parameter of the Phan-Thien-Tanner model on velocity distribution and showed an increment in elastic properties of fluid can increase the flow rate. Following works ([14] and [22]), for linear form of temperature distribution, showed in simplified Phan-Thien-Tanner viscoelastic fluids, heat which is generated via viscous dissipation has an important influence on fluid elasticity and heat transfer by imposed heating at the wall.  Forgoing method is applied to other problems, Coelho et all [23] extended the problem to flow and heat transfer in a dynamic and thermally fully developed channel flow of the SPTT fluid induced by combined electro-osmosis and pressure gradient. Also, Hashemabadi and Mirnajafizadeh [15] extended the method to investigate the effects of viscous dissipation on heat transfer by implementing Nahme-Griffith number as perturbation parameter. They observed that viscous dissipation may considerably alter isothermal flow characteristics and enhances Nusselt number. Unfortunately in all of the above mentioned works only effect of entropy elasticity were included in heat transfer analysis and the effect of energy elasticity which particularly illuminate the elastic effects of viscoelastic materials were omitted that can cause inaccuracy in this types of studies.
Motivated by the industrial relevance of viscoelastic flows, an important area which requires further analysis is the slip hydrodynamics in proximity to the pipe (conduit) boundary. Therefore, in the current study, exact analytical solutions for laminar viscoelastic fluid moving through a microchannel is developed to elucidate the effects of slip situation on flow and heat transfer characteristics for both linear and exponential forms of SPTT fluids. In flow part, in order to simulate the behavior of viscoelastic material the simplified form of Phan-Thien_Tanner constitutive equation is used. Using the definition of normal-stress differences, the coefficient of first normal-stress difference is calculate and the effect of this parameter is also in flow and heat transfer investigation are studied. In order to accurately have discussion over the effect of mechanical energy on heat transfer of the problem both energy elasticity and entropy part are considered on energy equation.