Conjugate thermal creep flow in a thin microchannel

In the present work, we analyze asymptotically and numerically the conjugate heat transfer between a rarified gas flow and the lower wall of a thin horizontal microchannel. The laminar motion of the gas is originated only by the thermal creep or transpiration effect on the lower wall of the microchannel. It is well known the need to impose, in general, a variable temperature regime at the lower wall to induce the transpiration effect. Usually, it can be reached by setting a linear temperature profile as a boundary condition. However, in our case, we prefer to avoid this simplification taking into account that in practical applications, the temperature profile at the lower wall can be unknown. This case can occur, for instance, in a heat sink or a similar device with a well defined heat dissipation rate. Under this physical configuration, we can assume then that the bottom or external face of this heat sink with finite thermal conductivity is exposed to a uniform heat flux. On the other hand, the upper wall of the microchannel is subjected to a prescribed condition. The above conditions are sufficient to consider the simultaneous or conjugate heat transfer analysis of the heat conduction equation for the heat sink and the mass, momentum and energy equations for the gaseous-phase. Resulting governing equations are written in dimensionless form, assuming that the Reynolds number associated with the characteristic velocity of the thermal creep and the aspect ratio of the microchannel, are both very small. The velocity and temperature profiles for the gas phase and the temperature profiles for the solid wall are predicted as functions of the involved dimensionless parameters and the main results confirm that the phenomenon of conjugate thermal creep exists whenever the temperature of the lower wall varies linearly or nonlinearly.