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The onset of thermal instability with hybrid nanoparticles Al₂O₃–Cu in nanofluid is investigated with the combined effects of variable gravity, variable heat source and Lorentz force in a porous medium. Its applications are in many areas like chemical engineering, geophysics, astrophysics, etc. Based on literature, many gravity variations are assumed in the present analysis, along with a space-dependent heat source/sink parameter that varies along the width of the channel. The finite difference-based Lobatto IIIa method has been applied to solve the system under no nanoparticle flux and rough boundary conditions, and approximate analytical results are generated for some special cases. The roughness parameters (${\lambda }_1,{\lambda }_2$), Darcy number (Da), gravity variation function ($G(z)$), and Chandrasekhar number (Q) delay the onset of convection and thus stabilize the system. It is also observed that an increase in the Lewis number Le, power index in variable heat source, and the nanoparticle Rayleigh concentration number Rn decreases the critical Rayleigh number $R_{\theta ,c}$ which destabilizes the system, and increases the critical wave number, which enlarges the convection cell size. In the case of exponential variation ($1-e^z$) in the gravity variation parameter, the system becomes stabilized due to a delay in the onset of convection. In addition, we have considered multilayer perceptron-artificial neural network (MLP-ANN) computation to predict the critical Rayleigh number as per function of important controlling parameters. The data set of 625 observations is chosen keeping 70% for testing, 15% for training and 15% for validation using efficient Levenberg−Marquardt back propagation algorithm with optimal accuracy measures i.e., root mean square deviation (RMSE), root mean relative error (RMRE) and $R^2$ (coefficient of determination). Finally, the regression plots are drawn that correlate, target and output data.

The effect of time-periodic two-frequency rotation modulation on Rayleigh–Bénard convection in water with either AA7072 or AA7075 nanoparticles is investigated. The single-phase description of the Khanafer–Vafai–Lightstone model is used for modeling the nanoliquids. An asymptotic expansion procedure is adopted in the case of the linear stability to obtain the correction (due to modulation) to the Rayleigh number at marginal stability of unmodulated convection. A nonlinear regime of convection is considered with a nonautonomous generalized Lorenz model as the governing equation. The method of multiscales is then employed to obtain the coupled nonautonomous Ginzburg–Landau equations with cubic nonlinearity from the Lorenz model. These equations are presented in the phase-amplitude form and the amplitude is used to quantify the heat transport. The modulation amplitude is considered to be small (of order less than unity) and moderate frequencies of modulation are considered. We found that there is a threshold frequency beyond which the system behavior reverses. At frequencies below the threshold, the mean Nusselt number increases with an increase in the amplitude of modulation while an opposite influence is seen for values above the threshold. Such a behavior is a consequence of what is analogously seen in the case of the critical Rayleigh number. The influence of two-frequency modulation is more pronounced on the results of the linear and nonlinear regimes compared to that of the single-frequency one. The heat transport is enhanced due to the presence of dilute concentration of suspended nanoparticles (either AA7072 or AA7075 nanoalloys) in water. The influence of nanoparticles is to modify the threshold values generating chaos but it does not qualitatively alter the dynamical behavior of the system. The plots of Lyapunov exponents reveal that there is no possibility of hyper-chaos in the generalized Lorenz model when there is a rotational modulation.