Pressure Drop and Flow Development in the Entrance Region of Microchannels With Second-Order Velocity Slip Condition and the Requirement for Development Length

In this investigation, $Lfd*$ and $Δp$ in the entrance region of circular and parallel plate microchannels have been determined for $10−2≤Re≤104$ and $10−4≤Kn≤0.2$⁠, employing the second-order velocity slip condition at the wall with $C1=1$ and $0≤C2≤0.5$⁠. Results indicate that although local velocity slip at the wall is always higher than that for the fully developed section, local wall shear stress for higher Kn and C₂ could be lower than its fully developed value, which is also more prominent for lower Re. Therefore, depending upon the operating condition, K(x) and K_fd could assume negative values, implying that pressure gradient in the developing region could even be less than that in the fully developed section. It has been further observed that both $Lfd*$ and K_fd are characterized by the low and the high Re asymptotes, using which extremely accurate correlations have been proposed for both geometries.