An effective slip boundary for micro-structured surfaces containing a second immiscible fluid

TL;DR

This paper derives effective slip boundary conditions for flows over micro-structured surfaces with a secondary immiscible fluid, accounting for viscosity differences and inertia, validated by analytical and numerical comparisons.

Contribution

It introduces a novel theoretical framework for effective slip boundaries on micro-structured surfaces containing a secondary fluid, applicable to arbitrary geometries.

Findings

Good agreement with prior analytical results for rectangular structures

Effective slip velocity matches two-phase numerical simulations

Flow fields align well with simulation results when applying the derived boundary conditions

Abstract

Effective slip boundary conditions for flows over periodic micro-structured surfaces containing a secondary immiscible fluid are derived. The primary fluid is in the Cassie state, while the geometries of the micro-structures can be arbitrary. We investigate the impact of the second immiscible fluid on external flow, introducing the effect caused by the viscosity difference between two fluids and the inertia effect of the second fluid into classic Navier slip condition. The effective slip length obtained from our theory for flows over rectangular micro-structures is in good agreement with prior analytical findings. We also apply the theory to mushroom-like micro-structures. The derived effective slip velocity also matches well with two-phase numerical simulations. Implementing the slip boundary conditions on micro-structured surfaces produces external flow fields that are aligned well with the simulation results. Employing a multi-scale homogenization method, we dispose of two-phase flows characterized by strong coupling at the fluid-fluid interface. By introducing the framework of lid-driven cavity approximation, our theory finds practical applications across various scenarios.