1. 35 th Conference Union of Bulgarian Mathematicians 5- 8 April 2006 Borovetc Elena Popova, Mariana Hadzhilazova, Ivailo Mladenov Institute of Biophysics Acad. G. Bontchev Str., Bl. 21, Sofia-1113, Bulgaria On Balloons, Membranes And Surfaces Representing Them

2. Plan  Surface Definition  Forces & Equilibrium Equations  Parameters  Surfaces of Delaunay - Unduloids - Nodoids  The Mylar Balloon

3. Equilibrium equations for an axisymmetric membrane.  The Generating Curve  The Surface where φ is the rotation angle, and e 3 = k const

4. Forces  Internal forces where, σ m - meridional stress resultant σ c - circumferential stress resultant. t - the tangent vector  External forces n- normal p- hydrostatic differential pressure w – the film weight density

5. Equilibrium equations where,

6. Shapes and Surfaces Delaunay Surfaces The Mylar Balloon

7. Delaunay Surfaces Equations  Mean curvature  Equilibrium Equations

8. Delaunay Surfaces Where, And C is a integration constant

9. Delaunay Surfaces Profile Curves  Cylinder H =1/2R  Sphere H = 1/R  Catenoid H = 0

10. Unduloids  C = 0.4  p 0 = 1.0 Consequently  k = 0.9241763715

11. Nodoids  C = -0.4  p 0 = 1.0 Consequently  k = 0.9892996329

12. The Mylar Balloon  Equilibrium Equations  Solution

13. The Mylar Balloon Profile and Shape

14. Future Goals  Studying other classes  Complete Solution of the Equilibrium Equation System