Nonlinear Stability of Double-diffusive Convection in a Porous Layer with Throughflow and Concentration based Internal Heat Source

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1 Transp Porous Med (2016) 111: DOI /s x Nonlinear Stability of Double-diffusive Convection in a Porous Layer with Throughflow and Concentration based Internal Heat Source N. Deepika 1 P. A. L. Narayana 1 Received: 29 September 2015 / Accepted: 5 January 2016 / Published online: 19 January 2016 Springer Science+Business Media Dordrecht 2016 Abstract In this stu, nonlinear stability analysis of double-diffusive convection in a horizontal fluid saturated porous layer has been investigated. Concentration based internal heat source and vertical throughflow effects are considered during investigation. Energy method has been implemented to develop the nonlinear stability analysis. Runge Kutta and shooting methods have been used to solve the eigenvalue problem. Critical thermal Rayleigh number is obtained for assigned values of governing parameters. Results of linear and nonlinear theories have been compared. It is observed that for downward throughflow, when Peclet number Pe is high, the effect of concentration based internal heat source γ is insignificant. Keywords Double-diffusive convection Nonlinear stability Concentration based internal heat source Vertical throughflow Porous medium 1 Introduction The stu of double-diffusive convection in fluid saturated porous media became most significant in the last few decades as it has applications in many fields, namely insulation of buildings, energy storage and recovery, disposal of nuclear wastage, geothermal reservoirs, chemical reactor engineering, oceanography, biological applications such as blood flow in tumors and bio-heat transfer in tissues. Contributions to the field of double-diffusive convection in porous media were incorporated in a book by Nield and Bejan (2013). The objective of the present article is to analyze the nonlinear stability analysis of doublediffusive convection in a horizontal porous layer under the effects of concentration based internal heat source and vertical throughflow. When concentration based internal heat source is introduced in the porous medium, it gives a system which is similar in model to the salt gradient layer in solar pond. In this problem, the numerical comparison between linear and energy theory is established. B P. A. L. Narayana ananth@iith.ac.in 1 Department of Mathematics, IIT Hyderabad, Kandi , Telangana, India

2 752 N. Deepika, P. A. L. Narayana The linear stability theory just gives sufficient condition for instability, but it cannot ensure whether the flow remains stable when it is disturbed under some conditions. Joseph (1966) concluded that energy method gives sufficient condition for stability by applying this method to stu the nonlinear stability of Boussinesq equations. Some of prior works give a conclusion that the linear and energy stability theories complement each other in determining the limits of stable and unstable regions in the parameter space. Davis (1969) employed the energy method to stu the stability of flow confined between two infinite horizontal plates by taking the buoyancy and surface tension effects into account. Implementation of energy method was elaborated in a book by Joseph (1976). Galdi and Straughan (1985) showed that if the linearized system of perturbation equations is symmetric, both the linear and nonlinear stability boundaries coincide with each other. The literature on the nonlinear stability analysis on convection in porous media was included in the books by Straughan (2004, 2008). There were several articles on linear and nonlinear stability analyses in porous media, viz. Siddheshwar and Sri Krishna (2003), Malashetty and Rajashekhar (2008), Saravanan and Brindha (2011)andRionero (2012). All the above works give a common idea that the linear instability boundary is close to the nonlinear stability boundary. Hence, a small band of Rayleigh number sets up in which subcritical instabilities arise. Two researchers Homsyand Sherwood (1976) examined convective instability in a thermally stratified porous layer with vertical throughflow by implementing the linear and nonlinear analysis. Later, Straughan (1989) studied the problem of convection with gravity variation by using the linear and nonlinear stability theories and compared both the results. Rionero and Straughan (1990) investigated the convection in a porous layer in which both the effects of variable gravity and internal heat source were present. Richardson and Straughan (1993) investigated the convection in a porous layer where the viscosity depends on temperature. The convection in a porous layer with vertical throughflow was studied in articles by Nield (1998)andNield and Kuznetsov (2013). Interestingly, when a porous layer is heated and salted from below, the heat given from below motivates the fluid to move in upward direction due to thermal convection, but heavier salt at lower part of porous layer shows precisely opposite effect by stopping the motion caused by thermal convection. Kaloni and Qiao (2000) examined the thermosolutal convection with horizontal mass flow by using energy method to stu the nonlinear stability analysis. Kaloni and Qiao (2001) analyzed the convection in a porous layer with variable gravity effect and compared the linear and nonlinear stability results. Hill (2005) examined the double-diffusive convection with concentration based internal heat source. Lombardo et al. (2001) studied the double-diffusive convection mixture in a horizontal porous layer by performing the nonlinear stability analysis via Lyapunov method. Hill (2008) analyzed the linear and nonlinear stability of penetrative double-diffusive convection in a porous layer, and the main idea of this work was to reduce the region of potential subcritical instabilities by using the operative method. Hill et al. (2007) investigated penetrative convection in a porous layer in the presence of throughflow. Hill and Malashetty (2012) used the operative method to generate sharp nonlinear stability threshold by analyzing the double-diffusive convection with concentration based internal heat source. Double-diffusive convection in porous media has been studied by many researchers, namely Rionero (2007)and Capone etal. (2011, 2013). 2 Mathematical Formulation An infinite horizontal fluid saturated porous layer with a height L, confined between two isothermal plates y = 0an = L, is considered. Let Ox yz be the Cartesian coordinate

3 Nonlinear Stability of Double-diffusive Convection in a 753 Fig. 1 Sketch of the porous medium system with y axis to be in vertical direction which opposes direction of gravity. The porous medium is modeled as homogeneous and subjected to internal heat source which varies linearly with concentration. The difference in vertical temperature and concentration from lower boundary to upper boundary is T 1 T 0 and C 1 C 0 respectively. The schematic diagram of the model is shown in Fig. 1. The Darcy law governs the fluid motion in porous layer. The Oberbeck Boussinesq approximation is invoked i.e. density variation comprises only in bo force term. And it can be expressed as ρ f = ρ 0 1 βt (T T 0 ) β C (C C 0 ), (1) where T is the temperature, C is the concentration, ρ f is the density of fluid, ρ 0 is the density at reference temperature T 0 and concentration C 0, β T is the thermal expansion coefficient, β C is the solutal expansion coefficient. Flow governing equations in dimensional form are v = 0, (2) μ K v = P + ρ f gĵ, (3) σ T + v T = α 2 T + κ(c C 0 ), (4) t φ C + v C = D 2 C, (5) t where v = (u, v, w) is the seepage velocity, P is the pressure, κ is proportionality constant of internal heat source.μ, K,φ,σ,αand D denote viscosity, permeability, porosity, specific heat ratio, thermal and solutal diffusivities of the medium, respectively. It is supposed that there is a vertical throughflow with constant velocity v 0. Then the boundary conditions are y = 0 : v = v 0, T = T 1, C = C 1, y = L : v = v 0, T = T 0, C = C 0. Nondimensional variables are set up as (x, y, z) = 1 α (x, y, z), t = L σ L 2 t, (u,v,w)= v = L α v, P = K (P + ρ 0gy), μα T = T T 0, T 1 T 0 C = C C 0, C 1 C 0 σ = (ρc) m. (ρc p ) f (6) (7)

4 754 N. Deepika, P. A. L. Narayana Using the scaling given in Eq. (7), Eqs. (2) (5) take the following nondimensional form as follows v = P + v = 0, (8) Ra T T + 1 Le Ra SC ĵ, (9) T + v T = 2 T + γ C, t (10) φ C + v C = 1 σ t Le 2 C, (11) and the corresponding boundary conditions are y = 0 : v = Pe, T = 1, C = 1, y = 1 : v = Pe, T = 0, C = 0. (12) In the above equations, the nondimensional parameters are defined as γ = κ L2 (C 1 C 0 ) α(t 1 T 0 ), Ra T = ρ 0gβ T LK(T 1 T 0 ) μα Pe = v 0L α, Le = α D,, Ra S = ρ 0gβ C LK(C 1 C 0 ), (13) μd where γ is measure of concentration based internal heat source, Pe is the vertical Peclet number, Leis Lewis number, Ra T is the thermal Rayleigh number, Ra S is the solutal Rayleigh number. The basic stea state solution to Eqs. (8) (11) subjected to boundary condition (12) is where u B = 0, v B = Pe, w B = 0, e LePe C B = (e LePe 1) elepey (e LePe 1), T B = A 2 B 2 e Pey 1 + γ Pe(e LePe e LePe y γ elepey 1) LePe 1 + γ Pe 2 (e LePe e LePe + γ elepey, (14) 1) (Le 1) A 2 = B 2 = γ e LePe 1 (e Pe 1)(e LePe 1)Pe Pe + 1 γ e Pe 1 + (e Pe 1)(e LePe 1)Pe 2 Le 1 + epe (e Pe 1) 1 (e Pe 1) 1 + γ e LePe (e LePe 1)Pe 2 γ (Le 1)LePe 2 + (Le 1)Pe 1. (Le 1) γ e LePe (e LePe 1)Pe + 1 LePe + 1,

5 Nonlinear Stability of Double-diffusive Convection in a 755 Temperature and concentration gradients along y axis direction are dt B = 1 γ (e Pe 1) (Le 1)LePe + Pe + γ elepe (e LePe 1) 1 ( + Pe(e LePe γ e LePe γ e LePey) + 1) e Pey γ e LePey Le Pe(Le 1)(e LePe 1), (15) dc B = LePe (e LePe 1) elepey. (16) When there is no vertical through flow (Pe = 0), basic stea state solution is u B = 0,v B = 0,w B = 0, C B = 1 y, y 3 T B = γ 6 y2 γ y + 1. (17) To determine the stability of stea state solution, perturbation is employed as of the form u = u B + U,v = v B + V,w = w B + W, T = T B + θ,c = C B +. (18) The resulting perturbation equations are U x + V y + W = 0, (19) z U = P + Ra T θ + 1 Le Ra S ĵ, (20) θ t + V T B y + Pe θ y + U θ = 2 θ + γ, (21) φ + V C B σ t y + Pe y + U = 1 Le 2, (22) and the corresponding boundaries are y = 0, 1 : V = θ = = 0. (23) The geometry of the system, boundary conditions and stea state solution for the governing equations are invariant under the rotation around an axis which is parallel to y axis. Hence, direction of propagation of disturbed plane wave can be in any horizontal (either x or z ) axis. Thus, the analysis of plane wave perturbation equations becomes two-dimensional, U = U(x, y, t), V = V (x, y, t), W = 0, θ = θ(x, y, t), = (x, y, t). (24) To stu the nonlinear stability analysis, we define energy functional as E(t) = ξ 2 θ 2 + ηφle 2σ 2, (25) where ξ and η are positive coupling parameters. Further, we define β = /Le. Multiplying Eq. (20)byU,Eq.(21)byθ and Eq. (22)byβ and integrate over V,whereV be period cell. After applying the divergence theorem, the following equations are obtained.

6 756 N. Deepika, P. A. L. Narayana U 2 = Ra T <θ V > + Ra S <βv >, (26) 1 d 2 dt θ 2 + <(U T B )θ> = θ 2 + γ Le<βθ>, (27) 1 φ 2 σ Le d dt β 2 + <(U C B )β> = β 2, (28) where <.> stands for integration over V,. denotes norm in L 2 (V ). FromEqs.(26) (28) together with Eq. (25), we can write as where Define de dt = I D, (29) I = ξ<(u T B )θ> + Leξγ<βθ> η<(u C B )β> + Ra T <θ V > + Ra S <βv >, D = ξ θ 2 + U 2 + η β 2. I m = Max H D, (30) where H is the space of admissible solutions such that U = 0. From Eqs. (25) (30) and by applying Poincare inequality yields, de dt 2π 2 (1 m) min ( 1, 1 ) σ E. (31) Le φ The integration of above inequality clearly shows E(t) 0 ast. Decay of θ and can be obtained from definition of E(t). E(t) does not contain the term U 2.Itisworthy checking what happens to U 2 as t. From arithmetic geometric mean inequality on Eq. (26), U 2 2( θ ), (32) Therefore, from Eqs. (32)and(25), it can be implied that decay of E(t) ensures the decay of U 2. Now, we return to the critical argument m = 1. The corresponding Euler Lagrange equations to the maximization problem are ξ T B θ + η C B β Ra T θ + Ra S β j + 2mU = λ, (33) ξ(u T B ) + ξ Leγβ + Ra T V + 2mξ 2 θ = 0, (34) η(u C B ) + ξ Leγθ + Ra S V + 2mη 2 β = 0, (35) where λ is Lagrange multiplier introduced since U is solenoidal. To get rid of λ in Eq. (33), second component of double curl of Eq. (33) is taken. Hence, the following equation is obtained. ξ dt B ( 2 θ z θ x 2 ( 2 β + Ra S z β x 2 ) η dc B ) 2m ( 2 V z 2 ( 2 β z β x V x 2 ) + Ra T ( 2 θ ) + 2m y ) z θ x 2 ) = 0. (36) ( U x + W z

7 Nonlinear Stability of Double-diffusive Convection in a 757 Putting stream function ψ such that W = 0, U = ψ y, Then Eq. (36) changes as 2 2 x x θ y 2 + Ra T x 2 + Ra S Now substituting normalmodes of the form V = ψ x, (37) 2 β x 2 ξ dt B 2 θ x 2 η dc B 2 β = 0, (38) x2 (x, y, t) = (y)e λt cos(ax), θ(x, y, t) = θ(y)e λt sin(ax), β(x, y, t) = β(y)e λt sin(ax), (39) into Eqs. (34), (35)and(38), the following eigenvalue problem and corresponding boundary conditions are obtained. (D 2 a 2 ) a Ra T θ + Ra S β 1 2 a ξ dt B θ + η dc B β, (40) (D 2 a 2 )θ 1 2 a dt B Leγβ + 1 2ξ ara T = 0, (41) (D 2 a 2 )β 1 2 a dc B + ξ 2η Leγθ + 1 2η ara S = 0, (42) y = 0, 1: = θ = β = 0. (43) 2.1 Derivation for Coupling Parameters The coupling parameters in the generalized energy functional play a very important role in the stability of the system. To this end, we have optimal values of these coupling parameters in the following way. I = ξ<(u T B )θ> + Leξγ<βθ> η<(u C B )β> +Ra T <θ V > + Ra S <βv >, D = ξ θ 2 + U 2 + η β 2, Rescaling temperature and concentration perturbations as θ = θ ξ, β = β η, the modified expressions for I and D are I = ξ ξ<(u T B ) θ>+ Le η γ< β θ> η<(u C B ) β> + Ra T ξ < θ V > + Ra S η < βv > + <p(x, t), U i,i >, D = θ 2 + U 2 + β 2, (44)

8 758 N. Deepika, P. A. L. Narayana where I D is maximum if δi mδd = 0. And is removed for the sake of convenience. The Euler Lagranges equations (as given in Straughan 2004) for the maximization problem of the form where Xθ ĵ + Yβ ĵ 2U = p, i, XV + Zβ θ = 0, YV + Zθ β = 0. (45) and X = Ra T dt B ξ, ξ Z = Ra S Y = dc B η η ξ η Leγ Let (U (1),θ (1),β (1) ), (U (2),θ (2),β (2) ) are solutions to (45) for(ξ (1),η (1) ), (ξ (2),η (2) ), respectively. Taking Eq. (45) with superscript (1), multiply each equation with U (2),θ (2),β (2), respectively, and integrate over V in each case. Repeat the same procedure with superscripts (1) and (2) reversed. After dividing by ξ (2) ξ (1) and then η (2) η (1) and taking the limit ξ (2) ξ (1) and η (2) η (1),wecanshow X Z <V θ>+ <θβ> = 0, ξ ξ Y η <Vβ> + Z <θβ> = 0. η Assuming coefficient of concentration based internal heat source γ is 0, implies Z = 0. Then Y η = X ξ = 0 Above expression implies ξ = Ra T ), η = Ra S ( ). dcb γ =0 γ =0 ( dtb These two serve as good approximations for the coupling parameters ξ,η, respectively. Here ( ) dtb γ =0 = Pe ( ) dcb (e Pe 1) epey, γ =0 = LePe (e LePe 1) elepey are negative definite functions. Therefore, ξ and η are always positive. 2.2 Eigenvalue Problem for Linear Stability In order to make numerical comparison between the linear stability threshold and energy threshold, an eigenvalue problem for linear stability is derived from Eqs. (19) (22) by neglecting the product of perturbations and performing normal mode analysis from Eq. (39). Thus, the yielded eigenvalue problem for linear theory is

9 Nonlinear Stability of Double-diffusive Convection in a 759 (D 2 a 2 ) + a Ra T θ + 1 Le Ra S = 0, (46) (D 2 a 2 )θ a dt B PeDθ + γ = λθ, (47) (D 2 a 2 ) LePeD ale dc B = φ λ, σ (48) y = 0, 1: = θ = = 0. (49) The above eigenvalue problem is derived by taking φ σ = 1 because the variation in this ratio effects only oscillatory modes (Nield et al. 1993). In the above system λ = 0 refers to stationary modes. Since, stationary longitudinal modes are preferred modes of stability (Deepika et al. 2015), only stationary modes have been considered. 3 Results and Discussion To solve the eigenvalue problems (40) (43) and(46) (49) numerically, fourth-order Runge Kutta method and shooting methods have been used. In order to apply Runge Kutta method, we must have an initial value problem. Therefore, by replacing Eqs. (43)and(49) with (0) = 0, (0) = 1, θ(0) = 0, θ (0) = χ, (0) = 0, (0) = ζ. (50) Here the normalization condition (0) = 1 shows the breaking scale invariance of eigenfunctions (,θ, ) in the Eqs. (40) (43) and(46) (49). And χ and ζ are unknown parameters; these can be evaluated by the shooting method and where the target condition is (1) = θ(1) = (1) = 0. For fixed parameters (a, Ra s, Pe, Le,γ), one can determine χ,ζ and Ra T by using boundary conditions. The thermal Rayleigh number Ra T is treated as eigenvalue. To implement explicit fourth-order Runge Kutta method, a built-in function, ode45 is used in the software package Matlab R2012b. Minimum of all Ra T by varying all flow governing parameters is the critical Rayleigh number Ra TC at the critical wave number a C. Therefore, critical Rayleigh number for the linear and nonlinear theory is given as Ra TC = min Ra T (a, Ra S, Le, Pe,γ) a Ra TC = max ξ max η min Ra T (a, Ra S,γ,Le, Pe,ξ,η). a Here ξ and η are the coupling parameters which are derived in Sect The critical Rayleigh number Ra TC is obtained for assigned values of flow governing parameters Ra S, Le,γ,Pe. In this section, the linear and nonlinear stability results have been compared. The gap between the linear and nonlinear stability curves is referred as region of subcritical instabilities, in which instability may occur before attaining the threshold in the linear stability theory. Here, Ra S > 0 refers to the concentration at the lower plate is higher than the concentration at the upper plate, whereas Ra S < 0 refers to the concentration at lower plate is lower than concentration at the upper plate. Throughout the discussion, the dashed lines represent the linear stability boundary and the solid lines represent the nonlinear stability boundary. Figure 2a, b shows response of the critical thermal Rayleigh number Ra TC against solutal Rayleigh number Ra S for the values of Le = 10, γ = 0, 2 with Pe = 5inFig.2a and Pe = 5inFig.2b. In the case of linear theory, solutal Rayleigh number Ra S shows insignificant destabilization effect from Ra S = 100 to Ra S = 100. But in the case of

10 760 N. Deepika, P. A. L. Narayana Fig. 2 Variation in Ra TC versus Ra S for Le = 10 Fig. 3 Variation in Ra TC versus Le for Pe = 5 nonlinear theory, the critical Rayleigh number is unaffected by negative Ra S,whereas positive Ra S shows destabilization effect. Region of subcritical instabilities is increasing when Ra S is increasing from 0 to 100. In the case of nonlinear theory, when γ is present, in the neighborhood of Ra S = 0, there is a extreme fall in the critical thermal Rayleigh number due to the singularity at Ra S = 0 in the nonlinear eigenvalue problem. For upward throughflow (Pe = 5), the flow with γ = 0 is slight unstable than the flow with γ = 2for both linear and nonlinear theories. But, for downward throughflow (Pe = 5), the curves for γ = 0andγ = 2 almost coincide except for positive Ra S in the nonlinear theory. Figure 3a, b shows behavior of the critical thermal Rayleigh number Ra TC to Lewis number Le for Pe = 5, γ = 0, 2 with Ra S = 10 in Fig. 3aandRa S = 10 in Fig. 3b. In both the cases, the critical thermal Rayleigh number Ra TC is unaffected with higher Lewis number Le. In Fig. 3a, small values of Le have stabilizing effect for the linear and nonlinear theories. In Fig. 3b, for the linear theory, small values of Le have destabilizing effect. But for nonlinear case, when γ = 2, small values of Le have stabilizing effect. In both the figures, flow with γ = 2 is slight unstable than the flow with γ = 0 for nonlinear theory. Figure 4 shows plot of critical thermal Rayleigh number Ra TC versus Peclet number Pe with Le = 10, Ra S = 10, γ = 0, 2. Peclet number Pe has stabilization effect. For large

11 Nonlinear Stability of Double-diffusive Convection in a 761 Fig. 4 Plot of Ra TC versus Pe for Le = 10, Ra S = 10 values of Pe, a boundary layer forms in the direction of throughflow. Thus, convection is restricted to thickness of boundary layer rather than length of the porous layer. Due to this, critical Rayleigh number increases as Pe increases; thus, stabilization takes place. This is consistent with the observations made by Nield (1998) and Nield and Kuznetsov (2013). When γ = 0, the curve of Ra TC is symmetric about Pe = 0 axis for both the linear and nonlinear theories. When Pe = 0, both of the theories show good agreement, but the region of subcritical instabilities increases as Pe increases for both the upward and downward throughflow. 4 Conclusions Nonlinear stability of the double-diffusive convection in a fluid saturated porous medium with concentration based internal heat source and vertical throughflow is examined. Comparison is made between linear (stationary longitudinal modes) and nonlinear results. Both stabilizing and destabilizing effect is insignificant as Lewis number Leis increased. When the strength of downward throughflow (Pe < 0) is high, the effect of γ is inappreciable. For the nonlinear theory, the critical thermal Rayleigh number is same for all negative Ra S except in the neighborhood of Ra S = 0. References Capone, F., Gentile, M., Hill, A.A.: Double-diffusive penetrative convection simulated via internal heating in an anisotropic porous layer with throughflow. Int. J. Heat Mass Transf. 54, (2011) Capone, F., De Luca, R., Torcicollo, I.: Longtime behavior of vertical throughflows for binary mixtures in porous layers. Int. J. Non Linear Mech. 52, 1 7 (2013) Davis, S.H.: Buoyancy-surface tension instability by the method of energy. J. Fluid Mech. 39, (1969) Deepika, N., Matta, A., Narayana, P.A.L.: Effect of throughflow on double-diffusive convection in porous medium with concentration based internal heat source. J. Porous Med. (2015) Galdi, G.P., Straughan, B.: Exchange of stabilities, symmetry, and nonlinear stability. Arch. Ration. Mech. Anal. 89, (1985)

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