# Mobility tensor of a sphere moving on a super-hydrophobic wall: application to particle separation

## TL;DR

This study models the hydrodynamics of a sphere near a patterned superhydrophobic wall, revealing how wall patterning influences particle motion and enabling potential particle separation techniques.

## Contribution

It introduces a systematic calculation of the mobility tensor near patterned superhydrophobic surfaces using boundary element methods, highlighting effects on particle trajectories and resonance-based sorting.

## Key findings

- Mobility tensor varies with particle position and size near patterned walls.
- Resonance induces a mean transversal drift in particle trajectories.
- Patterning can be exploited for particle separation applications.

## Abstract

The paper addresses the hydrodynamic behavior of a sphere close to a micro-patterned superhydrophobic surface described in terms of alternated no-slip and perfect-slip stripes. Physically, the perfect-slip stripes model the parallel grooves where a large gas cushion forms between fluid and solid wall, giving rise to slippage at the gas-liquid interface. The potential of the boundary element method (BEM) in dealing with mixed no-slip/perfect-slip boundary conditions is exploited to systematically calculate the mobility tensor for different particle-to-wall relative positions and for different particle radii. The particle hydrodynamics is characterized by a non trivial mobility field which presents a distinct near wall behavior where the wall patterning directly affects the particle motion. In the far field, the effects of the wall pattern can be accurately represented via an effective description in terms of a homogeneous wall with a suitably defined apparent slippage. The trajectory of the sphere under the action of an external force is also described in some detail. A resonant regime is found when the frequency of the transversal component of the force matches a characteristic crossing frequency imposed by the wall pattern. It is found that, under resonance, the particle undergoes a mean transversal drift. Since the resonance condition depends on the particle radius the effect can in principle be used to conceive devices for particle sorting based on superhydrophobic surfaces.