiring images than release the bead to capture the beads descent through the
H2 O
ˆ
Why It is important to start the program before the bead is dropped; else the bead
is likely not to be imaged since its velocity is signicantly greater than the aluminum
beads falling through glycerol.
13. Make sure to record the exposure time which is dierent than the frame rate!
ˆ
Why The exposure time is the amount of time that an individual frame is exposed
for this correlates to the amount of time the bead is allowed to travel in that frame
which will be less than the amount of time between frames which is determined by the
frame rate.
14. View the images and verify that the bead is present in
ˆ
Remark The bead will appear as a blur instead of a denable sphere.
15. Repeat step 9 through step 14 for a
total of 5 beads for all bead sizes.
16. Capture and save one image of a ruler at the
falling bead
ˆ
at least 1 image
same distance
from the cameras as the
Why This will serve as a conversion factor from pixels to units of length.
15
Depending on time, skip to Sec. 3.2 before proceeding to Sec. 3.1.3. Measurements for approaching terminal velocity at high Re must be completed before the end of day #2.
3.1.3 Image Processing
17. For one of the bead trials, open
all the images individually where the bead is fully present
in Fiji.
ˆ
Remember In science, the more data the better!
18. Adjust the brightness and contrast of the image to improve visualization of the blur from
→
the bead. (Image
Adjust
→
Brightness/Contrast)
19. Select the line drawing tool (Fig. 4).
Figure 4:
20. Measure from the
Screen shot of ImageJ with line draw tool highlighted.
center
of the bead at the beginning of the blur to the
center
of the
bead at the end of the blur (Fig.
ˆ
Remark This can be hard to determine from the image at times, but just do your
best.
(a)
Figure 5:
(b)
Example image of measuring the distance traveled by a 1/4 stainless steel bead
in a single frame.
(a)
Image after brightness and contrast adjusted to see path of bead.
Measuring the distance the bead traveled.
(b)
The red dotted lines are shows the approximate
initial position and nal position of the bead. The yellow solid line is the approximate distance
traveled based on the center of the bead positions.
16
21. Record the distance the bead traveled in your lab notebook or with ImageJ by sending
the measurement to the results table by pressing
ˆ
⟨Ctrl⟩+m.
Hint If sending measurements to the results table, verify that the correct measurement
is being recorded. If the incorrect measurement is recording, you can adjust this option
under Analyze
→
Set Measurements.
22. Repeat steps 17 through 21 for the additional trials of that bead size.
23. If you recorded the measurements with ImageJ, save the measured distances as a text le
for further analysis with MATLAB.
ˆ
Remember Record the le information in your lab notebook.
24. Repeat step 17 to step 23 for the remaining bead sizes.
3.1.4 Data Analysis
25. Transfer the tracking results for a single bead into MATLAB
26. Calculate the velocities in each frame (a) Concert the distance traveled in each frame from pixels to centimeters
(b) Calculate the velocity (cm/s) for each frame based on the exposure time
27. Copy and input the calculated velocities into the shared excel document of class data
found on Canvas
28.
29. Repeat step 25 through step 27 for all runs and bead sizes
Analyze and upload all velocities to the class data by the end of day
#2! Move on to the remaining analysis in this section only after velocities
are uploaded. Remaining analysis can be completed outside of if needed.
30. For your class data ˆ
Remark The MATLAB code for the following analysis can still be written if the class
data has not been completed then nalized once all class data is obtained.
(a) Calculate the average bead velocity with standard deviation for each bead size
(b) Use the CurveFit toolbox to t the average bead velocity as a function of bead radius
with the equation found in Prelim. Question #3c
ˆ
Contemplate Does this conrm the expected high Reynolds number model of the
1/2
movement of a falling sphere (vterm ∝ r )?
(c) On the same plot, plot the average bead velocity (cm/s) as a function of bead radius
(cm) using the standard deviation as error bars and plot the t found in step 30b
17
(d) Calculate the average Reynolds number and standard deviation for each bead size.
Refer to App. A for physical properties.
ˆ
Hint Make sure to use the average radius of each bead and to account for the
standard deviation of both the average velocities and radius by propagating the
uncertainty. Refer to App. B for commonly used uncertainty propagation formulas.
(e) Plot the average Reynolds number as a function of bead radius (cm) using the standard deviation as error bars
3.2
Approaching Terminal Velocity at High
Re
Since terminal velocity is not reached instantaneously at high Reynolds number (Sec. 1.2.2.1),
unlike low Reynolds number, the acceleration of a sphere as it reaches terminal velocity can be
measured.
3.2.1 Approaching Terminal Velocity Measure
ˆ
Adjust the set-up such that the majority of the cylinder is visible in the eld of view. The
upper portion of the cylinder is of particular interest.
Why The top portion must be imagined because the bead will be at terminal velocity
in the lower portion of the cylinder.
ˆ
Acquire a new calibration image with the ruler if the distance between the camera and
cylinder was changed
ˆ
Capture the descent of a single steel bead (any size you desire) as outlined in the previous
section (Sec. 3.1) such that there are
at least 3 images in a row of the falling bead.
Hint If struggling to get a minimum of 3 images in a row continue to make adjustments to the set-up. For example, try
1. Increasing the distance between the camera and the cylinder to capture a larger
portion of the cylinder. Remember to take a new calibration image with the ruler.
2. Increasing the number of frames per sec (max frame rate is 55
frames/sec) to capture
more images of the beads descent. However, make sure the bead still appears as
a blur and adjust the exposure time.
3. Using a smaller sized bead since it will have a lower velocity.
Remark If possible, try to capture more than 3 frames in a row. The more frames
capturing the beads descent the better.
ˆ
Plot the velocity of the bead over time.
ˆ
Hint Do your best to approximate time zero.
ˆ
Verify that the data follows the anticipated functional form of
Curve Fitting Toolbox.
18
v(t)
(Eq.
22) using the
3.3
Day #2 Clean Up
1. Wipe up all water.
2. Throw away any trash (i.e. dirty paper towels, paralm, etc.)
3. Unplug the camera and break down the set-up.
4. Leave extra steel beads and cylinder of water at the station.
References
[1] Vogel, S. Life in Moving Fluids: The Physical Biology of Flow ; Princeton University Press:
Princeton, NJ, 1994.
[2] Brown, P. P.; Lawler, D. F. Sphere Drag and Settling Velocity Revisited. J. Environ. Eng.
2003. 129, 222-231.
[3] White, F. Viscous Fluid Flow, 3rd ed.; McGraw-Hill Mechanical Engineering; McGrawHill:New York, 2005.
[4] Lommatzsch, T.; Meghar, M.; Mahe, E.; Devin, E. Conceptual Study of an Absolute
Falling-Ball Viscometer. Metrologia
2001, 38, 531-534.
19
Appendices
Reference of Physical Properties at 20o C
A
B
ˆ
Glycerol Viscosity,
ˆ
Glycerol Density,
ˆ
Water Viscosity,
ˆ
Water Density,
ˆ
Aluminum Density,
ˆ
Steel Density,
ˆ
Gravity,
g:
ηgly :
1412 cP
ρgly :
1.26 g/cm3
ηH2 O :
1.002 cP
ρH2 O :
0.998 g/cm3
ρAl :
ρsteel :
2.79 g/cm3
8.00 g/cm3
9.81 m/s2
Common Uncertainty Propagation Equations
Formula Name
Multiplication with a constant,
k
Function
Uncertainty Propagation
z = kx
σz = |k| σx
z =x+y
Addition or Subtraction
or
z =x−y
σz =
q
σx2 + σy2
r
z = xy
Multiplication or Division
Raised to the power of a constant,
k
or
z=
z = xk
20
x
y
σz = |z|
σx 2
x
σz = |k| xk−1 σx
or
+
σy
y
2
σz
σz = |kz| |x|