Post on 29-Dec-2015
Pendant Drop Experiments& the Break-up of a Drop
NJIT Math Capstone May 3, 2007
Azfar AzizKelly CroweMike DeCaro
Abstract
A liquid drop creates a distinct shape as falls Pendant drop, shape described by a system of equations Use of Runge-Kutta numerical methods to solve these
equations. An assessment of the experimental drop shape with the
simulated solution point by point agreement is found
Extract our computations in order to be able to calculate surface tension of a pendant drop
minimizing the difference between computed and measured drop shapes
High speed camera was used to analyze the breakup of a pendant drop.
Practical Applications Ink Jet Printers
Prevent splattering and satellite drops
Pesticide spray Drops that are too small with
defuse in the air and not apply to the plant
Fiber Spinning Opposite of break-up of drop – in
this case prevent the threads from breaking
The Experiment
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 Young-Laplace equation
Liquid used: PDMS Density: 0.971 g/cm3
The Experiment
The mean experimental surface tension was = 18.9.
The Experiment
Schematic drawing Used to find x and θ
Other measurements were taken in order for numerical computations 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
Runge-Kutta was used to approximate shape of a drop in Matlab.
Input data: x, z, and θ
1
2
3
( , , ) cos( )
( , , ) sin( )
sin( )( , , ) 2
f x z
f x z
f x z b c zx
1,
2, 1,1 1,2 1,3
3, 2,1 2,2 2,3
4, 3,1 3,2 3,3
1 3
( , , )
1 1 1( , , )
2 2 21 1 1
( , , )2 2 2
( , , )
i i
i i
i i
i i
i
k h f x z
k h f x k z k k
k h f x k z k k
k h f x k z k k
In this ODE, there exists two constants b and c b = curvature at the origin of coordinates
c = capillary constant of the system
c =
g
Constants Analysis
c = -1b = 2.8 (red)b = 3 (blue)
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
b = 2c = -2 (red)c = -1 (blue)
c = -.5 (green)
-2 -1.5 -1 -0.5 0 0.5 1 1.5 20
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Constant Analysis
b Analysis 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: 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 =18.9b=4.1422 c=-5.0348
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.50
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Calculating Gamma
Calculating surface tension from image 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 68f (dependant on temp.)
Pendant Drop Breakup
Use of high speed camera to compare theoretical predictions of breakup
Compared results to paper by Eggers 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
Confirmed experiments with theory through Matlab simulation 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
http://www.ksvltd.com/content/index/cam
http://www.rps.psu.edu/jan98/pinchoff.html
Nonlinear dynamics and breakup of free-surface flow, Eggers, Rev. Mod. Phys., vol. 69, 865 (1997)