Transport in Superhydrophobic Microchannels: A Porous Modeling Approach

Superhydrophobic surfaces combine roughness features with low energy surfaces to create materials with substantially decreased wettability and reduced drag resistance in laminar flows. These characteristics make superhydrophobic surfaces a promising technology for reducing the flow resistance of microchannels in a variety of applications, including thermal management and biofluidics. The presence of a gas layer that is trapped within the superhydrophobic surface, and which separates the majority of the microchannel wall from the working fluid, gives rise to a low shear-stress region responsible for the observed reduction in flow resistance. Although there have been numerous experimental and computational studies of fluid flow in superhydrophobic microchannels, to our knowledge no predictive analytical model capturing the essential features of the flow has been developed for the case of post-type surface roughness. In this work we propose the use of porous flow theory to predict the behavior of the fully-developed inertia-less flow of a constant viscosity Newtonian fluid in a parallel-plate, super-hydrophobic microchannel whose roughness features are composed of a square array of posts arranged transverse to the flow. The volume-averaged Navier-Stokes (VANS) equation is used to model the flow behavior in both the open and porous regions, taking into account the presence of a recirculating gas layer and the potential for partial liquid penetration into the porous region. The fluid motion in the porous and non-porous regions is coupled by imposing boundary conditions specifying the continuity of velocity and a stress jump at the interface between the two regions. An empirical factor, known as the stress jump coefficient β, appears in the stress jump boundary condition and is shown to be correlated to the geometric properties of the porous region via a scaling law inferred from non-dimensional analysis and observed in 3D computational fluid dynamics simulations. Finally, the predictions of the model are compared with existing experimental studies.