Asymptotic Analysis and Explicit Weierstrass Traveling Waves for a Darcy–Forchheimer Flow Model

ABSTRACT

The aim of the present work is to study the behavior of solutions to a Darcy–Forchheimer flow in a saturated porous medium. The profiles of solutions are provided as asymptotic expansions of solutions to a Jacobi‐type equation. This Jacobi‐type equation is solved explicitly, and a region of validity for our asymptotic expansions is presented. We further support our analytical findings by providing numerical approximations of the solutions, which serve to validate our theoretical work. Moreover, we delve into the symmetries inherent in the Darcy–Forchheimer flow, deriving the explicit form of the traveling wave solution. Additionally, we present a class of solutions characterized by infinitely many Weierstrass‐type traveling waves. These solutions are particularly intriguing, as they describe a scenario where the interaction between the fluid and the porous medium results in fluid accumulation to a point where the medium's porosity is no longer sufficient to sustain the flow.