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

TL;DR

The paper develops second-gradient incompressible viscous fluid theories to address hyperpressure indeterminacy in the Fried-Gurtin framework and to provide well-posed, physically meaningful extensions of the Navier–Stokes equations at small length scales and under high pressures. It specifies a concrete hyperstress constitutive form with $oldsymbol π=oldsymbol{ extell_1^2} abla p$, derives a fourth-order elliptic pressure equation, and extends the theory to include pressure-dependent viscosity. The authors derive explicit Poiseuille and Taylor–Couette solutions under strong and weak adherence, establish convergence of velocity fields to the classical Navier–Stokes profiles as the intrinsic length scales vanish, and discuss boundary conditions necessary to determine pressure. The results enhance predictive capability for micro- and high-pressure flows and provide a rigorous, elliptic formulation that improves upon limitations of classical continua, with potential applications to microfluidics and high-pressure fluid dynamics.

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.

Figures (6)

• Figure 1: Graph of $u(\sigma)$ for Poisuelle flow with strong adherence boundary conditions. The curve for $\lambda_1 = 0$ corresponds to the classical solution.
• Figure 2: Graph of $u(\sigma)$ for Poisuelle flow with weak adherence boundary conditions. Here, $\lambda_2, \lambda_3 \neq 0$.
• Figure 3: Graph of $u(\sigma)$ for Poisuelle flow with weak adherence boundary conditions. Here, $\lambda_2 = \lambda_3 = 0$.
• Figure 4: Graph of $\Phi$ with strong adherence boundary conditions and weak adherence boundary conditions, where $\lambda_2 = \lambda_3 = 0$.
• Figure 5: Graph of $u(\sigma)$ for Couette flow and strong adherence boundary conditions. The value $\lambda_1 = 0$ corresponds to the classical solution.