The study investigates the thermosolutal convection of a Casson fluid in a horizontal layer that is heated and salted from below. Both linear and non-linear analyses are performed using the method of normal modes to solve the governing equations. Interestingly, the study demonstrates that the linear and non-linear stability thresholds coincide. To solve the differential eigenvalue problem for linear theory, a one-term Galerkin approach is employed. Meanwhile, for the eigenvalue problem of non-linear instability, a numerical solution is obtained using the bvp4c routine in MATLAB. The results reveal some important findings. First, the Casson parameter is shown to destabilize the flow, leading to instability. However, the Darcy number and solutal Rayleigh number are found to have a stabilizing effect on the system. Furthermore, the study develops a weakly non-linear theory using multiple scale analysis to investigate heat and mass transport, offering valuable insight into these transport phenomena within the context of the system under consideration.

The effect of rotation and cross-diffusion on convection in a horizontal sparsely packed porous layer in a thermally conducting fluid is studied using linear stability theory. The normal mode method is employed to formulate the eigenvalue problem for the given model. One-term Galerkin weighted residual method solves the eigenvalue problem for free-free boundaries. The eigenvalue problem is solved for rigid-free and rigid-rigid boundaries using the BVP4c routine in MATLAB R2020b. The critical values of the Rayleigh number and corresponding wave number for different prescribed values of other physical parameters are analyzed. It is observed that the Taylor number and Solutal Rayleigh number significantly influence the stability characteristics of the system. In contrast, the Soret parameter, Darcy number, Dufour parameter, and Lewis number destabilize the system. The critical values of wave number for different prescribed values of other physical parameters are also analyzed. It is found that critical wave number does not depend on the Soret parameter, Lewis number, Dufour parameter, and solutal Rayleigh number; hence critical wave number has no impact on the size of convection cells. Further critical wave number acts as an increasing function of Taylor number, so the size of convection cells decreases, and the size of convection cells increases because of Darcy number.

The present investigation is aimed to study the joint effect of the variable gravity field and chemical reaction on the onset of convection in a sparsely packed porous layer. Linear, parabolic, cubic and exponential functions are considered to account for the variable gravity field. The normal mode technique is used to study the linear stability of the problem. The corresponding eigenvalue problem is solved for the free-free, rigid-free, and rigid-rigid boundaries. The influence of the Damkohler number Dm, the solutal Rayleigh number Rs, the Lewis number Le, the Darcy number Da and the variable gravity field parameter \({{{{\delta }}}_{1}}\) on stability of the system in a sparsely packed porous layer is investigated. The results show that the onset of convection is delayed due to increase in the solutal Rayleigh number, the variable gravity parameter, and the Lewis number. In contrast, the onset of convection is enhanced by increase in the Darcy number and the Damkohler number. It is also observed that the system becomes more stable for exponential variation in the gravity field and less stable for cubic variation in the gravity field.

This paper deals with the linear stability analysis and weakly non-linear analysis of Magneto-convection in a sparsely packed porous medium with constant vertical Magnetic field and gravity modulation. A linear stability analysis reported here and shows that the gravity modulation has significant effect on the stability limits of the system. The gravity modulation is know to have effect and is treated by a perturbation expansion in powers of the amplitude of modulation. The shift in the critical Rayleigh number is evaluated and depends on the prandtl number and frequency of modulation, using the Venezian method. It is also shown that the onset of convection can be advance or delay by the regulation of various parameters. Weakly nonlinear analysis is performed based on the method of power series, where the disturbance is expressed in terms of power series. A nonlinear Ginzburg-Landau equation to investigate the three different types of gravity modulation on heat transfer is derived as part of this work. Heat transfer have been shown to depend on Nusselt number, further the effect of different types of parameter on heat transport have been studied graphically. Nusselt number graph is also shown for different parameter and explain in detail. The effect of magnetic prandtl number and Chandrasekhar number are stabilize the system. The control of convection is a major issue in systems with fluids as a working media. This is all the more difficult if the fluid system is housed in a porous medium. The paper presents three mechanisms of controlling onset of convection and thereby the heat transfer in such fluid systems. In order the modulation effect is effective in its role, we have considered the system to be a fluid-saturated porous media.