Development of hydrodynamic and thermal boundary layer in the entrance region with hydrophobic surfaces

Entrance region plays a crucial role in micro-scale flow and heat transfer characteristics, significantly influencing boundary layer development and channel performance. While extensive research has focused on conventional no-slip conditions, the impact of hydrophobic surfaces that cause velocity slip and temperature jump on the entrance effect remains underexplored. This study systematically investigates the hydrodynamic and thermal boundary layer development in the microchannels with hydrophobic surfaces. Fitting correlations for hydrodynamic and thermal entrance lengths are derived as functions of Reynolds number (Re) and slip length. Several newly introduced parameters are quantitatively assessed for the entrance effect under various slip conditions. The results indicate that the hydrodynamic entrance length follows a nonlinear trend for Re < 15 and a linear trend for 15 < Re < 100, with slip conditions either increasing or decreasing depending on the flow regime. The thermal entrance length maintains a linear relationship with Re across different slip lengths and shows a maximum reduction of 12 %. The velocity slip reduces flow resistance by as much as 80.1 % while enhancing convective heat transfer efficiency with a maximum increase of 52.1 %. Considering multiple temperature jump models, a critical coefficient α_c, relating velocity slip length to temperature jump length is introduced to identify conditions where hydrophobic surfaces achieve both heat transfer enhancement and drag reduction. These findings offer a comprehensive framework for predicting slip-induced modifications in entrance region behavior, providing a theoretical guidance for the design of advanced microscale cooling systems.