Nonlinear stability and bifurcation in thermal flows with varying transport properties

The nonlinear stability and bifurcation characteristics of thermally driven flows with variable viscosity and thermal conductivity are investigated in this study. The analysis considers the coupled interaction of electro-osmotic forcing, magnetohydrodynamic effects, and Arrhenius-type chemical reactions in a thermally sensitive fluid system. The governing framework incorporates key dimensionless parameters, including the Brinkman number, Hartmann number, Prandtl number, and Frank–Kamenetskii parameter, which collectively control viscous dissipation, electromagnetic forcing, thermal diffusion, and chemical heat generation. The results reveal the existence of both subcritical and supercritical bifurcations depending on the thermal boundary conditions and geometric configuration. The nonlinear interactions between thermal and velocity perturbations give rise to multiple steady solutions and periodic oscillations, indicating complex flow behavior even in systems with varying transport coefficients. A key outcome of the study is the identification of bifurcation behavior governed by the Frank–Kamenetskii term, which controls the balance between exothermic heat generation and thermal dissipation. Beyond a critical threshold, the system undergoes a bifurcation leading to thermal runaway, where steady solutions cease to exist, and the temperature grows without bound. This indicates a transition from stable to unstable regimes driven by nonlinear reaction–diffusion interactions. The study provides fundamental insights into the dynamical transitions in thermally driven flows, offering benchmarks for validating numerical models and guiding experimental investigations in heat transfer and fluid stability problems.