Analytical solution of the Graetz problem for non-linear viscoelastic fluids in tubes of arbitrary cross-section

Graetz problem is solved analytically for steady laminar flow of non-linear viscoelastic fluids in straight tubes of arbitrary cross-section. The one-to-one mapping method used to model the cross-sectional geometry satisfies the no-slip and thermal boundary conditions for a wide range of arbitrary tube contours. Field variables are expanded in asymptotic series in terms of the Weissenberg number Wi, leading to a set of linear hierarchical equations which are solved successively up to and including the third order in Wi. Exact analytical solution for the Graetz problem for Newtonian fluids in round as well as arbitrary cross-sectional tubes is recovered at the lowest order. Secondary flows arise at the third order of the analysis as shown previously by Siginer and Letelier [1]. Their effect on the convection field is fully taken into account. The temperature distributions for triangular and square cross-sectional tubes are computed as particular cases of the general analysis together with the variation of the asymptotic Nusselt number for the triangular cross-section as a function of Wi to demonstrate the substantial enhancement of the heat transfer rates due to the non-linear viscoelastic constitutive structure of the fluid. The analytical algorithm presented is very versatile and can be easily applied to a wide spectrum of non-circular tube contours.