Thermal entry flow for a viscoelastic fluid: the Graetz problem for the PTT model

A theoretical study of the entrance thermal flow problem is presented for the case of a fluid obeying the Phan-Thien and Tanner (PTT) constitutive equation. This appears to be the first study of the Graetz problem with a viscoelastic fluid. The solution was obtained with the method of separation of variables and the ensuing Sturm–Liouville system was solved for the eigenvalues by means of a freely available solver, while the ordinary differential equations for the eigenfunctions and their derivatives were calculated numerically with a Runge–Kutta method. The scope of the present study was quite wide: it encompassed both the plane and axisymmetric geometries for channel and tube flows; two types of thermal boundary conditions with either an imposed wall temperature or an applied heat flux; inclusion of viscous dissipation; and elastic (through the Weissenberg number) and elongational (through the PTT parameter (epsilon)) effects. The main underlying assumptions were those of constant physical properties, negligible axial heat conduction, and fully developed hydrodynamic conditions. The results are discussed in terms of the main effects brought about by viscoelasticity and viscous dissipation on the Nusselt number variation and the bulk temperature.