Asymptotic solutions to an Eyring-Powell viscous flow with Darcy-Forchheimer non-linear reaction

Abstract We provide a mathematical treatment, analytical and numerical, for a fluid constructed as an hybrid of the Eyring-Powell and Darcy-Forchheimer fluid models. The Eyring-Powell model departs from the kinetic theory of liquids and it allows for a description of shear stresses and viscous terms. The Darcy-Forchheimer model permits to describe the fluid effects given in a porous media, and it provides non-linear reaction terms when considered as part of the momentum equations. Hence, it is natural to investigate mathematical characteristics of solutions for a fluid flow formulated as a combination of these two fluid models. First of all, we prove boundedness and uniqueness of solutions arising from rough (i.e., in $L^1(R) \cap L^\infty(R)$) initial data. This is physically relevant, since it means that we are considering general descriptions of the velocity distribution of the fluid, in a media with particular porosity distributions. Afterwards, stationary profiles are obtained by using a Hamiltonian description, and our construction is supported by numerical validating evidences. Furthermore, asymptotic solutions are explored based on an exponential scaling and a non-linear transport Jacobi equation. Finally, a region of validity for this asymptotic approach is provided, and a numerical validation of our asymptotic analysis is presented. Our main conclusion is that a fluid model combining Eyring-Powell and Darcy-Forchheimer characteristics is indeed possible to introduce, and that solutions of potential physical interest, can be obtained.