In this effort, the consequence of chemical reaction on the thermohaline convection in a permeable layer containing Casson fluid is inspected utilizing both linear and nonlinear stability procedures. To model the momentum equation, the Casson fluid version of Darcy's law is utilized. Applying the linear stability concept, the situation for the start of stationary and oscillatory pattern of convective motion is obtained, whereas the convective heat and mass transferences are determined using nonlinear stability theory. It is found that the oscillatory type of convective flow is probable only if the evaluation of the solutal Rayleigh–Darcy number is less than zero. The effect of the chemical reacting factor, the Casson factor, the Lewis number, and the solutal Rayleigh–Darcy number is to hurry the start of convection toward both stationary and oscillatory styles of convection and also surge the size of the convective cells. The convective heat and mass transfers enhance with increasing the Casson factor, the solutal and thermal Rayleigh–Darcy numbers while both decrease with enhancing the heat capacity quotient. The effect of chemical reacting factor is to increase the mass transmission in the system, while it has no effect on the heat transmission. It is also noted that the convective heat and mass transfers in the system enhance nearly 5.4% and 3.8%, respectively, with a 50% increase in the value of Casson factor.

In this paper, the joint impact of the interior heating and chemical reaction on the double diffusive convective flow in porous membrane enclosures soaked by a non-Newtonian Maxwell fluid is investigated applying linear and nonlinear stability techniques. The porous enclosures are square, slender and rectangular. Using the linear stability analysis, the expression for the critical thermal Rayleigh–Darcy number, above which the convective movement occurs, is derived analytically in terms of associated physical parameters. A nonlinear stability examination reliant on the Fourier double series is executed to calculate the convective heat and mass transports of the arrangement. It is observed that the pattern of convective activity is oscillatory only in the occurrence of a relaxation parameter and the threshold value of the relaxation parameter for the occurrence of the oscillatory pattern depends on the other physical parameters. The onset of convective instability accelerates with the increasing chemical reacting parameter, the interior heating parameter, the solute Rayleigh–Darcy number, the Lewis number, the Vadasz number, and the relaxation parameter, while it delays with the heat capacity ratio. The convective heat and mass transfers increase with the solute Rayleigh–Darcy number, the Vadasz number, the relaxation parameter, and the aspect ratio (for rectangular enclosure), while it decreases with the heat capacity ratio and the aspect ratio (for slender enclosure). Additionally, the convective heat transfer enhances with the interior heating parameter, while the convective mass transfer enhances with the chemical reacting parameter and the Lewis number. The effects of Vadasz number, heat capacity ratio, and relaxation parameter are witnessed only on the oscillatory pattern of convection and unsteady convective heat and mass transfers. Further, some existing literature results are compared with the current findings.

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.

Effects of the magnetic field and inertia on the onset of thermal convection in a horizontal bidispersive porous layer, rotating about a vertical axis, are analyzed. The Darcy equation with same temperature in the micro- and macrophases is used to characterize the fluid motion. The Vadasz number is taken into account in a generalized Darcy equation for the macrophase. The eigenvalue problem obtained from the linear stability analysis is solved analytically for free–free boundaries. Moving one step further from the traditional linear stability analysis, machine learning tools are introduced in this paper to include the effect of multiple parameters on the marginal state of the system. Machine learning techniques have been implemented to identify the mode of instability with respect to different parameters. In particular, classification algorithms, namely, Artificial Neural Networks (ANN) and Support vector machine, are used to examine the onset of oscillatory convection and stationary convection. The required data for training of the algorithms are generated from the results of linear stability analysis. It is found that ANN with the sufficient number of hidden layers along with good choice of training dataset can predict the mode of instability even on the small variation in a given parameter. The combined effect of rotation, magnetic field, and inertia is to reduce the oscillatory mode of instability; hence, the system exhibits the steady mode of instability for a significant region in the three dimensional space comprising the Taylor number, the Hartman number, and the Vadasz number.