Unsteady, Nonlinear Buoyancy‐Driven Convection in Temperature‐Sensitive Newtonian Liquids With Heat Source

ABSTRACT Rayleigh–Bénard convection (RBC) has important applications in studying heat transfer, fluid dynamics, and pattern formation, particularly in areas such as electronic cooling, crystal growth, underground waste management, and oil recovery. Understanding RBC is crucial for controlling and enhancing heat transport. In this study, we demonstrate an analytical study of unsteady, two‐dimensional, nonlinear RBC in Newtonian fluids with temperature‐dependent viscosity and volumetric heat sources/sinks. Linear theory predicts that the principle of exchange of stability is shown to be valid. It is demonstrated here that the heat source and temperature‐dependent viscosity together act to destabilize the system, whereas the heat sink stabilizes the same. The generalized Lorenz model derived in this study is essentially the classical Lorenz model, but its coefficients involve the internal Rayleigh number and the thermorheological parameter , which characterizes the heat source/sink and the temperature‐dependent viscosity, respectively. It is found that the trajectories of the Lorenz model are shown to be bounded within an ellipsoid, but its major and minor axes change with respect to the parameter variation. It is shown that the effect of increasing and is to expand the phase‐plane trajectories outward, whereas the effect of decreasing causes them to contract. The classical Lorenz model derived in this problem is analytically intractable; therefore, it is transformed into a tractable Ginzburg–Landau model to quantify heat transfer. This important result also establishes a connection between the two classical models.