Nanofluids are the liquids containing a suspension of nanoparticles (having a size typically in the range 1-100 nm) in a base fluid. The base fluid can be an organic solvent or water, and the choice of base fluid depends on the applications for which the nanofluid is prepared. A wide range of applications for nanofluid can be found in several sectors including transportation, commercial, residential, industrial, and so on. Owing to the importance of nanofluids in these sectors, the area nanofluid convection in porous media attracted the attention of various researchers. A benchmark study of convective transport in nanofluid was organized by Buongiorno . A new model was developed in his study, which consists of the effects of two important mechanisms, namely, Brownian diffusion and thermophoresis. There are several studies in which the phenomena related to the onset of nanofluid convection in porous media have been examined under different aspects. Using Buongiorno’s model, Nield and Kuznetsov  analytically studied the onset of convection in a layer of porous medium saturated by a nanofluid. The authors observed that the value of the critical thermal Rayleigh number depends on the distribution (i.e., top-heavy or bottom-heavy) of nanoparticles. They reported that the oscillatory convection may occur in case of a bottom-heavy nanoparticle distribution. Later, this problem was re-examined by the same authors for a revised set of boundary conditions . In their extended work, the authors considered the nanoparticle fraction in a way such that the nanoparticle flux is zero on the boundaries. The authors pointed out that the oscillatory convection can no longer occur with the choice of new boundary conditions. Using these more realistic boundary conditions, Yadav et al.  analyzed the thermal instability of rotating nanofluids. The authors mentioned that the
model selected in their study is more realistic physically than those used in the previous studies (i.e., models with non-zero nanoparticle flux at boundaries). Different aspects of natural convection in porous media have been examined thoroughly by many researchers (see, e.g., Ref. [5–9]).
An experimental investigation of the onset of convection in a stably stratified fluid layer because of selective absorption of radiation was conducted by Krishnamurti . The convection mechanism observed by Krishnamurti was a penetrative one, which was, stimulated by internal heating through absorption of radiation. Penetrative convection is a phenomenon that arises when buoyancy-driven motion penetrates into stratified layers . In the experimental work, Krishnamurti considered a layer of water that contains a pH indicator known as thymol blue. This model of Krishnamurti was again investigated by Straughan  with the bounding surfaces being fixed as they will be in the experiment. The author reported that the results obtained in his study supports the work of Krishnamurti and the model developed by Krishnamurti is a very effective one. Hill  studied a modification of the system introduced by Krishnamurti  for a fluid-saturated porous medium. He used Darcy’s model for the porous medium and performed both linear as well as nonlinear stability analysis. Later, the author again examined this system (i.e., Krishnamurti’s  model) by considering Brinkman’s model for the porous medium . Chang  extended the study of Krishnamurti by considering a two-layer system in which a fluid layer overlays a porous layer. The author examined the problem for two different configurations, namely, when the layer is heated from below and when the layer is heated from above.
The effect of magnetic field on nanofluid convection has its relevance and significance in numerous applications in biochemical engineering, geophysics, astrophysics, and chemical engineering . Yadav et al.  performed a linear stability analysis to investigate the effect of a uniform vertical magnetic field on the onset of nanofluid convection and discussed the case of both non-oscillatory as well as oscillatory convection. They reported that the parameters Le (nanofluid Lewis number), N (modified diffusivity ratio), and Rn A (concentration Rayleigh number) have a destabilizing effect on the system. Later, Gupta et al.  examined the onset of nanofluid convection subjected to an applied magnetic field. They considered a bottom-heavy nanoparticle distribution and discussed the stability analytically as well as numerically. The authors observed that the parameters Rn and Le decelerate the onset of convection for both oscillatory and stationary mode of convection, whereas, the parameter N advances the A onset of convection for stationary convection and delays the onset of convection for oscillatory convection. For more interesting studies related to nanofluid convection subject to an applied magnetic field, the reader may refer to [18, 19, 20, 21, 22] and references therein. Magnetic nanofluids (MNFs or ferrofluids) are the nanofluids that consist of magnetic nanoparticles suspended in a non-magnetic base fluid. One of the most important characteristic features of the MNF that distinguishes MNF from other fluids is that the fluid flow in MNF can be controlled by selecting an appropriate magnetic field. In many of the applications, it is desirable to control the fluid flow in porous media without directly accessing the fluid. These applications include emplacement of geophysically imageable liquids into specific zones for subsequent imaging, treatment chemicals, or controlled positioning of liquids . MNFs play a significant role in these applications. The study of ferrofluid convection in porous media subject to an applied magnetic field begins with the work of Vaidyanathan et al. . Mahajan and Sharma  studied the convective instability in a MNF-layer-saturated porous medium subject to a uniform magnetic field. The impact of Brownian motion, thermophoresis, magnetophoresis, and Darcy’s law on the MNF fluid flow is considered in their study. Sheikholeslami  investigated the free convection of MNF in a porous curved cavity subject to an external magnetic source. Several interesting problems related to the onset of magnetic nanofluid convection are investigated by [27, 28, 29, 30, 31, 32, 33] under different aspects.
The present literature survey confirms that there is no published work regarding the onset of MNF convection in porous media induced by selective absorption of radiation. In this article, a numerical investigation is performed to examine the onset of instability in a layer of MNF-saturated low-permeability porous media under the influence of an applied magnetic field. The results are discussed for various combinations of boundary conditions on impermeable surfaces, conducting surfaces, free surfaces, and surfaces with constant heat flux. In various engineering applications, the temperature of a wall is not uniform but, rather, is a result of the imposition of a constant heat flux . These applications required the study of constant heat flux boundaries. Experimental studies with the constant heat flux boundaries are conducted by various researchers including [35, 36]. In this article, we consider the following three different boundary conditions: (i) when both lower and upper boundaries are impermeable and conducting (I_C-I_C)
(ii) when lower boundary is impermeable and conducting while upper boundary is impermeable with constant heat flux (I_C-I_CHF), and (iii) when lower boundary is impermeable and conducting while upper boundary is free with constant heat flux (I_C-F_CHF). Moreover, the nanoparticle fraction is adjusted in a way such that the nanoparticle flux is zero on the boundaries. To derive the boundary conditions for nanoparticle volume fraction, the effect of both thermophoresis and magnetophoresis on the nanoparticle flux is taken into account. The results are calculated by using the Chebyshev pseudospectral method for two different configurations (i) when the layer is heated from below and (ii) when the layer is heated from above. The effects of several important parameters that affect the onset of instability are observed.
2 Formulation of the problem
The physical configuration of the problem is depicted in Figure 1. The model consists of a layer of incompressible, MNF-saturated, low-permeability, porous medium, subject to a uniform applied magnetic field
In order to derive the governing equations, the following assumptions are made :
- fluid flow is incompressible,
- suspension is dilute (φ □ 1),
- viscous dissipation is negligible,
- two components (magnetic nanoparticles and base fluid) are locally in thermal equilibrium,
- the particles are suspended in nanofluid using either a surfactant or a surface charge technology that prevents particles from agglomeration and deposition on the porous matrix.
- the Boussinesq approximation and Darcy’s law hold.
The equation of continuity is
where V represents the filter velocity.
The equation of momentum is
where ρf, E,t,p,μ,K,μ0, and M are the density of MNF, porosity parameter, time, pressure term, viscosity, permeability parameter, magnetic permeability of vacuum and magnetization, respectively. Moreover, the MNF density ρ is considered as ρ =ϕρp + (1 −ϕ) ρf (1 − α(T − TL )) . Here, we followed the model of Krishnamurti  and assumed that the density of fluid does not depend on the thymol blue concentration.
The equation of nanoparticle is
where ϕ is the volume fraction of magnetic nanoparticle. DB, DT, and DH stand for the Brownian diffusion, thermophoretic diffusion, and magnetophoretic coefficient, respectively. The term H0 represents the 0 uniform magnetic field of the MNF layer.
The equation of temperature is
where (ρc)m,(ρc)f, and (ρc)p are the effective volumetric heat capacity of the porous medium, volumetric heat capacity for the MNF, and volumetric heat capacity for the nanoparticles, respectively. The terms k1 and Q represent the MNF thermal conductivity and heat source (which depends on the amount of radiation absorbed) respectively. Following Krishnamurti , a relationship between Q and C is considered in the following form: Q = (ρc)fαtC, where trepresents a proportionality constant.
The equation of thymol blue concentration is
where C and k c are the concentration and thymol blue diffusivity, respectively.
The relevant Maxwell equations in the magnetostatic limit are taken as
where B represents the magnetic induction.
Following Kaloni and Lou , magnetization is assumed to be characterized as follows:
To determine the solution in the quiescent state, the magnetic equation is linearized in the following manner :
where Km = χH0 / T 0 and K p = χH0 / φ0 are the magnetic coefficients. The term M0 and χ are the constant mean value of magnetization and tangent magnetic susceptibility respectively.
The parameters χ and χ2 (chord magnetic susceptibility) can be estimated by using the Langevin parameter as
The boundary conditions are taken as
In addition, to derive the magnetic boundary conditions, the normal component of the magnetic induction and the tangential component of the magnetic field are assumed to be continuous across the boundary.
The dimensionless quantities are introduced as follows:
Note that J∣TL −TU ⊨ (TL −TU), where J = sign (TL−TU) which takes the value −1 when the layer is heated from above and +1 when the layer is heated from below.
are the square root of thermal Rayleigh number, concentration Rayleigh number, diffusivity ratio, Prandtl number, modified diffusivity ratios (NA ,NtA), ratio of internal heating to boundary heating, modified particle-density increment, MNF Lewis number, and magnetic parameters (M1,M1 t,M2,M2 t), respectively. addition, the parameters
In dimensionless form, the boundary conditions (equation (9)) take the following form:
3 The basic state
The basic state solution is assumed to be in the following form:
Figure 2 represents the base flow profiles of Tb, ϕb, Hb, and Mb . The figure is plotted for two different configurations, namely, when the layer is heated from below (J = +1) and when the layer is heated from above (J = −1). It can be seen from the figure that the base flow profiles of Tb, ϕb, Hb, and Mb for J = +1 are completely different from the base flow profiles plotted for J = −1.
4 The linear stability problem
Now, we have superimposed the infinitesimally small perturbation on the basic state solution in the following form:
where vt, pt ,ϕt ,θt, Ct, Ht, and Mt are the perturb variables that are considered to be small.
Equations (28)-(32) possesses a boundary value problem Page. We reset the present domain from [0, 1] to [−1,1] by applying a coordinate transformation from z to 2z −1 in equations (28)-(32). This is so because, here we have planned to apply the Chebyshev pseudospectral method in which the domain of the problem must be [−1,1]. Thereafter, we use the normal mode procedure in which the normal mode solution is considered to be in the following form:
where kx and ky represent the wave numbers in x -direction and y -direction, respectively.
The boundary conditions for the amplitudes take the form
with w =θ = 0 at z = ±1 for I _C − I _C boundaries; w =θ = 0 at z = −1 and w = Dθ = 0 at z = +1 for I _C − I _CHF boundaries; w =θ = 0 at z = −1 and Dw = Dθ = 0 at z = +1 for I _C − F_CHF boundaries.
5 Method of solution
The above-mentioned system of equations (34)-(38) - with the boundary conditions (equation (39)) constitutes an eigenvalue problem. The Chebyshev pseudospectral method is selected to solve this eigenvalue problem. We closely followed the same process and algorithm, as mentioned in Ref. . First, we applied QZ-algorithm to calculate the salient eigenvalue (say σ =σr + iσj ) for fixed values of k, αL, Le, Rn, Y, η, and several other dimensionless parameters. By the salient eigenvalue, we mean that eigenvalue which has the largest real part. Second, we applied the Regula Falsi method to determine the particular value of β corresponding to which the real part σr of the salient eigenvalue σ tends to zero. This procedure endows a single point in the neutral stability curve. We reiterate this procedure for several values of k to enlist the desired neutral stability curve. The critical temperature gradient βc with kc (the critical wave number) can be defined as
Moreover, the function FMINBND (a combination searching of gold section and parabolic method) of MATLAB is used to minimize equation (40).
To examine the nature of the stability, we applied a numerical approach by using the Chebyshev pseudospectral method. The nature of the stability is called stationary (or non-oscillatory) if the imaginary part of the salient eigenvalue tends to be zero at the same time when its real part approaches to zero. Otherwise, the nature of the stability is called oscillatory. In order to have a check on the nature of the stability, the salient eigenvalue σ =σr + iσ j is estimated one by one for all the dimensionless parameters and noticed that σj always tends to be zero simultaneously when σr tends to be zero. This behavior is found to be same for both the configurations (i.e., J = +1 and J = −1). Thus the nature of the stability for the present problem is stationary.
The present numerical solution is validated by comparing the results with those in Hill . For this purpose, we have solved our problem in the absence of a
magnetic field and magnetic nanoparticle concentration for I_C–I_C boundaries. It can be seen from Table 1 that the comparison is found to be very good.
Comparison of Ra c and k2.
|Hill ||Present study|
|J||η||k c2||Ra c||k c2||Ra c|
6 Results and discussion
In this section, the results are illustrated graphically in Figures 3-7 and in Table 2 for J = +1 and J = −1. The results are derived for the following non-dimensional parameters that significantly affect the onset of convection: porosity parameter, the ratio of internal heating to boundary heating, Lewis number, concentration Rayleigh number, Langevin parameter, the width of MNF layer, the diffusivity ratio, and modified diffusivity ratio. Following Kaloni and Lou  [Table 1, page 7] and Rosensweig  [Table 2.4, page 71], the values of pertaining parameters are taken as ρf = 1,180, k1 = 0.59, Ms = 15,900, μ = 0.007, α=5.2 e−4.
The values of the kc and Rac for three different boundary conditions.
In order to illustrate the influence of selective absorption of radiation at the onset of MNF convection, the neutral stability curves for different values of Y are plotted in Figure 3 for three different boundary conditions. For both the cases, namely, J = +1 and J = −1, it can be seen from the figure that as the value of Y increases, the value of Rac decreases for all the three boundary conditions. It elucidates that the parameter Y advances the onset of convection. The reason for such behavior of Y is the following: a higher value of Y implies that the internal heating is promoted, which in turn creates a disturbance in the MNF layer and leads to a lower value of Ra c . This finding agrees well with the result of Hill  in the absence of a magnetic field and nanoparticle concentration.
To examine the influence of porous medium and Lewis number on the stability of the system, we plot the variation of Rac as a function of porosity parameter E for several values of Le in Figure 4. It is clear from the figure that Rac increases as E increases. Thus, an increase in the parameter E decelerates the onset of MNF convection. Such behavior of epsilon is in agreement with the literature by Yadav et al. [7, 42] in the absence of an applied magnetic field. It is also observed from the figure 4 that the values of Ra c decreases as Le increases. As the Lewis number is directly proportional to the thermal diffusivity, therefore, as the value of Le increases, the value of thermal diffusivity also increase. For a higher value of thermal diffusivity, the amplitude of disturbance waves increases, which leads to a lower value of Rac . Thus the parameter Le advances the onset of MNF convection. A similar observation was made earlier by Yadav et al.  in the absence of an applied magnetic field.
In the primary study of nanofluid convection in porous media, Nield and Kuznetsov  reported that the stabilizing or destabilizing behavior of Rn depends on the type of nanoparticle distribution (i.e., whether nanoparticle distribution is bottom-heavy or top-heavy). Neutral stability curves for three different boundary conditions are presented in Figure 5 for several values of Rn . The graphs are plotted for two different configurations, namely, J = +1 (i.e., when the layer is heated from below) and J = −1 (i.e., when the layer is heated from above). It can be seen from Figure 5 that the value of Rac decreases as Rn increases for J = +1, whereas the value of Rac increases as Rn increases for J = −1. The reason behind this may be the change in the type of nanoparticle distribution, which is clear from Figure 2, where the nanoparticle distribution is found to be top-heavy when J = +1 and bottom-heavy for J = −1. A destabilizing behavior of Rn for top-heavy nanoparticle distribution was reported earlier by Nield and Kuznetsov . Moreover, for a bottom-heavy
In order to unfurl the key features of Langevin parameter αL and the width of nanofluid layer d, the influence of both αL and d on Rac is presented in Figure 6. We note that the value of Rac increases when αL increases. This is so because, in the process of MNF convection, the buoyancy forces together with the forces that originate because of the appearance of an internal heat source dominate the magnetic forces, and, hence as the intensity of magnetic field increases, the disturbance in the MNF layer decelerates. This leads to a higher value of Rac . As plotted in Figure 6, the value of Rac is found to be increasing with an increase in the value of d. As the value of d increase, the value of critical temperature ΔTc decreases which in turn suppresses the disturbance in the magnetic nanofluid layer and tends to a higher value of Rac . Thus both the parameters αL and d delay the onset of MNF convection. In the absence of magnetic nanoparticles, a similar behavior was observed for αL and d by Kaloni and Lou .
Figure 7 shows the effect of η and N A on Rac for three different boundary conditions. It can be seen that
Rac decreases as the value of N A increases. On the other hand, Rac increases with an increase in η. As the value of N A increases, the value of thermophoretic diffusivity also increases. At a higher value of thermophoretic diffusivity, thermophoresis promotes the growth of turbulence in the MNF layer, which gives a lower value of Rac . Thus the parameter η has a stabilizing effect and N A has a destabilizing effect on the system. An identical behavior of the parameter N A was reported earlier by Yadav et al.  in the absence of magnetic field. For the parameter η, a similar behavior was reported earlier by Hill  for a regular fluid.
Table 2 presents the values of kc and Rac for three different boundaries, namely, I_C-I_C, I_C-I_CHF, and I_C-F_CHF. These values are calculated for two different configurations, that is, J = +1 and J = −1. Our main interest in presenting this table is to examine the influence of Y and η on the critical wave number kc and the critical thermal Rayleigh number Rac . The table shows that the value of Rac increases as η increases, whereas, Rac decreases as Y increases. This behavior is noticed to be the same for both the two different configurations and all the three boundary conditions. Thus the parameter Y
always hastens the onset of convection and the parameter η always decelerates the onset of convection. These observations agree well with the findings of Hill  [see Table 2, pp. 462], where the author investigated this problem in the absence of nanoparticles and magnetic field. It is also noticed that the value of Rac is higher in case of I_C-I_C boundary condition and the least for I_C-F_ CHF boundary condition for both J = +1 and J = −1. Thus the system is most stable for I_C-I_C boundaries and least stable for I_C-F_CHF boundaries.
The linear stability theory is applied to study the magnetic nanofluid convection stimulated by selective absorption of radiation subjected to an applied magnetic field. The problem is investigated for two different configurations, namely, when the layer is heated from below (J = +1) and when the layer is heated from above (J = −1). The Chebyshev pseudospectral method is applied to solve the resulting eigenvalue problem for I_C-I_C, I_C-I_CHF, and I_C-F_CHF boundaries. The effect of the porosity parameter E, the parameter Y, the Lewis number Le, the concentration Rayleigh number Rn, the Langevin parameter αL, the width of nanofluid layer d, the diffusivity ratio η, and the modified diffusivity ratio N A is observed at the onset of convection. The following conclusions are drawn:
- The value of Rac increases as the value of E,αL, d, and η increases, and decreases as the value of Y,Le, and N A increases. Thus, E,αL, d, and η decelerate the onset of MNF convection, and Y,Le, and N promote the onset of MNF convection. A
- The effect of Rn on the onset of convection depends on the type of configuration (i.e., whether J = +1 or J = −1). For J = +1, Rn hasten the onset of MNF convection, whereas for J = −1, Rn delays the onset of MNF convection.
- Among all the three type of boundaries, the system is found to be most stable for I_C-I_C boundary condition and least stable for I_C-F_CHF boundary condition.
The work in this article is supported by Council of Scientific and Industrial Research (CSIR), New Delhi, in the form of Research and Development project [Ref. No. 25(0255)/16/EMR-II]. The authors gratefully acknowledge the support thus received.
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