Wall laws for viscous flows in 3D randomly rough pipes: optimal convergence rates and stochastic integrability

TL;DR

This paper advances the understanding of viscous flow approximations in 3D randomly rough pipes by establishing optimal convergence rates, improved stochastic integrability, and a refined Poiseuille's law using homogenization and regularity theories.

Contribution

It generalizes 2D results to 3D pipes, improves stochastic integrability, and introduces a refined Poiseuille's law for flows with random boundary roughness.

Findings

Established optimal convergence rates for wall laws in 3D pipes.

Improved stochastic integrability results for random boundary roughness.

Proved a refined version of Poiseuille's law in the stochastic setting.

Abstract

This paper is concerned with effective approximations and wall laws of viscous laminar flows in 3D pipes with randomly rough boundaries. The random roughness is characterized by the boundary oscillation scale $\varepsilon \ll 1 $ and a probability space with ergodicity quantified by functional inequalities. The results in this paper generalize the previous work for 2D channel flows with random Lipschitz boundaries to 3D pipe flows with random boundaries of John type. Moreover, we establish the optimal convergence rates and substantially improve the stochastic integrability obtained in the previous studies. Additionally, we prove a refined version of the Poiseuille's law in 3D pipes with random boundaries, which seems unaddressed in the literature. Our systematic approach combines several classical and recent ideas (particularly from homogenization theory), including the Saint-Venant's principle for pipe flows, the large-scale regularity theory over rough boundaries, and applications of functional inequalities.