Conformable 1D steady-state Navier-Stokes equations to describe flow through porous media

TL;DR

This paper introduces a generalized conformable derivative-based 1D Navier-Stokes model that accurately describes flow through porous media, matching classical models without explicit Darcy terms.

Contribution

It proposes a novel conformable derivative framework with analytical solutions for steady-state flow in porous media, bridging viscous and inertial-viscous models.

Findings

Conformal models match classical flow profiles in porous media.

Parameters optimized for Darcian and non-Darcian flows.

Models describe flow without explicit Darcy terms.

Abstract

From the definition of a generalized conformable spatial derivative, an exponential conformable function with three parameters $(a,b,\alpha)$ is proposed for a viscous and an inertial-viscous steady-state Navier-Stokes 1D models, obtaining analytical solutions for both generalized conformable models. The conformable models' parameters are optimized to compare the viscous model to a Darcian 1D flow and the inertial-viscous model to a non-Darcian 1D model for a specific range of Darcy numbers $(1\times10^{-2}<Da\leq 1)$. Velocity profiles for the porous medium and the conformable model are computed and compared, showing that the generalized conformable Navier Stokes 1D models describe the flow through a porous medium, for both Darcian and non-Darcian flow, without including a Darcy term or macroscopic porous characteristics.