Effective slip boundary conditions for arbitrary periodic surfaces: The surface mobility tensor

TL;DR

This paper derives a tensorial mobility relation for fluid slip over periodic surfaces, enabling better characterization of flow in microfluidic applications with complex surface patterns.

Contribution

It introduces a general formula for effective slip over arbitrary periodic surfaces, relating slip and stress via a mobility tensor, and provides an approximation method based on surface properties.

Findings

Derived a linear mobility tensor relating slip and stress.

Provided an approximate formula for the mobility law in terms of surface features.

Analyzed the impact of surface shape and randomness on slip behavior.

Abstract

In a variety of applications, most notably microfluidic design, slip-based boundary conditions have been sought to characterize fluid flow over patterned surfaces. We focus on laminar shear flows over surfaces with periodic height fluctuations and/or fluctuating Navier scalar slip properties. We derive a general formula for the "effective slip", which describes equivalent fluid motion at the mean surface as depicted by the linear velocity profile that arises far from it. We show that the slip and the applied stress are related linearly through a tensorial mobility matrix, and the method of domain perturbation is then used to derive an approximate formula for the mobility law directly in terms of surface properties. The specific accuracy of the approximation is detailed, and the mobility relation is then utilized to address several questions, such as the determination of optimal surface shapes and the effect of random surface fluctuations on fluid slip.