Hydromagnetic oscillatory instability in non-linear thermohaline convection of chemically reactive Casson fluid through a porous layer: Effect of open boundary

This work investigates oscillatory instability in non-linear thermohaline convection of Casson fluid flow through a porous medium open at the top. The problem uses a non-linear Oberbeck–Boussinesq approximation to describe how fluid density changes. It also includes heating due to viscous dissipation under an externally imposed vertical magnetic field, accompanied by a first-order chemical reaction. The flow field is non-dimensionalized with the help of the proper choice of scales, while the base flow is then perturbed with a small fluctuation to examine instability theory. The Runge–Kutta method, combined with the shooting technique, is employed for the numerical integration of eigenvalue problems that arise from stability analyses. The graphical illustration is made for critical thermal Rayleigh numbers concerning each flow-governing parameter, and quantitative values in tabular form are provided. Our results reveal that the non-linear solutal effect ${\gamma }_2$ acts to stabilize the solute distribution over the two boundaries when the lower boundary has a higher concentration of solute than the upper boundary ($R{a}_S<0$). Under zero solute transport conditions ($R{a}_s=0$), however, ${\gamma }_2$ is linear. Of note is the fact that the stabilized ${\gamma }_2$ has become destabilized because of the large amount of solute at the upper boundary relative to that at the lower ($R{a}_S>0$). Still, in every case, the non-linear thermal effect ${\gamma }_1$ acts to destabilize. Also, the Hartmann number $Ha$ delays the onset of convection and reduces the region of subcritical instabilities. At negative solutal Rayleigh numbers, the obliqueness of the basic flow stabilizes the thermal instability, while transverse rolls correspond to the most unstable mode for the onset of convection. A lower Casson parameter ($\beta$) causes the fluid to exhibit very extreme non-Newtonian behavior, which prevents oscillatory activity. Higher chemical reactivity shifts streamlines upward, directs isotherms towards the open boundary, and centralizes isohalines. Strong reactivity also weakens the transverse rolls and causes symmetrical isotherm vortices near the boundaries, whereas the isohalines begin by rising and then spreading through the porous layer.