Graetz Problem

The model

The Graetz equation describes coupled convection-diffusion problems in cylinders that are invariant along the z direction. Under this hypothesis, eigenmodes of a 2-dimensional operator provide the solution in the 3-dimensional space.

Our contribution: We have provided a numerical framework that allows to solve this problem in any geometrical configuration. This relies on a detailed analysis of the spectral properties of the Graetz operator.

The setting of the generalized Graetz model.

From left to right: the velocity in the fluid domain; first downstream eigenmode; 3rd downstream eigenmode; 5th downstream eigenmode.

More details:

J. Fehrenbach, F. de Gournay, C. Pierre, F. Plouraboué The generalized Graetz problem in finite domains, SIAM J. of Applied Mathematics, 72 (2012)

J. Fehrenbach, F. de Gournay Shape optimization for the generalized Graetz problem, Struct.Multidiscip.O., 49 (2014)

V. Debarnot, J. Fehrenbach, F. de Gournay, L. Martire The case of Neumann, Robin and periodic lateral conditions for the semi-infinite generalized Graetz problem, SIAM J. App. Math, 78 (2018)