Two-dimensional Carreau law for a quasi-newtonian fluid flow through a thin domain with a slightly rough boundary

TL;DR

This paper derives an effective two-dimensional nonlinear Reynolds model for quasi-Newtonian fluid flow in thin domains with slightly rough boundaries, using asymptotic analysis and rigorous mathematical techniques.

Contribution

It introduces a novel asymptotic approach to model quasi-Newtonian fluids in thin, rough-walled domains, incorporating boundary effects into the Reynolds model.

Findings

Derived the effective nonlinear Reynolds model for the flow.

Established sharp a priori estimates and compactness results.

Accounted for boundary oscillations in the limit model.

Abstract

This study investigates the asymptotic behavior of the steady-state quasi-Newtonian Stokesflow with viscosity given by the Carreau law within a thin domain, focusing on the effects of a slightly rough boundary of the domain. Employing asymptotic techniques with respect to the domain's thickness, we rigorously derive the effective nonlinear two-dimensional Reynolds model describing the fluid flow. The mathematical analysis is based on deriving the sharp a priori estimates and proving the compactness results of the rescaled functions together with monotonicity arguments. The resulting limit model incorporates contributions of the oscillating boundary and thus, it could prove useful in the applications involving this lubrication regime.