Linear and Weakly Nonlinear Stability Analysis: Transition to Chaos in Brinkman–Bénard Convection of a Nanofluid-Saturated Porous Medium with Variable Viscosity and Internal Heat Source or Sink

This paper emphasizes on linear and weakly nonlinear stability of Brinkman–Bénard convection in a horizontal porous layer saturated with a nanofluid, incorporating a temperature-dependent viscosity and a uniform internal heat source/sink. Linear stability analysis shows that a positive internal Rayleigh number and an increase in the thermorheological parameter stabilizes the motionless conductive state, while an increase in the porous parameter destabilizes it. A three-mode Lorenz-type model is derived through a weakly nonlinear expansion and truncated Fourier series. Equilibrium and linear stability analysis of the reduced model show that the conductive equilibrium is asymptotically stable when all eigenvalues are negative, convective equilibria lose stability at a critical Rayleigh number via a Hopf bifurcation, beyond which time-periodic and chaotic dynamics arise. Nanoparticles have a marginal effect, while internal heating slightly lowers the critical Rayleigh number. Increasing the inverse viscosity parameter from 0 to 0.5 raises the onset of convection by 50%, porosity by 36%, and combined effects by 87%. The Nusselt number rises by 3–86% with higher nanoparticle loading and viscosity variation, and the Hopf bifurcation point increases with thermorheological parameter, internal Rayleigh number, and volume fraction of the nanoparticle, enhancing system stability. Limiting-case results recover known benchmarks from the literature.