The steady laminar non-isothermal flow of a non-linear viscoelastic fluid in tubes or arbitrary contour is analyzed under constant wall heat flux. Viscous dissipation is taken into account. Non-linear viscoelastic behavior is modeled by means of the Phan-Thien-Tanner model. The equations of motion and energy are solved analytically through a perturbation approach coupled with a one-to-one mapping of the boundary to map the circular shape into various axially symmetric cross-sectional shapes with the Weissenberg number as the perturbation parameter Longitudinal and transversal velocity components are determined for several cross-sectional shapes, for which the temperature field and the Nusselt number variation with the Prandtl, Brinkman and Reynolds numbers are computed.

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