Rayleigh-Plateau Instability

1. Rayleigh-Plateau Instability By: Qiang Chen and Stacey Altrichter

2. Introduction • The Rayleigh-Plateau phenomenon is observed in daily life. • water dripping from a faucet • uniform water beads forming on a spider web during the night • ink-jet printing • Instead of remaining in cylindrical form, the fluid tends to break up into droplets due to surface tension.

3. Study of Instability • Joseph Plateau, in 1873, observed experimentally that a falling stream of water of length greater than approximately 3.13 times its diameter will form droplets while falling. • Rayleigh formed a theoretical explanation for a non-viscous liquid that is falling vertically. He stated that the liquid strand will break into drops once the length of the fall exceeds the circumference of the cross-sectional circle. • Rayleigh came up with an estimate of the wavelength of instability.

4. Procedure • Purpose • Observe the Rayleigh-Plateau Instability • Determine why liquids prefer to form drops on a string rather than remain in the initial cylindrical state • Suspended several strands horizontally • fishing wire • blue and red string • metal green wire • uncooked spaghetti pasta • Spread an even layer of fluid using a dropper.

5. Procedure • Substances we used included • compressor oil • corn syrup • canola oil • liquid soap • motor oil • Syrup • Each fluid kept in its own dropper to prevent mixing.

6. Procedure • High speed camera used to capture the evolution of liquid over time. • Suspended a ruler, in each frame above the strand to help measure the dimensions of the droplets. • For each trial with the blue string, red string, and pasta used a new strand to prevent contamination. Unfortunately, not done for the fishing wire and the metal green wire. • Pictures downloaded onto the computer. Picked frames.

7. Photographs and Data

8. Blue Thread with Motor Oil

9. Data of Blue String

10. Fishing Line with Motor Oil

11. Data of Fishing Wire

12. Green Metal Wire w/ Motor Oil

13. Pasta with Motor Oil

14. Theory on the Instability

15. Shape of Droplets • Determine what the shape of the droplets ought be. • Find the total energy equation of system and, apply a volume constraint. • Assume gravitational energy exerted on the droplets is zero which makes the surface energy equal to the total energy.

16. Shape of Droplets • Let z be the horizontal axis running through center of strand. • Define • droplet length to be from z=0 to z=L • R0 the radius of the strand • r(z) the height of the liquid at any given z value • Assume that droplets are perfectly symmetric about z=L/2. The max height occurs at z=L/2. • Take  to be the angle as shown below.

17. Energy Equation Volume Constraint Lagrange Multipliers  constant, V constraint minimize energy equation

18. Beltrami’s Identity where C0 is constant HERE WE HAVE: where

19. Assumption of Perfect Symmetry  when r=rmax (which occurs at z=L/2) r’=0. Hence

20. Euler-Lagrange Equation finding a condition on  Conditions:

21. Condition for  Find  so that the following conditions also hold From the boundary conditions and the data generated for rmax and  we can find all the undetermined constants and solve for r numerically to obtain the shape of the drops.

22. Wavelength of Instability • Consider the small perturbation of a long cylinder. • First, we consider the small axisymmetric perturbation case. Here the gravitational force on the drops has a very small effect on their shape compared to the surface tension force. • Experiments • Fishing wire and blue string • Drops long and slim • More affected by surface tension then gravity • Small axisymmetric perturbation useful • Pasta and metal wire • Drops short but very thick • More affected by gravity then surface tension • Non-axisymmetric perturbation useful

23. Take z to be the the axis running through the center axis of the strand • Let R0 to be the radius of the strand • Take r(z) to be the height of the fluid at position z • Consider

24. Volume Here we have a constant volume and it can be computed as follows: Due to periodicity, all sine functions are zero here.

25. Volume Since the volume of liquid spread on the strand is constant, all 2 terms will go to zero and which implies

26. Surface Area of Fluid Binomial Expansion:

27. According to the surface area eq, the potential energy eq is given by Potential Energy • Define the wavelength to be =2/k. • For >2R0 the potential energy P negative making the system is unstable. • For <2R0 the potential energy will be positive which makes the system stable.

28. Laplace-Young Law • Don't have an exact reason for our potential energy formula. We instead consider the Laplace-Young Law. • H is the mean curvature of the surface. We want to analyze the stability of the wavelength. To do so, we need to find the sign of P which is equivalent to finding the sign of H.

29. Mean Curvature We were unable to find a way to establish the sign of H with certainty.

30. Non-axisymmetric Perturbation of r becomes: where m0 is an integer that identifies the angular mode. The corresponding potential energy is of the form: Hence P >0 making the system always stable.

31. Analysis • The experiments we performed in lab are depicted in the figures. • Compared the theory found by papers and variation formulas with the experiments. • But since we do not have more accurate rules to measure our data, it was tricky using the experimental data to compute the numerical results. • But from the pictures, we still can see some results directly, such as the beads' shape, the instability of the drops and so on.

32. Conclusion • For the Rayleigh-Plateau instability, we need to consider the initial cylinder of the fluid, and to measure the radius of the string, fluid, radius of bead and so on. • But by the limitation of the tools we have, we can not the get the perfect initial values and the measurements, we can not the use the experiment's data to complete our model. • But for some types of line and fluid, such as fishing wire and motor oil, we still can see some phenomenon of the instability and the shape of beads very well.