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Heterogeneous Diffusion, Stability Analysis, and Solution Profiles for a MHD Darcy–Forchheimer Model

Author

Listed:
  • José Luis Díaz

    (Escuela Politécnica Superior, Universidad Francisco de Vitoria, Ctra. Pozuelo-Majadahonda Km 1800, Pozuelo de Alarcón, 28223 Madrid, Spain)

  • Saeed Rahman

    (Department of Mathematics, COMSATS University Islamabad, Abbottabad 22044, Pakistan)

  • Juan Miguel García-Haro

    (Escuela Politécnica Superior, Universidad Francisco de Vitoria, Ctra. Pozuelo-Majadahonda Km 1800, Pozuelo de Alarcón, 28223 Madrid, Spain)

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.

Suggested Citation

  • José Luis Díaz & Saeed Rahman & Juan Miguel García-Haro, 2021. "Heterogeneous Diffusion, Stability Analysis, and Solution Profiles for a MHD Darcy–Forchheimer Model," Mathematics, MDPI, vol. 10(1), pages 1-14, December.
    • Handle: RePEc:gam:jmathe:v:10:y:2021:i:1:p:20-:d:707937
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    References listed on IDEAS

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    1. Saif, Rai Sajjad & Muhammad, Taseer & Sadia, Haleema, 2020. "Significance of inclined magnetic field in Darcy–Forchheimer flow with variable porosity and thermal conductivity," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 551(C).
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    Cited by:

    1. Walid Aich & Fatih Selimefendigil & Badreddine Ayadi & Lotfi Ben Said & Badr M. Alshammari & Lioua Kolsi & Sid Ali Betrouni & Hatem Gasmi, 2022. "Application and CFD-Based Optimization of a Novel Porous Object for Confined Slot Jet Impingement Cooling Systems under a Magnetic Field," Mathematics, MDPI, vol. 10(15), pages 1-21, July.

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