Second-gradient models for incompressible viscous fluids and associated cylindrical flows

TL;DR

This paper develops second-gradient models for incompressible viscous fluids, ensuring physical and mathematical consistency, and demonstrates their application to cylindrical flows with convergence to classical solutions.

Contribution

It introduces a new constitutive relation for hyperpressure and extends the framework to pressure-dependent viscosities, guaranteeing well-posedness and ellipticity.

Findings

Second-gradient effects ensure ellipticity of the pressure equation.

Explicit solutions for cylindrical flows are derived.

Velocity profiles converge to Navier-Stokes solutions as length scales vanish.

Abstract

We introduce second-gradient models for incompressible viscous fluids, building on the framework introduced by Fried and Gurtin. We propose a new and simple constitutive relation for the hyperpressure to ensure that the models are both physically meaningful and mathematically well-posed. The framework is further extended to incorporate pressure-dependent viscosities. We show that for the pressure-dependent viscosity model, the inclusion of second-gradient effects guarantees the ellipticity of the governing pressure equation, in contrast to previous models rooted in classical continuum mechanics. The constant viscosity model is applied to steady cylindrical flows, where explicit solutions are derived under both strong and weak adherence boundary conditions. In each case, we establish convergence of the velocity profiles to the classical Navier-Stokes solutions as the model's characteristic length scales tend to zero.