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)