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 . 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 , and its recent revival has been inspired by the work of Serrin  and Joseph [1, 4, 5]. Rapid improvements of the classical energy theory have been made in recent years . 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  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 ). 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  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  applied the couple-stress fluid model to analyze the long slider bearing. Sharma and Thakur  and Sharma et al.  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.  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  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 . 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 . 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 , 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. R𝓁f ≡ Ref 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 K1 .
The fluid is assumed to occupy the layer
Here ρ, ρ 0 , q , g , t , p , μ , μ' ,
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
The boundary conditions are
with q',θ and γ satisfying the plane tiling periodicity.
3 Non-Linear Stability Analysis
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; D̃a is the Darcy–Brinkman number; Da is the Darcy number; Va is the Vadasz number (as named by Straughan ), F is the Couple-stress parameter, S is the solute Rayleigh–Darcy number and Le is the Lewis number.
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):
with λ1 and λ2 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
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
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
The maximum value of λ1 and λ2 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
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 ce f (see Tables 1–3).
The variation in the fluid-based critical thermal Rayleigh number Rcef with the couple-stress parameter F1 for Da=0.02, Vr =1 and Sf = 100.
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 xc ℓ and xce and critical thermal Rayleigh number R ef = R ce f depends on the parameters Vr , F1, S f and Da . The variation in x c and R cef with the variation in F 1 is given in Table 1, that with the variation in Da is given in Table 2 and that with the variation in S f is given in Table 3, and the
The variation in the fluid-based critical thermal Rayleigh number R cef with the couple-stress parameter (Da ) for F1=2, Vr=1 and Sf = 100.
The variation in the fluid-based critical thermal Rayleigh number R cef with the solute gradient (Sf) for Da, Vr =1.
results are further illustrated in Figs. 2–4, which represent the plot of critical thermal Rayleigh number R cef versus the parameter F 1, Da and S f , respectively. Figure 2 indicates that the parameter F 1 has the stabilizing effect on convection because as F 1 increases, the value of R cef 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.
This conclusion is further strengthened in Fig. 3, which shows the variation in Rcef with Darcy number Da at F = 2, Vr =1. Here, an increase in Da leads to a decrease in Rcef , rendering the system prone to instability. Figure 4 indicates that the solute gradient S f has a stabilizing effect because with the increase in S f , the values of Rcef 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 Da leads to a decrease in Rcef
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.
Joseph D.D. (1976). Stability of Fluid Motions Vols. I-II Springer Berlin.
Orr W. McF. (1907). Stability or instability of the steady motions of a perfect liquid. Proceedings of the Royal Irish Academy A 37 9.
Joseph D.D. (1966). Nonlinear stability of the Boussinesq equations by the method of energy. Archive for Rational Mechanics and Analysis 22 163.
Galdi G.P. Padula M. (1990). A new approach to energy theory in the stability of fluid motion. Archive for Rational Mechanics and Analysis 110 187.
Straughan B. Explosive Instabilities in Mechanics Springer Berlin 1998.
Straughan B. (2004). The Energy Method Stability and Nonlinear Convection Springer-Verlag New York.
Straughan B. (2005). A sharp nonlinear stability threshold in rotating porous convection. Proceedings of the Royal Society of London Series A 457 87.
Kaloni P.N. Qiao Z. (1997). Non-linear stability of convection in a porous medium with inclined temperature gradient. International Journal of Heat and Mass Transfer 40 1611.
Kaloni P.N. Qiao Z. (1997). Nonlinear convection with inclined temperature gradient and horizontal mass flow. International Journal of Engineering Science 35 299.
Kaloni P.N. Qiao Z. (2001). Non-linear convection in a porous medium with inclined temperature gradient and variable gravity effects. International Journal of Heat and Mass Transfer 44 1585.
Guo J. Kaloni P.N. (1995). Nonlinear stability problem of a rotating doubly diffusive porous layer. Journal of Mathematical Analysis and Applications 190 373.
Guo J. Qin Y. Kaloni P.N. (1994). Non-linear stability problem of a rotating doubly diffusive fluid layer. International Journal of Engineering Science 32 1207.
Payne L.E. Straughan B. (2000). Unconditional nonlinear stability in temperature – dependent viscosity flow in a porous medium. Studies in Applied Mathematics 105 59.
Sharma R. C. Thakur K. D. (2000). On couple-stress fluid heated from below in porous medium in hydromagnetics. Czechoslovak Journal of Physics 50 753.
Sharma R. C. Sunil Pal M. (2000). On couple-stress fluid heated from below in porous medium in presence of rotation. Applied Mechanical Engineering 5(4) 883.
Sunil Mahajan A. (2008a). A nonlinear stability analysis for magnetized ferrofluid heated from below. Proceedings of the Royal Society A 464 83.
Sunil Mahajan A. (2008b). A nonlinear stability analysis for rotating magnetized ferrofluid heated from below. Applied Mathematics and Computation 204 299.
Sunil Mahajan A. (2008c). A nonlinear stability analysis of a double-diffusive magnetized ferrofluid. Zeitschrift für Naturforschung 63a 797.
Sunil Mahajan A. (2008d). A nonlinear stability analysis in a double-diffusive magnetized ferrofluid layer saturating a porous medium. Journal of Geophysics and Engineering 5(3) 311.
Sunil Choudhary S. Bharti P. (2013). Global stability for thermal convection in a couple-stress fluid with temperature and pressure dependent viscosity. Studia Geotechnica et Mechanica 35(3) 85.
Sunil Choudhary S. Bharti P. (2012). Global stability for thermal convection in a couple-stress fluid saturating a porous medium with temperature and pressure dependent-dependent viscosity. International Journal of Applied Mechanics and Engineering 17(2) 583.
Hsu C. H. Lin J. R. Chiang H.L. (2003). Combined effects of couple stresses and surface roughness on the lubrication of short journal bearings. Industrial Lubrication and Tribology 55 233.
Lahmar M. (2005). Elastohydrodynamic analysis of double-layered journal bearings lubricated with couple-stress fluids. Proceedings of the Institution of Mechanical Engineers Part J: Journal of Engineering Tribology 219 145.
Nield D.A. Bejan A. (2006). Convection in Porous Media Springer New York.
Veronis G. (1968). Effect of a stabilizing gradient of solute on thermal convection. Journal of Fluid Mechanics 34 315.
Joseph D.D. (1970). Global stability of the conduction-diffusion solution. Archive for Rational Mechanics and Analysis 36 285.
Sunil Sharma D. Sharma R.C. (2004). Effect of rotation on ferromagnetic fluid heated and soluted from below saturating a porous medium. Journal of Geophysics and Engineering 1 116.
Sunil Sharma A. Sharma R.C. (2006). Effect of dust particles on ferrofluid heated and soluted from below. International Journal of Thermal Sciences 45 347.
Sunil Sharma A. Bharti P.K. Shandil R.G. (2007). Linear stability of double-diffusive convection in a micropolar ferromagnetic fluid saturating a porous medium. International Journal of Mechanical Sciences 49 1047.
Sunil Sharma P. Mahajan A. (2009). A nonlinear stability analysis of a rotating double-diffusive magnetized ferrofluid saturating a porous medium. Heat Transfer Research 40 351.
Sunil Sharma P. Mahajan A. (2010). Onset of Darcy– Brinkman double-diffusive convection in a magnetized ferrofluid layer using a thermal non-equilibrium model: a nonlinear stability analysis. Journal of Geophysics and Engineering 7 417.
Mahajan A. Nandal R. (2017). On the stability of penetrative convection in a couple-stress fluid. International Journal of Applied and Computational Mathematics 3(4) 3745-3758.
Nandal R. Mahajan A. (2018). Penetrative convection in couple-stress fluid via internal heat source/sink with the boundary effects. Journal of Non-Newtonian Fluid Mechanics 260 133-141.
Qin Y. Kaloni P.N. (1995). Nonlinear stability problem of a rotating porous layer. Quarterly of Applied Mathematics 53 129.
Chandrasekhar S. (1981). Hydrodynamic and Hydromagnetic Stability Dover New York.