Current density and joule heating of dual exothermic reaction in a Magneto-Williamson radiative fluid with biot-type convection

Reactive magnetohydrodynamic transport in non-Newtonian fluids exhibits complex thermo-fluid behavior when heat generation and dissipation interact within confined geometries. Unlike previous studies that separately examined MHD Williamson flow, radiative transport, or exothermic combustion, the present work simultaneously incorporates induced current density, Joule heating, quadratic Boussinesq buoyancy, dual-step exothermic reactions, and Biot-type cooling within a unified framework. This study investigates current density and the coupled effects of radiative cooling and Joule heating on dual exothermic diffusion in a Quadratic-Boussinesq Magneto-Williamson fluid governed by Biot-type bimolecular kinetics. A comprehensive mathematical model is formulated for coupled momentum and energy equations, incorporating nonlinear buoyancy, viscous dissipation, radiative absorption, electromagnetic damping, and shear-thinning rheology in steady two-dimensional channel flow. The resulting nonlinear boundary-value problem is solved using a Galerkin weighted-residual hybrid method, ensuring stable and accurate predictions. Results show that increasing the Brinkmann number, Williamson relaxation parameter, and two-step reaction kinetics enhances internal heat generation, steepens temperature gradients, and raises the likelihood of thermal runaway. Conversely, radiative transport, wall heat exchange characterized by the Biot number, and internal heat sinks improve thermal diffusion and suppress temperature rise. From a hydrodynamic perspective, the Hartmann number introduces a Lorentz force that reduces axial velocity, while the Grashof number promotes buoyancy-driven flow and vertical transport. These competing mechanisms create a delicate thermo-fluid balance controlling the onset of exothermic instability. The model integrates dual exothermic reactions, radiative heat transfer, nonlinear Boussinesq effects, Williamson rheology, and Joule heating within a unified reactive MHD framework. The numerical formulation is validated through comparison with available benchmark solutions, showing excellent agreement and confirming the accuracy of the proposed Galerkin weighted-residual hybrid scheme. The findings contribute to safer thermal management strategies in catalytic reactors, thermal batteries, metallurgical furnaces, polymer processing units, geothermal systems, and renewable-energy conversion technologies where thermal runaway remains a critical operational challenge.