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Equilibrium Equations for Surfaces of Delaunay: Balloons, Membranes, and Forces

Explore the formulation and solution of equilibrium equations for surfaces of Delaunay, focusing on balloons, membranes, and the forces affecting them. Delve into the parameters, shapes, and characteristics of various surface profiles. Future goals include a comprehensive solution to the equilibrium equation system.

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Equilibrium Equations for Surfaces of Delaunay: Balloons, Membranes, and Forces

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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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