Numerical analysis of a new mixed-formulation for eigenvalue convection-diffusion problems
Résumé
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 translationnaly 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 non-sectorial 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 being non sectorial, 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 axi-symmetrical situations, the convergence of which is found to be consistent with the numerical discretization.
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