A methodology for obtaining analytical solutions for the extended Graetz problem including the effects of axial conduction, viscous dissipation heating, and additional volumetric heating effects in an infinite domain has been developed. The domain extends infinitely in the directions parallel to the flow, being divided into a preparation upstream region, in which the walls are either thermally insulated or isothermal, and a downstream region, in which the walls are either uniformly heated or isothermal, at temperature that is different than that of the upstream walls; regardless of the selected configuration, there is always a step change in the wall heating condition at the origin — where both regions meet. The solution methodology is based on the Generalized Integral Transform Technique, in which eigenfunction expansions in terms of orthogonal bases are employed. Simple Sturm–Liouville eigenfunction bases in terms of Helmholtz problems are utilized for maintaining the calculation of eigenfunctions-related quantities as simple as possible, thus minimizing the amount of necessary computational work. A thorough error analysis is performed for verification purposes, demonstrating that larger truncation orders are required for cases with smaller Péclet number values, and with wall boundary conditions of different types in each region. It was also shown that positions near the origin or occurring near discontinuities in the Nusselt number, also require more terms in the eigenseries expansion. An interesting convergence phenomenon was observed for cases with the same type of wall conditions in both regions: as the Péclet number is large enough or at positions sufficiently far from the origin, the Nusselt number error decreases following a negative power of the truncation order; also, although smaller Péclet numbers were shown to have a worse convergence rate, they shift to the power-like form after a minimum truncation order is reached. The presented methodology was also verified through comparisons with literature studies. Finally, illustrative results were provided showing that the boundary condition in the upstream preparation region can have a prominent effect on the thermal developing Nusselt behavior, especially for lower Péclet number values.

已经开发了一种用于获得扩展 Graetz 问题的解析解的方法，包括轴向传导、粘性耗散加热和无限域中的附加体积加热效应的影响。该域在平行于流动的方向上无限延伸，分为制备上游区域，其中壁是绝热或等温的，以及下游区域，其中壁被均匀加热或等温，温度为与上游壁不同；无论选择何种配置，原点处的壁加热条件始终存在阶跃变化 - 两个区域相交处。求解方法基于广义积分变换技术，其中使用了正交基方面的特征函数展开。使用 Helmholtz 问题方面的简单 Sturm-Liouville 特征函数基来保持特征函数相关量的计算尽可能简单，从而最小化必要的计算工作量。出于验证目的进行了彻底的误差分析，表明对于具有较小 Péclet 数值的情况以及每个区域中不同类型的壁边界条件，需要更大的截断阶数。还表明，靠近原点的位置或发生在 Nusselt 数不连续点附近的位置，也需要在特征级数展开中使用更多项。对于两个区域具有相同类型壁面条件的情况，观察到了一个有趣的收敛现象：当 Péclet 数足够大或距离原点足够远时，Nusselt 数误差按照截断阶次的负幂减小；此外，虽然较小的 Péclet 数被证明具有更差的收敛速度，但在达到最小截断阶数后，它们会转变为类似幂的形式。所提出的方法也通过与文献研究的比较得到验证。最后，提供了说明性结果，表明上游制备区域的边界条件对热发展 Nusselt 行为具有显着影响，尤其是对于较低的 Péclet 数值。尽管较小的 Péclet 数被证明具有更差的收敛速度，但在达到最小截断阶数后，它们会转变为类似幂的形式。所提出的方法也通过与文献研究的比较得到验证。最后，提供的说明性结果表明上游制备区域的边界条件对热发展 Nusselt 行为具有显着影响，尤其是对于较低的 Péclet 数值。尽管较小的 Péclet 数被证明具有更差的收敛速度，但在达到最小截断阶数后，它们会转变为类似幂的形式。所提出的方法也通过与文献研究的比较得到验证。最后，提供的说明性结果表明上游制备区域的边界条件对热发展 Nusselt 行为具有显着影响，尤其是对于较低的 Péclet 数值。