Effective slip boundary conditions for arbitrary one-dimensional surfaces

TL;DR

This paper derives general expressions for effective slip boundary conditions on one-dimensional patterned surfaces, revealing a simple relation between transverse and longitudinal slip components, aiding flow characterization over such surfaces.

Contribution

It introduces a general framework for calculating effective slip-length tensors on 1D surfaces with arbitrary scalar slip distributions, simplifying flow analysis.

Findings

Transverse slip component is half of the longitudinal component with doubled local slip.

Flow along any direction can be determined from the longitudinal slip component.

Provides explicit formulas for eigenvalues of the effective slip-length tensor.

Abstract

In many applications it is advantageous to construct effective slip boundary conditions, which could fully characterize flow over patterned surfaces. Here we focus on laminar shear flows over smooth anisotropic surfaces with arbitrary scalar slip $b(y)$, varying in only one direction. We derive general expressions for eigenvalues of the effective slip-length tensor, and show that the transverse component is equal to a half of the longitudinal one with twice larger local slip, $2b(y)$. A remarkable corollary of this relation is that the flow along any direction of the 1D surface can be easily determined, once the longitudinal component of the effective slip tensor is found from the known spatially nonuniform scalar slip.