The paper deals with the theoretical investigation of the effect of dust/suspended particles on a layer of electrically conducting micropolar fluid heated and dissolved from below in the presence of a uniform vertical magnetic field in a porous medium. The presence of coupling between thermosolutal and micropolar effects and magnetic field brings oscillatory motions in the system. A dispersion relation governing the effects of solute gradient, magnetic field, and suspended particles is obtained for a fluid layer contained between two free boundaries using linear stability theory and normal mode technique. Graphs have been plotted by giving numerical values to various parameters involved to depict the stability characteristics for both cases of stationary convection and overstability. It has been found that, for permissible values of various parameters under consideration, the effect of magnetic field and solute gradient is stabilizing and that of medium permeability, suspended particles, and micropolar coefficient is destabilizing. Further it is found that the Rayleigh number for overstability is always less than that for stationary convection except for high values of suspended particle factor.
1. Introduction
A general theory of micropolar fluids was originally introduced by Eringen [1–3] in order to describe some physical systems which do not satisfy the Navier-Stokes equations. These fluids differ from the classical fluids in the sense that they can support couple stresses due to rotatory motion. The equations governing the flow of a micropolar fluid involve a spin vector (microrotation vector) and a microinertia tensor (gyration parameter) in addition to the velocity vector. Thus the model of micropolar fluid will have six degrees of freedom of rigid body (three corresponding to translation and three corresponding to microrotation). Physically speaking, a micropolar fluid may be thought of as the fluid containing elongated molecules, for example, plasma, polymeric fluids, suspension solutions, liquid crystals, animal blood, paints, colloidal solutions, and muddy fluids like crude oils. Micropolar fluid stabilities have become an important field of research due to its significant importance in industry. Ahmadi [4] and Pérez-García and Rubí [5] studied the effect of microstructures on thermal convection, and Lekkerkerker [6, 7], Bradley [8], and Laidlaw [9] investigated the existence of oscillatory motions. Chandrasekhar [10] presented the problem of thermal convection in a horizontal thin layer of Newtonian fluid heated from below under varying assumptions of hydrodynamics and hydromagnetics. This problem in the literature is popularly known as Rayleigh-Bénard convection problem. Datta and Sastry [11] investigated the Bénard problem in the micropolar fluid using the theory of Eringen. In connection with instability of micropolar fluids, one can refer to the papers by Siddheshwar and Pranesh [12, 13], Kazakia and Ariman [14], Sharma and Gupta [15], Sharma and Kumar [16, 17], Sunil et al. [18], Rani and Tomar [19], and Dragomirescu [20] including several others. The study of the flow of fluids through porous medium is of interest due to its natural occurrence. When the fluid permeates into a porous material, the gross effect is represented by Darcy’s law. As a result of this macroscopic law, the usual viscous term in the equations of motion of micropolar fluid is replaced by the resistance term [(-1/k1)(μ+κ)v→], where μ and κ are viscosity and dynamic microrotation viscosity, respectively, k1 is the medium permeability, and v→ is the Darcian (filter) velocity of the fluid. The effect of magnetic field on the stability of such fluids is of particular importance in geophysics, for example, in the study of earth’s core where the earth’s mantle which consists of conducting fluid behaves like porous medium.
The term “double-diffusive convection” applies to the convection in a fluid where there are two diffusing constituents having effect on buoyancy. For thermosolutal convection, buoyancy forces can arise not only from density differences due to variation in temperature gradient but also from those due to variation in solute concentration, and this double-diffusive phenomenon has importance and direct relevance in the field of chemical engineering, metallurgy, astrophysics, limnology, and oceanography. The study of double-diffusive convection problem for a layer of ordinary fluid took place in the mid sixties (Veronis [21]). In geophysical situations, the fluid is often not pure but contains suspended/dust particles. Motivation for the study of certain effects of particles immersed in the fluid such as particle heat capacity, particle mass friction, and thermal force is due to the fact that the knowledge concerning fluid-particle mixture is not commensurate with their industrial and scientific importance. Although several authors (like Sharma and Gupta [22, 23], Sunil et al. [24], Sharma and Rana [25], Gupta and Aggarwal [26, 27]) investigated the effect of suspended particles on various instability problems for Newtonian as well as viscoelastic fluids but very few namely, Sharma and Gupta [28] and Reena and Rana [29], have discussed the effect of suspended particles on micropolar fluids for Rayleigh-Bénard convection problem. Here it is worthwhile to mention that none of the authors have discussed the effect of dust particles on a micropolar fluid layer for double-diffusive convection problem in porous or nonporous medium. In the present paper, we investigate the double-diffusive convection problem for the micropolar fluid layer in porous medium with the further motivation to study the conflicting tendencies arising due to the stabilizing nature of magnetic field and solute gradient and destabilizing nature of suspended particles and permeability. The presence of coupling between thermosolutal and micropolar effects in the presence of magnetic field brings oscillatory motions in the system. Interestingly, with the increase in suspended particles factor, the mode of instability transforms from overstability to stationary convection. Some earlier known results have been recovered from the present formulation.
2. Formulation of the Problem
In the present problem, we have considered an infinite, horizontal, and incompressible electrically conducting micropolar fluid layer permeated with suspended particles and bounded by the planes z=0 and z=d, as shown in Figure 1. This layer is heated and dissolved from below such that steady adverse temperature gradient β(=dT/dz) and analogous solute concentration gradient β′(=dC/dz) are maintained. This is the Rayleigh-Bénard instability problem in the presence of salinity gradient for micropolar fluids. The fluid-particle layer is acted upon by a uniform external magnetic field H→=(0,0,H) and gravity force g→=(0,0,-g).
Geometrical configuration.
Within the framework of Boussinesq approximation which states that density differences are sufficiently small to be neglected, except where they appear in terms multiplied by g→ (the acceleration due to gravity), the field equations of micropolar fluid, in the absence of external body, couple and heat source densities are given by (Refer Chandrasekhar [10], Eringen [1], Pérez-García and Rubí [5], and Veronis [21])
(1)∇·v→=0,(2)ρ0ε∂v→∂t=-∇p-1k1(μ+κ)v→+κ∇×π→-ρge^z+K′Nε(u→-v→)+μe4π(∇×H→)×H→,(3)ρ0j∂π→∂t=(ε′+β′′)∇(∇·π→)+γ′∇2π→+κε∇×v→-2κπ→,(4)[ρ0cvε+ρscs(1-ε)]dTdt+ρ0cvv→·∇T+mNCpt×(ε∂∂t+u→·∇)T=kT∇2T+δ1(∇×π→)·∇T,(5)[ρ0cvε+ρscs(1-ε)]dCdt+ρ0cvv→·∇C+mNCpt×(ε∂∂t+u→·∇)C=kT′∇2C+δ1′(∇×π→)·∇C,(6)ρ=ρ0[1-α(T-T0)+α′(C-C0)],
where d/dt≡∂/∂t+(1/ε)v→·∇ is the convective derivative and the symbols v→, u→, π→, H→, T, C, ρ, ρs, p, g→, e^z, j, cv, cs, k1, Cpt, kT, kT′, μ, K′, ε, δ1, δ1′, α, α′, N, and mN denote fluid velocity, particle velocity, spin, magnetic field, temperature, solute concentration, density, density of solid matrix, pressure, acceleration due to gravity, unit vector in z-direction, microinertial constant, specific heat at constant volume, specific heat of the solid matrix, medium permeability, heat capacity of particles, thermal conductivity, solute conductivity, coefficient of viscosity, Stoke’s drag coefficient, porosity, coefficient accounting for the coupling between the spin flux and the heat flux, coefficient accounting for the coupling between the spin flux and the solute flux, coefficient of thermal expansion, analogous solvent coefficient, particle number density, and mass of suspended particles per unit volume, respectively. The parameters ε′, β′′, and γ′ stand for the micropolar coefficients of viscosity, and κ is dynamic microrotation viscosity. Also ρ0, T0, and C0 are reference density, reference temperature, and reference concentration, respectively.
In the present formulation, we have assumed that dust particles are of uniform size, spherical shape, and there are small relative velocities between the two phases (fluid and particles). As such, the net effect of the particles on the fluid is equivalent to an extra body force term per unit volume K′N(u→-v→)/ε, as has been taken in (2). This force exerted by the fluid on the particles is equal and opposite to that exerted by the particles on the fluid. The distance between the particles is assumed to be so large compared with their diameter that interparticle reactions are ignored. The equations of motion and continuity for the particles under these restrictions are
(7)mN[∂u→∂t+1ε(u→·∇)u→]=K′N(v→-u→),ε∂N∂t+∇·(Nu→)=0.
The well-known Maxwell equations are given by
(8)ε∂H→∂t=∇×(v→×H→)+εη∇2H→,∇·H→=0,
where η is the electrical resistivity of the fluid. The initial stationary state of the system is given by
(9)v→=0,u→=0,π→=0,T=T0-βz,C=β′z-C0,N=N0(constant),ρ=ρ0(1+αβz-α′β′z),p=p0-gρ0(z+αβz22-α′β′z22),
where p0 is the pressure at z=0 and β=(T0-T1)/d and β′=(C0-C1)/d are the magnitudes of uniform temperature and concentration gradient, respectively.
3. Perturbation Equations
To consider the stability of the system, we will apply small perturbations on the initial state and consider the reaction of the perturbations on the system. Let δp,δρ,θ,γ,v→(u,v,w),u→(l,r,s),Γ→,h→(hx,hy,hz), and N denote the perturbations in fluid pressure, fluid density, temperature, solute concentration, fluid velocity, particle velocity, spin, magnetic field, and particle number density N0, respectively. Then the perturbation equations of the fluid-particle layer are
(10)∇·v→=0,ρ0ε∂v→∂t=-∇δp-1k1(μ+κ)v→+κ(∇×Γ→)+ρ0g×(αθ-α′γ)e^z+K′N0ε(u→-v→)+μe4π(∇×h→)×H→,ρ0j∂Γ→∂t=(ε′+β′′)∇(∇·Γ→)+γ′∇2Γ→+κε∇×v→-2κΓ→,EH1dθdt=κT∇2θ+δ1ρ0cv(∇×Γ→)·∇θ-δ1ρ0cvβ(∇×Γ→)z+β(w+h1s),EH1dγdt=κT′∇2γ+δ1′ρ0cv(∇×Γ→)·∇γ-δ1′ρ0cvβ′(∇×Γ→)z+β′(w+h1s),mN0[∂u→∂t+1ε(u→·∇)u→]=K′N0(v→-u→),∂M∂t+∇·u→=0,ε∂h→∂t=∇×(v→×H→)+εη∇2h→,∇·h→=0,
where H1=1+h1, h1=fCpt/cv, f=mN0/ρ0, E=ε+(1-ε)ρscs/ρ0cv, and M=εN/N0. Also κT=kT/ρ0cv is thermal diffusivity, and κT′=kT′/ρ0cv is analogous solute diffusivity. Using the dimensionless parameters
(11)z=z*d,t=ρ0d2μt*,θ=βdθ*,γ=β′dγ*,u→=κTdu→*,v→=κTdv→*,p=μκTd2p*,Γ→=κTd2Γ→*,h→=μκTd2h→*,N2=K′N0d2μ,a=mK′d2μρ0
and then removing the stars for convenience, the non-dimensional forms of (10) become
(12)∇·v→=0,(13)1ε∂v→∂t=-∇δp-1k1(1+K)d2v→+K(∇×Γ→)+(Rθ-Sp1qγ)e^z+N2ε(u→-v→)+μe4π(∇×h→)×H→,(14)j∂Γ→∂t=c1∇(∇·Γ→)-c0∇×∇×Γ→+K[(1ε∇×u→)-2Γ→],(15)EH1p1dθdt=∇2θ+(w+h1s)+δ-[∇θ·(∇×Γ→)-(∇×Γ→)z],(16)EH1qdγdt=∇2γ+(w+h1s)+δ-′[∇γ·(∇×Γ→)-(∇×Γ→)z],(17)(addt+1)u→=v→,(18)∂M∂t+∇·u→=0,(19)ε∂h→∂t=1p1∇×(v→×H→)+εp2∇2h→,(20)∇·h→=0,
where various nondimensionalized parameters are
(21)K=κμ,j-=jd2,δ-=δ1ρ0cvd2,δ-′=δ1′ρ0cvd2,c0=γ′μd2,c1=ε′+β′′+γ′μd2,R=gαβρ0d4μκT,S=gα′β′ρ0d4μκT′,p1=μρ0κT,q=μρ0κT′,p2=μρ0η.
4. Linear Theory and Dispersion Relation
Since the disturbances applied on the system are assumed to be very small, the second order and higher order perturbation terms are neglected, and only linear terms are retained. Accordingly, the nonlinear terms (u→·∇)u→,(u→·∇)θ,(u→·∇)γ,∇θ·(∇×Γ→),∇γ·(∇×Γ→), and (u→·∇)Γ→ in (13)–(16) are neglected.
Eliminating u→ from (13) with the help of (17), we get
(22)L1′v→=εL2′[-∇δp-1k1(1+K)d2v→+K(∇×Γ→)+(Rθ-Sp1qγ)e^z+μe4π(∇×h→)×H→],
where L1′=a(d2/dt2)+F(d/dt),L2′=a(d/dt)+1, and F=f+1.
Eliminating s from (15)-(16) with the help of (17) applying curl operator twice to the resulting equations, and linearizing, we obtain
(23)L2[EH1p1ddt-∇2]θ=(a∂∂t+H1)w-δ-L2Ωz,(24)L2[EH1qddt-∇2]γ=(a∂∂t+H1)w+δ-′L2Ωz,
where L1=a(∂2/∂t2)+F(∂/∂t),L2=a(∂/∂t)+1.
Applying the curl operator twice to (22) and taking z-component, we get
(25)L1∇2w=εL2[R∇12θ-Sp1q∇12γ-1k1(1+K)d2∇2w+K∇2Ωz+μeH4π∂∂z∇2hz],
where ∇12=∂2/∂x2+∂2/∂y2, ∇2=∂2/∂x2+∂2/∂y2+∂2/∂z2, and Ωz=(∇×Γ→)z.
Applying curl operator to (14) and (19) and taking z-component, we obtain
(26)j-∂Ωz∂t=c0∇2Ωz-K(1ε∇2w+2Ωz),ε∂ςz∂t=Hp1∂ξz∂z+εp2∇2ςz,
where ςz=(∇×h→)z,ξz=(∇×v→)z.
Taking z-component of (19), we get
(27)ε∂hz∂t=Hp1∂w∂z+εp2∇2hz.
Applying curl operator to (22) and taking z-component, we get
(28)ε-1L1ξz=L2[-1k1(1+K)d2ξz+μeH4π∂ςz∂z].
Analyzing the disturbances into normal modes, let us assume that the perturbation quantities are of the form
(29)[w,Ωz,ξz,ζz,θ,γ,hz]=[W(z),Ω(z),Z(z),X(z),Θ(z),Γ(z),B(z)]×exp(ikxx+ikyy+nt),
where kx and ky are the wavenumbers along x- and y-directions and resultant wavenumber is given by k=(kx2+ky2)1/2, and n is the growth rate.
Using expression (29), (23)–(28) can be written as
(30)(an+1)[EH1p1n-(D2-k2)]Θ=(an+H1)W-(an+1)δ-(Ω),(an+1)[EH1qn-(D2-k2)]Γ=(an+H1)W-(an+1)δ-′(Ω),(D2-k2)[ε-1(an2+Fn)+1k-1(an+1)(1+K)]W=(an+1)[-Rk2Θ+Sp1qk2Γ+K(D2-k2)Ω+μeH4π(D2-k2)DB],[l1n+2A-(D2-k2)]Ω=-Aε-1(D2-k2)W,[n-1p2(D2-k2)]X=ε-1Hp1DZ,[n-1p2(D2-k2)]B=ε-1Hp1DW,[ε-1(an2+Fn)+1k-1(an+1)(1+K)]Z=μeH4π(an+1)DX,
where A=K/c0, l1=j-A/K, D≡d/dz, n=∂/∂t, ∇12=-k2, and k-1=k1/d2.
Consider the case in which both boundaries are free as well as maintained at constant temperatures while the adjoining medium is perfectly conducting. For the case of free boundaries, the appropriate boundary conditions are (Chandrasekhar [10])
(31)W=D2W=0,DZ=0,Θ=0,Γ=0,atz=0,1,
K=0, on perfectly conducting boundaries and hx, hy, and hz are continuous. Since the components of magnetic field are continuous and the tangential components are zero outside the fluid, we have
(32)DK=0,on the boundaries.
Using the above boundary conditions (31) and (32), it can be shown that all the even order derivatives of W must vanish for z=0 and 1. Hence, the proper solution of W characterizing the lowest mode is
(33)W=W0sinπz,
where W0 is a constant. Eliminating Θ,Γ,Z,B, and Ω between (30), we obtain
(34)Rk2(EH1qn+b)(n+bp2)×[(an+H1)(l1n+2A+b)-bAε-1δ-(an+1)]=b(EH1p1n+b)(EH1qn+b)(l1n+2A+b)×(n+bp2)×[ε-1(an2+Fn)+1k-1(an+1)(1+K)]+Sp1qk2(EH1p1n+b)(n+bp2)×[(an+H1)(l1n+2A+b)-bAε-1δ-′(an+1)]-KAε-1b2(an+1)(EH1p1n+b)(EH1qn+b)×(n+bp2)+μeH2πbε-14p1(an+1)(EH1p1n+b)×(EH1qn+b)(l1n+2A+b),
where b=π2+k2.
Equation (34) is the dispersion relation including the effects of magnetic field, dust particles, and permeability on the thermosolutal instability of micropolar fluid. The dispersion relation derived by V. Sharma and S. Sharma [30] is a particular case of this dispersion relation in the absence of suspended particles. Also, (34) reduces to the one derived by Sharma and Kumar [16] in the absence of solute gradient and suspended particles.
5. Case of Stationary Convection
For stationary convection, n=0, and the dispersion relation (34) reduces to
(35)R={μeH2πbp2ε-14p1Sp1q[2AH1+bH1-bAε-1δ-′]+1k2[μeH2πbp2ε-14p11k-1b2(1+K)(2A+b)-KAε-1b3+μeH2πbp2ε-14p1(2A+b)]}×[2AH1+bH1-bAε-1δ-]-1.
In the absence of coupling between spin and heat flux (δ-=0) and solute flux (δ-′=0), the above expression of R reduces to
(36)R={μeH2πbp2ε-14p1(2A+b)]Sp1q[(2A+b)H1]+1k2[μeH2πbp2ε-14p11k-1b2(1+K)(2A+b)-KAε-1b3+μeH2πbp2ε-14p1(2A+b)]}[(2A+b)H1]-1.
5.1. Special CasesCase 1.
In the absence of suspended particles (H1=1), (35) reduces to
(37)R={μeH2πbp2ε-14p1Sp1q[2A+b-bAε-1δ-′]+1k2[μeH2πbp2ε-14p11k-1b2(1+K)(2A+b)-KAε-1b3+μeH2πbp2ε-14p1(2A+b)]}×[2A+b-bAε-1δ-]-1,
which is in agreement with the expression of R given by V. Sharma and S. Sharma [30] in the absence of suspended particles.
Case 2.
In the absence of suspended particles and solute gradient, expression (35) reduces to
(38)R=1k2[1k-1b2(1+K)(2A+b)-KAε-1b3+μeH2πbp2ε-14p1(2A+b)]×[2A+b-bAε-1δ-]-1,
which is the same as that derived by Sharma and Kumar [16] for Rayleigh-Bénard problem.
Case 3.
In the absence of suspended particles, solute gradient, and magnetic field, (35) reduces to
(39)R=1k2{1k-1b2(1+K)(2A+b)-KAε-1b3}×[2A+b-bAε-1δ-]-1,
which coincides with the expression of Sharma and Gupta [15].
Case 4.
For S=0 and H=0, that is, in the absence of magnetic field for Rayleigh-Bénard problem, the expression of R reduces to
(40)R=1k2{1k-1b2(1+K)(2A+b)-KAε-1b3}×[2AH1+bH1-bAε-1δ-]-1,
which is in agreement with the result of Sharma and Gupta [28] in the absence of rotation.
Case 5.
For a Newtonian fluid in the absence of suspended particles and magnetic field for a nonporous medium (i.e., when δ-=δ-′=K=S=H=0, H1=1+h1=1, and k-1=1), (35) reduces to
(41)R=b3k2,
which agrees with the classical result of Chandrasekhar [10] for the relevant problem.
6. Overstability Motions
Let us write the complex quantity n as n=nr+ini, where nr,ni(∈ℜ) are the real and imaginary parts of n. Overstability motion corresponds to the case when n≠0 and nr=0 which means n=ini. Therefore, to determine the state at which the convection sets in as overstability motion, we separate the right hand side of dispersion relation (34) into real and imaginary parts by putting n=ini. Since, for overstability, we wish to determine critical Rayleigh number for the onset of overstability, it suffices to find conditions for which (34) will admit solution with real values of ni. The real and imaginary parts of (34) yield
(42)Rk2=〈b{(b2-E2H12p1qni2)(2Abp2+b2p2-l1ni2)-ni2EbH1(p1+q)(l1bp2+2A+b)(2Abp2+b2p2-l1ni2)}×{-ε-1ani2+1k-1(1+K)}-bni2{ε-1F+ak-1(1+K)}×{EbH1(p1+q)(2Abp2+b2p2-l1ni2)+(b2-E2H12p1qni2)(l1bp2+2A+b)(2Abp2+b2p2-l1ni2)}+Sp1k2q[(b2p2-EH1p1ni2)×(2AH1+bH1-al1ni2-bAε-1δ-′)-ni2(Ep1bH1p2+b)×(2aA+ab+H1l1-abAε-1δ-′)(b2p2-EH1p1ni2)(Ep1bH1p2+b)]-KAb2ε-1×[(b-EaH1p1ni2)(b2p2-EH1qni2)-ni2(ab+Ep1H1)(EqbH1p2+b)(b2p2-EH1p1ni2)]+μeH2πbε-14p1×[(b2-E2H12p1qni2)(2A+b-al1ni2)-ni2EbH1(p1+q)(2aA+ab+l1)](2Abp2+b2p2-l1ni2)〉×[(b2p2-EH1qni2)×(2AH1+bH1-al1ni2-bAε-1δ-)-ni2(EqbH1p2+b)×(2aA+ab+H1l1-abAε-1δ-)(EqbH1p2+b)]-1,(43)Rk2[b2p2(EqbH1p2+b)(2AH1+bH1-al1ni2-bAε-1δ-)+(b2p2-EH1qni2)×(2aA+ab+H1l1-abAε-1δ-)b2p2(b2p2-EH1qni2)]=b[-ni2EbH1(p1+q)(l1bp2+2A+b)}{ε-1F+ak-1(1+K)}×{(l1bp2+2A+b)(b2-E2H12p1qni2)(2Abp2+b2p2-l1ni2)-ni2EbH1(p1+q)(l1bp2+2A+b)}+{-ε-1ani2+1k-1(1+K)}×{EbH1(p1+q)(2Abp2+b2p2-l1ni2)+(b2-E2H12p1qni2)×(l1bp2+2A+b)(2Abp2+b2p2-l1ni2)}{ε-1F+ak-1(1+K)}]+Sp1qk2[(Ep1bH1p2+b)×(2AH1+bH1-al1ni2-bAε-1δ-′)+(b2p2-EH1p1ni2)×(2aA+ab+H1l1-abAε-1δ-′)(Ep1bH1p2+b)]-KAb2ε-1[(ab+Ep1H1)(b2p2-EH1qni2)+(b-EaH1p1ni2)(EqbH1p2+b)(ab+Ep1H1)(b2p2-EH1qni2)]+μeH2πε-1b4p1[EbH1(p1+q)(2A+b-al1ni2)+(b2-E2H12p1qni2)×(2aA+ab+l1)EbH1(p1+q)(2A+b-al1ni2)].
Eliminating R between (42)-(43), we get
(44)Bni8+Cni6+Dni4+Eni2+F=0,
where
(45)B=abq2E2H12l1ε-21k-1×[a(1+K)ε2EH1l1p1+k-1{aAbδ-EH1p1+εl1(ab+FEH1p1-E2H12p1)}],C=b4qk-1ε2p1p22×[+l1(-aH2πμp22+4p1(a2b2+Fp22)))〉}4(1+K)qεEH1p1×(-abq2EH1l1(aAδ-b+εl1)p22+a2b2εl12p1p22+q2εE2H13l12p1p22+aq2EH12×{a(2A+b)((2A+b)ε-Abδ-)Ep1p22+bεl12(abEp1+p22)})+k-1{×p1p22+l1(-aH2πμp22+4p1(a2b2+Fp22)))〉4a2b3qεl12p1p22+4aEH1l1p12×((b2Fq-ak2qSε+ak2p1Sε)aAb3qδ-+εl1×(b2Fq-ak2qSε+ak2p1Sε))p22-aq3εE3H14p1×(4(2A+b)2p1p22+4AbKl1p1p22+l12(4b2p1-H2πμp22))+aqEH12×(4ab(2A+b)q2×((2A+b)ε-Abδ-)Ep1p22+4AEl1p1×(-δ-(b2(-1+F)q2+ak2Sεp12)(b2Kq2+k2Sδ-′p12)+aε(b2Kq2+k2Sδ-′p12))p22+bεl12×(4ab2q2Ep1-aH2πμq2Ep22-4bp12p22))+q3E2H13p1×〈+l1(-aH2πμp22+4p1(a2b2+Fp22)))ε(p2(a2bH2πμE+(4bF-aH2πμE)p2))4a(2A+b)(2AF+bF-aAbK)Ep1p22+4aAbKEl1p1p22+l12(p2(a2bH2πμE+(4bF-aH2πμE)p2)4ab2FEp1+p2×(a2bH2πμE+(4bF-aH2πμE)p2)))+Abδ-E((a2b2+Fp22)4a(b-bF+A(2-2F+abK))×p1p22+l1×(-aH2πμp22+4p1(a2b2+Fp22)))〉}],F=b34qk-1ε2p1p22×[4(Ak2qSEδ-′-k2Sεl1)))ε(b2(2A+b)2(1+K)qεEH12-k-1((2A+b)qEH12(Ab3K+(2A+b)Sk2εH1))+Abk2Sδ-′(bH1(a(2A+b)-l1)-ab(2A+b)-(2A+b)qEH12+bH1(a(2A+b)-l1))(2A+b)qEH12(Ab3K))+Abδ-(Ak2qSEδ-′-k2Sεl1)))-b2(2A+b)(1+K)qEεH1+k-1×(-ab(2A+b)k2Sε+(2A+b)k2qSεEH12+bH1×(2aAk2Sε+abk2Sε+Ab2KqE-Ak2qSEδ-′-k2Sεl1)))p12+4k2Sk-1EH1(Abδ--(2A+b)εH1)×(Abδ-′-(2A+b)εH1)p13+bH2πqμk-1p2×(-b(a(2A+b)2ε+Aδ-(bl1-(2A+b)p2))(2A+b)2εH1(ab-p2)-b(a(2A+b)2ε+Aδ-(bl1-(2A+b)p2)))+bqp1((-a(2A+b)2ε+a(2A+b)2εH1-Abδ-l1))4b2(1+K)ε×(-a(2A+b)2ε+a(2A+b)2εH1-Abδ-l1)+k-1((2A+b)2H2+πμEH12p2)))-Ab(2A+b)δ-(4b2F+H2πμEH1p2)+ε((2A+b)2H2+πμEH12p2))4aAb3(2A+b)K+4b2H1×((2A+b)×(2AF+bF-aAbK)+AbKl1)×(2A+b)2H2+πμEH12p2)))].
The coefficients D and E being quite lengthy and not needed in the discussion of overstability have not been written here.
7. Numerical Results and Discussion
Computations are carried out using (35) for stationary convection and (42) satisfying (43) for the overstable case using the software Mathematica. This is to find out variation of the Rayleigh number R with wavenumber k for fixed values of the dimensionless parameters A=0.5, a=10, F=1.005, l=1, p1=3, p2=1, q=0.035, δ-=1, δ-′=0.02, ε=0.5, and μ=1. Let us denote Rayleigh number for stationary convection by Rs and that for overstability by Ro.
Figures 2(a)–2(d) correspond to the Rayleigh numbers Rs and Ro for H=50, H1=1.0, K=1.0, and k-1=2 and for four values of solute gradient S = 0, 50, 100, and 200, respectively. It is clear from the figures that Ro is always less than Rs; that is, in the presence of magnetic field, instability sets in as overstability for a layer of micropolar fluid in porous medium. From Table 1, it is seen that both Rs and Ro increase as S increases; therefore, it is concluded that solute gradient has a stabilizing effect on the micropolar fluid layer system.
Tabulated values of Rs and Ro for different values of S.
k
S=0
S=50
S=100
S=200
Rs
Ro
Rs
Ro
Rs
Ro
Rs
Ro
0.5
592982.0
122725.0
639898
147716.0
686813
169445.0
780645
208516.0
1.0
169909.0
33302.3
219975
54332.8
270040
72321.8
370172
106642.0
1.5
93017.5
16841.8
148333
35790.3
203649
52843.4
314280
86256.1
2.0
67824.9
11157.2
130491
28987.6
193156
45670.9
318488
78690.1
2.5
58050.7
8656.9
130166
25742.5
202282
42210.2
346514
74930.6
3.0
54742.7
7442.6
138408
23991.9
222074
40259.6
389406
72730.0
Variation of Rayleigh number R with wavenumber k for H=50, H1=1.0, K=1.0, and k-1=2 and for (a) S=0, (b) S=50, (c) S=100, and (d) S=200.
Figures 3(a)–3(d) correspond to the Rayleigh numbers Rs and Ro for S=50, H=50, H1=1.0, and K=1.0 and for four values of medium permeability k-1= 2, 5, 10, and 20, respectively. We can see that the instability sets in as overstability as Ro is always less than Rs for a particular set of values of various parameters. It is clear from Table 2 that Rs and Ro both decrease as k-1 increases, confirming the destabilizing effect of medium permeability.
Tabulated values of Rs and Ro for different values of k-1.
k
k-1=2
k-1=5
k-1=10
k-1=20
Rs
Ro
Rs
Ro
Rs
Ro
Rs
Ro
0.5
639898
147716.0
637141
146378.0
636222
145980.0
635762
145781.0
1.0
219975
54332.8
219126
54013.1
218843
53906.8
218702
53874.6
1.5
148333
35790.3
147816
35613.9
147643
35571.5
147557
35550.3
2.0
130491
28987.6
130059
28868.8
129915
28835.2
129843
28818.4
2.5
130166
25742.5
129737
25652.0
129594
25621.8
129522
25606.7
3.0
138408
23991.9
137934
23896.9
137777
23861.1
137698
23846.3
Variation of Rayleigh number R with wave number k for S=50, H=50, H1=1.0, and K=1.0 and for (a) k-1=2, (b) k-1=5, (c) k-1=10 and (d) k-1=20.
Figures 4(a)–4(d) correspond to Rayleigh numbers Rs and Ro for S=50, H=50, H1=1.0, and k-1=2 and for four values of K= 0.5, 0.75, 1.0, and 1.5, respectively. Here again, instability sets in as overstability (Ro<Rs). From Table 3, it is seen that Rs decreases as K increases showing the destabilizing effect of micropolar coefficient of coupling between vorticity and spin effects for the stationary convection. For overstable case, the stabilizing/destabilizing effect of K depends upon wave number k. The said effect is destabilizing for k≥1 and stabilizing for k=0.5.
Tabulated values of Rs and Ro for different values of K.
k
K=0.5
K=0.75
K=1.0
K=1.5
Rs
Ro
Rs
Ro
Rs
Ro
Rs
Ro
0.5
640840
147711.0
640369
147714.0
639898
147716.0
638955
147859.0
1.0
220269
54370.6
220122
54351.6
219975
54332.8
219680
54337.2
1.5
148516
35800.8
148425
35795.5
148333
35790.3
148150
35780.0
2.0
130646
28993.1
130568
28990.4
130491
28987.6
130335
28964.0
2.5
130324
25763.1
130245
25760.2
130166
25742.5
130008
25736.8
3.0
138586
23999.7
138497
23995.8
138408
23991.9
138231
23971.4
Variation of Rayleigh number R with wavenumber k for S=50, H=50, H1=1.0, and k-1=2 and for (a) K=0.5, (b) K=0.75, (c) K=1.0, and (d) K=1.5.
Figures 5(a)–5(d) correspond to the Rayleigh numbers Rs and Ro for S=50, H=50, K=1.0, and k-1=2 and for four values of suspended particles factor H1= 1.0, 1.2, 1.5, and 2.0, respectively. Very interestingly, it can be seen from the figures that, as the value of H1 increases, the mode of instability changes from overstability to stationary convection. When H1=1.0 and H1=1.2, instability sets in as overstability, as R for overstability is lower than that for stationary convection (Figures 5(a) and 5(b)). For values of H1=1.5 and H1=2.0, instability sets in as stationary convection since R is lower for stationary convection than for overstability. Thus, as H1 increases, the mode of instability shifts from overstability to stationary convection. Further, it is clear from Table 4 that the effect of suspended particles is destabilizing when the instability sets in as stationary convection, and the effect is stabilizing when the instability sets in through overstability.
Tabulated values of Rs and Ro for different values of H1.
k
H1=1.0
H1=1.2
H1=1.5
H1=2.0
Rs
Ro
Rs
Ro
Rs
Ro
Rs
Ro
0.5
639898
147716.0
201070
181832.0
100957.0
235185.0
56599.5
332323.0
1.0
219975
54332.8
68099.0
65503.4
35320.1
82922.2
21005.8
114850.0
1.5
148333
35790.3
43968.2
42587.4
23301.9
53194.4
14465.7
72672.0
2.0
130491
28987.6
35995.5
34231.2
19221.2
42353.0
12223.1
57141.7
2.5
130166
25742.5
32701.2
30237.6
17432.5
37180.0
11221.3
49675.5
3.0
138408
23991.9
31218
28050.8
16534.4
34318.7
10703.1
45489.7
Variation of Rayleigh number R with wavenumber k for S=50, H=50, K=1.0, and k-1=2 and for (a) H1=1.0, (b) H1=1.2, (c) H1=1.5, and (d) H1=2.0.
Figures 6(a)–6(d) correspond to Rayleigh numbers Rs and Ro for S=50, H1=1.0, K=1.0, and k-1=2 and for four values of magnetic field H= 50, 100, 200, and 500, respectively. The Rayleigh number for stationary convection Rs is greater than the Rayleigh number for overstability Ro establishing the onset of instability as overstability in the presence of magnetic field. Further from Table 5, one can see that both Rs and Ro increase with the increase in magnetic field parameter H, confirming the stabilizing effect of magnetic field on the system.
Tabulated values of Rs and Ro for different values of H.
k
H=50
H=100
H=200
H=500
Rs
Ro
Rs
Ro
Rs
Ro
Rs
Ro
0.5
639898
147716.0
2417610
498969.0
9528440
1891740
59304300
11600000
1.0
219975
54332.8
729344
157840.0
2766820
566224
17029200
3400000
1.5
148333
35790.3
427189
89162.9
1542610
299001
9350570
1760000
2.0
130491
28987.6
333820
63801.9
1147140
201019
6840370
1150000
2.5
130166
25742.5
304193
51632.1
1000300
153609
5873060
860962
3.0
138408
23991.9
302517
44867.4
958953
127186
5554000
699101
Variation of Rayleigh number R with wavenumber k for S=50, H1=1.0, K=1.0, and k-1=2 and for (a) H=50, (b) H=100, (c) H=200, and (d) H=500.
8. Conclusion
Thermosolutal convection of a dusty micropolar fluid layer in the presence of magnetic field saturating a porous medium has been analyzed. The effects of magnetic field, salinity gradient, suspended particles, permeability, and micropolar coefficient of coupling between vorticity and spin effects on Rayleigh number have been studied. It is concluded that the following hold.
The Rayleigh number for overstability is always less than the Rayleigh number for stationary convection except for high values of suspended particles factor H1.
Increase in solute gradient and magnetic field results in an increase in Rayleigh number for both stationary convection as well as overstability establishing the fact that solute gradient and magnetic field have stabilizing effect on the micropolar fluid layer system.
For stationary convection, increase in medium permeability and suspended particles factor results in the decrease in Rayleigh number which shows the destabilizing influence of permeability and suspended particles. For overstability, Rayleigh number R decreases with increase in permeability and increases with an increase in suspended particles factor.
The effect of micropolar coefficient of coupling between vorticity and spin effects is destabilizing for stationary convection while, for overstable case, its stabilizing/destabilizing effect depends upon wavenumber k.
As the value of suspended particles factor H1 increases, the mode of instability changes from overstability to stationary convection.
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