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Stability of Thermosolutal Free Convection on Electrically Conducting Superposed Fluid and Porous Layer

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Stability of Thermosolutal Free Convection on Electrically Conducting Superposed Fluid and Porous Layer 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 Convection 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 Chandrasekhar number, thermosolutal convection in a rectangular annulus which is magnetic Prandtl number, Schmidt number, 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 magnetic Prandtl number 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. (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] 13022 International Journal of Applied Engineering Research ISSN 0973-4562 Volume 13, Number 17 (2018) pp. 13022-13030 © Research India Publications. http://www.ripublication.com ∂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 ρ [ ( ) + (u⃗ . )]= −∇p − u⃗ + gρ + ∇ u⃗ + (∇ × H⃗⃗ ) × H⃗⃗ (2) 0 ∂t ϕ ϕ ϕ k μ 4π f 2 2 2 ∂ 1 2 1 2 ∂ T ∂ C 1 4 ∂ ∂ hz ( ∇ w) = − ∇ w + Gr + Gr + ∇ w + Q ( ) (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 = gβT0∆Td ⁄ν (Thermal Grashof number) ∂u⃗⃗ 3 2 u⃗ = = ⃗T⃗ = C⃗ = H⃗⃗ = 0 at GrC = gβC0∆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 13023 International Journal of Applied Engineering Research ISSN 0973-4562 Volume 13, Number 17 (2018) pp. 13022-13030 © Research India Publications. http://www.ripublication.com 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 .
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