1. The Rayleigh-Taylor Instability By: Paul Canepa and Mike Cromer Team Leftovers

2. The Rayleigh-Taylor Instability Outline ● Introduction ● Experiment ● Data ● Theory ● Model ● Data Analysis ● Interface ● Conclusion

3. The Rayleigh-Taylor Instability Introduction The Rayleigh-Taylor instability occurs when a light fluid is accelerated into a heavy fluid. The acceleration causes perturbations at the interface, which are the cause of the instability. The study of liquid layer dynamics is important in many applications, for example, coating non-uniformities, flotation and filtration, even using the motion induced by instabilities to provide rational models for predicting continental drift and volcanic activity. Our goal is to develop a model which will give us the wavelength of the most unstable mode and reconstruct the early onset of perturbations of the interface. The model will depend heavily on two ideas: the fluid-fluid interface and the energy in the fluids. We take a dynamical approach in order to be able to determine the wavelength which dominates the time evolution of the velocity field. Finally, we compare our model predictions with the experimental results to assess the validity, as well as attempt to recreate the evolution of the interface over time.

4. The Rayleigh-Taylor Instability Experiment Setup: 1) We filled a rectangular box first with a thin film of the heavy fluid (molasses), then poured the lighter fluid (water) on top, completely filling the box, and then allowed time for the two fluids to separate. 2) We filled half of a rectangular box with corn syrup then poured silicon oil on top until the box was completely filled. 3) Poured a thin layer of silicon oil on a piece of glass then flipped it over. Procedure: We flipped the box over so as to have the lighter fluid accelerated into the heavier fluid by gravity. For (1) we took pictures from the top to see the bubbles that formed; for (2) we used the pixel camera to capture the interfacial motion; for (3) we used the high- speed camera to capture the interfacial motion.

5. The Rayleigh-Taylor Instability Data Data collection*: Our goal is to find the wavelength of the most unstable mode. For (1) we measured the distance between centers of neighboring drops; for (2) we measured the distance between neighboring spikes; for (3) we measured the distance between peaks of drops. For (3) we also measured the height of the drops, to compare with our interface reconstruction. * All data collected from the experiment can be found found on the wiki.

6. The Rayleigh-Taylor Instability Theory Note: The following theory and model proposal follow an approach due to Chang and Bankoff (reference on wiki). We first assume that the fluid is incompressible and the motion is irrotational, and that the fluid motion in the horizontal direction is sinusoidal. We can then assume that the velocity potentials have the form:

7. The Rayleigh-Taylor Instability Theory We will consider two systems. First, we assume infinitesimally thin layers of fluid (i.e. y --> 0), for which the velocity field becomes: Second, we will look at infinite layers of fluid, for which the velocity field is:

8. The Rayleigh-Taylor Instability Theory In order to follow a particle we must find the particle path slopes: Now, let the initial position of the particle be given by, then integrating the above, and noting that at the interface, we find the equation of the interface:

9. The Rayleigh-Taylor Instability Model Now that we have an expression for how a particle moves along the interface, all we need to do is find q. In order to do this we use an energy balance over one wavelength. For this system there are several energies which must be considered – kinetic, potential and surface. However, we believe that viscosity affects the rate of deformation of the fluid.

10. The Rayleigh-Taylor Instability Model 1 First we consider infinitesimally thin layers of fluid, say height:. After computing the integrals we arrive at the following equation for q, F & K are on the wiki:

11. The Rayleigh-Taylor Instability Model 1 Now we consider q small:

12. The Rayleigh-Taylor Instability Model 1 Now we look at what dominates the time evolution of the interface:

13. The Rayleigh-Taylor Instability Model 1 In an attempt to find q, we consider the next order equation: We are currently unsure of the appropriate initial conditions, but for now we will go with:

14. The Rayleigh-Taylor Instability Model 2 For the infinite layers we first let h represent the height of the fluid. Computing the integrals results in:

15. The Rayleigh-Taylor Instability Model 2 Once again we consider small q:

16. The Rayleigh-Taylor Instability Model 2 To the leading order, q satisfies the equation: Now, to account for an infinite height of fluid:

17. The Rayleigh-Taylor Instability Model 2 There are two unknowns in the previous equation, k & q; in order to deal with this problem we let q assume a particular form: n = n(k) is the growth rate, which satisfies the equation: Our goal now is to find the wavenumber, k, associated with maximum growth, i.e.:

18. The Rayleigh-Taylor Instability Data Analysis Experiment vs Thin Layer Theory Experiment Average measured wavelength: Silicon Oil – Air: 12.17 mm Molasses – Water: 15.66 mm Model Most unstable wavelength: Silicon Oil – Air: 13.23 mm Molasses – Water: 7.11 mm Corn Syrup – Silicon Oil: 10.08 mm ~ 3.78 Dynes/cm

19. The Rayleigh-Taylor Instability Data Analysis Experiment vs Infinite Layer Theory Experiment Average measured wavelength: Corn Syrup - Silicon Oil: 4.85 mm Model Most unstable wavelength: Corn Syrup - Silicon Oil: 9.74 mm Molasses – Water: 7.14 mm Silicon Oil – Air: 8.14 mm ~ 4.84 Dynes/cm

20. The Rayleigh-Taylor Instability Interface 1 Molasses–Water Interface

21. The Rayleigh-Taylor Instability Interface 2 Corn syrup–Silicon oil Interface

22. The Rayleigh-Taylor Instability Interface 3 Silicon oil–Air Interface

23. The Rayleigh-Taylor Instability Conclusion There is most likely a lot of error in our liquid-liquid data due to the difficulty of finding good experimental methods and the inability to determine which spikes were on the same line. Our thin layer model appears to be a good approach; it compares very well with the good data that we have. The infinite layer model may be good, as can be seen by comparing to thin layer predictions (except for silicon oil-air). Also, for the liquid-liquid system, the interface takes on odd shapes, ones that could only be described by using a nonlinear ode for q – for future work we could either use an experimental wavelength or the thin layer approximation to use for k, then solve the original ode for q numerically.