Nonlinear Reynolds equations for non-Newtonian thin-film fluid flows over a rough boundary

TL;DR

This paper derives nonlinear Reynolds equations for non-Newtonian thin-film flows over rough boundaries, showing that classical regimes persist and identifying new equations for each regime.

Contribution

It extends classical Reynolds equation theory to non-Newtonian fluids with power-law viscosity in thin domains with rough boundaries, revealing regime-specific nonlinear equations.

Findings

Classical flow regimes remain valid for non-Newtonian fluids.

Different nonlinear Reynolds equations are derived for each regime.

The analysis covers a range of flow indices p from 9/5 to infinity.

Abstract

We consider a non-Newtonian fluid flow in a thin domain with thickness $\eta_\varepsilon$ and an oscillating top boundary of period $\varepsilon$. The flow is described by the 3D incompressible Navier-Stokes system with a nonlinear viscosity, being a power of the shear rate (power law) of flow index $p$, with $9/5\leq p<+\infty$. We consider the limit when the thickness tends to zero and we prove that the three characteristic regimes for Newtonian fluids are still valid for non-Newtonian fluids, i.e. Stokes roughness ($\eta_\varepsilon\approx \varepsilon$), Reynolds roughness ($\eta_\varepsilon\ll \varepsilon$) and high-frequency roughness ($\eta_\varepsilon\gg \varepsilon$) regime. Moreover, we obtain different nonlinear Reynolds type equations in each case.