Pressure Drop and Flow development in the Entrance Region of Micro-Channels with Second Order Slip Boundary Conditions and the Requirement for Development Length

In the present investigation, the development of axial velocity profile, the requirement for development length ($L_{fd}^{\ast }=L\mathrm{/}{D}_h$) and the pressure drop in the entrance region of circular and parallel plate micro-channels have been critically analysed for a large range of operating conditions ($10^{-2}\le Re\le 10^{4}$,$10^{-4}\le Kn\le 0.2$and$0\le C_2\le 0.5$). For this purpose, the conventional Navier-Stokes equations have been numerically solved using the finite volume method on non-staggered grid, while employing the second-order velocity slip condition at the wall with $C_1=1$. The results indicate that although the magnitude of local velocity slip at the wall is always greater than that for the fully-developed section, the local wall shear stress, particularly for higher $Kn$ and $C_2$, could be considerably lower than its fully-developed value. This effect, which is more prominent for lower $Re$, significantly affects the local and the fully-developed incremental pressure drop number $K(x)$ and $K_{fd}$, respectively. As a result, depending upon the operating condition, $K_{fd}$, as well as $K(x)$, could assume negative values. This never reported observation implies that in the presence of enhanced velocity slip at the wall, the pressure gradient in the developing region could even be less than that in the fully-developed section. From simulated data, it has been observed that both $L_{fd}^{\ast }$ and $K_{fd}$ are characterised by the low and the high $Re$ asymptotes, using which, extremely accurate correlations for them have been proposed for both geometries. Although owing to the complex nature, no correlation could be derived for $K(x)$ and an exact knowledge of $K(x)$ is necessary for evaluating the actual pressure drop for a duct length $L^{\ast }<L_{fd}^{\ast }$, a method has been proposed that provides a conservative estimate of the pressure drop for both $K_{fd}>0$ and $K_{fd}\le 0$.