Laminar boundary layers over small-scale textured surfaces

TL;DR

This work develops a homogenised slip-length framework for steady, 2D laminar boundary layers over small-scale textured surfaces, replacing the texture with a slip condition of length $\lambda$ derived from an inner-region homogenisation. A three-region matched asymptotic expansion (outer inviscid, middle boundary layer, inner textured) connects scales and yields a slip boundary condition for the boundary-layer equations; an asymptotic solution is obtained for $\lambda/\sqrt{x}\ll 1$, and a Chebyshev collocation plus finite-difference scheme solves the problem for $\lambda = O(1)$. The approach is applied to canonical small-scale textures (SHSs and riblets), deriving explicit slip-length expressions and quantifying effects on velocity profiles, wall shear stress, and displacement thickness, with implications for drag reduction in microfluidic and marine applications. The framework offers computationally efficient prediction of near-wall flow over textured surfaces while avoiding explicit resolution of microtextures, enabling practical design and optimization of textured coatings and devices in laminar regimes.

Abstract

We develop a model for steady, laminar boundary layers over small-scale textured surfaces. Although the texture is small relative to the boundary-layer thickness, it modifies the flow via a slip length. We use matched asymptotic expansions to simplify the problem, dividing the flow into outer, boundary-layer and inner regions. The far-field behaviour of the inner problem yields a slip boundary condition for the boundary layer. We derive an asymptotic solution valid when the slip length is small. For arbitrary slip lengths, we develop a numerical method combining Chebyshev collocation and finite differences. We apply this framework to canonical small-scale textured surfaces, including superhydrophobic surfaces and riblets, and utilise existing analytical slip formulae. We demonstrate how slip can effect the boundary layer's velocity field, wall shear stress and displacement thickness. Our approach enables computationally inexpensive modelling of small-scale textured surfaces in applications ranging from microfluidics to marine transport.

Figures (5)

• Figure 1: (a) A ship with a potential superhydrophobic coating. (b) Schematic of the boundary layer forming over a transversely ridged SHS with texture period $\hat{P}$ and length $\hat{L}$. (c) An aeroplane that could be equipped with riblets. (d) Schematic of the boundary layer forming over a transversely ribleted surface with texture amplitude $\hat{A}$, texture period $\hat{P}$ and length $\hat{L}$. Schematics in (b) and (d) illustrate the outer inviscid region, boundary layer and inner viscous region.
• Figure 2: Leading-order and first-order similarity solutions for the velocity profile in the presence of slip. (a) The Blasius profile $f$, satisfying the no-slip boundary condition (\ref{['eq:blasius_1']}--\ref{['eq:blasius_2']}). (b) The first-order correction $g$, arising at $O(\lambda/\sqrt{x'})$ from the slip boundary condition at the wall (\ref{['eq:blasius_4']}--\ref{['eq:blasius_5']}).
• Figure 3: Streamwise velocity $u$ and normal velocity $v$ for (a--b) $\lambda = 0.1$ and (c--d) $\lambda = 1$. Dashed lines indicate asymptotic solutions using (\ref{['eq:expansion']}, \ref{['eq:blasius_1']}, \ref{['eq:blasius_4']}); solid lines show numerical results using \ref{['eq:numerical_sol']}. The plots illustrate how slip modifies the boundary-layer development, with larger deviations near the leading edge for larger $\lambda$.
• Figure 4: Displacement thickness $\delta$ and wall shear stress $\tau$ for (a--b) $\lambda = 0.1$ and (c--d) $\lambda = 1$. Dotted lines depict asymptotic solutions using (\ref{['eq:delta_corrected']}, \ref{['eq:shear_corrected']}); solid red lines show numerical results obtained from (\ref{['eq:numerical_sol']}, \ref{['eq:disp_1']}, \ref{['eq:stress_1']}); solid blue lines indicate the Blasius solution (\ref{['eq:blas_disp']}, \ref{['eq:blas_stress']}). The plots illustrate how slip modifies boundary-layer thickness and wall shear, with larger deviations near the leading edge for higher $\lambda$.
• Figure 5: Slip length $\lambda$ for (a) transverse SHSs as a function of gas fraction $\phi$ and period $\epsilon$ and (b) transverse riblets as a function of texture amplitude $A$ and period $\epsilon$, where $\text{Re} = 100$. The plots show how geometry and texture parameters influence slip behaviour in laminar boundary layers.