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In the present investigation, the development of axial velocity profile, therequirement for development length ($L^*_{fd}=L/D_{h}$) and the pressure dropin the entrance region of circular and parallel plate micro-channels have beencritically analysed for a large range of operating conditions ($10^{-2}\leRe\le 10^{4}$, $10^{-4}\le Kn\le 0.2$ and $0\le C_2\le 0.5$). For this purpose,the conventional Navier-Stokes equations have been numerically solved using thefinite volume method on non-staggered grid, while employing the second-ordervelocity slip condition at the wall with $C_1=1$. The results indicate thatalthough the magnitude of local velocity slip at the wall is always greaterthan that for the fully-developed section, the local wall shear stress,particularly for higher $Kn$ and $C_2$, could be considerably lower than itsfully-developed value. This effect, which is more prominent for lower $Re$,significantly affects the local and the fully-developed incremental pressuredrop number $K(x)$ and $K_{fd}$, respectively. As a result, depending upon theoperating condition, $K_{fd}$, as well as $K(x)$, could assume negative values.This never reported observation implies that in the presence of enhancedvelocity slip at the wall, the pressure gradient in the developing region couldeven be less than that in the fully-developed section. From simulated data, ithas been observed that both $L^*_{fd}$ and $K_{fd}$ are characterised by thelow and the high $Re$ asymptotes, using which, extremely accurate correlationsfor them have been proposed for both geometries. Although owing to the complexnature, 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$L^*<L^*_{fd}$, a method has been proposed that provides a conservativeestimate of the pressure drop for both $K_{fd}>0$ and $K_{fd}\le0$.

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