Plane Poiseuille flow through a porous medium - an analytical solution

TL;DR

This paper presents an exact analytical solution for the Darcy velocity in plane Poiseuille flow through a porous medium using Weierstrass elliptic functions, enabling precise calculation of flow characteristics.

Contribution

It introduces a novel closed-form solution expressing Darcy velocity with Weierstrass elliptic functions, providing new analytical tools for porous media flow analysis.

Findings

Derived a closed-form analytical expression for Darcy velocity.

Provided formulas for volume flux and dissipation rate.

Developed an algorithm to compute flow profiles from parameters.

Abstract

This paper develops a closed-form, analytical expression for the Darcy velocity of a Newtonian fluid flowing through a channel filled with a porous medium, bound by rigid walls and driven by a constant pressure gradient. We express the Darcy velocity as a Weierstrass elliptic function of the transverse coordinate. It allows us to get analytical expressions for volume flux and the rate of dissipation. The Weierstrass elliptic function is a doubly-periodic, complex-valued function defined over the complex plane. It takes real values only on certain segments in the complex plane. We show how to align the transverse coordinate axis along these segments to ensure real values of Darcy velocity. Lastly, we give an algorithm to generate velocity profile and compute volume flux given the flow parameters.