Conformable Derivatives In Viscous Flow Describing Fluid Through Porous Medium With Variable Permeability

Two new conformable spatial derivatives are defined and introduced to a classical viscous steady-state Navier–Stokes 1D model. The functions for the conformable derivatives have parameters a, b and the fractional parameter α. Analytical solutions for the velocity profile and flow rate are obtained from the conformable models and a fractional model with Caputo’s derivative. The parameters in the conformable derivatives are optimized to fit a classical Darcy–Brinkman 1D model with constant and variable permeability, showing that the conformable models reproduce quite accurately the flow through a porous medium. The ψ1-conformable model describes with high accuracy the flow in a porous media with constant permeability, and also it was compared with experimental information for a flow through plates containing an aligned cylindrical fiber preforms. The other conformable model is the best representation for a medium with variable permeability. Both conformable models are better to depict the velocity profile than the fractional model. Additionally, an expression for the permeability, a classical function of the porosity, the tortuosity, and the size distribution, is given as an explicit function of the parameters in the conformable derivative. Finally, a geometrical interpretation is given, the new conformable derivatives have the potential to describe qualitatively a deformed space that seems like a porous medium.