## 1 Introduction

Conventional hydrodynamic stability theory is mainly concerned with the determination of critical values of Rayleigh number, demarcating a region of stability from that of instability. The potentials of linear theory of stability and the energy method are complementary to each other in the sense that the linear theory gives conditions under which the hydrodynamic system is definitely unstable. It cannot with certainty conclude stability. On the other hand, the energy theory gives conditions under which the hydrodynamic system is definitely stable. It cannot with certainty conclude instability. Suffering from its basic assumptions, the validity of the linearized stability theory becomes questionable.

Hence, the non-linear approach becomes inevitable to investigate the effect of finite disturbances. The formulation and derivation of the basic equation of a layer of fluid heated and soluted from below in the porous medium using Boussinesq approximation has been given in a treatise by Joseph [1]. When a fluid flows in an isotropic and homogenous porous medium, the gross effect is represented by the Darcy’s law. The study of a layer of fluid heated and soluted from below in the porous media is motivated both theoretically and by its practical application in engineering. Among the application in engineering disciplines, one can find the food process industry, chemical process industry and solidification and centrifugal casting of metals. The oldest method of non-linear stability analysis that can deal with finite disturbances is the energy method, originated by Orr [2], and its recent revival has been inspired by the work of Serrin [3] and Joseph [1, 4, 5]. Rapid improvements of the classical energy theory have been made in recent years [6]. The approach adopted in the present article is by the application of the energy method, pioneered and developed in its modern use way by Straughan [7, 8]. Straughan [9] developed a sharp non-linear energy stability analysis for the saturated porous medium, and the results obtained are the best possible showing that subcritical instabilities are not possible. By selecting the optimal, it has been possible to sharpen the stability bound in many physical problems (Straughan [8]). A nonlinear stability analysis of fluids by using generalized energy stability theory has been considered by many authors[10, 11, 12, 13, 14, 15]).

There are a lot of analyses of performance and experiment in the couple-stress lubricant. Stokes [16] proposed a simplest theory called the Stokes microcontinum theory, which could be used for the simulation of couple-stress fluid. One of the applications of couple-stress fluid is its use in the study of the mechanism of lubrication of synovial joints, which has become the object of scientific research. A human joint is a dynamically loaded bearing that has auricular cartilage as the bearing and synovial fluid as the lubricant. When a fluid film is generated, squeeze-film action is capable of providing considerable protection to the cartilage surface. Ramanaian [17] applied the couple-stress fluid model to analyze the long slider bearing. Sharma and Thakur [18] and Sharma et al. [19] have studied the problems of couple-stress fluid heated and soluted from below in the hydromagnetic porous medium and rotation separately. Recently, Sunil and Mahajan [20, 21, 22, 23] studied the non-linear stability analysis for thermal convection in a magnetized ferrofluid heated from below saturating a porous medium. Sunil et al. [24, 25] studied the non-linear stability analysis for thermal convection in a couple-stress fluid heated from below saturating a porous medium. Hsu et al. [26] studied the combined effects of couple stress and surface roughness using journal bearings lubricated with the non-Newtonian fluid. It was found that the combined effects of couple stress and surface roughness can improve the load carrying capacity and decrease the attitude angle and friction parameter. Recently, Lahmar [27] also found that the lubricants with couple-stress fluid would increase the load carrying capacity and stability and decrease the friction factor and the attitude angle.

The purpose of the present article was to study the non-linear stability analysis of couple-stress fluid heated and soluted from below, saturating a porous medium of high permeability [28]. The really interesting situation from a mathematical viewpoint arises when the layer is simultaneously heated and salted from below. In the standard Bènard problem, the instability is driven by a density difference caused by a temperature difference between the upper and lower surfaces bounding the fluid. If, additionally, the fluid layer has salt dissolved in it, then there are potentially two destabilizing sources for the density difference, the temperature field and the salt field. When there are two effects such as this, the phenomenon of convection that arises is called double-diffusive convection. The driving force for many studies in double-diffusive or multicomponent convection is largely physical applications. The double-diffusive convection problems have been studied by many authors [13-14, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39,]. In porous media, an alternative to Darcy’s equation is what is known as Brinkman’s equation [28]. It is believed that for the flow of a high-porosity porous medium, the Brinkman equation removes some of the deficiencies and gives preferable result. In the work of Qin and Kaloni [40], it was remarked that for high porosity materials and when boundary layer effects need to be taken into account, the Brinkman model is superior to Darcy’s model. Here, we establish the optimal result, that is the linear instability and non-linear stability of Rayleigh numbers are the same, i.e. *R*_{𝓁f} ≡ *R _{ef}*

^{}We also find that the critical value of thermal Rayleigh number for the couple-stress fluid is higher than the critical value of thermal Rayleigh number for the ordinary fluid; hence, the couple-stress fluid is thermally more stable than the ordinary fluid. This problem, to the best of my knowledge, has not been investigated yet.

## 2 Mathematical Formulation Of The Problem

Here, we consider an infinite, horizontal layer of thickness ‘ *d* ’ of incompressible thin couple-stress fluid with constant viscosity that is heated and soluted from below, saturating an isotropic homogeneous porous medium of porosity ε and medium permeability *K*_{1} .

The fluid is assumed to occupy the layer **g** = (0,0,-*g* ) pervading the system in the negative *z-*direction.

The equations governing the flow of an incompressible couple-stress fluid (utilizing the Boussinesq approximation) are given as follows [13, 24, 41]:

Mass balance:

Momentum balance:

Temperature equation:

Solute equation:

Here *ρ*, ρ _{0} , **q** , **g** , *t* , *p* , μ , μ^{'} , ^{'} , *K*_{1} , ε ,α , , α' and *C*_{0} are the fluid density, reference density, filter velocity, acceleration due to gravity, time, pressure, coefficient of viscosity, coefficient of visco-elasticity, effective viscosity, thermal diffusivity, solute diffusivity, permeability of porous medium, porosity, thermal expansion coefficient, concentration expansion coefficient analogous to the thermal expansion coefficient and specific heat at constant pressure, respectively. The subscripts ‘*m*’ and ‘*f*’ refer to the fluid– solid mixture and the fluid, respectively. *T _{a}* and

*C*are the average temperature and solute concentration given by

_{a}*T*,

_{L}*T*and

_{U}*C*

_{L}_{}

*C*are the constant average temperatures and solute concentrations of the lower and upper surfaces of the layer, respectively, and

_{U}The basic state is assumed to be the quiescent state and is given by

where the subscript ^{'}*b*^{'} denotes the basic state.

We shall analyze the stability of the basic state by introducing the following perturbations:

The non-linear equations for the perturbations **q**' = (*u*,*v*,*w*), *p*', ρ' ', θ and γ , which represent perturbations in velocity **q** , pressure *p*, density ρ, , temperature *T* and concentration *C* , respectively, are given by

where

The boundary conditions are

with **q**^{'},*θ and**γ* satisfying the plane tiling periodicity.

## 3 Non-Linear Stability Analysis

To investigate the non-linear stability analysis, the governing equations (7), (8), (9), (10) in the non-dimensional form (dropping asterisk) can be written as

where the following non-dimensional quantities and parameters are introduced:

Here, *R* is the Rayleigh–Darcy number, which is the product of Darcy number and the usual Rayleigh number for a clear viscous fluid; ^{D̃}* _{a}* is the Darcy–Brinkman number;

*D*is the Darcy number;

_{a}*V*is the Vadasz number (as named by Straughan [9]),

_{a}*F*is the Couple-stress parameter,

*S*is the solute Rayleigh–Darcy number and

*L*is the Lewis number.

_{e}On multiplying (12) by **q**, 14) byθ , (15) by γ and integrating over *V*, we get (after using equation (11), the boundary conditions and the divergence theorem):

where *V*, *L*^{2} (*V* ) norm and *V* denotes a typical periodicity cell.

To study the non-linear stability of the basic state (5), an *L*^{2} energy, *E*(*t*), is constructed using equations (17) – (19), and the evolution of *E* (*t*) is given by

where

with λ_{1} and λ_{2} as two positive coupling parameters.

Here, the negative sign with the

We now define,

where *H* is the space of admissible solutions.

Then, we require ^{m}^{<}^{1} so that

where *a*_{0} =1-*m*(> 0) .

From the Poincaré inequality, we have

where *k*^{*} = 2 π ^{2} min

This gives

implying

Thus, *E* decays at least exponentially fast, and non-linear stability is assured for all values of *E* (*0*). It is important to note that the result holds for all initial data.

## 4 Variational Problem

We now return to equation (24) and use calculus of variation to find the maximum problem at the critical argument *m* =1. The associated Euler–Lagrange equations after taking transformations

where *p* is a Lagrange’s multiplier introduced, since **q** is solenoidal.

On taking curl curl of equation (29) and then taking the third component of the resulting equation, we find

Now, we assume a plane tiling form

where *a* being the wave number [9, 42].

The wave number is found a posteriori to be non-zero; thus, from equations (29), (30), (31), we see that *W* and Θ satisfy

Thus, the exact solution to the equations (34), (35), (36) subject to boundary conditions

is written in the form

where *A*_{0} , *B*_{0} and *C*_{0} are constants. Substituting solution (37) in equations (34), (35), (36), we get the equations involving coefficients of *A*_{0} , *B*_{0} and *C*_{0} . For the existence of non-trivial solutions, the determinant of the coefficients of *A*_{0} , *B*_{0} and *C*_{0} must vanish. This determinant on simplification yields

where

The maximum value of λ_{1} and λ_{2} is determined by the conditions

Using (40) in equation (39), we have

We obtain the fluid-based thermal Rayleigh number as

As a function of *x* , *R _{ef}* given by equation (42) attains its minimum when

where

The thermal Rayleigh number *R _{e f}* is minimized with respect to

*x*, and we use the Newton–Raphson iterative scheme to obtain the value of critical wave number and the corresponding critical thermal Rayleigh number

*R*(see Tables 1–3).

_{ce f}The variation in the fluid-based critical thermal Rayleigh number *R _{cef}* with the couple-stress parameter

*F*

_{1}for

*D*=0.02

_{a}*, V*=

_{r}*1*and

*S*= 100.

_{f}F_{1} | x_{ce} | R _{cef} |
---|---|---|

01 | 0.9636 | 307.90 |

2 | 0.6154 | 661.40 |

3 | 0.5682 | 1001.9 |

4 | 0.5483 | 1340.8 |

5 | 0.5374 | 1679.0 |

6 | 0.5305 | 2017.0 |

7 | 0.5258 | 23537 |

8 | 0.5223 | 2696.0 |

9 | 0.5197 | 3030.3 |

10 | 0.5176 | 3367.9 |

0.5159 | 3705.6 |

For analyzing the linear instability results, we return to the perturbed equations (7), (8), (9), (10), neglecting the nonlinear terms. We again perform the standard stationary normal mode analysis and look for the solution of these equations in the form (38). The boundary conditions in the present case are same, i.e. (37) (here, the thermal Rayleigh number).

In the absence of solute ( *S _{f}* = 0 ), this further simplifies to

i.e., in both the case, the linear instability boundary ≡ the non-linear stability boundary, and so no subcritical instabilities are possible for the case of couple-stress fluid. This result is equivalent to the result given by Joseph [4, 5] for the standard Bénard problem.

## 5 Discussion Of Results And Conclusion

The critical wave numbers *x _{c}*

_{ℓ}and

*x*and critical thermal Rayleigh number R

_{ce}*=*

_{ef}^{}

*R*depends on the parameters

_{ce f}*V*,

_{r}*F*

_{1},

*S*. The variation in

_{f}and D_{a}*x*and

_{c}*R*with the variation in

_{cef}*F*

_{1}is given in Table 1, that with the variation in

*D*is given in Table 2 and that with the variation in

_{a}*S*is given in Table 3, and the

_{f}The variation in the fluid-based critical thermal Rayleigh number *R _{cef}* with the couple-stress parameter (

*D*) for

_{a}*F*

_{1}=2,

*V*=1 and

_{r}*S*= 100.

_{f}D_{a} | x_{ce} | R_{cef} |
---|---|---|

0.01 | 0.5684 | 1897.1 |

0.02 | 0.5682 | 1001.9 |

0.03 | 0.5679 | 7017 |

0.04 | 0.5676 | 5532 |

0.05 | 0.5673 | 4636 |

0.06 | 0.5671 | 405.2 |

0.07 | 0.5668 | 365.0 |

0.08 | 0.5665 | 330.6 |

0.09 | 0.5663 | 305.7 |

0.10 | 0.5660 | 285.8 |

0.11 | 0.5667 | 269.5 |

The variation in the fluid-based critical thermal Rayleigh number *R _{cef}* with the solute gradient (

*S*) for

_{f}*D*,

_{a}*V*=1.

_{r}S | |||
---|---|---|---|

100 | 661.4 | 1001.9 | 1340.8 |

200 | 761.4 | 1101.9 | 1440.8 |

300 | 861.4 | 1201.9 | 1540.8 |

400 | 961.4 | 1301.9 | 1640.8 |

500 | 1061.4 | 1401.9 | 1740.8 |

600 | 1161.4 | 1501.9 | 1840.8 |

700 | 271.4 | 1601.9 | 1940.8 |

800 | 1361.4 | 1701.9 | 2040.8 |

900 | 1461.4 | 1801.9 | 2140.8 |

results are further illustrated in Figs. 2–4, which represent the plot of critical thermal Rayleigh number *R _{cef}* versus the parameter

*F*

_{1},

*D*and

_{a}*S*, respectively. Figure 2 indicates that the parameter

_{f}*F*

_{1}has the stabilizing effect on convection because as

*F*

_{1}increases, the value of

*R*increases. We also note that the value of critical thermal Rayleigh number remains the same for both the theories (linear theory and non-linear theory) and no subcritical instabilities are possible.

_{cef}This conclusion is further strengthened in Fig. 3, which shows the variation in *R _{cef}* with Darcy number

*D*at

_{a}*F*= 2,

*V*=

_{r}^{}1. Here, an increase in

*D*leads to a decrease in

_{a}*R*, rendering the system prone to instability. Figure 4 indicates that the solute gradient

_{cef}*S*has a stabilizing effect because with the increase in

_{f}*S*, the values of

_{f}*R*also increase. Here, two diffusing components heat and salt are present that produce the density differences required to derive the motion. The components make opposing contributions to the vertical density gradient as motion is encouraged due to heating and solute acts to prevent motion through convection overturning. Thus, these two physical effects are competing against each other. We also note that the value of critical thermal Rayleigh number remains the same for both the theories (linear theory and non-linear theory) and no subcritical instabilities are possible. In other words, medium permeability destabilizes the flow. To investigate our result, we must review the results and its physical explanation. When the fluid layer is assumed to be flowing through an isotropic and homogenous porous medium, then the medium permeability has a destabilizing effect. This is because, as medium permeability increases, the void space increases, and as a result of this, the flow quantities perpendicular to the planes will clearly be increased. Thus, an increase in heat transfer is responsible for early onset of convection. Hence, an increase in

_{cef}*D*leads to a decrease in

_{a}*R*

_{cef}The principal conclusion from the above analysis is as follows:

- – The result we establish is that boundaries of nonlinear stability and linear instability analyses coincide with each other. So, no subcritical instabilities are possible.
- – The couple stress has the tendency to slow down the motion of the fluid in the boundary layer, thus reducing the heat transfer from bottom to top. The a decrease in heat transfer is responsible for delaying the onset of convection. Thus, the couple-stress parameter
*F*_{1}promotes stabilization. - – The medium permeability is found to have destabilizing effect on the system.
- – It is observed that solute gradient delays the onset of convection and thus has a stabilizing effect on the system.

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