Pendant Drop Experiments & the Break-up of a Drop NJIT Math Capstone May 3, 2007 Azfar Aziz Kelly Crowe Mike 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: g/cm 3

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 = g/cm 3 = 9.8 m/s 2

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

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

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

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

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 = θ=23 =18.9 b= c=

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 = = Goniometer =18.9 true = 19.8 mN/m at 68f (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  ex/cam ex/cam  hoff.html hoff.html  Nonlinear dynamics and breakup of free-surface flow, Eggers, Rev. Mod. Phys., vol. 69, 865 (1997) Nonlinear dynamics and breakup of free-surface flow