Onset of magneto convection of Casson fluid in an inclined porous layer using local thermal non-equilibrium model: A linear and nonlinear analyses

The Casson fluids, with their distinct rheological properties, are prevalent in biological, industrial, and material processing systems and the local thermal non-equilibrium (LTNE) model realistically captures heat transfer in porous media by considering separate temperature fields for fluid and solid phases. Incorporating inclination and magnetic effects adds critical complexity and control, making this study essential for optimizing heat and mass transfer in systems. The LTNE model has been used to examine the magneto convection of a Casson fluid in an inclined porous medium under the influence of a magnetic field. Darcy’s law is applied, and we assume the validity of the Oberbeck–Boussinesq approximation. The onset of convection is investigated by two methods: the linear instability analysis and the nonlinear stability analysis. The linear instability analysis is conducted using the normal mode method. The nonlinear stability analysis is studied by utilizing the energy stability method. The resulting eigenvalue problem is solved numerically through the bvp4c routine in MATLAB R2023a. The impact of essential parameters, including the Hartmann number ($H{a}^2$), the inter-phase heat transfer parameter ($H$), the porosity-modified conductivity ratio ($\tau$), the Casson fluid parameter ($\beta$), and an inclination angle ($\gamma$), is thoroughly investigated. A comparison of linear instability and nonlinear stability theories reveals a notable difference in the critical Rayleigh numbers ($R{a}_c$), suggesting the potential existence of a subcritical instability region. The major result of the study indicates that an increase in $H{a}^2$, $H$, and $\gamma$ leads to significant stabilization of the system, whereas an increase in $\beta$ tends to destabilize it. Moreover, the disappearance of transverse rolls is strongly influenced by the parameter $\beta$, making it a critical factor in determining the stability of the system.