Isoflux Nusselt number and slip length formulae for superhydrophobic microchannels

We analytically and numerically consider the hydrodynamic and thermal transport behavior of fully-developed flow through a superhydrophobic parallel-plate channel. Constant hydrodynamic slip, thermal slip and heat flux are prescribed at each surface. We first derive a general expression for the Nusselt number valid for asymmetric velocity profiles. Next, we demonstrate that both thermal and hydrodynamic slip lengths in the local Stokes flow limit can be found by redefining existing solutions for conduction spreading resistances, assuming an adiabatic and shear-free liquid-gas interface. Expressions for the thermal slip length for pillar and ridge surface topographies are determined. Comparison of fundamental half-space solutions for the Laplace and Stokes equations yields an expression for the hydrodynamic slip length over pillar-structured surfaces based on existing solutions for the conduction spreading resistance from an isothermal source. Numerical validation is performed and an analysis of the idealized thermal transport behavior suggests conditions under which superhydrophobic microchannels may enhance heat transfer.