Maclaurin-series Assisted Linear and Nonlinear Stability Analyses of Rayleigh-Bénard Convection with Variable Gravity Fields

The effect of variable gravity fields that vary through the height of a viscous fluid layer in a Rayleigh-Bénard convection (RBC) is investigated in the paper. A minimal Fourier-series expansion leads to a boundary eigenvalue problem with variable coefficients. Using the Maclaurin-series approach, the recurrence relations for six cases of gravity fields are generated. The eigenvalue of the problem is located using the Newton-Raphson method with an error tolerance of 10-8. The main novelty of the present work is studying the influence of variable gravity fields on the nonlinear dynamics of the problem. A weakly nonlinear stability analysis is performed by first identifying the convective mode and by further arriving at the scaled Lorenz model. Comparison is made with the results of three boundary conditions viz., free-free, rigid-free and rigid-rigid. In the absence of the gravity variation parameter, the results of RBC with constant gravity are recovered. The influence of varying gravity fields on the dynamics is studied using indicators: rH-plots, bifurcation-diagram, periodicity-diagram and trapping-region of the trajectories. It is found that the effect of increasing the strength of gravity is to delay the appearance of chaos for all cases of gravity fields except for the positively linear case. Furthermore, by varying the gravity fields, one can witness shifts in the chaotic and periodic regimes. Moreover, the first largest periodic burst almost overlaps for cubic and biquadratic cases of gravity fields. The shrinking in the size of the ellipsoid with change in the gravity fields is highlighted in the paper.