The Generalized Graetz problem in finite domains
Résumé
We consider the generalized Graetz problem associated with stationary convection-diffusion inside a domain having any regular three dimensional translationally invariant section and finite or semi-infinite extent. Our framework encompasses any previous ''extended'' and ''conjugated'' Graetz generalizations and provides theoretical bases for computing the orthogonal set of generalized two-dimensional Graetz modes. The theoretical framework both includes heterogeneous and possibly anisotropic diffusion tensor. In the case of semi-infinite domains, the existence of a bounded solution is shown from the analysis of a two-dimensional operator eigenvectors which form a basis of $L^2$. In the case of finite domains a similar basis can be exhibited and the mode's amplitudes can be obtained from the inversion of newly defined finite domain operator. Our analysis both includes the theoretical and practical issues associated with this finite domain operator inversion as well as its interpretation as a multi-reflection image method. Error estimates are provided when numerically truncating the spectrum to a finite number of modes. Numerical examples are validated for reference configurations and provided in non-trivial cases. Our methodology shows how to map the solution of stationary convection-diffusion problems in finite three dimensional domains into a two-dimensional operator spectrum, which leads to a drastic reduction in computational cost.
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