Analysis and profiles of travelling wave solutions to a Darcy-Forchheimer fluid formulated with a non-linear diffusion

The intention along the presented analysis is to explore existence, uniqueness, regularity of solutions and travelling waves profiles to a Darcy-Forchheimer fluid flow formulated with a non-linear diffusion. Such formulation is the main novelty of the present study and requires the introduction of an appropriate mathematical treatment to deal with the introduced degenerate diffusivity. Firstly, the analysis on existence, regularity and uniqueness is shown upon definition of an appropriate test function. Afterwards, the problem is formulated within the travelling wave domain and analyzed close the critical points with the Geometric Perturbation Theory. Based on this theory, exact and asymptotic travelling wave profiles are obtained. In addition, the Geometric Perturbation Theory is used to provide evidences of the normal hyperbolicity in the involved manifolds that are used to get the associated travelling wave solutions. The main finding, which is not trivial in the non-linear diffusion case, is related with the existence of an exponential profile along the travelling frame. Eventually, a numerical exercise is introduced to validate the analytical solutions obtained.

Automated summary

The intention of this study is to explore the existence, uniqueness, regularity of solutions, and profiles of travelling waves in a Darcy-Forchheimer fluid flow with nonlinear diffusion. The formulation of such flow is a novel aspect of this study and requires appropriate mathematical treatment to handle the introduced degenerate diffusivity. Firstly, the existence, regularity, and uniqueness are analyzed using an appropriate test function. Then, the problem is formulated in the travelling wave domain and analyzed near critical points using the Geometric Perturbation Theory. Based on this theory, precise and asymptotic profiles of travelling waves are obtained. Additionally, the Geometric Perturbation Theory is used to provide evidence of the normal hyperbolicity in the involved manifolds that are used to obtain the associated travelling wave solutions. The main finding, which is non-trivial in the case of nonlinear diffusion, is the existence of an exponential profile along the travelling frame. Lastly, a numerical exercise is conducted to validate the obtained analytical solutions.