The Generalized Graetz Problem in Finite Domains

International audienceWe consider the generalized Graetz problem associated with stationary convection- diusion inside a domain having any regular three-dimensional translationally invariant section and nite or semi-innite 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 includes both heterogeneous and possibly anisotropic diusion tensors. In the case of semi-innite domains, the existence of a bounded solution is shown from the analysis of two-dimensional operator eigenvectors which form a basis of L2 . In the case of nite domains a similar basis can be exhibited, and the mode's amplitudes can be obtained from the inversion of newly dened nite domain operator. Our analysis includes both the theoretical and practical issues associated with this nite domain operator inversion as well as its interpretation as a multireection image method. Error estimates are provided when numerically truncating the spectrum to a nite number of modes. Numerical examples are validated for reference congurations and provided in nontrivial cases. Our methodology shows how to map the solution of stationary convection-diusion problems in nite three-dimensional domains into a two-dimensional operator spectrum, which leads to a drastic reduction in computational cost