Heat transfer enhancement in steady pressure gradient driven laminar flow of a class of nonlinear viscoelastic fluids in straight tubes of noncircular cross-section at constant temperature is discussed together with the flow structure, and the physics is clarified. The variation of the average Nusselt number Nu with the Weissenberg Wi and Reynolds Re numbers in cross-sections with n axes of symmetry is analyzed. A continuous one-to-one mapping is used to obtain arbitrary tube contours from a base tube contour partial derivative D-0. The analytical method presented is capable of predicting the velocity and temperature fields in tubes with arbitrary cross-section. The base flow is the Newtonian field and is obtained at the lowest order. Heat transfer enhancements represented by average Nusselt numbers of an order of magnitude larger as compared to their Newtonian counterparts are predicted as a function of the Reynolds and Weissenberg numbers even for slightly non-Newtonian, dilute fluids. The asymptotic independence of Nu =f(Pe,Wi) -> Nu = f(Pe) with increasing Wi is shown analytically for the first time. The implications on the heat transfer enhancement of the change of type of the vorticity equation is discussed in particular for slight deviations from Newtonian behavior where a rapid rise in enhancement seems to occur as opposed to the behavior for larger values of the Weissenberg number where the rate of increase is much slower. The coupling between viscoelastic and inertial nonlinearities is crucial to enhancement. Fluid vorticity will change type when the velocity in the centre of the tube is larger than a critical value defined by the propagation of the shear waves. The asymptotic independence of Nu from elasticity with increasing Wi is related to the extent of the supercritical region away from the wall controlled by the viscoelastic Mach number M and the Elasticity number E and its interaction with the growing strength of the secondary flows with Wi. The physics of the intertwined effects of the Elasticity E, Viscoelastic Mach M, Reynolds Re and Weissenberg Wi numbers on generating the heat transfer enhancement is discussed.