Non-linear magnetoconvection in a bidispersive porous layer: a brinkman model

This study examines the magnetic effect on Darcy Brinkman convection in a Bidispersive horizontal porous layer, considering the importance of convective motions of electrically conducting porous media accompanying a magnetic field in real-life applications such as geophysics, metallurgical field and solidification structures. In order to conduct a thorough study, the boundaries are classified as free-free, rigid-free, and rigid-rigid. The fluid motion is described using the Brinkman-Darcy equation with a single temperature in the macropores and micropores. The eigenvalue problem is solved analytically for the free-free case by employing linear stability theory. A non-linear analysis using the energy method is undertaken to prove that linear instability and global non-linear stability thresholds are the same. The eigenvalue problem for rigid-free and rigid-rigid boundaries is numerically solved with the bvp4c routine in MATLAB R2020 with the Rayleigh number as the eigenvalue. It is found that the Hartmann number \(M^2\), Darcy number Da, permeability ratio \(\kappa _r\), and momentum transfer coefficient \(\gamma\) stabilize the system. Rigid-rigid boundaries are found to be the most stable ones, followed by rigid-free and free-free, which are the least stable boundaries.