Nonlinear evolution of weak discontinuity waves in Darcy-type porous media

TL;DR

This paper analyzes the nonlinear evolution of weak discontinuity waves in Darcy-type porous media, deriving evolution equations and examining how porosity influences wave steepening and shock formation.

Contribution

It introduces a new theoretical framework for understanding wave evolution in porous media, including critical amplitude criteria for shock formation.

Findings

Critical amplitude for shock formation identified

Porosity significantly affects wave steepening

Wave behavior varies with symmetry type

Abstract

The propagation of nonlinear waves in one dimensional space, unsteady and compressible flow in Darcy-type porous media is analyzed. It is assumed that the weak discontinuity propagate long the characteristic path using the characteristics of the governing quasilinear system as the reference coordinate system. Evolution equation in the characteristic plane is derived. As an application of the theory the breaking point at the wave front is determined. It is assessed as to how the porosity of the medium affects the process of steepening and flattering of acceleration waves with planar, cylindrical, and spherical symmetry. The critical amplitude of the initial disturbance has been determined such that any compressive disturbance with initial amplitude greater than the critical one always grows into a shock wave, while the initial amplitude less than the critical one always decays.