Analysis of travelling wave solutions for Eyring-Powell fluid formulated with a degenerate diffusivity and a Darcy-Forchheimer law

José Luis Díaz Palencia; Saeed ur Rahman; Antonio Naranjo Redondo; José Luis Díaz Palencia; Saeed ur Rahman; Antonio Naranjo Redondo

doi:10.3934/math.2022834

AIMS Mathematics

2022, Volume 7, Issue 8: 15212-15233. doi: 10.3934/math.2022834

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Research article

Analysis of travelling wave solutions for Eyring-Powell fluid formulated with a degenerate diffusivity and a Darcy-Forchheimer law

1.
Department of Mathematics, COMSATS University Islamabad, Abbottabad Campus, Abbottabad 22060, Pakistan
2.
Department of Mathematics and Education, Universidad a Distancia de Madrid, 28400 Madrid, Spain
3.
Technology Programs, Schiller International University, Calle Serrano 156, 28002, Madrid, Spain

Received: 24 April 2022 Revised: 31 May 2022 Accepted: 06 June 2022 Published: 16 June 2022
MSC : 35B65, 35Q35

The goal of this paper is to provide analytical assessments to a fluid flowing in a porous medium with a non-linear diffusion linked to a degenerate diffusivity. The viscosity term is formulated with an Eyring-Powell law, together with a non-homogeneous diffusion typical of porous medium equations (as known in the theory of partial differential equations). Further, the equation is supplemented with an absorptive reaction term of Darcy-Forchheimer, commonly used to model flows in porous medium. The work starts by analyzing regularity, existence and uniqueness of solutions. Afterwards, the problem is transformed to study travelling wave kind of solutions. An asymptotic expansion is considered with a convergence criteria based on the geometric perturbation theory. Supported by this theory, there exists an exponential decaying rate in the travelling wave profile. Such exponential behaviour is validated with a numerical assessment. This is not a trivial result given the degenerate diffusivity induced by the non-linear diffusion of porous medium type and suggests the existence of regularity that can serve as a baseline to construct numerical or energetic approaches.
- Eyring-Powell fluid,
- Darcy-Forchheimer,
- porous medium equation,
- travelling waves,
- geometric perturbation theory
Citation: José Luis Díaz Palencia, Saeed ur Rahman, Antonio Naranjo Redondo. Analysis of travelling wave solutions for Eyring-Powell fluid formulated with a degenerate diffusivity and a Darcy-Forchheimer law[J]. AIMS Mathematics, 2022, 7(8): 15212-15233. doi: 10.3934/math.2022834

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Abstract

The goal of this paper is to provide analytical assessments to a fluid flowing in a porous medium with a non-linear diffusion linked to a degenerate diffusivity. The viscosity term is formulated with an Eyring-Powell law, together with a non-homogeneous diffusion typical of porous medium equations (as known in the theory of partial differential equations). Further, the equation is supplemented with an absorptive reaction term of Darcy-Forchheimer, commonly used to model flows in porous medium. The work starts by analyzing regularity, existence and uniqueness of solutions. Afterwards, the problem is transformed to study travelling wave kind of solutions. An asymptotic expansion is considered with a convergence criteria based on the geometric perturbation theory. Supported by this theory, there exists an exponential decaying rate in the travelling wave profile. Such exponential behaviour is validated with a numerical assessment. This is not a trivial result given the degenerate diffusivity induced by the non-linear diffusion of porous medium type and suggests the existence of regularity that can serve as a baseline to construct numerical or energetic approaches.

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