LAUSR

Abstract 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 Peclet 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 Peclet 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 Peclet 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 Peclet number values.