Analysis and profiles of travelling wave solutions to a Darcy-Forchheimer fluid formulated with a non-linear diffusion.
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2022
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AIMS Mathematics
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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
di usion. 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 di usivity. 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 di usion 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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Existence, Uniqueness, Asymptotic, Travelling waves, Geometric Perturbation Theory, Porous media equation, Darcy-Forchheimer