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.

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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.

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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.

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Photographs and Data
8
Blue Thread with Motor Oil
9
Data of Blue String
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Fishing Line with Motor Oil
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Data of Fishing Wire
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Green Metal Wire w/ Motor Oil
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Pasta with Motor Oil
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Theory on the Instability
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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.

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Shape of Droplets

• Let z be the horizontal axis running through
  center of strand.
• Define
• droplet length to be from z0 to zL
• 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 zL/2. The max height occurs at zL/2.
• Take ? to be the angle as shown below.

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Energy Equation

• Volume Constraint

Lagrange Multipliers
? constant, V constraint minimize energy equation
18
Beltramis Identity
where C0 is constant

• HERE WE HAVE

where
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Assumption of Perfect Symmetry ?

• when rrmax (which occurs at zL/2) r0. Hence

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Euler-Lagrange Equation

• finding a condition on ?

Conditions
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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.
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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

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• 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

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Volume

• Here we have a constant volume and it can be
  computed as follows

Due to periodicity, all sine functions are zero
here.
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Volume

• Since the volume of liquid spread on the strand
  is constant, all ?2 terms will go to zero and
  which implies

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Surface Area of Fluid
Binomial Expansion
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Potential Energy
According to the surface area eq, the potential
energy eq is given by

• Define the wavelength to be ?2?/k.
• For ?gt2?R0 the potential energy P negative making
  the system is unstable.
• For ?lt2?R0 the potential energy will be positive
  which makes the system stable.

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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.

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Mean Curvature
We were unable to find a way to establish the
sign of H with certainty.
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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 gt0 making the system always stable.
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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.

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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.