Rayleigh-Plateau Instability

Rayleigh-Plateau Instability

By:Qiang Chen and Stacey Altrichter

By:Qiang Chen and Stacey Altrichter

IntroductionIntroduction 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.

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.

Study of InstabilityStudy 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.

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.

ProcedureProcedure 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.

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.

ProcedureProcedure 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.

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.

ProcedureProcedure 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.

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.

Photographs and DataPhotographs and Data

Blue Thread with Motor OilBlue Thread with Motor Oil

Data of Blue StringData of Blue String Drop size (Pixel

) Drop size (mm)

Drops distance (Pixel)

Drops distance (mm)

1 8.00 0.8163 2 7.75 0.7908 20.88 2.131 3 6.00 0.6122 28.63 2.921 4 5.50 0.5612 5 4.50 0.4592 28.12 2.869 6 4.75 0.4847 26.63 2.717 7 8.00 0.8163 29.63 3.023 8 8.25 0.8418 26.88 2.743 9 6.03 0.6153 38.63 3.942 10 9.70 0.9898 46.37 4.732 11 7.80 0.7959 33.88 3.457 12 6.75 0.6888 33.12 3.380 13 5.50 0.5612 26.78 2.733 14 8.05 0.8214 23.88 2.437 string 2.25 0.2296

Fishing Line with Motor OilFishing Line with Motor Oil

Data of Fishing WireData of Fishing Wire Drop size (Pixel

) Drop size (mm)

Drops distance (Pixel)

Drops distance (mm)

1 9.50 0.9694 2 7.50 0.7653 21.50 2.1939 3 9.25 0.9439 32.75 3.342 4 8.00 0.8163 33.00 3.367 5 9.25 0.9439 41.25 4.209 6 6.25 0.6378 26.00 2.653 7 7.50 0.7653 28.00 2.857 8 8.50 0.8673 59.50 6.071 9 7.00 0.7143 35.50 3.622 10 8.75 0.8929 59.50 6.071 11 8.00 0.8163 62.25 6.352 string 3.76 0.3837 47.50 4.847

Green Metal Wire w/ Motor Oil

Green Metal Wire w/ Motor Oil

Pasta with Motor OilPasta with Motor Oil

Theory on the InstabilityTheory on the Instability

Shape of DropletsShape 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.

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.

Shape of DropletsShape of Droplets Let z be the horizontal axis running through

center of strand. Define

droplet length to be from z=0 to z=L 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 z=L/2. The max height occurs at z=L/2.

Take to be the angle as shown below.

Let z be the horizontal axis running through center of strand.

Define droplet length to be from z=0 to z=L 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 z=L/2. The max height occurs at z=L/2.

Take to be the angle as shown below.

Energy EquationEnergy Equation

Volume ConstraintVolume Constraint

E[r(z)] r 1 (r')2

0

L0

2 dzd

V r2 R02

0

L dz

E[r(z)]2 r 1 (r')2

0

L dz

Lagrange Multipliers

Lagrange Multipliers

constant, V constraint

minimize energy equation

constant, V constraint

minimize energy equation

E *[r(z)]E V

E *[r(z)] 2r 1 (r')2

0

L (r2 R02)dz

Beltrami’s IdentityBeltrami’s Identity

HERE WE HAVE:HERE WE HAVE:

F r'Fr'C0

F 2r 1 (r')2 (r2 R02)

r2

2

r1 (r')2

C1

wherewhere

C1 C0

2R0

2

2

where C0 is constant

where C0 is constant

Assumption of Perfect Symmetry Assumption of Perfect Symmetry

when r=rmax (which occurs at z=L/2) r’=0. Hencewhen r=rmax (which occurs at z=L/2) r’=0. Hence

rmax

2

2rmax C1

2

(rmax2 r2) r

1 (r')2 rmax

0

Euler-Lagrange EquationEuler-Lagrange Equation

finding a condition on finding a condition on

Frd

dz

Fr'

0

F 2r 1 (r')2 (r2 R02)

r' '

(1 (r')2)3 / 2 1 (r')2

rConditions:Conditions:

r(0)R0

r'(0)

1 (r'(0))2tan

Condition for Condition for Find so that the following conditions also holdFind so that the following conditions also hold

r(L)R0

r'(L)

1 (r'(L))2tan

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.

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.

Wavelength of InstabilityWavelength 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

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

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

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

r(z)R0 R1 cos(kz)2R2

VolumeVolume

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

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

V r2(0

z )d

V R02 2R0R1 cos(k ) 22R0R2

2R12

2(1 cos(2k))

0

z d O(3)

V R02 2R0R1

ksin(k) 22R0R2

2R12

2

2

4ksin(2k)

0

z

O(3)

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

V z R02 2 2R0R2

R12

2

O(3)

VolumeVolume

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

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

R2 R1

4R0

Surface Area of FluidSurface Area of Fluid

SA r 1 (r')2

0

z d

Binomial Expansion: Binomial Expansion:

r(1 (r')2) r 11

2(r')2

SA R0 R1 cos(k ) 2 R12

4R0

1

2k 2R12

2sin2(k)

0

z d

SAz 2R0 2R1

2

2R0

(k 2R0

2 1)

O(3)

Potential EnergyPotential Energy

Define the wavelength to be =2/k. For >2R0 the potential energy P

negative making the system is unstable. For <2R0 the potential energy will be

positive which makes the system stable.

Define the wavelength to be =2/k. For >2R0 the potential energy P

negative making the system is unstable. For <2R0 the potential energy will be

positive which makes the system stable.

P 2R1

2

2R0

(k 2R02 1)T

According to the surface area eq, the potential energy eq is given by

According to the surface area eq, the potential energy eq is given by

Laplace-Young LawLaplace-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.

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.

P 2H

Mean CurvatureMean Curvature2

2 3/ 2

1 '' (1 ( ') )

2 (1 ( ') )

r r rH

r r

H 1

2

k 2R0R1 cos(kz) 2k 2R12 1

r(12k 2R12 sin2(kz))3 / 2

We were unable to find a way to establish the sign of H with certainty.We were unable to find a way to establish the sign of H with certainty.

Non-axisymmetricNon-axisymmetricPerturbation of r becomes:Perturbation of r becomes:

r(z,)R0 R1 cos(kz)cos(m)2R2

where m0 is an integer that identifies the angular mode. The corresponding potential energy is of the form:

where m0 is an integer that identifies the angular mode. The corresponding potential energy is of the form:

P 2R1

2

2R0

(k 2R02 1m2)T

Hence P >0 making the system always stable.

Hence P >0 making the system always stable.

AnalysisAnalysis 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.

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.

ConclusionConclusion 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.

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.