Transversal field and heat transfer in the flow of non-linear viscoelastic fluids in tear-drop shaped tubes including viscous dissipation

Steady, laminar flow of a class of non-linear viscoelastic fluids in tear-drop shaped ducts is investigated for constant heat flux at the wall including viscous dissipation. Field equations are solved analytically through expanding the field variables in double asymptotic series in terms of the Weissenberg number and a mapping parameter coupled with the shape factor method, a continuous and one-to-one mapping, to map the circular contour onto the tear-drop shaped tube boundary. Analytical solution for the transversal and axial velocity fields is derived for the first time, and the influence of the viscoelastic constitutive parameters in shaping the transversal vortex structure is discussed in detail. The temperature field and the dependence of the Nusselt Nu = f (Br, Pe, Re, We) number on the Brinkman Br, Péclet Pe, Reynolds Re and Weissenberg We numbers as well as combinations of the material parameters are investigated. At the lowest order in the Weissenberg number the velocity and temperature fields of Newtonian fluids in tear-drop shaped straight tubes as well as in straight round tubes with viscous dissipation are recovered. The approach the analysis is based on can be easily adapted to yield the velocity and temperature fields of Newtonian and constitutively non-linear viscoelastic fluids in non-circular tubes other than tear-drop shaped including viscous dissipation.