The Rayleigh-Taylor Instability

Jenna BratzRachel Bauer

Introduction

• Rayleigh Taylor Instability occurs when a denser fluid is being accelerated by a lighter fluid

• This project deals with silicone oil as the denser fluid, and air as the less dense fluid

• Theory will attempt to explain the most unstable wavelength, critical wavelength, and 2-D pattern formation

Literature

• Fermigier et al. discussed patterns, dominant wavelength, investigated dripping

• Sir Geoffrey Taylor describes instability and imposes a vertical acceleration larger than gravity on the fluid (Verified by Lewis)

• Ott described a model assuming zero thickness

• Chen and Fried investigated a liquid-liquid interface and showed viscosity doesn’t change the ultimate formation of the instability

Procedure

- A glass plate was placed on a picture frame with graph paper placed underneath

- 2 or 3 mL of silicon oil were poured at varying viscosities

- 6 Trials were conducted, each allowing the oil to spread for 5 minutes

- Glass was flipped over and pictures were taken as the instability evolved

Procedure (cont’d)

-One trial (3 mL at 1000 cst) was left for 4 hours to spread

- Pictures were taken until dripping was observed

Data (cont’d)

• 350 cst, 3 mL, sitting for 5 minutes before flipping

Data

• Matlab was used for measurements

• Measurements were taken (when visible):– Between drops in each ‘layer’– Diameter of drops– Distance between ‘layers’– Thickness of ‘layers’

Data (cont’d)

• Diameter of drops were measured for up to 10 drops in each trial

• Std. Dev.= .0416 cm• Mean= 1.014 cm

Data (cont’d)

• The outer ‘layer’ is the first visible ring formed in the fluid

• Distance between the centers of the drops was measured

• Std. Dev.= .2747 cm• Mean= 1.6431 cm

Data (cont’d)

• Distance between the centers of the drops in the second layer were measured

• Std. Dev.= .19794 cm

• Mean= 1.5377 cm

Theory

• Most Unstable Wavelength (wavelength you actually see)

• Wavelength for which the system is stable/unstable

• 6-Axis symmetry in the 2-D pattern that forms

Most Unstable Wavelength

• Dimensional Analysis– depends on surface tension ( ), density ( ),

and gravity (g).

– Matching up the units yields the following equation:

Most Unstable Wavelength (cont’d)

• Matching up the exponents gives a system of equations, which leads to a result which still has a dimensionless constant:

• Though we could not obtain the value of C, it is expected to be 2*Pi*sqrt(2). (Fermigier )

Most Unstable Wavelength (cont’d)• Use data to estimate c.

Most Unstable Wavelength (cont’d)

• Observed C was measured using

• Known C was

Average Observed C was 8.8821

• Error only .0036

Stability

• Consider the question, for what wavelength does the system become unstable?

• Compare energy of initial cylinder with perturbed cylinder

• First perturb the height by only looking at a radial perturbation

Stability (cont’d)

Stability (cont’d)

• First, introduce a volume constraint since the volume is constant.

• Since volume is constant, all epsilon terms go to zero.

Stability (cont’d)• Want to find difference between the energy of the

perturbed (E[u(r)]) and the unperturbed system (E[h]).

• Energy is proportional to the surface area minus

gravitational potential energy. 

Stability (cont’d)

• Want to look at E[h]-E[u(r)]:

• But the volume constraint result gave:

Stability (cont’d)

• Since we expect the same outcome for each wavelength, we can just look at one wavelength and when the system is stable for this wavelength. So let

• Then

Stability (cont’d)

• Simplify to obtain:

• This is >0 when , and so when

the system is unstable.And, the system is stable for

Stability (cont’d)• Surprising result?

– This critical value of lambda is approximately 2.3468 cm, which is greater than all of the experimental wavelengths measured (including the theoretically calculated most unstable wavelength which was 1.324 cm)

– Would expect the most unstable wavelength to be greater than the critical wavelength since the system reaches instability after wavelength reaches this critical length

Stability (cont’d)

• Now consider perturbing in the theta direction, as well as with respect to the radius of the cylinder

• Set-Up is similar except now the perturbed height becomes

• And the Energy becomes

Stability (cont’d)

• Now,

• Similar to just perturbing the radius, we should consider E[h]-E[u(r,theta)], this would give a condition on both w and gamma for stability

Conclusion

• Data validated that the most unstable wavelength is equal to:

Where is capillary length

Conclusion (cont’d)

• Although it may not coincide with measured data, when

The system becomes unstable. So for wavelengths exceeding this wavelength, ‘rings’ will form, so the ‘pancake’ of fluid will not retain its shape.

Conclusion (cont’d)

• Considering the energy difference for a perturbation in both the direction of the radius and theta should give another stability condition, showcasing the symmetry of the 2-D pattern.

Further Work

• Over what time period does the instability occur?

• Verify the critical wavelength with more data.

• Theoretically obtain the constant C in the expected wavelength

• Explore directing the fluid into different shapes, and looking at the effect on the instability

The End.

Questions??