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35 th Conference Union of Bulgarian Mathematicians 5- 8 April 2006 Borovetc

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

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35 th Conference Union of Bulgarian Mathematicians 5- 8 April 2006 Borovetc

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  1. 35th Conference Union of Bulgarian Mathematicians5- 8 April 2006 Borovetc On Balloons, Membranes And Surfaces Representing Them Elena Popova, Mariana Hadzhilazova, Ivailo Mladenov Institute of Biophysics Acad. G. Bontchev Str., Bl. 21, Sofia-1113, Bulgaria

  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 e3 = k const

  4. Forces • Internal forces where, σm- meridional stress resultant σc- circumferentialstress 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 SurfacesEquations • 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 • p0 = 1.0 Consequently • k = 0.9241763715

  11. Nodoids • C = -0.4 • p0 = 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

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