Diffusion on Surfaces of Revolution

Conference paper
Part of the Mathematics in Industry book series (MATHINDUSTRY, volume 17)

Abstract

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.

Keywords

Variational Principle Fundamental Solution Liouville Equation Industrial Device Axisymmetric Shape 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Carslaw, H.S., Jaeger, J.C.: Conduction of Heat in Solids. Clarendon Press, London (1959)Google Scholar
  2. 2.
    Courant, R., Hilbert, D.: Methoden der Mathematischen Physik. Springer, Berlin (1993)Google Scholar
  3. 3.
    Lew, A., Marsden, J.E., Ortiz, M., West, M.: Asynchronous variational integrators. Arch. Ration. Mech. Anal. 167, 85–146 (2003)Google Scholar
  4. 4.
    Ronveaux, A. (ed.): Heun’s Differential Equations. Oxford University Press, Oxford (1995)Google Scholar
  5. 5.
    Tung, M.M.: Basics of a differential-geometric approach to diffusion: Uniting Lagrangian and Eulerian models on a manifold. In: Bonilla, L.L., Moscoso, M.A., Platero, G., Vega, J.M. (eds.) Progress in Industrial Mathematics at ECMI 2006, Mathematics in Industry, vol. 12, pp. 897–901. Springer, Berlin (2007)Google Scholar
  6. 6.
    Tung, M.M., Hervás, A.: A differential-geometric approach to model isotropic diffusion on circular conic surfaces in uniform rotation. In: Fitt, A.D., Norbury, J., Ockendon, H., Wilson, E. (eds.) Progress in Industrial Mathematics at ECMI 2008, Mathematics in Industry, vol. 15, pp. 1053–1060. Springer, Berlin (2010)Google Scholar
  7. 7.
    Zwillinger, D.: Handbook of Differential Equations. Academic Press, San Diego (1998)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

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

Personalised recommendations