The Rayleigh-Taylor Instability
Jenna Bratz
Rachel 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 (
and gravity (g).
), density ( ),
– 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
the system is unstable.
And, the system is stable for
so when
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??