Shear work contribution to convective heat transfer of dilute gases in slip flow regime

In the literature some researchers highlighted that for a dilute gas in slip flow regime and in presence of a non negligible viscous heating, the analysis of the gas micro-convection has to be tackled by modifying the thermal boundary conditions to account for the shear work due to the slip at the wall. Although in the recent past a specific modified boundary condition has been proposed, theoretically justified and applied to investigate the effect of the shear work on the convective heat transfer in presence of dilute gases, in this paper is demonstrated that there is not a need of a modified boundary condition in order to take into account the effect of the shear work in the analysis of forced convection. In the present work by means of a comprehensive theoretical analysis is demonstrated that the modified boundary condition is useless for the analysis of the effect of shear work on the evaluation of the convective heat transfer coefficients in presence of a dilute gas with non-negligible viscous dissipation. Moreover, to evaluate the inaccuracy of the results obtained by using the modified boundary conditions the difference, in terms of Nusselt number, between the exact and the approximate solution has been numerically estimated for elliptical microchannels. The numerical outcomes point out that the adoption of the modified boundary condition leads to an underestimation or an overestimation of the Nusselt numbers depending on the values of Brinkman number and on the channel cross section geometry.