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

S. Rahman; J. L. Díaz Palencia; J. Roa González; S. Rahman; J. L. Díaz Palencia; J. Roa González

doi:10.3934/math.2022383

AIMS Mathematics

2022, Volume 7, Issue 4: 6898-6914. doi: 10.3934/math.2022383

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

1.
Department of Mathematics, COMSATS University Islamabad, Abbottabad, Pakistan
2.
Universidad a Distancia de Madrid. Vía de Servicio A-6, 15, 28400 Collado Villalba, Madrid, Spain
3.
Escuela Politécnica Superior, Universidad Francisco de Vitoria, Ctra, Pozuelo-Majadahonda Km 1800, 28223, Pozuelo de Alarcón, Madrid, Spain

Received: 05 December 2021 Revised: 14 January 2022 Accepted: 23 January 2022 Published: 28 January 2022
MSC : 35K92, 35K91, 35K55

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.
- existence,
- uniqueness,
- asymptotic,
- travelling waves,
- Geometric Perturbation Theory,
- porous media equation,
- Darcy-Forchheimer
Citation: S. Rahman, J. L. Díaz Palencia, J. Roa González. Analysis and profiles of travelling wave solutions to a Darcy-Forchheimer fluid formulated with a non-linear diffusion[J]. AIMS Mathematics, 2022, 7(4): 6898-6914. doi: 10.3934/math.2022383

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Abstract

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.

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