International Journal of Applied Engineering Research ISSN 0973-4562 Volume 13, Number 17 (2018) pp. 13022-13030 © Research India Publications. http://www.ripublication.com Stability of Thermosolutal Free on Electrically Conducting Superposed Fluid and Porous Layer

K.Sumathi1*, T.Arunachalam2 and S.Aiswarya3 Department of Mathematics, 1,3PSGR Krishnammal College for Women, Coimbatore – 641 004, TamilNadu, India. 2Kumaraguru College of Technology, Coimbatore – 641 049, TamilNadu, India. (*Corresponding Author)

Abstract investigated the onset of convection in a horizontal sparsely packed porous medium with imposed thermal and The effect of thermosolutal convection on linear stability of an concentration gradient at both the boundaries and found that incompressible fluid in a horizontal fluid/porous interface in Darcy number destabilizes the system in case of stationary the presence of a vertical magnetic field has been analyzed. and oscillatory convection. Asymptotic solution is obtained for rigid boundaries using normal mode analysis and the analysis is restricted to long Jena et al. [6] by employing Darcy-Brinkman-Forchheimer wave approximations. The influence of various non- model studied about the linear stability of onset of dimensional parameters such as , thermosolutal convection in a rectangular annulus which is magnetic , , Prandtl number, highly heated and soluted from within and exposed to low porosity, wave number and depth ratio on stability thermal and solutal gradients from outside in a non-Darcian characteristics of flow field are represented numerically. region. Abdullah et al. [2] studied the effect of thermosolutal convection of a viscous incompressible fluid in the presence Keywords: Thermosolutal convection, magnetic field, fluid- of rotation on the stability of the system and observed that porous interface, stability analysis. solute concentration and rotation stabilizes the system.

Hirata et al. [4] using linear stability analysis studied the INTRODUCTION thermal convection of a viscous incompressible fluid applying The combined effect of thermosolutal convection occurs in normal mode analysis. These earlier works concentrated on many practical situations. Nield [8] analyzed the linear the marginal stability of thermosolutal convection under stability of thermohaline convection induced by thermal and different physical conditions and very less emphasis has been concentration gradients in a horizontal saturated porous given to analyze the linear stability of thermosolutal medium by applying Fourier series. The effect of chemical convection. Hence in this paper, the work of Hirata et al. [4] reaction on stability of a binary mixture in a porous medium has been extended to study the linear stability of thermosolutal heated from below or above has been studied by Steinberg [9]. convection on an incompressible viscous fluid in the presence of vertical magnetic field using method of small oscillations Gobin et al. [3], Zhao and Chen [12], Hirata et al.[5] analyzed and the analysis is restricted to long wave approximations. the combined effect of thermal and solutal gradients on stability of fluid layer overlying a porous layer which is of great importance in thermal insulation, drying processes, MATHEMATICAL FORMULATION ground water analysis etc. Consider a viscous incompressible electrically conducting Murthy and Lee [7] investigated the onset of convection due fluid which is bounded by a horizontal infinite parallel plate. to temperature and solute concentration differences at the The porous medium is assumed to be saturated, isotropic and walls using linear stability analysis. The effect of magnetic homogeneous governed by Darcy – Brinkman model. The field on onset of thermosolutal convection in a viscous impermeable horizontal walls are maintained at different incompressible fluid which is heated and soluted at both the temperatures and concentrations such that 푇푢, 푆푢 at the upper walls has been examined by Abdullah [1] and he found that walls and 푇푙, 푆푙 at the lower walls. It is assumed that the fluid variable has no effect on stationary obeys Boussinesq approximation and variations of density due convection but induces overstability of the system by applying to temperature and solute concentration is assumed to be Chebyshev polynomials. 𝜌(푇, 퐶) = [1 − 훽푇(푇 − 푇0) − 훽퐶(퐶 − 퐶0)] where thermal Pritchard and Richardson [10] carried out a numerical study to expansion coefficient 훽푇 ≥ 0 and solute expansion coefficient analyze the effect of temperature solvability on stability of 훽퐶 ≤ 0. thermosolutal convection in a saturated horizontal porous layer. Employing Brinkman model, Wang and Tan [11]

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∂T′ + (u⃗ ′. ∇T∗) = ∇. (α∇T′) (11) ∂t ∂C′ ϕ + (u⃗ ′. ∇C∗)= 퐷 ∇. (ϕ∇C′) (12) ∂t 푓 ∇. h⃗ = 0 (13)

∂h⃗⃗ ϕ = ∇ × (u⃗ ′ × H⃗⃗ ∗) + ϕη∇2h⃗ (14) ∂t By eliminating pressure term and introducing the non- dimensional variables for length, velocity, time, temperature, concentration and magnetic field respectively as follows Figure 1. Physical Description of the Flow ν d2 z = z∗d, w = w∗ , t = t∗ , T′ = ΔT T∗, Under the flow assumptions the governing equations takes the d ν form H νh ∗ C′ = ΔC C∗, h = 0 z ∇. u⃗ = 0 (1) z η

∂ u⃗ 1 ∇u⃗ μ μeff 2 μe The system of linearized perturbed equations becomes ρ0 [ ( ) + (u⃗ . )]= −∇p − u⃗ + gρ + ∇ u⃗ + (∇ × H⃗⃗ ) × H⃗⃗ (2) ∂t ϕ ϕ ϕ k μf 4π ∂ 1 1 ∂2T ∂2C 1 ∂ ∂2h ( ∇2w) = − ∇2w + Gr + Gr + ∇4w + Q ( z) (15) ∂t ϕ Da T ∂x2 C ∂x2 ϕ ∂z ∂x2 ∂T⃗⃗ + (u⃗ . ∇⃗T⃗ )= ∇. (α∇T⃗⃗ ) (3) ∂t ∂T d̂ + w= ∇2T (16) ∂t P ∂C⃗⃗ r ϕ + (u⃗ . ∇C⃗ ) = 퐷 ∇. (ϕ∇C⃗ ) (4) ∂t 푓 ∂C ϕ ϕ + w= ∇2C (17) ∂t Sc ∇. H⃗⃗ = 0 (5) ∂hz 1 ∂w 2 pm = + ∇ hz (18) ∂H⃗⃗ ∂t ϕ ∂z ϕ = ∇ × (u⃗ × H⃗⃗ ) + ϕη∇2H⃗⃗ (6) ∂t where Da = k⁄d2 (Darcy number), with boundary conditions 3 2 GrT = gβT𝜌0∆Td ⁄ν (Thermal ) ∂u⃗⃗ 3 2 u⃗ = = ⃗T⃗ = C⃗ = H⃗⃗ = 0 at GrC = gβC𝜌0∆Cd ⁄ν (Mass Grashof number), Pr = ν⁄αf ∂z 휈 μ H 2d2 (Prandtl number), 푆푐 = (Schmidt Number), Q = e 0 푧 = 0 and 푧 = 푑 (7) 퐷푓 4πρ0ην (Chandrasekhar number), pm = ν⁄η (Magnetic Prandtl where 푢⃗ , ρ, p, g , T, C, H⃗⃗ (0,0, H0), ϕ, μe, η, ρ0, μeff, μf, α number) respectively denote the velocity vector, density, pressure, acceleration due to gravity, temperature, concentration, Also the special variations of α, ϕ and 푘 are taken as null. constant vertical magnetic field, porosity, magnetic Applying normal mode analysis to the dependent variables permeability, electrical resistivity, density at reference level, (ikx+σt) effective viscosity of the porous medium, dynamic viscosity (w, T, C, hz) = (W(z), θ(z), S(z), H(z))e of the fluid and thermal diffusivity. where 푘 is the nondimensional wave number and σ represents In quiescent state, basic state flow field are given by u⃗ ∗ = the growth rate. Substituting the above expression into ∗ ∗ ∗ ∗ equations (15) – (18) we get (0, 0, 0), P = P(z), H⃗⃗ = (0,0, H0), T = T(z) and C = C(z)and so the temperature field and concentration field using ∂2 ∂2 ϕ ∂ ( − k2) ( − k2 − − σ) W = k2ϕGr θ + k2ϕGr S + k2Qϕ H (19) the boundary conditions become ∂z2 ∂z2 Da T C ∂z

Tl−T0 Cl−C0 ∂2 pr pr T = z + and C = z + (8) ( − k2 − σ ) θ = W (20) Tu−Tl Cu−Cl ∂z2 d̂ d̂ Let the small disturbance in the initial states of velocity, ∂2 Sc temperature, concentration, pressure and magnetic field ( − k2 − σSc) S = W (21) ∂z2 휙 respectively be denoted by u⃗ ′(x, z, t), T′(x, z, t), ′ ′ C (x, z, t), P (x, z, t) and h⃗ (h , 0, h ), then the linearized ∂2 1 ∂ x z ( − k2 − σ p ) H =− W (22) perturbed equation (1) – (6) becomes ∂z2 m ϕ ∂z ′ ∇. u⃗ = 0 (9) with the corresponding boundary condition ′ ∂ u⃗ 1 ′ ν ′ ′ 1 2 ′ μe ∗ ( )= − ∇p − u⃗ − gβTT k̂ + ∇ u⃗ + (∇ × h⃗ ) × H⃗⃗ (10) ∂W ∂t ϕ ρ0 k ϕ 4πρ0 W = = θ = S = H = 0 at z = 0 and z = 1 (23) ∂z

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EIGEN VALUES AND EIGEN FUNCTIONS W1 = A15 + A16z + A17 cosh(rz)

Now we expand W, σ, θ , S and H in powers of k +A18 sinh(rz) + A19z sinh(rz) 2 4 W = W0 + k W1 + k W2 + ⋯ +A20z cosh (rz)+C24 cosh(r1z) 2 4 3 σ = σ0 + k σ1 + k σ2 + ⋯ +C25 sinh(r1z) + C26 + C27z + C28 cosh(r2z) +C29 cosh(r2z) 2 4 + C30 sinh(r3z) + C31 cosh(r3z) θ = θ0 + 푘 θ1 + 푘 θ2 + ⋯ (24) 2 4 S = S0 + k S1 + k S2 + ⋯ 2 4 θ1 = A21 cosh(r1z) + A22 sinh(r1z) + C53 H = H0 + k H1 + k H2 + ⋯ Substituting (24) in equations (19) to (23) and collecting the +C54z + C55cosh(rz) + +C56 sinh(rz) like powers of k we get +C57 z sinh(rz) + C58z cosh(rz) ∂2 ∂2 ϕ ( − − σ ) W = 0 (25) +C59 z sinh(r1z) + C60z cosh ( r1z) ∂z2 ∂z2 Da 0 0 3 ( ) ∂2 Pr Pr +C61z + C62 cosh r2z + C63 sinh ( r2z) ( − σ ) θ = W (26) ∂z2 0 d̂ 0 d̂ 0 +C64 sinh(r3z) + C65 cosh(r3z) ∂2 Sc ( − σ Sc) S = W (27) ∂z2 0 0 ϕ 0

∂2 1 S = A cosh(r z) + A sinh(r z) + C ( − σ p ) H = − DW (28) 1 23 2 24 2 81 ∂z2 0 m 0 ϕ 0 +C82z + C83 cosh(rz) + +C84 sinh(rz) ∂2 ∂2 ϕ ∂2 ∂2 ϕ ( − − σ ) W =(1 + σ ) W + ( − − σ ) W + ∂z2 ∂z2 Da 0 1 1 ∂z2 0 ∂z2 Da 0 0 +C85 z sinh(rz) + C86z cosh(rz) ϕGr θ + ϕGr S + QϕDH T 0 C 0 0 3 +C87 cosh(r1z) + C88 sinh ( r1z) + C89z (29) +C90 z sinh(r2z) + C91 z cosh ( r2z ∂2 Pr Pr Pr ( − σ0 ) θ1 = W1 + (1 + σ1 ) θ0 (30) ∂z2 d̂ d̂ d̂ +C92 sinh(r3z) + C93 cosh(r3z) ∂2 Sc ( − σ Sc) S = W + (1 + σ Sc)S (31) ∂z2 0 1 ϕ 1 1 0 H1 = A25 cosh(r3z) + A26 sinh(r3z) + C107 ∂2 1 ( 2 − σ0 pm) H1=− DW1 + (1 + σ1 pm)H0 (32) ∂z ϕ +C108 sinh(rz) + C109 cosh(rz) The corresponding boundary conditions are +C110z cosh(rz) + C111 z sinh ( r z)

W0 = DW0 = θ0 = S0 = H0 = 0 +C112 sinh(r1z) + C113 cosh(r1z) at z = 0 and z = 1 2 +C114 z + C115 sinh(r2z) + C116 cosh(r2z) (33) +C117z sinh(r3z) + C118z cosh(r3z) W1 = DW1 = θ1 = S1 = H1 = 0 The zeroth order eigen values are given by the following at z = 0 and z = 1 transcendental equation On solving equations using boundary conditions we get A3[cosh(r) − 1] + A4[sinh(r) − r] = 0

W0 = A1 + A2z + A3 cosh(rz) + A4sinh (rz) A3sinh (r) + A4[cosh(r) − 1] = 0 The solution of above expression will not give explicit values of 𝜎 . Hence the values of 𝜎 are obtained using Mathematica θ0 = A5 cosh(r1z) + A6 sinh(r1z) + A7 + A8푧 0 0 8.0. + A9 cosh(rz) + A10sinh (rz) The first order approximations of the growth rate is given by

퐶37 S0 = A11 cosh(r2z) + A12 sinh(r2z) + C2 𝜎1 = − 퐶36 +C3z + C4 cosh(rz) + C5 sinh(rz) For brevity constants are given in appendix.

H0 = A13 cosh(r3z) + A14 sinh(r3z) + C7 RESULTS AND DISCUSSION To get physical insight into linear stability of thermosolutal +C8 sinh(rz) + C9 cosh(rz) convection of an electrically conducting viscous fluid bounded by saturated porous medium in presence of vertical

13024 International Journal of Applied Engineering Research ISSN 0973-4562 Volume 13, Number 17 (2018) pp. 13022-13030 © Research India Publications. http://www.ripublication.com magnetic field, the effects of various non-dimensional number decreases the magnetic field intensity while increase parameters such as Chandrasekhar number Q, magnetic in magnetic Prandtl number increases the magnetic field Prandtl number pm, Schmidt number Sc, Prandtl number Pr, effect. Darcy number Da, depth ratio d and porosity ϕ on temporal growth rate, velocity, temperature, concentration and CONCLUSION magnetic field has been discussed numerically and plotted in figures (2) – (21). We have fixed the values of parameters as We have investigated stability of thermosolutal convection in Da = 0.2, ϕ = 0.6, Pr = 0.71, d = 0.08, Sc = 0.66, an electrically conducting viscous fluid under influence of GrT = 5.0, GrC = −5.0, 푞 = 50.0, 푝푚 = 5.0, vertical magnetic field confined between infinite horizontal k = 0.9 throughout the entire study of the problem. plates using method of small oscillation and the effects of various non-dimensional parameters on characteristics of the Figures (2) – (3) illustrate the effect of Chandrasekhar number flow has been analysed. The following interpretations were Q on growth rate and it is found that increase in made from the findings. Chandrasekhar number stabilizes the system for small Darcy numbers and induces instability for large Darcy numbers.

The influence of porosity ϕ, magnetic Prandtl number 푝푚 and  Increase in Chandrasekhar number stabilizes the Schmidt number Sc on growth rate is depicted in figures (4) – system for small Darcy numbers and induces (6). It is observed that increase in porosity creates instability for large Darcy numbers. stability/instability in the system, while magnetic Prandtl  Increase in Darcy number and Chandrasekhar number increases the growth rate thereby inducing instability. number destabilizes the system as porosity increases. Effect of Schmidt number destabilizes the system but it has  It is shown that increase in Chandrasekhar number less significant on temporal growth. reduces the velocity flow field and that increase in magnetic Prandtl number enhances the flow field. Figures (7) – (10) show the influence of wave number k,  It is found that increase in Chandrasekhar number magnetic Prandtl number pm, Chandrasekhar number Q and and Prandtl number decreases the temperature profile Darcy number Da with respect to porosity ϕ on frequency. It near the lower plate and increases as we move depicts that with increase in wave number system becomes towards the upper wall. unstable near the lower plate and stabilizes the system as we  It may be inferred that concentration decreases with moves towards the upper plate, whereas increase in magnetic increase in Chandrasekhar number and Schmidt Prandtl number and Darcy number increases instability. number increases the concentration profile. Increase in Chandrasekhar number destabilizes the system for  Increase in magnetic Prandtl number increases the small values for Darcy number. magnetic field effect. Increase in Chandrasekhar number Q and magnetic Prandtl number pm destabilizes the system with increase in Darcy number as shown in figures (11) and (12). -44.9 Figure (13) represent the effect of wave number k on -45 0 0.20.40.60.8 1 frequency parameter and it is found that increase in wave -45.1 number decreases the growth rate thereby creating stability in the system. -45.2 Q=50.0 The influence of Chandrasekhar number Q and magnetic -45.3 Q=100.0 Prandtl number pm on velocity field is depicted in figures (14) σ -45.4 – (15). It is shown that increase in Chandrasekhar number -45.5 Q=150.0 reduces the velocity flow field and that increase in magnetic -45.6 Prandtl number enhances the flow field. Q=200.0 -45.7 Figures (16) – (17) illustrate the variations of temperature field due to variation in Chandrasekhar number Q and Prandtl -45.8 number Pr. It is found that increase in Chandrasekhar number -45.9 and Prandtl number decreases the temperature profile near the k lower plate and increases as we move towards the upper wall. Figure 2. Effect of Chandrasekhar number on The effect of Chandrasekhar number Q and Schmidt number growth rate for small Darcy number Sc on concentration profile is plotted in figures (18) – (19). It may be inferred that increase in Chandrasekhar number decreases concentration profile whereas Schmidt number increases the concentration profile. In figures (20) – (21), the variations of Chandrasekhar number Q and magnetic Prandtl number pm on magnetic field is depicted and it is observed that increase in Chandrasekhar

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14 14 12 12 10 10 8 Q=50.0 8 Sc=0.5 σ σ 6 Q=100.0 6 Sc=0.6 4 Q=150.0 4 Sc=0.63 2 Q=200.0 2 Sc=0.66 0 0 0 0.20.40.60.8 1 0 0.2 0.4 0.6 0.8 1 k k

Figure 3. Effect of Chandrasekhar number on growth rate Figure 6. Effect of Schmidt number on growth rate for large Darcy number

40 15 20

10 0 0 0.2 0.4 0.6 0.8 1 k=1.0 -20 5 휙=0.5 σ k=2.0 휙=1.0 -40 σ k=3.0 0 휙=1.5 -60 k=4.0 0 0.2 0.4 0.6 0.8 1 휙=2.0 -5 -80 -100 -10 Φ k Figure 7. Effect of wave number on growth rate Figure 4. Effect of porosity on growth rate

14 15 12 10 10 5 8 pm=2.5 0 pm=1.0 6 pm=3.0 0 0.20.40.60.8 1 σ σ -5 pm=2.0 4 pm=3.5 -10 pm=3.0 2 pm=4.0 -15 pm=4.0 0 -20 0 0.2 0.4 0.6 0.8 1 -2 -25 Φ k Figure 8. Effect of magneticPrandtl number on growth rate Figure5. Effect of magnetic Prandtl number on growth rate

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16 20

14 10 12 0 10 Q=50.0 0 0.2 0.4 0.6 0.8 1 pm=2.5 σ 8 -10 Q=100.0 σ pm=3.0 6 -20 pm=3.5 4 Q=150.0 -30 pm=4.0 2 Q=200.0 -40 0 0 0.2 0.4 0.6 0.8 1 -50 Da Φ

Figure 12. Effect of magnetic Prandtl number on growth Figure 9. Effect of Chandrasekhar number on growth rate rate

16

14 0 12 0 0.2 0.4 0.6 0.8 1 -5 10 Da=0.2 σ 8 -10 Da=0.3 k=0.5 6 -15 Da=0.4 4 σ k=1.0 -20 2 Da-0.5 k=1.5 0 -25 k=2.0 0 0.2 0.4 0.6 0.8 1 -30 Φ -35 d Figure 10. Effect of Darcy number on growth rate

Figure 13. Effect of wave number on growth rate 20

10 0.05 0 0 0 0.2 0.4 0.6 0.8 1 Q=50.0 0 0.20.40.60.8 1 -10 -0.05 σ Q=100.0 -20 -0.1 Q=50.0 Q=150.0 W -0.15 Q=100.0 -30 Q=200.0 -0.2 Q=150.0 -40 -0.25 Q=200.0 -50 Da -0.3 -0.35 Figure 11. Effect of Chandrasekhar number on growth rate z

Figure 14. Effect of Chandrasekhar number on velocity field

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0.1 0.05 0 0 0.20.40.60.8 1 -0.05 0 0.20.40.60.8 1 -0.1 -0.2 pm=2.0 -0.15 Q=50.0

W -0.3 pm=2.5 C -0.25 Q=100.0 -0.4 pm=3.0 Q=150.0 -0.35 -0.5 pm=3.5 Q=200.0 -0.45 -0.6 -0.7 -0.55 z z

Figure 15. Effect of magnetic Prandtl number on velocity Figure 18. Effect of Chandrasekhar number on concentration field field

14 0.02

12 0 10 0 0.20.40.60.8 1 -0.02 8 Q=50.0 Sc=0.5 6 -0.04 θ Q=100.0 C Sc=0.6 4 -0.06 Q=150.0 Sc=0.63 2 Q=200.0 -0.08 Sc=0.66 0 -2 0 0.2 0.4 0.6 0.8 1 -0.1 -4 -0.12 z z

Figure 16. Effect of Chandrasekhar number on temperature Figure 19. Effect of Schmidt number on concentration field field

1 12 0 10 -1 0 0.2 0.4 0.6 0.8 1 8 -2 Q=50.0 6 Pr=0.71 -3 4 H Q=100.0 θ Pr=1.0 -4 2 Q=150.0 Pr=1.5 -5 0 Q=200.0 Pr=2.0 -6 -2 0 0.2 0.4 0.6 0.8 1 -7 -4 -8 -6 z z Figure 20. Effect of Chandrasekhar number on magnetic Figure 17. Effect of Prandtl number on temperature field field

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medium”, The Journal of Chemical Physics, vol.78, pp. 4 2655, 1983. 2 [10] D. Pritchard, and C. N. Richardson, “The effect of 0 temperature - dependent solubility on the onset of -2 0 0.2 0.4 0.6 0.8 1 thermosolutal convection in a horizontal porous layer”, pm=2.0 J. Fluid Mech., vol.57, no.1, pp.59–95, 2007. -4 H pm=2.5 [11] S. Wang and W. Tan, “The onset of Darcy-Brinkman -6 thermosolutal convection in a horizontal porous media, pm=3.0 -8 Physics Letters A”, vol. 373, no.7, pp. 776–780, 2009. pm=3.5 -10 [12] P. Zhao and C. F. Chen, “Stability analysis of double diffusive convection in superposed fluid and porous -12 layers using a one equation model”, Int. J. heat and -14 mass transfer, vol. 44, no.24, pp. 4625-4633,2001. z

Figure 21. Effect of magnetic Prandtl number on magnetic field APPENDIX

REFERENCES [1] A. Abdullah, “Thermosolutal Convection in a Non- √ϕ r = ⁄Da + σ0; A1 = 1 ; Linear Magnetic Fluid”, International Journal of −r (cosh(r)−1) Thermal Sciences, vol.39, no.2, pp. 273-284, 2000. A = −1 ; A = ; 3 2 (sinh(r)−r) [2] A. Abdullah, S.D. Ahmari, and A.J. Chamkha, cosh(r)−1 σ Pr “Thermosolutal instability in a horizontal fluid layer A = ; r = √ 0 ⁄ ; affected by rotation”, Int. J. Mathematics Trends and 4 (sinh(r)−r) 1 d Technology, vol. 20, no.1, pp. 55, 2015. A Pr A = −(A + A ) ; A = − ( 1 ) ; 5 7 9 7 r2 d [3] D. Gobin, B. Goyeau and J.P. Songbe, “Double 1 diffusive natural convection in a composite fluid −1 A6 = ( ) (A5 cosh(r1) + A7 + A8 + A9 cosh(r) + porous layer”, Transactions of the ASME, vol.120, sinh(r1) no.1, pp.234-242, 1998. A10 sinh(r)); A Pr [4] S. C. Hirata, B. Goyeau, D. Gobin, and R.M. r = √σ sc ; A = − ( 2 ); 2 0 8 r2 d Cotta,“Stability of Natural Convection in Superposed 1 Fluid and Porous Layers Using Integral Transforms”, A3 Pr A4 Pr A9 = 2 2 ( ) ; A10 = 2 2 ( ) ; Numerical , Part B, vol.50, pp. 409 – (r −r1) d (r −r1) d 424,2006. sc −C1A1 C1A2 C1A3 C1A4 C1 = ; C2 = 2 ; C3 = − 2 C4 = 2 2 ; C5 = 2 2 ; A11 = ∅ r2 r2 r −r2 r −r2 [5] S. C. Hirata, B. Goyeau and D. Gobin, “Stability of −(C2 + C4); thermosolutal natural convection in superposed fluid −1 and porous layer”, Transport porous Med., vol.78, pp. A12 = ( ) (A11 cosh(r2) + C2 + C3 + C5sinh (r) + 525-536, 2009. sinh(r2) C4 cosh(r)); [6] S. K. Jena, S. K. Mahapatra, and A. Sarkar, 1 r = √σ p ; C = − ; “Thermosolutal Convection in a Rectangular 3 0 m 6 ϕ Concentric Annulus: A Comprehensive Study”, C A C rA Transport in Porous Media, vol.98, no. 1, pp. 103 – C = − ( 6 2); C = ( 6 3 ); 7 r2 8 (r2−r2) 124, 2013. 3 3 C6rA4 C9 = ( 2 2 ); A13 = −(C7 + C9); [7] J. Murthy, and P. Lee, “Thermosolutal Convection in a (r −r3) Floating Zone: The Case of an Unstable Solute −1 Gradient”, International Journal of Heat and Mass A14 = {A13 cosh(r3) + C7 + C8 sinh(r) + C9cosh (r)} Transfer, vol.31, no.9, pp. 1923 – 1932, 1988. sinh (r3)

[8] D. A. Nield, “Onset of thermohaline convection in a C10 = GrTϕ; C11 = Greϕ; C12 = Qϕ ; porous medium”, Water Resource Research, vol.4, C = r2A + C A + C C + C rC no.3, pp. 553 -560, 1968. 13 3 10 9 11 4 12 8 2 [9] V. Steinberg, “Convective instabilities of binary C14 = r A4 + C10A10 + C11C5 + C12rC9; C15 = C10A5 ; C16 = mixtures with fast chemical reaction in a porous C10A5 ;

13029 International Journal of Applied Engineering Research ISSN 0973-4562 Volume 13, Number 17 (2018) pp. 13022-13030 © Research India Publications. http://www.ripublication.com

C = C A + C C ; C = C A + C C ; C = C A ; −C48 C49 C50 17 10 7 11 2 18 10 8 11 3 19 11 11 C61 = 2 ; C62 = 2 2 ; C63 = 2 2 ; r1 r2−r1 r2−r1 C20 = C11A12 ; C C C = 51 ; C = 52 ; 1 C15 64 2 2 65 2 2 r3−r1 r3−r1 C21 = C12r3A13 ; C22 = C12r3A14 ; C23 = 3 ; C24 = 2 2 2 ; 2r r1(r1−r ) C 16 A21 = −(C53 + C55 + C62 + C65) ; C25 = 2 2 2 r1(r1−r ) −1 { ( ) ( ) −C17 −C18 C16 C20 A22 = A21 cosh r1 + C53 + C54 + C55 cosh r + sinh (r1) C26 = 2 ; C27 = 2 ; C28 = 2 2 2 ; C29 = 2 2 2 ; C30 = r 6r r2(r2−r ) r2(r2−r ) C56 sinh(r) + C57 sinh(r) + C58 cosh(r) + C59 sinh(r1) + C21 2 2 2 ; C cosh(r ) + C + C cosh(r ) + C sinh(r ) + r3(r3−r ) 60 1 61 62 2 63 2 C64 sinh(r3) + C65cosh (r3)} ; C22 2 2 C31 = 2 2 ; C32 = r A4C23 − r A3C23 sinh(r) − r (r −r2) Sc 3 3 C = ; C = (1 + σ Sc) ; 2 1 ∅ 67 1 r A4C23cosh (r) ; C = C (A + C ) + C C ; C33 = {−C13C23 sinh(r) + C14C23(1 − cosh(r)) + C24[1 − 68 1 15 26 67 2 cosh (r1)]+C25[r1 − sinh (r1)] − C27 + C28[1 − cosh (r2)] + C69 = C1A16 + C67C3 ; C70 = C1A17 + C67C4 ; C71 = C1A18 + C29[r2 − sinh (r2)] + C30[r3 − sinh (r3)] + C31[1 − cosh (r3)]} ; C67C5;C72 = C1A19 ; C = r2A C − r2A C [rcosh(r) + sinh (r)] − 34 4 23 3 23 C = C A ; C = C C ; C = C C ; C = C C ;C = 2 73 1 20 74 1 24 75 1 25 76 1 26 77 r A4C23[rsinh(r) + cosh (r)] ; C1C28 + C67A11 ; C = {C C [1 − rsinh(r) − cosh (r)] − r C sinh(r ) + 35 14 23 1 24 1 C68 C78 = C1C29 + C67A12 ; C79 = C1C30 ; C80 = C1C31;C81 = − 2 ; r1C25[1 − cosh (r1)] − C13C23[rcosh(r) + sinh (r)] − 3C27 − r2 C C 2rC r2C28 sinh(r2) + r2C29[1 − cosh (r2)] + r3C30[1 − cosh (r3)] − 69 70 72 C82 = − 2 ; C83 = 2 2 − 2 2 2 ; r2 r −r2 (r −r2) r3C31sinh (r3)} ; C71 2rC73 C84 = 2 2 − 2 2 2 ; C36 = C32r sinh(r) − C34(cosh (r) − 1); r −r2 (r −r2)

C = C r sinh(r) − C (cosh (r) − 1); C72 C73 C74 C75 37 33 35 C85 = 2 2 ; C86 = 2 2 ; C87 = 2 2 ; C88 = 2 2 ; C89 = r −r2 r −r2 r1−r2 r1−r2

σ1C32+C33−(cosh(r)−1) C76 C77 C78 C79 C80 A17 = 1; A18 = − 2 ; C90 = 2 ; C91 = 2 ; C92 = 2 2 ; C93 = 2 2 ; sinh(r)−r r2 2r2 2r2 r3−r2 r3−r2

2 A19 = (σ1r A3 + C13)C23 ; A23 = −(C81 + C83 + C87 + C93) ;

2 A20 = (σ1r A4 + C14)C23 ;

Pr Pr C = (1 + σ ) ; C = ; 38 d 1 39 d

C40 = [C39(A15 + C26) + C38A7];

C41 = [C39A16 + C38A8] ;

C42 = [C39A17 + C38A9] ;

C43 = [C39A18 + C38A10] ;

C = C A ; C = C A ; 44 39 19 45 39 20 C46 = C39C24 + C38A5 ;

C47 = C39C25 + C38A6 ;

C48 = C39C27 ;

C49 = C39C28 ; C50 = C39C29 ;

C40 C51 = C39C30 ; C52 = C39C31 ;C53 = − 2 ; r1

−C41 6C48 C12 2rC44 C54 = 2 − 4 ; C55 = 2 2 − 2 2 2 ; r1 r1 r −r1 (r −r1)

C43 2rC45 C44 C56 = 2 2 − 2 2 2 ; C57 = 2 2 ; r −r1 (r −r1) r −r1

C45 C46 C47 C58 = 2 2 ; C59 = ; C60 = ; r −r1 2r1 2r1

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