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Rayleigh-Plateau Instability

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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
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Page 1: Rayleigh-Plateau Instability

Rayleigh-Plateau Instability

Rachel and Jenna

Page 2: Rayleigh-Plateau Instability

Overview Introduction to Problem Experiment and Data Theories

1. Model2. Comparison to Data

Conclusion More Ideas about the Problem

Page 3: Rayleigh-Plateau Instability

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.

Page 4: Rayleigh-Plateau Instability

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

Page 5: Rayleigh-Plateau Instability

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).

Page 6: Rayleigh-Plateau Instability

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.

Page 7: Rayleigh-Plateau Instability

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.

Page 8: Rayleigh-Plateau Instability

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.

Page 9: Rayleigh-Plateau Instability

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.

Page 10: Rayleigh-Plateau Instability

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

Page 11: Rayleigh-Plateau Instability

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

Page 12: Rayleigh-Plateau Instability

Data (Motor Oil on Fishing String)

Page 13: Rayleigh-Plateau Instability

Data(Syrup on Red Thread)

Page 14: Rayleigh-Plateau Instability

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)

Page 15: Rayleigh-Plateau Instability

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.

Page 16: Rayleigh-Plateau Instability

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.

Page 17: Rayleigh-Plateau Instability

Theory (cont.)

Where is the surface tension. Use the Method of Lagrange multipliers to

minimize the energy with the volume constraint.

Page 18: Rayleigh-Plateau Instability

Theory (cont.)

The function F for the Euler-Lagrange formula

We first use the Beltrami identity to find some relationships between our variables.

Page 19: Rayleigh-Plateau Instability

Theory (cont.) Simplifying and combining the

constants into a new constant C0 we get

Now using the perfect wetting assumptions, we have that when

Page 20: Rayleigh-Plateau Instability

Theory (cont.) Therefore we get the relationship

between .

Then our equation becomes

Page 21: Rayleigh-Plateau Instability

Theory (cont.) Next we know that when r(z) is a

maximum, r’(z) = 0. So we can find the value of rmax.

Page 22: Rayleigh-Plateau Instability

Theory (cont.) Now we want to find the actual solution for

r(z). Use the Euler-Lagrange equation to do

this.

Page 23: Rayleigh-Plateau Instability

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’’.

Page 24: Rayleigh-Plateau Instability

Theory (cont.) Therefore our system of first order

ODEs is

The initial conditions are

Page 25: Rayleigh-Plateau Instability

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

Page 26: Rayleigh-Plateau Instability

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).

Page 27: Rayleigh-Plateau Instability

Theory (cont.) The numerical solution to our system is

given by the following plot of points (z, r(z)).

Page 28: Rayleigh-Plateau Instability

Theory (cont.) A least squares curve of best fit was fitted to

these points. The equation of best fit was

Page 29: Rayleigh-Plateau Instability

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))

Page 30: Rayleigh-Plateau Instability

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.

Page 31: Rayleigh-Plateau Instability

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

Page 32: Rayleigh-Plateau Instability

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

Page 33: Rayleigh-Plateau Instability

Analysis (cont.) Drop 1 Drop 2

Page 34: Rayleigh-Plateau Instability

Analysis (cont.) Drop 3

Page 35: Rayleigh-Plateau Instability

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.

Page 36: Rayleigh-Plateau Instability

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

Page 37: Rayleigh-Plateau Instability

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

Page 38: Rayleigh-Plateau Instability

Theory (cont.) The wavelength, is given by . We can compute the volume of the

perturbed cylinder:

Page 39: Rayleigh-Plateau Instability

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.

Page 40: Rayleigh-Plateau Instability

Theory (cont.) Using this condition from the

constant volume, we can calculate the surface area of the perturbed cylinder.

Page 41: Rayleigh-Plateau Instability

Theory (cont.) Now using a binomial expansion we get an

approximation for the surface area.

Again the sine terms cancel off and we get

Page 42: Rayleigh-Plateau Instability

Theory (cont.) Now we want to use the Laplace-

Young Law to find a condition for k. We have where

and

Page 43: Rayleigh-Plateau Instability

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 .

Page 44: Rayleigh-Plateau Instability

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

Page 45: Rayleigh-Plateau Instability

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

Page 46: Rayleigh-Plateau Instability

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.

Page 47: Rayleigh-Plateau Instability

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.

Page 48: Rayleigh-Plateau Instability

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

Page 49: Rayleigh-Plateau Instability

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

Page 50: Rayleigh-Plateau Instability

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.

Page 51: Rayleigh-Plateau Instability

More Thoughts… More consistent way to apply the

liquid. Investigate other parameters

Angle of string Time Gravity


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