The generalized Graetz problem refers to stationary convection–diffusion in uni-directional flows. In this contribution we demonstrate the analyticity of generalized Graetz solutions associated with layered domains: either cylindrical (possibly concentric) 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 eigenfunctions (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 computation is henceforth made explicit and can be performed 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.

中文翻译：

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Graetz 模式在平行和同心配置中的解析特性

广义 Graetz 问题是指单向流动中的固定对流扩散。在这个贡献中，我们展示了与分层域相关的广义 Graetz 解的分析性：圆柱形（可能是同心的）或平行的。这种配置被认为是热交换器装置的原型，并出现在涉及传热或传质的众多应用中。Graetz 模式的既定框架允许将传热的 3D 分辨率重新转换为 2D 甚至 1D 光谱问题。相关的特征函数（称为 Graetz 模式）是在一系列递归计算的闭包函数的帮助下获得的。频谱由显式分析系列的零点给出，其截断允许通过求解多项式方程来近似特征值。Graetz 模式计算从此变得明确并且可以使用形式微积分的标准软件来执行。它允许对各种配置的结果解决方案进行直接和无网格计算。说明了一些解决方案，以展示 Graetz 解决方案的无网格分析推导的兴趣，可用于验证其他数值方法。