An analytical study of the forced convection of an elastoviscoplastic fluid in tubes of arbitrary cross section is presented. The constitutive structure of the fluid is described by a frame indifferent linear combination of the Modified Phan-Thien-Tanner model of non-linear viscoelastic fluids and the Bingham model of non-linear plastic fluids. Arbitrary tube cross sections are modeled by a continuous one-to-one mapping of the circular base contour into a wide spectrum family of non-circular tube contours. Field variables are expanded into double asymptotic series in terms of the elasticity measure Weissenberg number WI, and a mapping parameter leading to a set of linearized hierarchical momentum balance, constitutive structure and thermal field equations which are solved successively up to and including the third order in Wi for the velocity and temperature fields. The general algorithm developed is applied to the study of forced convection in tubes with exact equilateral triangular and approximately square cross sections. The analysis also yields the solution of the forced convection of linear (Newtonian) fluids in non-circular tubes and the forced convection in circular tubes of the family of non-linear fluids (viscoplastic, viscoelastic, elastoviscoplastic) described by the constitutive structure under consideration thereby providing as well validation for the computations carried out for non-linear fluids in non-circular cross-sections. A thorough comparison of the velocity and thermal fields of the Newtonian, viscoplastic, viscoelastic and elastoviscoplastic fluids in tubes of equilateral triangular and pseudo square cross-sectional tubes is presented as specific cases. (C) 2019 Elsevier Ltd. All rights reserved.