Diffusion on Surfaces of Revolution

  • Michael M. Tung
Conference paper
Part of the Mathematics in Industry book series (MATHINDUSTRY, volume 17)


Diffusion processes play a fundamental role in mathematical models and simulation for particle and fluid dynamics, or heat transport. Many industrial devices have simple design, in particular the shape of surfaces of revolution. For sourceless diffusion on surfaces of revolution, we introduce a Hamiltonian system on a manifold and derive the governing diffusion equation from a variational principle. For axisymmetric shapes these equations of motion reduce to self-adjoint eigenvalue problems. This property is important as it guarantees completeness of the corresponding fundamental solutions. Also, we prove that the dependence on the height is given by a Sturm–Liouville equation. As an example, we tackle diffusion on a circular paraboloid as found in mirror telescopes or antennas among other applications. These analytical results may provide a valuable theoretical tool for related diffusion models.


Variational Principle Fundamental Solution Liouville Equation Industrial Device Axisymmetric Shape 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Instituto Universitario de Matemática MultidisciplinarUniversidad Politécnica de ValenciaValenciaSpain

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