We present an analytical study of secondary flows in tubes of teardrop shape for a viscoelastic fluid described by the Phan-Thien-Tanner (PPT) model. The analytical development is based on a one-to-one mapping (shape function) to describe the teardrop contour and the expansion of field variables into asymptotic series in terms of the Weissenberg number Wi. Secondary flows and the temperature field are determined solving the hierarchical set of equations resulting from the asymptotic series expansions.