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Rayleigh-Plateau Instability. Rachel and Jenna. Overview. Introduction to Problem Experiment and Data Theories 1. Model 2. Comparison to Data Conclusion More Ideas about the Problem. Introduction. The Rayleigh-Plateau Instability is apparent in nature all the time. - PowerPoint PPT Presentation
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Rayleigh-Plateau Instability
Rachel and Jenna
Overview Introduction to Problem Experiment and Data Theories
1. Model2. Comparison to Data
Conclusion More Ideas about the Problem
Introduction 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 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 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 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.) 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.) 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 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.) Red Thread
Drop # DR In Btwn. W
Corn Syrup 1 0.0124 1 to 2 0.0650
2 0.0124 2 to 3 0.0836
3 0.0124 4 to 5 0.0712
4 0.0108
5 0.0124
AVG 0.0121 0.0733
Dish Soap 1 0.0125 1 to 2 0.1028
2 0.0125 2 to 3 0.0997
3 0.0140 4 to 5 0.1153
4 0.0125 5 to 6 0.1090
5 0.0109
6 0.0140
AVG 0.0127 0.1067
Drop # DR In Btwn. W
Syrup 1 0.0426 1 to 2 0.2791
2 0.0465 3 to 4 0.3178
3 0.0426 4 to 5 0.2713
4 0.0426 5 to 6 0.3101
5 0.0504 6 to 7 0.2868
6 0.0388
7 0.0388
AVG 0.0432 0.2930
Syrup 1 0.0310 1 to 2 0.2558
2 0.0388 2 to 3 0.2403
3 0.0388 3 to 4 0.2248
4 0.0349 4 to 5 0.2171
5 0.0388 6 to 7 0.2713
6 0.0349 7 to 8 0.2791
7 0.0310
8 0.0310
AVG 0.0349 0.2481
Data (cont.) Fishing String
Drop # DR In Btwn. W
Syrup 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 0.0805
5 0.0847
AVG 0.0847 0.7006
Motor Oil 1 0.0423 1 to 2 0.3269
2 0.0423 3 to 4 0.3192
3 0.0423 4 to 5 0.3115
4 0.0404
5 0.0423
AVG 0.0419 0.3192
Drop # DR In Btwn. W
Motor Oil 1 0.0367 1 to 2 0.2857
2 0.0367 2 to 3 0.2896
3 0.0367 4 to 5 0.3282
4 0.0405
5 0.0386
AVG 0.0378 0.3012
Canola Oil 1 0.0423 1 to 2 0.2846
2 0.0404 2 to 3 0.2308
3 0.0404
AVG 0.0410 0.2722
Data (Motor Oil on Fishing String)
Data(Syrup on Red Thread)
Theory (Shape of Drop) We first want to model the shape of
one of the drops on the string after the liquid has stabilized.
Assumptions 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.) We let the string be
oriented in the z-direction 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.) 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.)
Where is the surface tension. Use the Method of Lagrange multipliers to
minimize the energy with the volume constraint.
Theory (cont.)
The function F for the Euler-Lagrange formula
We first use the Beltrami identity to find some relationships between our variables.
Theory (cont.) Simplifying and combining the
constants into a new constant C0 we get
Now using the perfect wetting assumptions, we have that when
Theory (cont.) Therefore we get the relationship
between .
Then our equation becomes
Theory (cont.) Next we know that when r(z) is a
maximum, r’(z) = 0. So we can find the value of rmax.
Theory (cont.) Now we want to find the actual solution for
r(z). Use the Euler-Lagrange equation to do
this.
Theory (cont.) 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.) Therefore our system of first order
ODEs is
The initial conditions are
Theory (cont.) 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.) 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.) The numerical solution to our system is
given by the following plot of points (z, r(z)).
Theory (cont.) A least squares curve of best fit was fitted to
these points. The equation of best fit was
Theory (cont.) 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 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 1
1 cm 260 pixels
z_experiment r_experiment(z) r_theory(z) Error
0.0000 0.0000 0.0076 0.0076
0.0269 0.0327 0.0233 0.0094
0.0500 0.0365 0.0308 0.0057
0.0808 0.0404 0.0323 0.0081
0.1115 0.0327 0.0239 0.0087
0.1346 0.0269 0.0113 0.0156
0.1577 0.0000 -0.0068 0.0068
Average Error 0.0089
Drop 2
1 cm 261 pixels
z_experiment r_experiment(z) r_theory(z) Error
0.0000 0.0000 0.0076 0.0076
0.0230 0.0307 0.0215 0.0092
0.0421 0.0383 0.0289 0.0095
0.0651 0.0421 0.0327 0.0094
0.0996 0.0383 0.0283 0.0100
0.1341 0.0287 0.0116 0.0171
0.1571 0.0000 -0.0063 0.0063
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.) Drop 1 Drop 2
Analysis (cont.) Drop 3
Analysis (cont.) 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.) 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 Drop # DR (rmax - DR)
rmax Motor Oil 1 0.0423 0.0201
0.0222 2 0.0423 0.0201
3 0.0423 0.0201
4 0.0404 0.0182
5 0.0423 0.0201
AVG 0.0419 0.0197
Motor Oil 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
AVG 0.0378 0.0156
Canola Oil 1 0.0423 0.0201
2 0.0404 0.0182
3 0.0404 0.0182
AVG 0.0410 0.0188
Average 0.0181
Theory (Instability) 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.
We let the perturbation be described by
Theory (cont.) The wavelength, is given by . We can compute the volume of the
perturbed cylinder:
Theory (cont.) 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.) Using this condition from the
constant volume, we can calculate the surface area of the perturbed cylinder.
Theory (cont.) Now using a binomial expansion we get an
approximation for the surface area.
Again the sine terms cancel off and we get
Theory (cont.) Now we want to use the Laplace-
Young Law to find a condition for k. We have where
and
Theory (cont.) 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
Red Thread In
Btwn. W (W-P) In
Btwn. W (W-P)
Thread Radius (R_0)Corn Syrup 1 to 2 0.0650 -0.0008 Syrup 1 to 2 0.2791 0.2133
0.0105 2 to 3 0.0836 0.0178 3 to 4 0.3178 0.2520
2*Pi*R_0=P 4 to 5 0.0712 0.0054 4 to 5 0.2713 0.2055
0.0658 5 to 6 0.3101 0.2443
6 to 7 0.2868 0.2210
AVG 0.0733 0.0075
Dish Soap 1 to 2 0.1028 0.0370 AVG 0.2930 0.2272
2 to 3 0.0997 0.0339 Syrup 1 to 2 0.2558 0.1900
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
AVG 0.1067 0.0409 6 to 7 0.2713 0.2055
7 to 8 0.2791 0.2133
AVG 0.2481 0.1823
Analysis (cont.)Fising String In Btwn. W (W-P) Syrup 1 to 2 0.5763 0.4135
St. Radius (R_0) 3 to 4 0.7797 0.6169
0.0259 4 to 5 0.7458 0.5830
2*Pi*R_0=P
0.1627
AVG 0.7006 0.5378
Motor Oil 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
Motor Oil 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
Canola Oil 1 to 2 0.2846 0.1219
2 to 3 0.2308 0.0680
AVG 0.2722 0.1094
Analysis (cont.) 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.) 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. W abs(W-W_0) In Btwn. W abs(W-W_0
Thread Radius (R_0) Corn Syrup 1 to 2 0.0650 0.0280 Syrup 1 to 2 0.2791 0.1860
0.0105 2 to 3 0.0836 0.0095 3 to 4 0.3178 0.2248
PI*sqrt(2)*2*R_0=W_0 4 to 5 0.0712 0.0218 4 to 5 0.2713 0.1783
0.0931 5 to 6 0.3101 0.2170
6 to 7 0.2868 0.1938
AVG 0.0733 0.0198
0.0931 AVG 0.2930 0.2000
Dish Soap 1 to 2 0.1028 0.0097 Syrup 1 to 2 0.2558 0.1628
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 0.1067 0.0136
AVG 0.2481 0.1550
Analysis (cont.)Fising String In Btwn. W abs(W-W_0)
Syrup 1 to 2 0.5763 0.3461
St. Radius (R_0) 3 to 4 0.7797 0.5495
0.0259 4 to 5 0.7458 0.5156
PI*sqrt(2)*2*R_0=W_0
0.2301 AVG 0.7006 0.4704
Motor Oil 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
Motor Oil 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
Canola Oil 1 to 2 0.2846 0.0545
2 to 3 0.2308 0.0006
AVG 0.2722 0.0420
Conclusion 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… More consistent way to apply the
liquid. Investigate other parameters
Angle of string Time Gravity