Inclined flow of a second-gradient incompressible fluid with pressure-dependent viscosity

TL;DR

This paper introduces a second-gradient extension of the incompressible Navier-Stokes model to account for pressure-dependent viscosity in inclined flows, providing new insights into flow behavior under high-pressure conditions.

Contribution

It develops and analyzes a novel second-gradient model with pressure-sensitive viscosity, addressing well-posedness and flow characteristics in inclined plane scenarios.

Findings

Flow profiles depend on inclination angle and ambient pressure.

Viscosity sensitivity significantly affects flow behavior.

Higher-order derivatives influence boundary conditions and flow stability.

Abstract

Many viscous liquids behave effectively as incompressible under high pressures but display a pronounced dependence of viscosity on pressure. The classical incompressible Navier-Stokes model cannot account for both features, and a simple pressure-dependent modification introduces questions about the well-posedness of the resulting equations. This paper presents the first study of a second-gradient extension of the incompressible Navier-Stokes model, recently introduced by the authors, which includes higher-order spatial derivatives, pressure-sensitive viscosities, and complementary boundary conditions. Focusing on steady flow down an inclined plane, we adopt Barus' exponential law and impose weak adherence at the lower boundary and a prescribed ambient pressure at the free surface. Through numerical simulations, we examine how the flow profile varies with the angle of inclination, ambient pressure, viscosity sensitivity to pressure, and internal length scale.