Variational formulation on Joule heating in combined electroosmotic and pressure driven microflows

The present study attempts to analyze the extended Graetz problem in combined electroosmotic and pressure driven flows in rectangular microchannels, by employing a variational formulation. Both the Joule heating and axial conduction effects are taken into consideration. Since assuming a uniform inlet temperature profile is not consistent with the existence of these effects, a step change in wall temperature is considered to represent physically conceivable thermal entrance conditions. The method of analysis considered here is primarily analytical, in which series solutions are presented for the electrical potential, velocity, and temperature. For general treatment of the eigenvalue problem associated with the solution of the thermal field, an approximate solution methodology based on the variational calculus is employed. An analytical solution is also presented by considering thin electrical double layer limits. The results reveal non-monotonic behaviors of the Nusselt number such as the occurrence of singularities in the local Nusselt number values when the fluid is being heated from the wall. Moreover, the effect of increasing the channel aspect ratio is found to be increasing both the temperature difference between the wall and the bulk flow and the Nusselt number. In addition, higher wall heat fluxes are obtained in the entrance region by increasing the Peclet number.