Graetz Problem Extended to Mixed Electroosmotically and Pressure Driven Flow

Abstract

Thermally developing mixed electroosmotically and pressure-driven flow in a parallel plate microchannel with a step change in wall temperature is considered in the framework of an extended Graetz problem. Both Joule heating and viscous dissipation effects are taken into consideration. Expressions for the dimensionless temperature and Nusselt number in the form of infinite series are presented. In general, the associated eigenvalue problem is solved numerically. Nevertheless, an analytical solution is also presented for axial locations close to the entrance. Comparisons aremade between the present results and those obtained by approximating the electroosmotic velocity with the Helmholtz–Smoluchowski velocity. It is revealed that, for a pressure-assisted flow with a sufficiently high magnitude of pressure gradient, the Nusselt number is weakly dependent on the dimensionless Debye–Huckel parameter. Accordingly, the approximate solution is valid for such cases.However, unless a sufficiently large value of the dimensionless Debye–Huckel parameter is considered, the incurred error rapidly increases when the pressure gradient is absent or acts adversely. The approximate solution also fails to predict accurate results near the singularity points that occur in theNusselt numberswhen thewall temperature in the downstream is higher than that of upstream.