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The present study investigates the thermal convection of a power-law fluid in a horizontal porous layer that is heated from below. The study of flow in a porous medium is important because of its applications in various fields such as agriculture, geothermal sciences, and engineering. Linear instability analysis is performed using the normal mode method to solve the governing equations after non-dimensionalization. The bvp4c routine in MATLAB R2020a has been used to solve the raised problem for linear instability. The impact of gravity parameter, Peclet number, and power-law index on linear instability has been investigated. Linear and quadratic variations of gravity field are considered. From the results, it is evident that the critical Rayleigh number exhibits a non-monotonic relationship with the Peclet number. Increasing the gravity variation parameter leads to a more stable system, particularly in the case of linear gravity variation.

The present study investigates the impact of a magnetic field on double‐diffusive convection in an inclined porous layer, employing linear instability theory. The perturbed state is solved using the normal mode technique, and the stability eigenvalue problem is numerically analyzed for longitudinal and traveling rolls using the Runge‐Kutta method coupled with the shooting method. Various dimensionless physical parameters, including solutal and thermal Rayleigh numbers, inclination angle, Hartmann number, and Lewis number, are examined for their influence on system stability. The findings reveal that, for Le<1 $Le\lt 1$, the Hartmann number, solute Rayleigh number, and inclination angle act as stabilizing factors, with greater stability observed for traveling rolls compared to longitudinal rolls. In the case of Le=1 $Le=1$, the critical Rayleigh number shows a monotonic relationship with the solute Rayleigh number and inclination angle, while its relationship with the Hartmann number is non‐monotonic for traveling rolls. Moreover, for Le>1 $Le\gt 1$, the Hartmann number stabilizes the system by raising the onset threshold value, favouring longitudinal modes. The solute Rayleigh number also contributes to system stability. The impact of the inclination angle on system stability is contingent upon its magnitude, with small angles favouring the stability of longitudinal rolls and larger angles leading to an initial transition from traveling to longitudinal rolls, indicating a complex non‐monotonic behavior. [ABSTRACT FROM AUTHOR]

In this inspection, the control of the magnetic power on the onset of Casson fluid convection formed by purely inner warming in a porous medium layer is examined. The modified Darcy model is employed to designate the rheological arrival of Casson liquid flow in a porous matrix. Two types of thermal boundaries are exploited, namely, type (I) both isothermal and type (II) lower insulated and top isothermal boundaries. Using the linear stability inspection and Galerkin technique, the approximate analytical solution and numerical solution correct to one decimal place are offered. It is detected that for type (I) boundary conditions, the convective wave concentrates in the upper layer if it occurs, whereas for type (II) boundary conditions, it emphases in the whole layer. The magnetic Chandrasekhar number postpones the convection movement while the Casson constraint accelerates it. The facet of the convective cells drops with enhancing the magnetic strength and the Casson constraint. In the absenteeism of magnetic field, the Casson constraint has no regulation on the dimension of convective cells. It is also found that the presented analytical result with two term Galerkin process has overall 5% error, while with one term Galerkin process the error was overall 19%. [ABSTRACT FROM AUTHOR]

AbstractThe study investigates the linear and weakly nonlinear stability of Jeffrey fluids in an anisotropic porous medium with throughflow. In the linear stability analysis, the study examines how convective motion changes with variations in the Jeffrey parameter, Peclet number, and anisotropic parameters. The study finds that thermal anisotropy parameter, positive Peclet number, and mechanical anisotropy parameter delay the onset of fluid convection, whereas the Jeffrey parameter enhances the onset of convection in the presence of throughflow. In the nonlinear analysis, heat transfer is calculated using the Ginzburg–Landau Equation (GLE). The study observes that heat transfer increases with the Vadasz number and positive Peclet number, making the system unstable, whereas heat transfer decreases with an increased value of the Jeffrey parameter, thermal anisotropy parameter, mechanical anisotropic parameter, and negative Peclet number, making the system stable. [ABSTRACT FROM AUTHOR]

The qualitative influence of internal heat and variable gravity field on the onset of convection in a sparsely packed porous layer with horizontal fluid-saturated are investigated. Linear stability analysis is performed using the normal mode technique. The dimensionless governing equations with four cases of gravity field variation: (1) F (z) = − z , (2) F (z) = − z 2 , (3) F (z) = − z 3 , and (4) F (z) = − (e z − 1) are solved using bvp4c in MATLAB R2020a and the corresponding eigenvalue problem is calculated for three types of boundaries, namely free-free, rigid-free, and rigid-rigid, respectively. The effect of an internal heat parameter (Q) , Darcy number (D a) , and gravity variation parameter (δ) for the stability of the system is investigated graphically. The critical value of the Rayleigh number and corresponding wavenumber are calculated for other fixed parameters. It is observed that internal heat parameters enhance the onset of convection, whereas the gravity variation parameter delays the same. It is also observed that the system becomes more stable for the exponential variable gravity function and less stable for the cubic variable gravity function. [ABSTRACT FROM AUTHOR]

The flow through porous medium accounts for numerous applications in various fields namely, agriculture, geothermal sciences, and engineering. Furthermore, dissolution‐driven convection in porous media has grabbed great attention in recent years due to its practical applications in long‐term geological storage of carbon dioxide, in the production of mineral deposits, and other industrial applications. In this regard, the current numerical analysis focuses on addressing the thermal instability of dissolution‐driven convective phenomena of a power‐law fluid through a porous horizontal domain with a first‐order chemical reaction. For linear stability analysis, the method of normal modes has been employed to solve governing dimensionless equations which give rise to an eigenvalue problem. The bvp4c routine in MATLAB R2020a has been used to solve the raised problem for the onset of convection. The impact of Damköhler number, Péclet number, and power‐law index on the onset of convection has been investigated. The role of these critical parameters is found to be highly significant in stabilizing the system. An increase in the power‐law index causes stabilization or destabilization in the system, depending on the Péclet number. An enhancement in the magnitude of Damköhler number makes the system stable for all values of the Péclet number. Also, Damköhler and critical Rayleigh number are inter‐related, that is, an increment in Damköhler number results in the enhancement of critical Rayleigh numbers, which in turn leads to stabilization of the system. The critical wave number is observed to have a remarkable influence on Damköhler number as well as power‐law index. [ABSTRACT FROM AUTHOR]

This article aims to study the effect of the vertical rotation and magnetic field on the dissolution-driven convection in a saturated porous layer with a first-order chemical reaction. The system’s physical parameters depend on the Vadasz number, the Hartmann number, the Taylor number, and the Damkohler number. We analyze them in an in-depth manner. On the other hand, based on an artificial neural network (ANN) technique, the Levenberg–Marquardt backpropagation algorithm is adopted to predict the distribution of the critical Rayleigh number and for the linear stability analysis. The simulated critical Rayleigh numbers obtained by the numerical study and the predicted critical Rayleigh numbers by the ANN are compared and are in good agreement. The system becomes more stable by increasing the Damkohler and Taylor numbers.

The joint influence of the Coriolis effect and a variable gravity field on the stability of convective phenomena iii a porous layer is investigated numerically iii the presence of a heat source. Three types of gravity variations, such as linear, parabolic, and cubic functions. are considered. In linear theory, the method of normal modes is employed to solve governing dimensionless equations. which leads to an eigenvalue problem. The effects of the internal heat source parameter (Q), gravity variation parameter (δ). and Taylor number (Ta) on the stability of the system are investigated. It is found that the onset of convection is delayed by increasing the Taylor number and gravity variation parameters. The system becomes more stable for linear variation and less stable for cubic variation. [ABSTRACT FROM AUTHOR]

Linear and weakly nonlinear stability analyses of thermosolutal convection in a couple‐stress fluid with effects of helical force and rotation are performed. The governing nondimensional equations are solved using the normal modes. We have shown the effect of the helical force parameter, solutal Rayleigh number, Couple stress parameter, Lewis number, Taylor number, and Prandtl number on stationary and oscillatory convection regions and presented graphically. Solutal Rayleigh number, Couple stress parameter, Lewis number, and Taylor number have a stabilizing effect on the system whereas the helical force parameter has a destabilizing effect on the system. To study heat transport by convection we have derived the Ginzburg–Landau equation. [ABSTRACT FROM AUTHOR]

This effort investigates the arrival of magnetothermal convection of ethylene glycol‐silver Jeffrey nanofluid in a Hele‐Shaw cell utilizing the linear stability concept. The model practiced for the Jeffrey nanofluid includes the impacts of Brownian movement and thermophoresis. The norms for both marginal and overstable modes of convections are developed analytically. The impact of magnetic Chandrasekhar number QM, magnetic Prandtl number Prm, Jeffrey parameter δ, Hele‐Shaw number HS, and a variety of nanofluid parameters such as the volumetric fraction of nanoparticles ϕ, nanoparticle Rayleigh number Rp, adjusted diffusive ratio NA, and Lewis number Le on the beginning of convective motion are investigated, and results are illustrated graphically. It is observed that the overstable approach of convection is probable below the certain threshold estimate of the magnetic Prandtl number Prm. This threshold estimate of the magnetic Prandtl number Prm upturns with a rise in the rate of QM, HS, and δ, while it drops with a surge in the nanofluid parameters. The system was found to be more stable by decreasing the Hele‐Shaw number HS, the Jeffery parameter δ, and nanofluid parameters, while it was unstable by decreasing the magnetic Chandrasekhar number QM and the magnetic Prandtl number Prm. [ABSTRACT FROM AUTHOR]

Linear and weakly nonlinear properties of magnetoconvection in a sparsely packed porous medium are investigated. We have obtained the values of Takens-Bogdanov bifurcation points and codimension two bifurcation points by plotting graphs of neutral curves corresponding to stationary and oscillatory convection for different values of physical parameters relevant to magnetoconvection in a sparsely packed porous medium near a supercritical pitchfork bifurcation. We have derived a nonlinear two-dimensional Ginzburg-Landau equation with real coefficients by using Newell-Whitehead (1969) method. The effect of the parameter values on the stability mode is investigated and shown the occurrence of secondary instabilities namely, Eckhaus and Zigzag instabilities. We have studied Nessult number contribution at the onset of stationary convection. We have also derived two nonlinear one-dimensional coupled Ginzburg-Landau-type equations with complex coefficients near the onset of oscillatory convection at a supercritical Hopf bifurcation and discussed the stability regions of standing and travelling waves.

The thermal instability of internally heated convection in a porous medium saturated by a power‐law (PL) fluid is studied. The governing dimensionless equations are solved using the normal mode approach, which leads to an eigenvalue problem for the linear stability theory. The neutral curves are obtained for different prescribed values of the other physical parameters. The effect of the PL index, internal heat source parameter, and Peclet number on the onset of instability was analyzed. Furthermore, an analytical solution is obtained for the regime of small wave numbers by using asymptotic analysis. [ABSTRACT FROM AUTHOR]

In this paper, the linear and nonlinear instability of magnetoconvection in a Darcy-Benard setup saturated by a Maxwell fluid with chemical reactions is studied. The governing non-dimension equations are solved using the normal modes, and we obtain the expressions for steady and oscillatory thermal Rayleigh numbers. The effects of different physical parameters such as the Damkohler number ( 0 < D a < 20 ), Hartmann number ( 0 < H a < 1 ), Solute Rayleigh number ( 0 < R S < 1000 ), relaxation parameter ( 0 < λ < 1 ), Magnetic Prandtl number ( 0 < P r < 10 ), Lewis number ( 0 < L e < 100 ) on stationary and oscillatory critical thermal Rayleigh numbers are presented and described. Enhancing the values of the Solute Rayleigh number and Lewis number makes the system unstable. Also, the Hartmann number and Damkohler number have a contrasting effect on stationary and oscillatory convection. Enhancing the value of the relaxation parameter makes the system more stable. In order to study heat transport by convection, the well-known equation, the Landau-Ginzburg equation, is derived. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]