Rayleigh-Plateau Instability
Rachel and Jenna
Overview
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Introduction to Problem
Experiment and Data
Theories
1. Model
2. Comparison to Data
Conclusion
More Ideas about the Problem
Introduction
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The Rayleigh-Plateau Instability is apparent in
nature all the time.
This instability occurs when a thin layer of liquid
is applied to a surface and beads up into evenly
spaced droplets of the same size.
Lord Rayleigh, a physicist of the 19th century,
observed and modeled this particular instability.
He calculated that the most unstable wavelength
(the wavelength that is seen) is about nine
times the radius of the liquid.
Introduction
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In this project we studied this instability
that was discovered by Lord Rayleigh.
Many different aspects to model
- Shape of Drops
- Under what conditions does the
instability occur
- What is the expected wavelength
between drops
Literature
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There is a lot of literature on the Rayleigh-Plateau
Instability and other related topics.
Lord Rayleigh wrote journals concerned with capillary
tubes and the capillary phenomena of jets.
A book by Chandrasekhar modeled the conditions under
which the instability will occur using the change in
pressure (Laplace-Young Law)
Campana and Saita concluded that surfactants (a
coating which cuts down on surface tension of a liquid)
had no impact on the final shape, size or spacing of the
drops in the instability.
Most articles considered a cylindrical jet which was
vertical (not this model).
Procedure
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7 different liquids (motor oil, canola oil,
syrup, corn syrup, dish soap, Windex and
water)
4 different types of string or wire
The string was attached horizontally with
magnets to two upright poles.
The height of the string was checked by a
ruler to maker sure it was level.
Procedure (cont.)
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A centimeter length was marked on the string
for a reference length in the pictures.
For consistency, Rachel took the pictures and
Jenna placed the fluid on the string.
The motor oil, canola oil, dish soap, Windex, and
water were put onto the string with an eye
dropper, and the more viscous fluids, such as
syrup and corn syrup were put onto the string
with a popsicle stick.
This was chosen to ensure the most consistent
initial cylinder on the string.
Procedure (cont.)
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The data was measured in MATLAB.
The wavelength is the distance between
each drop, which was measured from the
top of one drop to the top of the next.
The diameter of the droplets was defined
to be the distance from the top to the
bottom of the largest part of the drop.
The radius of the drop is half of this
distance.
Data
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The data was collected from our
experiments.
Only certain droplets with similar shapes and
sizes in a row were measured.
The table shows the data for the red thread
and the fishing string with several types of
liquid.
Many other pictures were taken, but because
of human error, only select data was used.
Data (cont.)
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Red Thread
Drop #
Corn Syrup
DR
Drop #
In Btwn.
W
Syrup
W
1
0.0426
1 to 2
0.2791
2
0.0465
3 to 4
0.3178
0.0124
1 to 2
0.0650
3
0.0426
4 to 5
0.2713
2
0.0124
2 to 3
0.0836
4
0.0426
5 to 6
0.3101
3
0.0124
4 to 5
0.0712
5
0.0504
6 to 7
0.2868
4
0.0108
6
0.0388
5
0.0124
7
0.0388
0.0121
AVG
0.0733
Syrup
AVG
In Btwn.
1
AVG
Dish Soap
DR
0.0432
0.2930
1
0.0310
1 to 2
0.2558
2
0.0388
2 to 3
0.2403
1
0.0125
1 to 2
0.1028
3
0.0388
3 to 4
0.2248
2
0.0125
2 to 3
0.0997
4
0.0349
4 to 5
0.2171
3
0.0140
4 to 5
0.1153
5
0.0388
6 to 7
0.2713
4
0.0125
5 to 6
0.1090
6
0.0349
7 to 8
0.2791
5
0.0109
7
0.0310
6
0.0140
8
0.0310
0.0127
0.1067
AVG
0.0349
0.2481
Data (cont.)
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Fishing String
Drop #
Syrup
DR
In Btwn.
W
1
0.0890
1 to 2
0.5763
2
0.0975
3 to 4
0.7797
3
0.0720
4 to 5
0.7458
4
5
AVG
Motor Oil
AVG
DR
In Btwn.
W
1
0.0367
1 to 2
0.2857
2
0.0367
2 to 3
0.2896
3
0.0367
4 to 5
0.3282
0.0805
4
0.0405
0.0847
5
0.0386
0.0847
Motor Oil
0.7006
1
0.0423
1 to 2
0.3269
2
0.0423
3 to 4
3
0.0423
4 to 5
4
0.0404
5
0.0423
0.0419
Drop #
AVG
Canola Oil
0.0378
0.3012
1
0.0423
1 to 2
0.2846
0.3192
2
0.0404
2 to 3
0.2308
0.3115
3
0.0404
AVG
0.3192
0.0410
0.2722
Data
(Motor Oil on Fishing String)
Data
(Syrup on Red Thread)
Theory (Shape of Drop)
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We first want to model the shape of one
of the drops on the string after the liquid
has stabilized.
Assumptions
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Perfect wetting of the string
Gravity does not affect the drops
Drop is axisymmetric (so we can find a model
that describes the curve of the drop above
the string)
Theory (cont.)
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We let the string be
oriented in the zdirection and have
radius R0.
The equation for the
drop that we want to
model is r(z), and the
drop width goes from
0 to L.
Theory (cont.)
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We begin by looking at the energy of the
drop.
When the liquid has stabilized the energy
will be minimized, but the volume of the
liquid will not change.
Minimize the energy, with a volume
constraint.
Assuming no gravity, therefore the energy
is proportional to the surface area.
Theory (cont.)
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Where  is the surface tension.
Use the Method of Lagrange multipliers to minimize
the energy with the volume constraint.
Theory (cont.)
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The function F for the Euler-Lagrange
formula
We first use the Beltrami identity to find
some relationships between our variables.
Theory (cont.)
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Simplifying and combining the constants
into a new constant C0 we get
Now using the perfect wetting
assumptions, we have that when
Theory (cont.)
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Therefore we get the relationship between
.
Then our equation becomes
Theory (cont.)
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Next we know that when r(z) is a
maximum, r’(z) = 0. So we can find the
value of rmax.
Theory (cont.)
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Now we want to find the actual solution for r(z).
Use the Euler-Lagrange equation to do this.
Theory (cont.)
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To begin to solve this second order
nonlinear ODE, we rewrite it as a system
of first order ODE.
Let w = r’, and therefore w’ = r’’.
Theory (cont.)
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Therefore our system of first order ODEs
is
The initial conditions are
Theory (cont.)
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This system is not easily computed, so we need
to solve it numerically.
We used the MATLAB function ode15s in order
to do this.
Since  is the surface tension constant, we
varied
in order to find the
that meets
the conditions
Theory (cont.)
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Using the numeric values of R0=.01 cm
and L=.14 cm, we find the value that
satisfies these conditions is
These values of L and R0 are taken from
the fishing string data (they are average
values for that data).
Theory (cont.)
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The numerical solution to our system is given by
the following plot of points (z, r(z)).
Theory (cont.)
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A least squares curve of best fit was fitted to these
points. The equation of best fit was
Theory (cont.)
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We also fit a cosine curve to the points, and found the
curve of best fit.
The equation of this fit is r(z) = .034*cos(20(z-.07))
Analysis of Drop Shape
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From the theory we have found a model
that gives the equation for the shape of a
drop.
We now want to compare our
experimental data with the theory.
We compared our equation to motor oil
and canola oil drops on the fishing string.
Analysis (cont.)
Drop 2
Drop 1
1 cm
260 pixels
z_experiment
r_experiment(z)
r_theory(z)
Error
1 cm
261 pixels
z_experiment
r_experiment(z)
r_theory(z)
Error
0.0000
0.0000
0.0076
0.0076
0.0000
0.0000
0.0076
0.0076
0.0269
0.0327
0.0233
0.0094
0.0230
0.0307
0.0215
0.0092
0.0500
0.0365
0.0308
0.0057
0.0421
0.0383
0.0289
0.0095
0.0808
0.0404
0.0323
0.0081
0.0651
0.0421
0.0327
0.0094
0.1115
0.0327
0.0239
0.0087
0.0996
0.0383
0.0283
0.0100
0.1346
0.0269
0.0113
0.0156
0.1341
0.0287
0.0116
0.0171
0.1577
0.0000
-0.0068
0.0068
0.1571
0.0000
-0.0063
0.0063
Average Error
0.0089
Average Error
0.0099
Analysis (cont.)
Drop 3
1 cm
261 pixels
z_experiment
r_experiment(z)
r_theory(z)
Error
0.0000
0.0000
0.0076
0.0076
0.0230
0.0326
0.0215
0.0111
0.0536
0.0402
0.0315
0.0088
0.0766
0.0421
0.0326
0.0095
0.1149
0.0345
0.0224
0.0121
0.1303
0.0268
0.0141
0.0127
0.1456
0.0000
0.0034
0.0034
Average Error
0.0093
Analysis (cont.)
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Drop 1
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Drop 2
Analysis (cont.)
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Drop 3
Analysis (cont.)
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The average error between our model and
actual data is .0094 cm.
Overall, the data seems to match our
theoretical model for drops of the same
string and similar drop width.
Analysis (cont.)
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We also found the
theoretical maximum
value of the drop
height (rmax).
The rmax value was
the radius of the drop
in our data. This is
compared to the
theoretical rmax value.
The average error is
relatively small, only
.0181 cm.
Fishing
String
rmax
Drop #
Motor Oil
0.0222
DR
1
0.0423
0.0201
2
0.0423
0.0201
3
0.0423
0.0201
4
0.0404
0.0182
5
0.0423
0.0201
0.0419
0.0197
1
0.0367
0.0145
2
0.0367
0.0145
3
0.0367
0.0145
4
0.0405
0.0183
5
0.0386
0.0164
0.0378
0.0156
1
0.0423
0.0201
2
0.0404
0.0182
3
0.0404
0.0182
0.0410
0.0188
AVG
Motor Oil
AVG
Canola Oil
AVG
(rmax - DR)
Average
0.0181
Theory (Instability)
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We now want to find the perturbations to which
the cylinder of liquid is unstable.
We will again take the z-axis to be through the
thread, and r(z) to be the perturbed surface of
the cylinder.
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We let the perturbation be described by
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Theory (cont.)
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The wavelength, is given by
.
We can compute the volume of the perturbed
cylinder:
Theory (cont.)
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Since we are looking a unit length and r(z) is periodic,
the sine terms will go to zero.
The volume must be constant, so all epsilon terms must
go to zero.
Theory (cont.)
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Using this condition from the constant
volume, we can calculate the surface area
of the perturbed cylinder.
Theory (cont.)
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Now using a binomial expansion we get an
approximation for the surface area.
Again the sine terms cancel off and we get
Theory (cont.)
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Now we want to use the Laplace-Young
Law to find a condition for k.
We have
where
and
Theory (cont.)
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Putting this back into the Laplace-Young
Law we get
We know that they cylinder will be
unstable when
. This occurs when
. Therefore the cylinder will be
unstable when
.
Analysis of Unstable Wavelength
In
Btwn.
Red Thread
Thread Radius (R_0)
Corn
Syrup
0.0105
2*Pi*R_0=P
W
In
Btwn.
(W-P)
Syrup
(W-P)
1 to 2
0.2791
0.2133
1 to 2
0.0650
-0.0008
2 to 3
0.0836
0.0178
3 to 4
0.3178
0.2520
4 to 5
0.0712
0.0054
4 to 5
0.2713
0.2055
5 to 6
0.3101
0.2443
6 to 7
0.2868
0.2210
AVG
0.2930
0.2272
1 to 2
0.2558
0.1900
0.0658
Dish Soap
W
AVG
0.0733
0.0075
1 to 2
0.1028
0.0370
2 to 3
0.0997
0.0339
4 to 5
0.1153
0.0495
2 to 3
0.2403
0.1745
5 to 6
0.1090
0.0432
3 to 4
0.2248
0.1590
4 to 5
0.2171
0.1513
6 to 7
0.2713
0.2055
7 to 8
0.2791
0.2133
AVG
0.2481
0.1823
AVG
0.1067
0.0409
Syrup
Analysis (cont.)
Fising String
In Btwn.
Syrup
St. Radius (R_0)
0.0259
W
(W-P)
1 to 2
0.5763
0.4135
3 to 4
0.7797
0.6169
4 to 5
0.7458
0.5830
AVG
0.7006
0.5378
1 to 2
0.3269
0.1642
3 to 4
0.3192
0.1565
4 to 5
0.3115
0.1488
AVG
0.3192
0.1565
1 to 2
0.2857
0.1230
2 to 3
0.2896
0.1268
4 to 5
0.3282
0.1655
AVG
0.3012
0.1384
1 to 2
0.2846
0.1219
2 to 3
0.2308
0.0680
AVG
0.2722
0.1094
2*Pi*R_0=P
0.1627
Motor Oil
Motor Oil
Canola Oil
Analysis (cont.)
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As seen in the last column, our data
supports this theory.
The values of W-P are all positive except
for the first one.
Analysis (cont.)
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The expected wavelength from theory to will be seen in
our experiment is defined as W0=2*Pi*sqrt(2)*R0.
This expected wavelength was compared to each of the
measured wavelengths.
The error was very good on less viscous fluids, which
spread onto the wire or string more evenly, such as
canola or motor oil. However, error was much higher on
syrup and corn syrup. This is most likely due to a human
error when applying the liquid (due to ‘clumping up’).
Without the thicker substances, the average error for the
wavelength was only .0464 cm.
Analysis (cont.)
Red Thread
In Btwn.
Thread Radius (R_0)
Corn Syrup
0.0105
PI*sqrt(2)*2*R_0=W_0
W
abs(W-W_0)
1 to 2
0.0650
0.0280
2 to 3
0.0836
4 to 5
0.0712
In Btwn.
Syrup
0.0733
0.2791
0.1860
0.0095
3 to 4
0.3178
0.2248
0.0218
4 to 5
0.2713
0.1783
5 to 6
0.3101
0.2170
6 to 7
0.2868
0.1938
AVG
0.2930
0.2000
1 to 2
0.2558
0.1628
0.0198
0.0931
Dish Soap
abs(W-W_0
1 to 2
0.0931
AVG
W
1 to 2
0.1028
0.0097
2 to 3
0.0997
0.0066
2 to 3
0.2403
0.1473
4 to 5
0.1153
0.0222
3 to 4
0.2248
0.1318
5 to 6
0.1090
0.0160
4 to 5
0.2171
0.1240
6 to 7
0.2713
0.1783
7 to 8
0.2791
0.1860
AVG
0.2481
0.1550
AVG
0.1067
Syrup
0.0136
Analysis (cont.)
Fising String
In Btwn.
Syrup
W
abs(W-W_0)
1 to 2
0.5763
0.3461
3 to 4
0.7797
0.5495
0.0259
4 to 5
0.7458
0.5156
0.2301
AVG
0.7006
0.4704
1 to 2
0.3269
0.0968
3 to 4
0.3192
0.0891
4 to 5
0.3115
0.0814
AVG
0.3192
0.0891
1 to 2
0.2857
0.0556
2 to 3
0.2896
0.0594
4 to 5
0.3282
0.0980
AVG
0.3012
0.0710
1 to 2
0.2846
0.0545
2 to 3
0.2308
0.0006
AVG
0.2722
0.0420
St. Radius (R_0)
PI*sqrt(2)*2*R_0=W_0
Motor Oil
Motor Oil
Canola Oil
Conclusion
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Overall, the theory was verified by our
experimental data.
Human error had a large impact on the
validity of the theory (when applying the
liquid it was difficult to obtain an even
layer of liquid)
Numerical model is only valid for a
particular string radius.
More Thoughts…
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More consistent way to apply the liquid.
Investigate other parameters
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Angle of string
Time
Gravity