1. Galina Lyutskanova Kiril Mihaylov Vasil Kolev Instructor: Tihomir Ivanov

2. Surface tension Surface tension is a property of fluids that makes them take a shape that minimizes their surface area.

3. Experiment A drop of a given fluid is subjected to rotation or to gravitational forces. Then, a photo of the drop is taken. This is the experimental profile of the drop.

4. Problem Picture Experimental Profile Surface tension Mathematical model Theoretical profile Surface tension Material Software that is used to control the measuring devices finds the value of the surface tension for which the theoretical and experimental profiles coincide.

5. Drop shape analysis methods The drop shape analysis are : easy to do; can be used in difficult experimental conditions; require only a small amount of the liquid material; can be used for real-time estimations.

6. Axisymmetric drop shape analysis (ADSA) ADSA is a powerful drop shape method. It is fast, easy to handle and produce accurate results. The corner stone of this method is the fact that any given drop is axisymmetric. Using this assumption with the help of the Young-Laplace equation we can efficiently analyze the shape of any given drop. ADSA is used in various systems such as tensiometers.

7. Implementation of ADSA Two possible settings are pendant drop and rotating drop. In both the settings we approach the problem in the same way. We create a program that processes the image of the drop that we are going to examine in order to obtain the cloud of points. Then, we acquire differential equations that describe the influence of the interfacial tension and the gravity on the shape of the drop. These equations are dependent on parameters with the help of which we can identify the interfacial tension.

8. Give approximate discrete solutions of the differential equations using the Euler and RK methods. By optimizing the error of our approximations we identify the parameters on which our equations depend and find the interfacial tension. The only difference is in the differential equations but it has no substantial effects on any of the steps.

9. Image Processing

10. Steps for Contour Extraction

11. Solution of the Young – Laplace Equation

12. The Young – Laplace Equation of Capillarity

13. The model for the pendant drop

15. Substituting (2) and (3) into (1) and parameterizing the curve with the arc length s, we obtain z With initial conditions b

16. Euler method Runge-Kutta method Numerical Solution of the ODE System

17. ODE45 – One-step solver, based on a Runge-Kutta method ODE113 – Multistep solver, based on the Adams-Bashforth Methods big numerical errors

18. Solutions with RK4 b = 1 c = [ 0.05 – 2.5]

19. Optimization

20. Parametric identification algorithm

21. Optimization Gradient descent Gauss-Newton

22. Gradient descent Gauss-Newton

23. Gradient descent Gauss-Newton

24. Thank you for your attention!