Pendant Drop Experiments & the Break-up of a Drop

1. Pendant Drop Experiments& the Break-up of a Drop NJIT Math Capstone May 3, 2007 Azfar Aziz Kelly Crowe Mike DeCaro

2. 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.

3. 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

4. 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

5. The Experiment • The mean experimental surface tension was = 18.9.

6. 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

7. 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

8. Runge-Kutta for System of Equations • Runge-Kutta was used to approximate shape of a drop in Matlab. • Input data: x, z, and θ

9. Constants Analysis • In this ODE, there exists two constants b and c • b = curvature at the origin of coordinates • c = capillary constant of the system c =

10. c = -1 b = 2.8 (red) b = 3 (blue)

11. b = 2 c = -2 (red) c = -1 (blue) c = -.5 (green)

12. 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

13. Numerical vs. Experiment Results x =0.0943 θ=23 =18.9 b=4.1422 c=-5.0348

14. 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

15. Determining Gamma • b = 3.73 • c = -5.90 • = 16.1285 • Goniometer =18.9 • true = 19.8 mN/m at 68f (dependant on temp.)

16. 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)

17. Pendant Drop Breakup

18. Before Breakup Left: Experiment Right: Eggers

19. At Breakup Left: Experiment Right: Eggers

20. 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

21. 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)