Rayleigh-Plateau Instability Rachel and Jenna

Overview Introduction to Problem Introduction to Problem Experiment and Data Experiment and Data Theories Theories 1. Model 2. Comparison to Data Conclusion Conclusion More Ideas about the Problem More Ideas about the Problem

Introduction The Rayleigh-Plateau Instability is apparent in nature all the time. 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. 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 19 th century, observed and modeled this particular instability. Lord Rayleigh, a physicist of the 19 th 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. 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. In this project we studied this instability that was discovered by Lord Rayleigh. Many different aspects to model Many different aspects to model - Shape of Drops - Shape of Drops - Under what conditions does the instability occur - 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. 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. 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) 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. 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). 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) 7 different liquids (motor oil, canola oil, syrup, corn syrup, dish soap, Windex and water) 4 different types of string or wire 4 different types of string or wire The string was attached horizontally with magnets to two upright poles. 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. 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. 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. 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. 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. This was chosen to ensure the most consistent initial cylinder on the string.

Procedure (cont.) The data was measured in MATLAB. 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 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 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. The radius of the drop is half of this distance.

Data The data was collected from our experiments. The data was collected from our experiments. Only certain droplets with similar shapes and sizes in a row were measured. 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. 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. Many other pictures were taken, but because of human error, only select data was used.

Data (cont.) Red Thread Red Thread Drop #DRIn Btwn.W Corn Syrup to to to AVG Dish Soap to to to to AVG Drop #DRIn Btwn.W Syrup to to to to to AVG Syrup to to to to to to AVG

Data (cont.) Fishing String Fishing String Drop #DRIn Btwn.W Syrup to to to AVG Motor Oil to to to AVG Drop #DRIn Btwn.W Motor Oil to to to AVG Canola Oil to to AVG

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. We first want to model the shape of one of the drops on the string after the liquid has stabilized. Assumptions Assumptions Perfect wetting of the string Perfect wetting of the string Gravity does not affect the drops 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) 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 R 0. We let the string be oriented in the z- direction and have radius R 0. The equation for the drop that we want to model is r(z), and the drop width goes from 0 to L. 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. 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. 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. Minimize the energy, with a volume constraint. Assuming no gravity, therefore the energy is proportional to the surface area. Assuming no gravity, therefore the energy is proportional to the surface area.

Theory (cont.) Where is the surface tension. Where is the surface tension. Use the Method of Lagrange multipliers to minimize the energy with the volume constraint. Use the Method of Lagrange multipliers to minimize the energy with the volume constraint.

Theory (cont.) The function F for the Euler-Lagrange formula The function F for the Euler-Lagrange formula We first use the Beltrami identity to find some relationships between our variables. We first use the Beltrami identity to find some relationships between our variables.

Theory (cont.) Simplifying and combining the constants into a new constant C 0 we get Simplifying and combining the constants into a new constant C 0 we get Now using the perfect wetting assumptions, we have that when Now using the perfect wetting assumptions, we have that when

Theory (cont.) Therefore we get the relationship between. Therefore we get the relationship between. Then our equation becomes 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 r max. Next we know that when r(z) is a maximum, r’(z) = 0. So we can find the value of r max.

Theory (cont.) Now we want to find the actual solution for r(z). Now we want to find the actual solution for r(z). Use the Euler-Lagrange equation to do this. 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. 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’’. Let w = r’, and therefore w’ = r’’.

Theory (cont.) Therefore our system of first order ODEs is Therefore our system of first order ODEs is The initial conditions are The initial conditions are

Theory (cont.) This system is not easily computed, so we need to solve it numerically. This system is not easily computed, so we need to solve it numerically. We used the MATLAB function ode15s in order to do this. 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 Since is the surface tension constant, we varied in order to find the that meets the conditions

Theory (cont.) Using the numeric values of R 0 =.01 cm and L=.14 cm, we find the value that satisfies these conditions is Using the numeric values of R 0 =.01 cm and L=.14 cm, we find the value that satisfies these conditions is These values of L and R 0 are taken from the fishing string data (they are average values for that data). These values of L and R 0 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)). 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 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. 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)) 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. 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 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. We compared our equation to motor oil and canola oil drops on the fishing string.

Analysis (cont.) Drop 1 1 cm260 pixels z_experimentr_experiment(z)r_theory(z)Error Average Error Drop 2 1 cm261 pixels z_experimentr_experiment(z)r_theory(z)Error Average Error0.0099

Analysis (cont.) Drop 3 1 cm261 pixels z_experimentr_experiment(z)r_theory(z)Error Average Error0.0093

Analysis (cont.) Drop 1 Drop 1 Drop 2 Drop 2

Analysis (cont.) Drop 3 Drop 3

Analysis (cont.) The average error between our model and actual data is.0094 cm. 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. 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). 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 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. The average error is relatively small, only.0181 cm. Fishing String Drop #DR(rmax - DR) rmaxMotor Oil AVG Motor Oil AVG Canola Oil AVG Average0.0181

Theory (Instability) We now want to find the perturbations to which the cylinder of liquid is unstable. 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 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 We let the perturbation be described by

Theory (cont.) The wavelength, is given by. The wavelength, is given by. We can compute the volume of the perturbed cylinder: 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. 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. 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. 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. Now using a binomial expansion we get an approximation for the surface area. Again the sine terms cancel off and we get 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. Now we want to use the Laplace-Young Law to find a condition for k. We have where We have where and and

Theory (cont.) Putting this back into the Laplace-Young Law we get Putting this back into the Laplace-Young Law we get We know that they cylinder will be unstable when. This occurs when We know that they cylinder will be unstable when. This occurs when. Therefore the cylinder will be unstable 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 Syrup 1 to to to *Pi*R_0=P 4 to to to to AVG Dish Soap 1 to AVG to Syrup 1 to to to to to to AVG to to AVG

Analysis (cont.) Fising String In Btwn.W(W-P) Syrup1 to St. Radius (R_0) 3 to to *Pi*R_0=P AVG Motor Oil1 to to to AVG Motor Oil1 to to to AVG Canola Oil1 to to AVG

Analysis (cont.) As seen in the last column, our data supports this theory. As seen in the last column, our data supports this theory. The values of W-P are all positive except for the first one. 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 W 0 =2*Pi*sqrt(2)*R 0. The expected wavelength from theory to will be seen in our experiment is defined as W 0 =2*Pi*sqrt(2)*R 0. This expected wavelength was compared to each of the measured wavelengths. 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’). 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. Without the thicker substances, the average error for the wavelength was only.0464 cm.

Analysis (cont.) Red Thread In Btwn.Wabs(W-W_0) In Btwn.Wabs(W-W_0 Thread Radius (R_0)Corn Syrup1 to Syrup1 to to to PI*sqrt(2)*2*R_0=W_0 4 to to to to AVG AVG Dish Soap1 to Syrup1 to to to to to to to to to AVG AVG

Analysis (cont.) Fising String In Btwn.Wabs(W-W_0) Syrup1 to St. Radius (R_0) 3 to to PI*sqrt(2)*2*R_0=W_ AVG Motor Oil1 to to to AVG Motor Oil1 to to to AVG Canola Oil1 to to AVG

Conclusion Overall, the theory was verified by our experimental data. 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) 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. Numerical model is only valid for a particular string radius.

More Thoughts… More consistent way to apply the liquid. More consistent way to apply the liquid. Investigate other parameters Investigate other parameters Angle of string Angle of string Time Time Gravity Gravity