A BIO-THERMAL CONVECTION IN A POROUS MEDIUM SATURATED BY NANOFLUID CONTAINING GYROTACTIC MICROORGANISMS UNDER AN EXTERNAL MAGNETIC FIELD

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INTRODUCTION
Many theoretical and practical study in fields such as soil mechanics, groundwater hydrology, petroleum engineering, industrial filtration, powder metallurgy, nuclear energy, and so on have been based on the study of the physics of flow through a porous media. Such flows through porous media are of great interest to petroleum engineers and geophysical fluid dynamicists. The problem of thermal instability of a liquid layer in a porous medium is of special importance. Ingham and Pop [1] as well as Nield and Bejan [2] provide excellent reviews of the most of the studies on convection in porous media. The issues of fluid flow and heat transfer in rotating porous media have been studied in detail in a recent review by Vadasz [3].
Objects with dimensions of less than one hundred nanometers have developed as a result of the progress of nanotechnology. Such nanosized objects are called nanoparticles. Choi [4] proposed suspending these nanoparticles in a base fluid (known as nanofluid) to improve heat conductivity and convective heat transfer of the base fluid. Thus, nanofluids began to be intensively used in industry, such as coolants, lubricants, heat exchangers, microchannel radiators, and many others. Buongiorno [5] extensively studied convective transport in nanofluids and concentrated on explaining the additional heat transfer increases observed under convective flows. Tzou [6] used Buongiorno transport equations to study the onset of convection in a horizontal layer uniformly heated from below for a nanofluid and discovered that due to Brownian motion and thermophoresis of nanoparticles, the critical Rayleigh number is much lower, by one to two orders of magnitude, than that of an ordinary fluid.
Because of the remarkable characteristics of nanofluids in heat transfer phenomena, research involving nanofluids in a porous media has become required. Kuznetsov and Nield [7]- [8] used the Brinkman model to investigate the onset of thermal instability in a porous media saturated with a nanofluid, taking into account Brownian motion and nanoparticle thermophoresis. They discovered that the presence of nanoparticles may significantly lower or increase the critical thermal Rayleigh number, depending on whether the basic nanoparticle distribution is top-heavy or bottomheavy. Further, Bhadauria and Agarwal [9] and Yadav et al. [10] extended the thermal instability problem by including the Coriolis force term (rotation) in momentum equations for porous and non-porous materials, concluding that including the Coriolis force component made the overall system more stable. Bhadauria and Agarwal [9] made a nonlinear investigation of nanofluid convection in a rotating porous layer in terms of Nusselt number. Chand and Rana [11] used the Darcy-Brinkman model to investigate the influence of rotation on nanofluid convection. For stationary convection, an equation for the thermal Rayleigh number was found. Gupta et al. [12] studied the onset of convection in a horizontal nanofluid layer in the presence of a vertical magnetic field and discovered that stability increases as the magnetic field value increases. The effects of a magnetic field on a horizontal layer of nanofluid have several significant characteristics, making it necessary to explore the effects of a magnetic field in a porous medium. Ahuja et al. [13] used the Darcy-Brinkman model to explore the thermal convection of a nanofluid layer in the presence of an applied vertical magnetic field saturated by a porous medium for both-free, rigid-free, and both-rigid barriers. The effect of an externally applied magnetic field on the stability of a binary nanofluid layer in porous medium is considered by Sharma et al. [14]. It was established that semiconducting nanofluids are found to be more stable than metallic nanofluids. Furthermore, porosity destabilizes the layer while solute difference (at the boundaries of the layer) stabilizes it and the magnetic field stabilizes the fluid layer system significantly. Yadav et al. [15] investigated the effect of Hall current on the criterion for the onset of MHD convection in a porous medium layer saturated by a nanofluid. They were shown that the increase in the Hall current parameter, the Lewis number, the modified diffusivity ratio, and the concentration Rayleigh Darcy number is to hasten the onset of convection while the magnetic Darcy number, the porosity parameter and the Darcy number have stabilized on the onset of convection. Ahuja and Gupta [16] investigated the onset of thermal convection of a porous nanofluid layer under the combined influence of rotation and magnetic field using the Darcy-Brinkman model. As shown in [16], the stability of the system enhances with the rise in Chandrasekhar number and Taylor number whereas it falls with the rise in porosity. According to [16] silver nanoparticles stabilized water-based fluids better than copper nanoparticles, while 2 TiO nanoparticles in semiconductors improve system stability better than 2 SiO nanoparticles. A detailed review of the current state of the problem of MHD convection in nanofluids presents in [17]. The article [18] discusses a new review devoted to the subject of Rayleigh-Benard instability in a nanofluid. In addition to the above, a relatively new area of research: bioconvection inside a nanofluid is of great importance in applied problems of creating biosensors, microdevices for measuring the toxicity of nanoparticles. Bioconvection is the movement of a fluid medium caused by a directed flow of microorganisms that results in a redistribution of the medium's density. As a result, hydrodynamic processes in the medium emerge, analogous to natural convection in the presence of temperature gradients. Bioconvection takes into account the movement of bacteria and algae, with microorganisms having a higher density than water. Microorganisms can move due to gravity forces (gyrotactic microorganisms), oxygen concentration gradients (oxytactic microorganisms), light radiation (phototaxis microorganisms), nutrition gradients (chemotaxis microorganisms), and other factors. The number of self-propelled microorganisms can be quite considerable, varying from 10 7 cm -3 in a low concentratin regime to 10 11 cm -3 in a turbulent regime with almost densely packed microorganisms. A linear theory of the stability of bioconvection of gyrotactic microorganisms in a finite depth layer of an ordinary fluid was developed by Pedley et al. [19]- [21]. In these works, the criteria for the onset of a bioconvective flow were determined. Avramenko [22] based on the Lorenz approach [23] a nonlinear theory of bioconvection for gyrotactic microorganisms in a layer of ordinary liquid was developed. In [22], the boundaries of various hydrodynamic regimes of two-dimensional bioinvection were determined.
Unlike microorganisms, nanoparticles are not self-propelled, move due to Brownian motion and thermophoresis, and are driven by fluid flow. Kuznetsov's [24], [25] works are devoted to the interaction of nanofluids with bioconvection. The paper [24] investigated the possibility of oscillatory instability in a nanofluid suspension containing oxitate bacteria. In [25] analyzes the combined influence on the onset of biothermal convection in suspension, nanoparticles, gyrostatic microorganisms, and algae. A linear analysis of the instability is performed, allowing the bounds of the oscillatory instability to be determined. The destabilizing effect of Brownian motion and thermophoresis of nanoparticles, vertical throughflow, and gyrotactic microorganisms on biothermal instability was discovered by Saini and Sharma [26].
Because of the practical nature of the problem, a significant number of works (for example, [27][28][29][30][31]) have recently arisen that investigate two-dimensional magnetohydrodynamic flows and the processes of heat and mass transfer of a watery nanofluid, including gyrotactic microorganisms. The effects of buoyancy, Brownian motion, thermophoresis, and chemical reactivity on bioconvection of nanofluidic gyrostatic microorganisms are investigated in these papers. The aim of using microorganisms is to stabilize nanoparticle suspensions created by bioconvection caused by the combined effects of buoyancy and magnetic field forces.
There are a lot of publications on the effect of gyrotactic microorganisms on nanofluid flows in bounded porous media. The study of biological processes in porous media has recently rapidly progressed. Kuznetsov, Nield, and Avramenko [32][33][34][35][36] made a significant contribution to the dynamics of biological processes in porous media. A linear analysis of biothermal convection based on the Darcy-Brinkman model in a suspension of gyrotactic microorganisms in a highly porous medium heated from below was carried out in [37]. The following conclusions [37] were drawn there: a suspension containing faster-swimming cells is more unstable; gyrotaxis contributes to the development of bioconvection instability; the modified Darcy number associated with the effective viscosity stabilizes the slurry.
Recently, the flow of boundary layers of natural convection nanofluids over a vertical flat plate embedded in a saturated porous Darcy medium containing gyrotactic microorganisms was studied in studies [38]- [40]. The effects of dimensionless parameters, such as the Lewis number of bioconvection, the Rayleigh number of bioconvection, the Peclet number of bioconvection, the Brownian motion parameter, the buoyancy coefficient, the thermophoresis parameter, the index of change of the power law, and the Lewis number, on the flow characteristics of the bioconvection process are analyzed during a numerical solution.
Despite significant progress in the study of bioconvection in porous media, the impact of gyrotactic organisms on thermal instability in a layer of a porous media saturated by a nanofluid under the action of a magnetic field has yet to be investigated. Of particular interest is the study of the linear and nonlinear stages of bio-thermal convection in a porous medium saturated with nanofluid for various types of nanoparticles such as metal, metal oxides, and semiconductor nanoparticles under an external magnetic field. The aim of this paper is to investigate the abovementioned problem.
The content of the paper is presented in the following sections. The Introduction (section 1) presents a review of the literature on this issue. In section 2, we describe the problem statement and obtain evolution equations in the Boussinesq-Oberbeck approximation in a layer of a Darcy-Brinkman porous medium saturated by an electrically conducting nanofluid with gyrotactic microorganisms. In section 3, we obtained equations for the basic state and small perturbations in dimensionless form. The equation for the vertical component of the perturbed velocity was obtained using the elimination of variables technique. This equation is supplemented with conditions for free-free boundaries. In section 4, we derive a general dispersion equation taking into account the effects of a vertical magnetic field, gyrotactic microorganism bioconvection, Brownian diffusion of nanoparticles, and thermophoresis. The oscillatory and stationary regimes of mixed convection are considered. In section 5, we study the non-linear theory based on the representation of the Fourier series method used to find the temporary behavior of heat and mass transports. In section 6, we analyze the linear regime of stationary convection and calculate the expressions for heat and mass transports. In section 7, we perform a numerical/graphical analysis of the development of stationary convection carried out for oxide metal, metallic, and semiconductor nanofluids. Convective heat and mass transfers in terms of Nusselt number ( ) Nu t and Sherwood number ( ) Sh t were calculated. The Conclusions present the main conclusions of this paper.

STATEMENT OF THE PROBLEM AND BASIC EQUATIONS
Let us consider an infinite horizontal layer of incompressible electrically conductive nanofluid containing nanoparticles and gyrotactic microorganisms in a homogenous porous medium. Initially, a homogeneous porous layer with a thickness of h is assumed to be at rest. Heating from below the layer causes disturbance, where , respectively (as shown in Fig. 1). The presence of nanoparticles is considered to not affect the direction and speed with which microorganisms swim. Physically, this is possible if a flow generated by bioconvection exists for a small concentration of nanoparticles. The terminology for nanoparticles are written using Buongiorno's [5] theory, whereas those for gyrotactic microorganisms are written using Pedley's [19]- [21] approach.  • All thermophysical characteristics are constant except for density in the buoyant force (Boussinesq approximation).
• Because the fluid phase, microorganisms, and nanoparticles are in a condition of thermal equilibrium, the heat flow may be described using a one equation model.
• Each boundary wall is assumed to be impermeable and perfectly thermal conducting.
• The porous material is completely saturated with a nanofluid in which microorganisms can swim.
• The material's pores must be big enough to accommodate biological organisms and their movement.
• The solid porous matrix does not absorb microorganisms, and each one has the same volume and form, as well as swimming at the same speed.
• The heating from below has no influence on the cells' gyrotactic activity and does not kill them. According to the above assumptions, the bioconvective motion of gyrotactic microorganisms in a porous medium saturated with nanofluid in the presence of a magnetic field will be described using the Darcy-Brinkman model. Let's suppose that a suspension of swimming microorganisms and nanoparticles is incompressible, and therefore = 0, The conservation equation for the nanoparticles given by 2 2 = , The conservation equation of cells is given by [20,32,33] The momentum equation can be written using the Boussinesq approximation [7] as here the buoyancy force is made up of three different components: the fluid's temperature change, the distribution of nanoparticles (nanoparticles are heavier than water), and the concentration of microorganisms (microorganisms are also heavier than water). In equation (4), the external magnetic field is taken into account with the help of an additional force, the Lorentz force, which affects the movement of the electrically conductive nanofluid. The thermal energy conservation equation is The inductive magnetic fields caused by convective flows of an electrically conductive nanofluid is described by the following equations [41]    We assumed that random motions of microorganisms are simulated by a diffusion process. c W l  is the average microorganism swimming velocity ( c W is constant, l  is a unit vector of movement of microorganisms). P is the pressure,  is the thermal expansion coefficient, = (0, 0,1) e  is a unit vector in the direction of the axis z .  is the porosity of the porous medium, K is the permeability of the porous medium,   is the effective viscosity,  ,  and e  are the viscosity, magnetic viscosity and magnetic permeability of nanofluid, respectively.
We assume that the temperature and volume fraction of nanoparticles have fixed values at the boundaries of the porous layer.
is the flux of microorganisms.
Let us introduce the following non-dimensional parameters N const is average concentration of microorganisms in the layer. Using expressions (10) and omitting the asterisks, we get the following system of dimensionless equations In Eqs. (11)-(17), we introduced the following dimensionless parameters: is the modified nanoparticles density, Eq. (14) has been linearized using a small temperature gradient in a dilute suspension of nanoparticles and microorganisms using the Boussinesq approximation. Equations (11)-(17) are supplemented with boundary conditions in non-dimensional form: EQUATIONS FOR PERTURBATIONS Let's start with the assumption that the fluid layer is at rest. The physical system is then moved slightly out of its equilibrium position. We assume that all variables in Eqs. (11)- (17) can describe the sum of the steady and perturbed components: The steady profiles of temperature Taking into account the experimental data for most nanofluids [5], a good approximation of the base state (21) is a The pressure distribution in the ground state satisfies the equation from which the explicit form ( ) b P z may be found by integration. The stationary profile of the concentration of microorganisms ( ) b n z is determined by the following expression [19] For the case of small Peclet numbers Pe , it follows from (24) that 0 ( ) = n z N const  . Following the approach as in article [22], to simplify the, we will focus on this case.
According to publications [19],[ analysis 32], the equation for the perturbation of a unit vector indicating the direction of swimming of microorganisms has the following form: where i  and j  are the unit vectors in the x -and y -directions, respectively.
is a dimensionless parameter characterizing the reorientation of microorganisms under the action of a gravitational moment against viscous resistance, d is the displacement of the center of mass of the cell from the center of buoyancy. In Eq.
(25), the parameters  and  in the x -and y -components of vector  is the cell eccentricity which is given by the following equation [19,32]: where max r and min r are the semi-major and semi-minor axes of the spheroidal cell.
Taking into account Eqs. (25)-(26), we get the following equation for the perturbation of the cell number density: is a dimensionless orientation parameter [22].
After substituting (20) into the equations (12)-(16), we get equations for the variables ' ' ' ' ' ' , , , , , , which are also linearized. Further, the analysis of equations for the perturbation will be investigated by the method of normal modes, assuming that the perturbing quantities have the following form: where , x y k k are the wave number along the x and y directions, and 2 2 = x y a k k  is the horizontal wave number of the disturbances. The growth rate parameter is denoted by  .
Using (29), we get the linearized equations in dimensionless form after a few simple but cumbersome transformations: The system of equations (30) where the operators are Eq. (33) is supplemented with boundary conditions for free boundaries Equation (35) describes bioconvection in a layer of a porous medium saturated by an electrically conducting nanofluid in an external vertical magnetic field. Further, we will use a single term approximation of Galerkin method to solve (35).

OSCILLATING AND STATIONARY CONVECTION REGIMES
The eigenfunctions W of equation (35) take the form of a simple harmonic for free boundary conditions (39): where 0 W is constant. By substituting (40) for (38) and integrating across the layer thickness = (0,1) z , the characteristic equation is obtained, with Rayleigh number Ra as the eigenvalue: For most nanofluids, the quantities , In the equation (39) Ra R PeG is the modified bioconvection Rayleigh-Darcy number.
Since the value Ra is real, then the imaginary part in (40) and the frequency of neutral oscillations = i   , satisfying the following equation: where the following notation has been introduced: Equation (44)

Stationary convection regime
The marginal state will be characterized by The last term in (45) We get an equation for determining cr a by substituting the expression (45) into the condition (46). This equation has a rather cumbersome form, so we do not present it here. However, we can conclude that the critical wave number cr a does not depend on the nanofluid parameters ( , , )    [20]. In addition, we will be doing a numerical study of the dispersion equation (45) using the physical characteristics of nanofluids from metallic and semiconductor nanoparticles. Yang et al. [42] found that the physical characteristics of nanofluids change with the form, size, and volumetric percentage of nanoparticles. Metallic nanofluids, as the name implies, are metallic nanoparticles dispersed in a base fluid. Because metallic nanoparticles have a high thermal conductivity, they increase the thermal conductivity of nanofluids. We consider nanoparticles of metals ( Cu and Ag ) dispersed in the base fluid water for analyzing the stability of metallic nanofluids. SiO -water nanofluid [16].

WEAK NONLINEAR STABILITY ANALYSIS
We explored linear stability analysis using the normal mode method in the previous section. Although linear stability analysis is appropriate for studying the stability condition of the motionless solution describing convective flow. However, this approach cannot offer information regarding convection amplitudes and hence heat and mass transfer rates. In this section, we consider the situation of two-dimensional rolls, assuming that all physical variables are independent of y . In this case, Eqs. (1) and (7) for velocity and magnetic field perturbations will take the form We may introduce two scalar functions using Eq. (49), the hydrodynamic function of the current  and the magnetic function  , for which the following relationships hold: Then equations (2)-(6) for the perturbed quantities, taking into account nonlinear effects, take the following form for dimensionless variables The boundary conditions considered for solving the given system of equations (51) = ( ) cos( ) sin( ) ( ) sin (2 ).
The minimum order Fourier expansion was first used by Lorentz to model atmospheric convection [23]. The system of ordinary differential equations obtained by Lorentz is a low-order spectral model, but it is quite capable of qualitatively reproducing convective processes, in particular, the transition to a weakly turbulent state (chaos) through a series of bifurcations. In this study, we also use the Lorentz approach in describing the weakly nonlinear stage of convective instability. However, we do not touch upon issues related to the chaotic behavior of the obtained system of ordinary differential equations (51)-(55) since it requires a particular study. So we apply the weakly nonlinear theory EEJP. 4 (2022) [43]- [45] to the problem of determining the characteristics of heat and mass transfer: nonstationary Nusselt ( ) Nu t and Sherwood ( ) Sh t numbers. Substituting (58) in (51)-(55) and taking into account the orthogonality condition, we get the evolution equations for amplitudes: In phase space, the eight-mode differential Eqs. (59)-(66) have an interesting property that indicates that the system is dissipative: 02  02  02  11  11  11  11  11   11  11  11  02  11  02  11  02   2  2  2  2  2  2  2 As a result, the impact of parameters , , , , , R R L N L  on trajectories is to attract them to a set of measures zero, or a fixed point. For the time-dependent variables, the nonlinear system of differential equations cannot be solved analytically and must be solved numerically. In the case of steady motions, Eqs. (59)-(66) become: Ra Ra k k L L M   .
In the limit of small amplitudes 11 ( 0) A  , Eq. (69) transforms into dispersion equation (45) for the stationary convection regime when the wave number a is replaced by k . It should be noted that the amplitude of the stream function must be real, hence we must only consider positive signs in the roots of Eq. (69). If we determine the value of 11 A , we can find the value of heat and mass transfer in a stationary regime.

RESULTS AND DISCUSSIONS 6.1. Analysis of the linear regime of stationary convection
Now, we will study the effects of axial magnetic field, Darcy number, medium porosity, Lewis number, modified diffusivity ratio, concentration Rayleigh-Darcy number, modified bioconvection Rayleigh-Darcy number = Eq. (70) shows that the derivative / st dRa dQ is always positive, i.e. the vertical magnetic field has a stabilizing effect on stationary convection in the porous medium. As a consequence, it is discovered that increasing the magnetic field delays the initiation of convection, which is agreement with the results derived by Ahuja et al. [13].
Eq. (71) implies that the Darcy number has a stabilizing influence on the stationary convection of the system, which is in a good agreement with the results derived by Kuznetsov and Nield [7], Rana and Chand [11], Ahuja et al. [13]. With an increase in Darcy number, st Ra increases, indicating that the heat transmission characteristics of the nanofluid will improve. Therefore, the effect of the Darcy number, as well as the magnetic field, delays the onset of convection.
Eq. (72) shows that porosity can have both a stabilizing and destabilizing effect. If inequality   appears with a negative sign, it is obvious that the suspension of nanoparticles in ordinary fluids decreases the critical value of Rayleigh number. As a result, the system with nanoparticle distribution at the top of the fluid layer is less stable than the system with regular fluid and bottom heavy nanoparticle distribution.
Finally, we proceed to study the impact of gyrotactic microorganism bioconvection on magnetic convection. For this purpose, we calculate the following derivatives Eq. (74) shows that the spherical shape of microorganisms 0 = 0  contributes has a destabilizing effect since

Heat and mass transports
The determination of heat and mass transport is critical in the study of fluid convection. This is because the onset of convection, when the Rayleigh number increases, is more easily observed through its influence on heat and mass transport. Consequently, heat and mass fluxes of nanoparticles are important in identifying thermal-and bioconvective motion in its early stages. Heat transfers can be calculated and described using the Nusselt number ( ) Nu t (see, for example [45]) In the next section, we will consider the numerical/graphical investigation of the equations (70)-(79) by considering the numerical values of various parameters of the system.

NUMERICAL RESULTS AND DISCUSSION
In this section, we use the standard Maple computer environment programs for the numerical analysis of equations  2 3 Al O -water nanofluid Figures 2 and 3 depict the role of the magnetic field, Darcy parameter, porosity, Lewis number, modified diffusivity ratio, nanoparticles concentration Rayleigh number, modified bioconvection Rayleigh-Darcy number, and cell (or microorganism) eccentricity 0  on Rayleigh number for metal oxide ( 2 3 Al O ) nanoparticles in water based nanofluid. The physical properties of aluminum-water nanofluids are given in (47). As a consequence, the axial magnetic field can stabilize stationary convection, which is consistent with the analytical conclusion obtained from Eq. (70).

Stationary MHD bioconvection in
For different values of the Darcy number, the Rayleigh-Darcy number is displayed against the dimensionless wavenumber in Fig. 2b  This demonstrates that when Darcy's number rises, so does the Rayleigh-Darcy number. As a result, the Darcy number has a stabilizing impact on stationary convection, which is in agreement with the analytical result obtained from Eq. (71).
For different values of medium porosity, the Rayleigh-Darcy number is plotted against dimensionless wavenumber in Fig. 2c  This shows that when porosity increases, the values of Rac decrease significantly. As a result, medium porosity has a destabilizing impact on stationary convection, which is consistent with the analytical result obtained from Eq. (72).  [19] demonstrated that a suspension of gyrotactic microorganisms ( > 0 G ) is unstable under the same conditions. As a result, gyrotaxis plays a role in the emergence of bioconvection instability. As can be seen from Fig. 3c, with an increase in the parameter B Ra , the threshold for the occurrence of magnetic convection decreases. This is because the movement of microorganisms leads to a redistribution of the density of the nanofluid, reducing the process of heat transfer in the nanofluid. As a consequence, the cell's swimming can destabilize stationary magnetoconvection, which is consistent with the analytical conclusion obtained from Eq. (74). Al O -water nanofluid are shown in Table 1. Let's notice that results in Table 1 were obtained numerically using (46) for 2 3 Al O -water nanofluid and are in good agreement with the graphical results in Figs  Ra is plotted against the wavenumber for several nanofluids in Fig. 4. From Fig. 4, we can see that the 2 SiO -water nanofluid exhibits the highest stability compared to 2 3 Al O -water, 2 TiO -water, Cu -water and Ag -water nanofluids. The phenomena for this behavior is not only the different density of nanoparticles but also different thermophysical properties. We can conclude that semiconductor and metal oxide nanoparticles have a more destabilizing effect on stationary convection than metallic nanoparticles. SiO ) in water-based nanofluids. These figures show that when the value of the Chandrasekhar parameter Q increases, the values of the thermal Rayleigh-Darcy number for both forms of convection increase, indicating that magnetic field has a stabilizing impact. The curves depicting the influence of Chandrasekhar number for Cu -water nanofluid are above those for Ag -water nanofluid, indicating that Cu -water nanofluid is more stable than Ag -water nanofluid. When the situation of semiconductors is considered (Fig. 5b), it is found that 2 SiO -nanoparticles improve the stability of nanofluid more than 2 TiO -nanoparticles as the Chandrasekhar number increases. Thus, it is also interpreted from the figures that semiconductors inhibit the onset of convection as compared to metals under the influence of magnetic field. SiO -water nanofluids exhibit higher stability than Ag -water and 2 TiO -water nanofluids. Further, let us study the influence of the effect of porosity on the system. Figures 7a and 7b show that the increase in  porosity stimulates the onset of instability. The critical Rayleigh-Darcy numbers for nonmetallic nanofluids are higher than that for metallic nanofluids. Ra on the stationary convective instability for metallic and semiconductors nanofluids. We can observe that the cell's swimming can destabilize stationary magnetoconvection since the threshold for the occurrence of magnetic convection decreases when the parameter B Ra is increased. As can be seen from Fig. 8 nonmetallic nanoparticles delay the onset of convection even in the absence of bioconvection = 0 B Ra (or movement of microorganisms).    for metallic nanofluids ( Cu -water, Ag -water ) are shown in Table 2. Take note that the results in Table 2 were derived numerically using (46) for metallic nanofluids ( Cu -water, Ag -water) and correspond well with the graphical results in Figs. 5a, 6a, 7a, 8a, 9a. SiO -water) are shown in Table 3. We notice that the numerical results in Table 3 for semiconductor nanofluids ( 2 TiO -water, 2 SiO -water) agree well to the graphical results in Figs. 5b, 6b, 7b, 8b, 9b. In the limited case of no microorganisms ( =0) B Ra , the results from Tables 2 and 3 agree well with the results of paper [13]. It is noteworthy that copper-water nanofluid is more stable than silver-water nanofluid (refer to Figures 5a, 6a, 7a, 8a, 9a) and silicon oxide-water nanofluid is more stable than titanium oxide-water nanofluid (refer to Figures 5b, 6b, 7b, 8b, 9b). These conclusions are consistent with the results of paper [14].

The numerical/graphical results for Nusselt
( ) Nu t and Sherwood ( ) Sh t numbers In general, the transition from linear to non-linear convection can be complex. The study of Eqs. (59)-(66), whose solution provides a full description of the two dimensional non-linear evolution issues, helps to understand the transition. The Runge-Kutta technique is used to solve the autonomous system of unstable finite-amplitude equations. Convective heat and mass transfers were calculated using Nusselt number (

CONCLUSIONS
Under a vertical magnetic field, we investigated linear stability in a horizontal porous media saturated by nanofluid and gyrotactic microorganisms, heated from below and cooled from above, using the Darcy-Brinkman model, which incorporates Brownian motion and thermophoresis. The system with nanoparticle distribution at the top of the fluid layer has also been proposed. The influence of gyrotaxes on the orientation of swimming microorganisms was used in this study. The normal mode method was used for the linear analysis. The impact of various factors on the development of thermal instability was then established. The results are graphically represented. The following are our conclusions for 2 3 Al O -water nanofluid with gyrotactic microorganisms:  The vertical magnetic field and Darcy number enhance the stability of the system.  Spherical gyrotactic microorganisms contribute more effectively to the development of thermal instability. Similar conclusions are also valid for metallic and semiconductor nanofluids. It has been determined that coppernanofluid is more stable than silver-water, whereas silicon oxide-water nanofluid is more stable than titanium oxidewater nanofluid, according to a comparative investigation of thermal instability using metallic and semiconducting nanofluids.
Graphic representation of the nonlinear theory results for 2 3 Al O -water nanofluid containing gyrotactic microorganisms. The following conclusions may be taken from these research results: L are changed, the stationary value of mass transfer is established faster than convective heat transfer and is approximately ( ) 3.9 Sh t  at an initial value of (0) = 1 Sh . The results of the theoretical studies presented in this work can be applied in geophysics, especially in the study of sea flows through a porous medium (the ocean crust) containing nanoparticles and gyrotactic microorganisms, as well as in designing biosensors.