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

Abstract

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 [Formula: see text] and the fractional parameter [Formula: see text]. 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 [Formula: see text]-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.

Surface dynamics on Jeffrey nanofluid flow with Coriolis effect and variable Darcy regime

“…It is worth mentioning that Equation (10) will be substituted into Equation ( 8) to obtain appropriate temperature flow equation. Following [36][37][38][39], the variable Darcy model defined in Equation ( 11) is incorporated into the momentum equations ( 6) and (7),…”

“…It is worth mentioning that Equation () will be substituted into Equation () to obtain appropriate temperature flow equation. Following [36–39], the variable Darcy model defined in Equation () is incorporated into the momentum equations () and (), $$$$\begin{equation} K(z) = \frac{m_p^2 \; \lambda (z)^3}{175{\left(1-\lambda (z) \right)}^2 }; \qquad \text{Such that } \qquad \lambda (z) = \xi _0 {\left(1+ \xi _1 e^{-\frac{\xi _2 \; z}{m_p}} \right)}, \end{equation}$$$$where ξ 1 and ξ 2 are the empirical constants subject to porous particle diameter, ξ 0 represent the ambient porosity and $$m_p$$ is the particle diameter. The experimental value of the embedding parameters ξ 0 , ξ 1 , ξ 2 , and $$Pr$$ are used (fixed value) for simulation.…”

“…Experimental value of the thermophysical parameter in accordance with blood as the based fluid[36][37][38][39][40][41].…”