Analytical solution for the temperature distributions, heat transfer coefficients, and Nusselt numbers of steady electroosmotic flows with an arbitrary pressure gradient are obtained for two-dimensional straight micro-channels. The thermal analysis considers interaction among inertial, diffusive and Joule heating terms in order to obtain the thermally developing behavior of mixed electroosmotic and pressure driven flows. In mixed flow cases, the governing equation for energy is not separable in general. Therefore, we introduced a new method that considers the extended Graetz problem. Heat transfer characteristics are presented for low Reynolds number micro-flows where the viscous and electric field terms are very dominant. Analytical results show that the heat transfer coefficient of mixed-electroosmotic and pressure driven flow is smaller than that of pure electroosmotic flow. For the parameter range studied here (Re<0.7), the fully developed Nusselt number is independent of the thermal Peclet number and pressure gradient. Moreover, in mixed electroosmotic and pressure driven flows, the thermal entrance length increases with the imposed pressure gradient.

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