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Rayleigh–Bénard convection is the heat transfer process due to buoyancy effect involved that occurred in a horizontal plane of nanofluids layer heated from below. The model for nanofluids includes the mechanisms of Brownian motion and thermophoresis. The onset of Rayleigh–Bénard convection in a horizontal rotating nanofluids layer and in a horizontal nanofluids layer saturated in a rotating porous medium with feedback control, internal heat source, magnetic field, double–diffusive coefficients, porosity, anisotropic, viscosity variation and thermal conductivity variation parameters are investigated theoretically. The confining lower and upper boundary conditions of the nanofluids layer are assumed to be free–free, rigid–free and rigid–rigid. A linear stability analysis of Rayleigh–Bénard convection is used, then the eigenvalue is obtained numerically using the Galerkin method and solved using Maple software. The impact of the feedback control, rotation, internal heat source, magnetic field, double–diffusive coefficients, porosity, anisotropic, viscosity variation and thermal conductivity variation parameters on the onset of convection in nanofluids system are analyzed and presented graphically. It is found that the impact of increasing the effects of feedback control, rotation, magnetic field, Dufour, porosity, anisotropic and thermal conductivity variation parameters help to delay the onset of convection in the system, meanwhile elevating the effects of internal heat source, Soret and viscosity variation parameters hasten the instability of the system. Further, the lower and upper boundary conditions in the present investigation are obviously found to be more stable in rigid–rigid boundaries compared to free–free and rigid–free boundaries.

Effect of rotation on the onset of Rayleigh-Benard convection in a horizontal nanofluid layer with vertical magnetic field and internal heat source is investigated. Linear stability analysis based upon normal mode method is employed to find solution of the horizontal nanofluid layer bounded between free-free, rigid-free and rigid-rigid boundaries. Rayleigh number has been determined using the galerkin method. Graphs have been plotted to study the efficiency of rotation, magnetic field, internal heat source and other nanofluid parameters to the system.

A linear stability analysis has been carried out to examine the effect of internal heat source on the onset of Rayleigh–Bénard convection in a rotating nanofluid layer with double diffusive coefficients, namely, Soret and Dufour, in the presence of feedback control. The system is heated from below and the model used for the nanofluid layer incorporates the effects of thermophoresis and Brownian motion. Three types of bounding systems of the model have been considered which are as follows: both the lower and upper bounding surfaces are free, the lower is rigid and the upper is free, and both of them are rigid. The eigenvalue equations of the perturbed state were obtained from a normal mode analysis and solved using the Galerkin method. It is found that the effect of internal heat source and Soret parameter destabilizes the nanofluid layer system while increasing the Coriolis force, feedback control, and Dufour parameter helps to postpone the onset of convection. Elevating the modified density ratio hastens the instability in the system and there is no significant effect of modified particle density in a nanofluid system.

The effect of feedback control on the onset of double-diffusive convection in a horizontal binary fluid layer is studied analytically subjected to cross diffusion coefficients which are the Soret and Dufour parameters. The confined boundaries of the binary fluid layer are considered to be free-free, rigid-free and rigid-rigid which described the lower and upper surfaces respectively. The linear stability theory is applied and the eigenvalue solution is obtained numerically using Galerkin technique. Focusing on the stationary convection, it is shown that there is a positive thermal resistance in the presence of feedback control on the onset of double-diffusive convection in binary fluid mixture.

The influences of feedback control and internal heat source on the onset of Rayleigh–Bénard convection in a horizontal nanofluid layer is studied analytically due to Soret and Dufour parameters. The confining boundaries of the nanofluid layer (bottom boundary–top boundary) are assumed to be free–free, rigid–free, and rigid–rigid, with a source of heat from below. Linear stability theory is applied, and the eigenvalue solution is obtained numerically using the Galerkin technique. Focusing on the stationary convection, it is shown that there is a positive thermal resistance in the presence of feedback control on the onset of double-diffusive convection, while there is a positive thermal efficiency in the existence of internal heat generation. The possibilities of suppress or augment of the Rayleigh–Bénard convection in a nanofluid layer are also discussed in detail.

The influence of coriolis force on the onset of steady Rayleigh-Benard convection subjected to Soret parameter in a horizontal nanofluid layer is considered analytically. The confined lower and upper boundary conditions of the nanofluid layer are considered to be free-free, rigid-free and rigid-rigid respectively. The model used for the nanofluid incorporates the effects of Brownian motion and thermophoresis diffusion. Following the usual linear stability theory, the eigenvalue solution is obtained numerically by using Galerkin technique. From the investigation, the presence of coriolis force due to the rotation inhibits the onset of convection in nanofluid layer and have a stabilizing effect. Further, the instability of the system get advanced with the increased values of the Soret parameter.

This investigation reports on a stability analysis of Rayleigh-Benard convection in a horizontal of micropolar fluid layer heated from below. The effect of a feedback control strategy on the onset of steady convection in the presence of internal heat source is investigated theoretically using Galerkin technique. The eigenvalues are obtained for free-free, rigid-rigid, free-rigid boundary combination with isothermal temperature boundary condition. The influence of various micropolar parameters on the onset of convection has also been analyzed. The onset of motion is found to depend on the feedback control parameter, K and internal heat source, Q and the micropolar parameter Ni.

The effect of controller (feedback control) is applied to the steady Rayleigh-Benard convection in a horizontal micropolar fluid layer. The bounding surfaces of the liquids are considered to be free-free, rigid-free, and rigid-rigid with combination isothermal on the spin-vanishing boundaries. A linear stability analysis is used and the Galerkin method is employed to find the critical stability parameters numerically. It is shown that the critical Rayleigh number increases as the value of feedback control increases.

Linear stability analysis is performed to study the Soret induced convection in a doubly diffusive fluid layer heated from below. The effect of a feedback control on the onset of steady convection is investigated theoretically using Galerkin technique. The eigenvalues are obtained for Free-Free, Rigid-Rigid, Rigid-Free boundaries combined with isothermal temperature boundary condition. The influence of various doubly diffusive parameters on the onset of convection has also been analyzed. It is found that the onset of motion can be stabilized by using the feedback control in all cases.

The effect of uniform distribution of internal heat generation on the linear stability analysis of the Benard-Marangoni convection in an Eringen's micropolar fluids with feedback control is investigated theoretically. The upper free surface is assumed to be non-deformable and the lower boundary is taken to be rigid and isothermal with fixed temperature and span-vanishing boundaries. The eigenvalue is solved numerically using the Galerkin method. The influence of the internal heat generation; Q and feedback control; K in micropolar fluids with various parameters on the onset of stationary convection has been analysed.

Combined effect of magnetic field and internal heat generation on the onset of Rayleigh–Bénard convection in a horizontal micropolar fluid layer is studied. The bounding surfaces of the liquids are considered to be rigid-free, rigid-rigid, and free-free with combination isothermal on the spin-vanishing boundaries. A linear stability analysis is used and the Galerkin method is employed to find the critical stability parameters numerically. The influence of various parameters on the onset of convection has been analyzed. It is shown that the presence of magnetic field always has a stability effect on the Rayleigh–Bénard convection in micropolar fluid.

The effect of feedback control on the criterion for the onset of Rayleigh-Benard convection in a horizontal micropolar fluid layer is studied theoretically. The bounding surfaces of the liquid are considered to either rigid on the upper and lower boundaries or upper boundary free and lower boundary rigid. A linear stability analysis is used and the Galerkin method is employed to find the critical stability parameters numerically. It is found that the onset of instability can be delayed through the use of feedback control.

Rayleigh-Bénard convection is the heat transfer process which due to buoyancy forces that occurred in a plane horizontal layer of micropolar fluids heated from below. The effect of feedback control and magnetic field on the onset of Rayleigh-Bénard convection in a horizontal micropolar fluids layer in the presence of heat generation has been studied. The fluid layer with various boundary conditions at the lower and upper boundaries, and perfectly heat conduction are investigated theoretically based on the linear stability analysis. The various boundary conditions are assumed to be rigid-rigid, rigid-free and free-free. The resulting eigenvalue problems are solved analytically using the Galerkin method. The critical Rayleigh numbers are obtained and the influences of various parameters have been analyzed. It is found that the rigid-rigid surface is the most stable system, and the effect of feedback control as well as the magnetic field always stabilized the system is shown.