Abstract

A mixed formulation is proposed and analyzed mathematically for coupled convection-diffusion in heterogeneous medias. Transfer in solid parts driven by pure diffusion is coupled with convection-diffusion transfer in fluid parts. This study is carried out for translation-invariant geometries (general infinite cylinders) and unidirectional flows. This formulation brings to the fore a new convection-diffusion operator, the properties of which are mathematically studied: its symmetry is first shown using a suitable scalar product. It is proved to be self-adjoint with compact resolvent on a simple Hilbert space. Its spectrum is characterized as being composed of a double set of eigenvalues: one converging towards $-\infty$ and the other towards $+\infty$, thus resulting in a nonsectorial operator. The decomposition of the convection-diffusion problem into a generalized eigenvalue problem permits the reduction of the original three-dimensional problem into a two-dimensional one. Despite the operator being nonsectorial, a complete solution on the infinite cylinder, associated to a step change of the wall temperature at the origin, is exhibited with the help of the operator's two sets of eigenvalues/eigenfunctions. On the computational point of view, a mixed variational formulation is naturally associated to the eigenvalue problem. Numerical illustrations are provided for axisymmetrical situations, the convergence of which is found to be consistent with the numerical discretization.

MSC codes

  1. 75R99
  2. 46-99
  3. 35A15
  4. 65M60

Keywords

  1. convection-diffusion
  2. variational formulation
  3. Hilbert space
  4. mixed formulation

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Published In

cover image SIAM Journal on Applied Mathematics
SIAM Journal on Applied Mathematics
Pages: 658 - 676
ISSN (online): 1095-712X

History

Submitted: 26 September 2008
Accepted: 27 March 2009
Published online: 22 July 2009

MSC codes

  1. 75R99
  2. 46-99
  3. 35A15
  4. 65M60

Keywords

  1. convection-diffusion
  2. variational formulation
  3. Hilbert space
  4. mixed formulation

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