rayleigh-plateau instability
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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.
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