Abstract
In the presented analysis, a heterogeneous diffusion is introduced to a magnetohydrodynamics (MHD) Darcy–Forchheimer flow, leading to an extended Darcy–Forchheimer model. The introduction of a generalized diffusion was proposed by Cohen and Murray to study the energy gradients in spatial structures. In addition, Peletier and Troy, on one side, and Rottschäfer and Doelman, on the other side, have introduced a general diffusion (of a fourth-order spatial derivative) to study the oscillatory patterns close the critical points induced by the reaction term. In the presented study, analytical conceptions to a proposed problem with heterogeneous diffusions are introduced. First, the existence and uniqueness of solutions are provided. Afterwards, a stability study is presented aiming to characterize the asymptotic convergent condition for oscillatory patterns. Dedicated solution profiles are explored, making use of a Hamilton–Jacobi type of equation. The existence of oscillatory patterns may induce solutions to be negative, close to the null equilibrium; hence, a precise inner region of positive solutions is obtained.
Highlights
The mathematical formulation of non-Newtonian fluids is of relevance to model complex scenarios emerging in engineering and physics
This is the case of the Darcy–Forchheimer model that arises in MHD
The proposed generalized diffusion to a classical second-order Darcy–Forchheimer flow was treated in the presented study, leading to an fourth-order operator, defining the extended Darcy–Forchheimer flow
Summary
The mathematical formulation of non-Newtonian fluids is of relevance to model complex scenarios emerging in engineering and physics. The non-Newtonian fluids are considered under dedicated descriptions subjected to particular applications This is the case of the Darcy–Forchheimer model that arises in MHD. As an example of this approach to biological applications, in [15], the authors study a diffusive structure, accounting for a diffusive gradient Such a diffusive gradient was obtained from a generalized Landau–Ginzburg free energy model that ends in a fourth-order diffusion. Note that the intention along this analysis is to study some heterogeneous patterns close to the critical points for a MHD flow of the Darcy–Forchheimer type To this end, a perturbation term, in the form of third-order diffusion, was introduced ad-hoc, so that (2). The presented analysis provides insight into the characterization of such oscillatory profiles for an extended Darcy–Forchheimer flow with a non-homogeneous diffusive perturbation. A region of validity for positive solutions with monotone behavior is explored
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