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

The Rayleigh-Taylor Instability Outline ● Introduction ● Experiment ● Data ● Theory ● Model ● Data Analysis ● Size & Shape of Bubble ● Conclusion

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. We would like to develop a model which will give us the wavelength of the most unstable mode. In order to determine the accuracy of the model we must first have some data with which to compare our results. The case we will examine in our experiment is when the heavier fluid (silicon oil) is supported against gravity by the lighter fluid (air). The model will depend heavily on two ideas: the liquid-air interface and the energy in the heavy fluid. We take a dynamical approach in order to be able to determine the wavelength associated with maximum growth. Finally, we compare our model predictions with the experimental results to assess the validity, as well as use the wavelength to find both the size of the 'circles' as well as show the 2D formation.

The Rayleigh-Taylor Instability Experiment Setup: We poured silicon oil of different viscosities onto a sheet of glass and then placed a rectangular box around the fluid. We then let the liquid spread until it hit the entire boundary. We then flipped the glass over and proceded to photograph the ensuing instability. Procedure: We tried our best to capture the initial rolls, but failed. However, after several trials we discovered that the bubbles actually formed ON the rolls we were trying to film. We also saw that the initial rolls formed in both directions, which can be seen on the next slide.

The Rayleigh-Taylor Instability Data Data collection*: Our goal is to find the wavelength of the most unstable mode. As noted on the previous slide, we assumed the experimental wavelengths are measured between centers of bubbles both horizontally and vertically. * All data collected from the experiment can found found on the wiki.

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 potential has the form: From which we get the velocity field:

The Rayleigh-Taylor Instability Theory We now consider an infnitessimally thin layer of fluid (i.e. y --> 0), so the velocity field becomes: Although we assume ”zero” thickness, we will still consider a change in the y-direction as a function of x, this will allow us to follow a particle along the interface.

The Rayleigh-Taylor Instability Theory In order to follow a particle we must find the particle path slope: Now, let the initial position of the particle be given by, then integrating the above we get:

The Rayleigh-Taylor Instability Theory Next, we would like to eliminate the term from the y-equation, so we do the following: Then, expanding about gives us: From which we get that the interface can be described by:

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. Hence, we believe that the rate of change of kinetic, potential and surface energy is balanced by viscous dissipation:

The Rayleigh-Taylor Instability Rate of Kinetic Energy

The Rayleigh-Taylor Instability Rate of Potential Energy

The Rayleigh-Taylor Instability Rate of Surface Energy

The Rayleigh-Taylor Instability Viscous Dissipation

The Rayleigh-Taylor Instability Model Combining the previous four equations we arrive at the following non-linear ODE: Which we can rewrite as:

The Rayleigh-Taylor Instability Model Unfortunately, we do not know what to do with the natural log, so for now we are going to ignore it, and proceed with some hopefully useful results. Let, where n is the linear growth constant, and we arrive at the following equation for n: From which we get: The next step in our procedure is to linearize this ODE, which is given by the following:

The Rayleigh-Taylor Instability Model Finally, we want to find the fastest growing wavenumber (i.e. find k such that n(k) is maximized). So we solve the following for k: Thus, given the density, surface tension and viscosity of the liquid we can find the most unstable wavelength: As we will see later, this gives us a wavelength not even close to the one measured, thus the natural log must be important. The next few sides are our attempt at dealing with this ugly term.

The Rayleigh-Taylor Instability Model

The Rayleigh-Taylor Instability Model For small q: Substituting into our nonlinear ode, we have: or,

The Rayleigh-Taylor Instability Model For q small we let: Substituting we get: where

The Rayleigh-Taylor Instability Model After a Taylor expansion we arrive at the equation:

The Rayleigh-Taylor Instability Model Now, multiply through by epsilon and take the limit as epsilon goes to zero to arrive at the leading order equation: Thus, we get that the wavenumber which satisfies the energy balance to the leading order must be: Hence the wavelength is:

The Rayleigh-Taylor Instability Data Analysis Experiment vs Theory Experiment Average measured wavelength is: 14.4 mm Incorrect Model Most unstable wavelength is: mm 2 nd attempt Most unstable wavelength is: mm

The Rayleigh-Taylor Instability 2D shape & formation Looking down at the glass it seems clear that the cross-section of the bubbles is a circle. We also see, with our rectangular setup, equal spacing between these circles horizontally and vertically. Thus, since the wavelength can be seen as the distance between centers of neighboring circles, if we can find the diameter of the circle we can obviously draw a circle, but we will also be able to recreate the 2D formation. Knowing the wavelength will allow us to set up a grid of equally spaced dots (representing the centers of circles). Now all we have to do is find the diameter. To find the diameter we take the difference between the wavelength and the shortest distance between neighboring circles (which must come from the experiment), leaving us with 2*radius = diameter.

The Rayleigh-Taylor Instability 2D shape & formation From our measured data we have that the shortest distance between two circles is approximately 2.9 mm. If we define the capillary wavelength to be: then, with our data: So, we can say:

The Rayleigh-Taylor Instability Conclusion Taking a dynamical approach to the problem, we were able to find the wavelength of the instability by first determining the velocity field, including the the effects of time, q(t). We then described the interface as a function of x. Next we set up a balance between the change in energy in the system and the rate of deformation of the fluid due to viscosity. From this energy balance we were able to determine a differential equation from which we will be able to describe q. But, for small q we noticed terms of different orders, so taking an asymptotic expansion for small q, we find that to the leading order, the dominant features are the gravitational and surface energy, which is expected. To this order, we find two wavenumbers: 0, which corresponds to the undisturbed fluid, and which corresponds to the wavelength of the instability.

The Rayleigh-Taylor Instability Conclusion Knowing the wavelength we can now set up a grid that shows the equal spacing between centers of neighboring circles in the 2D formation. In order to draw the actual circles we need to know the diameter. Comparing our measurements to the capillary wavelength, we notice that the shortest distance between neighboring circles is approximately. Since, theoretically, the circles are evenly spaced, we find that the diameter of each circle satisfies: In conclusion, we were able to find the most unstable wavelength, the diameter of the drops at the plate, and the rectangular 2D pattern that these drops form.