A Study of Regular and Chaotic Dynamics of Rayleigh–Bénard Convection in a Casson Fluid with Vertical Throughflow

This study investigates the impact of a Casson fluid on convective heat transfer in the presence of permeable boundaries using a dynamical systems approach. A weakly nonlinear analysis leads to the derivation of the cubic Stuart–Landau amplitude equation, and a simplified Lorenz-type model is constructed via a Galerkin truncation method. The transition from steady to chaotic convection is examined through bifurcation diagrams, Lyapunov exponents, and phase space projections. Key parameters such as the Péclet number and the Casson fluid parameter are analyzed for their roles in the onset of chaos. Results show that increasing the Péclet number raises the Hopf–Rayleigh number and stabilizes the system by reducing heat transfer, while higher values of the Casson parameter significantly lower the critical Rayleigh number, enhancing instability. The largest Lyapunov exponent is computed across various aspect ratios and driving intensities, highlighting the complex interplay between flow parameters and chaotic behavior. These results may help guide the optimization of heat transfer in industrial systems involving Casson fluids, aiding in the design of stable reactors and efficient heat exchangers.