Dissolution-Driven Convection in a Porous Medium Due to Vertical Axis of Rotation and Magnetic Field

: 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.


Introduction
Dissolution-driven convection occurs in the host phase of a partially miscible system when a buoyantly unstable density stratification develops upon dissolution.The onset of convection in a porous layer has received considerable interest in science, engineering, and technology, such as food engineering, oil recovery, chemical reactor design, and plastic processing.Dissolution-driven convection in porous media has received recent interest in the context of the long-term geological storage of carbon dioxide in the underground, natural, brine-filled caverns, often referred to as saline aquifers, in the production of mineral deposits, and a variety of other applications.Following injection into the saline aquifer, dissolution of supercritical carbon dioxide in the host brine causes a local density increase, leading to gravitational instability of the diffusive boundary layer and the formation of convective fingers [1][2][3][4][5][6].In addition, Benard and chemical instabilities were studied for the dissociation of a horizontal layer of Navier-Stokes fluid due to the Boussinesq approximation.Dissolution-driven convection of a binary fluid in a reactive porous layer was foremost studied [7,8], then secondary instabilities [9], and constant temperatures and chemical equilibrium in binary fluid at the boundary surfaces while the solubility of the dissolved issue relies upon temperature [10][11][12][13][14][15][16][17][18].The diffusive boundary layer becomes unstable in anisotropic porous media where both the capillary transition zone and dispersion are considered, even if the geochemical reaction is significantly large.While the reaction enhances stability by consuming the solute, porous media anisotropy, hydrodynamic dispersion, and capillary transition zone destabilize the diffusive boundary layer that is unstably formed in a gravitational field [19][20][21].Stability techniques that look at and broaden the way the solute's dissolution influences the thermal convection and prolongs this evaluation by the use of an asymptotic energy method, Galerkin and spectral techniques, are expecting the structure of the preliminary bifurcation.Darcy, Darcy Brinkmann, and Darcy Lapwood Brinkmann's models were used to study porous, anisotropic porous, and sparsely packed porous medium over multiple diffusive convection [22][23][24][25].
Exhausting the magnetic field is an adequate method to regulate a thermally induced flow.The magnetic field will propagate a Lorentz force to permeate the convective flow.The penetration effect depends on the strength of the applied magnetic field and its assimilation into the convective flow direction.The magnetic field is significant for engineering applications such as magnetohydrodynamics, cooling of nuclear reactors, micropump electronic packages, and microelectronic devices.The density can be enhanced or reduced depending on the magnetic field and electrode configuration.The magnetic field effect on formally charged transfer-controlled active dissolution and the Lorentz force reduces the field gradient force, which boosts active dissolution.The convective cavities of various aspect ratios in the magnetohydrodynamics of fluids are broadly studied [26][27][28][29][30][31][32][33][34][35][36][37][38].Due to the simultaneous action of buoyancy and induced magnetic forces, heat transfer to liquid metals may be significantly affected by the presence of a magnetic field, but very small effects are experienced by other fluids.The Coriolis and centrifugal buoyancy forces arising from rotation have a remarkable influence on the local heat transfer when compared with the nonrotating results.A series of interferograms, stream functions, and isotherm plots demonstrated the strong effect of rotation on the flow field and heat transfer.A correlation of the Nusselt number as a function of Taylor and Rayleigh numbers is presented [15,19,[39][40][41][42].
Various machine learning techniques, in particular artificial neural networks (ANN), have been widely used in different research areas for predicting data.Recently, many researchers have used ANN to predict the data and compare them with their results.Neural networks are used to solve different types of large data-related problems and solve the Navier-Stokes equations for turbulence by using the Bayesian cluster.The combination of ANN and gene expression programming compares the local Nusselt number with their numerical results [43][44][45].The investigation of bifurcating fluid phenomena using a reduced-order modeling setting was aided by artificial neural networks, ANNs, to study the flow and thermal fields of the onset of convection in a rectangular channel.From their results, they found that the ANN can precisely predict the Nusselt number with less computational time and cost compared to the DNS [46][47][48][49][50][51].
The purpose of this article is to explore the magnetic effect, the Coriolis effect, and chemical reaction effects on the onset of convection in a porous medium.To the best of our knowledge, linear stability theory and ANN prediction of threshold Rayleigh number for the onset of magneto-rotating convection in a porous medium with first-order chemical reaction have not been studied so far.The plan for this article is as follows: Section 2 describes the mathematical modeling under consideration, Section 3 presents the ANN methodology, and Section 4 discusses the results.The paper ends with a conclusion in Section 5.

Basic Equations
Consider an electrically conducting fluid-saturated porous layer of thickness d that is salted from below and confined between two parallel horizontal planes at z = 0 and z = d.The horizontal coordinate x and vertical coordinate z increase upwards in a Cartesian coordinate with the origin at the bottom of the porous medium.The surfaces are extended infinitely in x and y directions and a constant salinity gradient C is maintained across the porous layer.Let Ω = Ω êz be the constant angular velocity of the layer.To make the Boussinesq approximation valid, the physical properties of the fluid are assumed to be constant, except for density in the buoyancy term.The porous medium is considered homogeneous and isotropic.Based on [15,19,39,40], with the physical configuration recalled in Figure 1, the governing equations are subject to the following boundary conditions Here, Ca, µ, K, p, ρ, β c , g, t, , D v , and β are the acceleration coefficient, dynamic viscosity, permeability, dynamic pressure, reference density, solute expansion coefficient, gravity acceleration, time, porosity, solute diffusion coefficient, and reaction rate of the solute, respectively.The dimensionless quantities are given as follows: as well as the non-dimensional quantities where Ra, Ta, Ha, Dm, and Va are the Rayleigh number, Taylor number, Hartmann number, Damkohler number, and Vadasz number, respectively.The non-dimensional form of the governing Equations ( 1)-( 3) and the corresponding boundary conditions (4) are given by 1 Va subject to the boundary conditions Porous medium

Basic Flow
The basic stationary flow of Equations ( 7)-( 10) is as follows:

Linear Stability Analysis
The perturbation of the basic state for the Equations ( 7)-( 10) is By substituting Equation ( 13) into Equations ( 7)-( 10), one obtains subject to the boundary conditions By taking the third component of the curl of Equation ( 15) and curl of curl of Equation ( 15), one obtains From Equations ( 16), ( 18) and ( 19), we obtain where Let us introduce the normal mode by writing that the perturbation is in the form of w = e i(lx+my)+σt sin(πz), (24) where l and m are the wave numbers along x and y directions and σ is a complex parameter.Substituting Equation (24) into Equation ( 20), one obtains where

Stationary Mode
To study the stationary stability, take σ = 0 in the Rayleigh number for the exchange of the stabilities at the onset of stationary convection, say Ra sc .It is given as The critical Rayleigh number at the onset of stationary convection Ra sc c is The above stationary Rayleigh number reduces to Ra sc = δ 4 q 2 with the critical values Ra sc c = 4π 2 , q sc c = π in the absence of a magnetic field, Coriolis effect, and chemical reaction effect, which agrees with the results of Horton and Rogers [41] and Lapwood [42] for the onset of convection in a porous layer.

Oscillatory Mode
To study the oscillatory stability, take σ = iω.The Rayleigh number at the onset of oscillatory convection is where

Artificial Neural Network Modeling
Let us now present some basis for the ANN modeling.An ANN is a computing system based on biological neural networks (which are interconnected) that resemble a brain.In general, ANN can be used to predict data.In this study, we used a network with three layers: input, hidden, and output, as well as other components, such as feed-forward propagation, an optimal number of neurons, and backpropagation (update weights and biases) (see Figures 2 and 3).To train the suggested network, we use the Levenberg-Marquardt backpropagation algorithm, as proposed by Yu and Wilamowski [46].To prepare the organization, data are first divided into three groups.A total of 650 datasets were utilized to train, test, and validate the ANN model, with 70%, 15%, and 15% of the data being randomly allocated for preparing and assessing.The optimal number of neurons (Nn) for the best performing artificial neural network architecture is determined by examining three different statistical values: coefficient of determination (R 2 ), root mean square error (RMSE), and root mean relative error (RMRE), which are defined by Here, Ra c,s is the simulated critical Rayleigh number, Ra c,a is the ANN critical Rayleigh number, the index i refers to the i-th experiment, bar denotes average value, and N is data size or number.See Seo et al. [45] for further details on these measures.The regression plots of training, testing, and validation for these three different sets are illustrated in Figure 4.The values of R 2 , RMSE, and RMRE for different values of Va, Ha, Ta, and Dm are illustrated in Tables 1 and 2. From these tables, it is clear that the present ANN model can predict the critical Ra for linear stability analysis for different Va, Ha, Ta, and Dm.

Discussion
The numerical results and discussion are presented in this section.In this results part, we evaluated a numerical study of the effect of the magnetic field and rotation on the onset of dissolution-driven convection saturated porous layer with ANN prediction.The critical Rayleigh number at the onset of stationary (Ra sc c ) and oscillatory (Ra oc c ) convection is obtained for the prescribed values of the other parameters.The investigations are performed for various values of the Hartmann number, Taylor number, Vadasz number, and Damkohler number.In Figures 5-10, solid and dotted lines represent the stationary and oscillatory convection, respectively.The following physically realistic range of these parameters is considered: 0 ≤ Va ≤ 20 [37], 0 ≤ Ta ≤ 50 [22], 0 ≤ Ha ≤ 0.9 [40], and 0 ≤ Dm ≤ 20 [23].
First, we shall discuss the theory of bifurcation points in Figures 5-7, the results obtained numerically by linear and weakly nonlinear stability analysis.Takens-Bogdanov and codimension two bifurcation points are identified in these figures.Takens-Bogdanov bifurcation point is the point at which the oscillatory neutral curve intersects the stationary neutral curve and approaches zero as the intersection point is approached.At the Takens-Bogdanov bifurcation point, we have The codimension two bifurcation point is the intersection between a Hopf and Pitchfork bifurcation with distinct wave numbers.At the codimension two bifurcation point, we have The effect of the Vadasz number Va on the neutral curves is presented in Figure 8.We find that the Ra sc c is independent of the Vadasz number Va, whereas the Ra oc c decreases with a decrease in the value of the Vadasz number Va.This reports the porosity effects on driven convection in a Newtonian-fluid-saturated porous layer.Furthermore, from this figure, one can notice that for Va = 1, there exists a threshold Ta * ∈ (1.9, 2) such that for Ta < Ta * , stationary convection sets in, while for Ta * ∈ (1.9, 2), there is a switch from stationary to oscillatory convection.Similar behavior can be observed for the other values of Va.    Figure 9 illustrates the effect of the magnetic field on the onset of convection.From this figure, one can observe that the Hartmann number has a stabilizing effect on stationary convection.On the contrary, the Hartmann number has a stabilizing effect on oscillatory convection.We find that the minimum value of the stationary Rayleigh number for stationary mode increases with increasing Hartmann number Ha.On the other hand, the minimum value of the oscillatory Rayleigh number decreases with an increase in the value of the Hartmann number Ha.Thus, Ha has a contrasting effect on the stability of the system in the case of stationary and oscillatory modes.From this figure, we notice that for Ha = 0.2, there exists a threshold Ta * ∈ (1.9, 2) such that for Ta < Ta * , oscillatory convection sets in, while for Ta * ∈ (1.9, 2), there is a switch from oscillatory to stationary convection.Similar behavior can be observed for the other values of Va.Similarly, Figure 10 depicts the effect of Dm on the system.From this figure, we see that the effect of increasing Dm is to increase the Ra sc c and Ra oc c , implying that Dm has a stabilizing effect on the onset of dissolution-driven convection in a porous medium.This can be explained as follows.An increase in the value of Dm promotes the dissolution reaction to absorb some of the heat energy, causing the surrounding environment to feel cold.Hence, a larger solute gradient is required for the onset of convection so that the system is stabilized.We find that for fixed values of other physical parameters, there exists a critical Taylor number Ta c such that when Ta < Ta c , convection begins as an oscillatory type, and when Ta > Ta c , the convection switches to stationary.Further, when Ta = Ta c , the stationary and oscillatory modes occur simultaneously.Furthermore, Figures 8-10 demonstrate the Coriolis effect on the onset of convection.All of these figures show that the Ra sc c and Ra oc c increase as the Taylor number increases.Hence, the Taylor number has a stabilizing effect on the system.This can be explained as follows: in the fluid, the rotation creates vorticity.As a result, the fluid has a faster velocity in horizontal planes.Hence, the perpendicular velocity of the fluid decreases.Therefore, the convection does not start right away.
The comparison of numerical and predicted ANN data of the critical Ra with different values of Va, Ha, Ta, and Dm is shown in Figures 11 and 12. From all these figures, it is obvious to see that the trained predictive ANN model holds well with the numerical results.

Conclusions
In the present analysis, the onset of dissolution-driven convection in a porous layer with the effect of the magnetic field and rotation is studied.The behavior of various physical parameters is investigated.The results can be summarized as follows: Takens-Bogdanov and codimension two bifurcation points are identified.The Vadasz number does not show any effect on stationary convection, whereas it has a destabilizing effect on oscillatory convection.The Hartmann number has destabilizing and stabilizing effects on stationary and oscillatory convection, respectively.The Damkohler number has a stabilizing effect on the system.Furthermore, an artificial neural network (ANN) is used to model and predict the critical Rayleigh numbers.The simulated and predicted values of the proposed ANN model were found to be highly close, indicating that the expected critical Rayleigh number and the observed critical Rayleigh number are quite similar.
In future work, we plan to study linear instability and nonlinear stability.Another interesting problem is to investigate the stationary or oscillatory convection at the onset of instability using Brinkmann's law.

Figure 3 .Figure 4 .
Figure 3. Flow chart of the artificial neural network.

Figure 8 .
Figure 8. Plots of the critical Ra as the function of Ta for Va = 1, 5, 10, 15.

Figure 11 .
Figure 11.Comparison of the simulated and ANN-predicted critical Rayleigh number values for (a) Dm, (b) Ha, and (c) Ta.

Figure 12 .
Figure 12.Comparison of the simulated and ANN-predicted critical Rayleigh number values for (a) Dm, (b) Ha, and (c) Ta.
Schematic representation of a multilayer feed-forward network consisting of two inputs, one hidden layer, and two outputs.

Table 1 .
Calculated stationary values of R 2 , RMSE, and RMRE at various values of Ta, Dm, and Ha.