Pendant Drop Experiments & the Break-up of a Drop NJIT Math Capstone May 3, 2007 Azfar Aziz Kelly Crowe Mike De. Caro
Abstract ¡ A liquid drop creates a distinct shape as falls l l ¡ An assessment of the experimental drop shape with the simulated solution l ¡ point by point agreement is found Extract our computations in order to be able to calculate surface tension of a pendant drop l ¡ Pendant drop, shape described by a system of equations Use of Runge-Kutta numerical methods to solve these equations. 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 l ¡ Pesticide spray l ¡ Prevent splattering and satellite drops Drops that are too small with defuse in the air and not apply to the plant Fiber Spinning l 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 cam 101 goniometer in order to find The software calculated the surface tension by curve fitting of the Young. Laplace equation Liquid used: PDMS l Density: 0. 971 g/cm 3
The Experiment ¡ The mean experimental surface tension was = 18. 9.
The Experiment ¡ Schematic drawing l ¡ Used to find x and θ Other measurements were taken in order for numerical computations l l l determined by experiment = 0. 971 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
Constants Analysis ¡ In this ODE, there exists two constants b and c l b = curvature at the origin of coordinates l c = capillary constant of the system c=
c = -1 b = 2. 8 (red) b = 3 (blue)
b=2 c = -2 (red) c = -1 (blue) c = -. 5 (green)
Constant Analysis ¡ b Analysis l l l ¡ 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: l l l 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. 9 b=4. 1422 c=-5. 0348
Calculating Gamma ¡ Calculating surface tension from image l l Obtain image from CAM 101 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 m. N/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 ¡ l 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 l l ¡ 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/ind ex/cam ¡ http: //www. rps. psu. edu/jan 98/pinc hoff. html ¡ Nonlinear dynamics and breakup of free-surface flow, Eggers, Rev. Mod. Phys. , vol. 69, 865 (1997) ¡
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