Transcript Practical Applications - New Jersey Institute of Technology

Pendant Drop Experiments
& the Break-up of a Drop
NJIT Math Capstone
May 3, 2007
Azfar Aziz
Kelly Crowe
Mike DeCaro
Abstract
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A liquid drop creates a distinct shape as falls
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An assessment of the experimental drop shape with
the simulated solution
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point by point agreement is found
Extract our computations in order to be able to
calculate surface tension of a pendant drop

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Pendant drop, shape described by a system of equations
Use of Runge-Kutta numerical methods to solve these
equations.
minimizing the difference between computed and
measured drop shapes
High speed camera was used to analyze the breakup of
a pendant drop.
Practical Applications
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Ink Jet Printers

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Pesticide spray
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Prevent splattering and satellite
drops
Drops that are too small with
defuse in the air and not apply to
the plant
Fiber Spinning
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Opposite of break-up of drop – in
this case prevent the threads from
breaking
The Experiment
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Experimental procedures were done to
determine the surface tension
The cam101 goniometer in order to find
The software calculated the surface
tension by curve fitting of the YoungLaplace equation
Liquid used: PDMS
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Density: 0.971 g/cm3
The Experiment
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The mean experimental surface tension
was = 18.9.
The Experiment
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Schematic drawing
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Used to find x and θ
Other measurements
were taken in order for
numerical computations
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determined by
experiment
= 0.971 g/cm3
= 9.8 m/s2

Numerical Experiment
The profile of a drop can be described by the
following system of ordinary differential
equations as a function of the arc length s
Runge-Kutta for System of Equations
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Runge-Kutta was used to approximate
shape of a drop in Matlab.
Input data: x, z, and θ
i 1 3
f1 ( x, z,  )  cos( )
k1,i  h  f i ( x, z ,  )
f 2 ( x, z,  )  sin( )
1
1
1
k2,i  h  f i ( x   k1,1 , z   k1,2 ,    k1,3 )
2
2
2
1
1
1
k3,i  h  f i ( x   k2,1 , z   k 2,2 ,    k 2,3 )
2
2
2
k4,i  h  f i ( x  k3,1 , z  k3,2 ,   k3,3 )
sin( )
f 3 ( x, z ,  )  2  b  c  z 
x
Constants Analysis

In this ODE, there exists two constants b
and c
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b = curvature at the origin of coordinates
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c = capillary constant of the system
c=
   g

1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
c = -1
b = 2.8 (red)
b = 3 (blue)
0.4
0.6
0.8
1
2
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
-2
-1.5
-1
-0.5
0
0.5
b=2
c = -2 (red)
c = -1 (blue)
c = -.5 (green)
1
1.5
2
Constant Analysis
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b Analysis
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Varying b causes the profile to become larger or
smaller depending on how b is affected.
The shape remains the same.
The size of the drop is inversely proportional to b
c Analysis:
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Varying c causes the profile to curve greater at the
top
The initial angles of the profile are the same, yet at
the top of the drop, the ends begin to meet.
The curvature of the drop is proportional to c
Numerical vs. Experiment Results
x =0.0943 θ=23
=18.9
b=4.1422 c=-5.0348
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
Calculating Gamma
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Calculating surface tension from
image
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Obtain image from CAM101 and
extracted points (via pixel correlation)
Minimize difference between theoretical
points and those from the image
Determine constants b,c
Calculate surface tension from c
Determining Gamma
b = 3.73
c = -5.90
= 16.1285
• Goniometer
=18.9
• true
= 19.8 mN/m at
68f (dependant on temp.)
Pendant Drop Breakup
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Use of high speed camera to
compare theoretical predictions of
breakup
Compared results to paper by
Eggers
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Nonlinear dynamics and breakup of
free-surface flow, Eggers, Rev. Mod.
Phys., vol. 69, 865 (1997)
Pendant Drop Breakup
Before Breakup
Left: Experiment
Right: Eggers
At Breakup
Left: Experiment
Right: Eggers
Conclusion
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Confirmed experiments with theory
through Matlab simulation
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Determination of drop shape given size
and surface tension
Determination of surface tension given
shape of drop
Compared break-up experiment
with Eggers results
References
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http://www.ksvltd.com/content/ind
ex/cam
http://www.rps.psu.edu/jan98/pinc
hoff.html
Nonlinear dynamics and breakup of
free-surface flow, Eggers, Rev. Mod.
Phys., vol. 69, 865 (1997)