Pendant Drop Experiments& the Break-up of a Drop

NJIT Math Capstone May 3, 2007

Azfar AzizKelly CroweMike DeCaro

AbstractA liquid drop creates a distinct shape as fallsPendant 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 foundExtract our computations in order to be able to calculate surface tension of a pendant dropminimizing the difference between computed and measured drop shapes High speed camera was used to analyze the breakup of a pendant drop.

Practical ApplicationsInk Jet PrintersPrevent splattering and satellite dropsPesticide sprayDrops that are too small with defuse in the air and not apply to the plantFiber SpinningOpposite of break-up of drop in this case prevent the threads from breaking

The ExperimentExperimental procedures were done to determine the surface tension The cam101 goniometer in order to findThe software calculated the surface tension by curve fitting of the Young-Laplace equationLiquid used: PDMSDensity: 0.971 g/cm3

The ExperimentThe mean experimental surface tension was = 18.9.

The ExperimentSchematic drawingUsed to find x and Other measurements were taken in order for numerical computations determined by experiment = 0.971 g/cm3 = 9.8 m/s2

Numerical ExperimentThe 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 EquationsRunge-Kutta was used to approximate shape of a drop in Matlab.Input data: x, z, and

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

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

Constant Analysisb AnalysisVarying 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 bc Analysis:Varying c causes the profile to curve greater at the topThe 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 Resultsx =0.0943 =23 =18.9b=4.1422 c=-5.0348

Calculating GammaCalculating surface tension from imageObtain image from CAM101 and extracted points (via pixel correlation) Minimize difference between theoretical points and those from the imageDetermine constants b,cCalculate surface tension from c

Determining Gammab = 3.73c = -5.90 = 16.1285 Goniometer =18.9 true = 19.8 mN/m at 68f (dependant on temp.)

Pendant Drop BreakupUse of high speed camera to compare theoretical predictions of breakupCompared results to paper by EggersNonlinear dynamics and breakup of free-surface flow, Eggers, Rev. Mod. Phys., vol. 69, 865 (1997)

Pendant Drop Breakup

Before BreakupLeft: ExperimentRight: Eggers

At BreakupLeft: ExperimentRight: Eggers

ConclusionConfirmed experiments with theory through Matlab simulationDetermination of drop shape given size and surface tensionDetermination of surface tension given shape of dropCompared break-up experiment with Eggers results

Referenceshttp://www.ksvltd.com/content/index/cam http://www.rps.psu.edu/jan98/pinchoff.htmlNonlinear dynamics and breakup of free-surface flow, Eggers, Rev. Mod. Phys., vol. 69, 865 (1997)