The Graetz problem in tubes of arbitrary cross section

The Graetz problem is investigated analytically for the steady laminar flow of Newtonian fluids in tubes of arbitrary cross section. The one-to-one mapping introduced satisfies the no-slip condition and allows the determination of the velocity field in a wide spectrum of arbitrary tube contours. The energy equation is solved for the temperature field in the corresponding tube contours, and the temperature distribution for the triangular, square and circular cross-sectional tubes is presented as particular case. Furthermore, in order to illustrate its relevance for a moderate Péclet number (Pe) regime, the solution applied to the square cross section is compared to numerical simulations for two scenarios, Case I with Pe = 100 and Case II with Pe = 500. It is found that for Case I the relative error does not exceed 2.9 %, being maximum at the center of the tube, while for Case II both analytical and numerical solutions match rather precisely, with less than 1 % of difference near the edges.