In this work we solve numerically the conjugated heat transfer problem of a non-Newtonian fluid and solid walls in a microchannel under the influence of pressure and electro-osmotic forces. The velocity field is determined taking into account a hydrodynamically fully-developed flow and a constitutive relation based in a viscoelastic rheological model with a simplified Phan-Thien Tanner fluid. The numerical process results in solid and fluid temperature distributions. Is shown the influence of nondimensional parameters involved in the analysis on the conjugated heat transfer problem: an indicator of viscoelastic behavior, the Peclet number, a normalized power generation term being the ratio of heat flow from the external wall to the Joule heating, a conjugation term which determines the basic heat transfer regimes between fluid and solid sections in the microchannel. For the flow field: the ratio of pressure forces to the electro-osmotic forces acts on flow as a drag reducer and drag increaser under favorable and adverse pressure gradients, respectively, moreover, for increasing values of the viscoelastic parameter, the velocity of the fluid increases with respect the Newtonian fluid flow case. These velocity perturbations resulting in cross-sectional variations of temperature.

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