Analytical properties of Graetz modes in parallel and concentric configurations
Résumé
The generalized Graetz problem refers to
stationary convection–diffusion in uni-directional
flows. In this contribution we demonstrate the analyt-
icity of generalized Graetz solutions associated with
layered domains: either cylindrical (possibly concen-
tric) or parallel. Such configurations are considered as
prototypes for heat exchangers devices and appear in
numerous applications involving heat or mass transfer.
The established framework of Graetz modes allows to
recast the 3D resolution of the heat transfer into a 2D
or even 1D spectral problem. The associated eigen-
functions (called Graetz modes) are obtained with the
help of a sequence of closure functions that are
recursively computed. The spectrum is given by the
zeros of an explicit analytical series, the truncation of
which allows to approximate the eigenvalues by
solving a polynomial equation. Graetz mode compu-
tation is henceforth made explicit and can be per-
formed using standard software of formal calculus. It
permits a direct and mesh-less computation of the
resulting solutions for a broad range of configurations.
Some solutions are illustrated to showcase the interest
of mesh-less analytical derivation of the Graetz
solutions, useful to validate other numerical
approaches.
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