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Physical and Numerical Simulation of Geotechnical Engineering 10th Issue, Mar. 2013 Stability Analysis of Thermosolutal Convection in a Horizontal Porous Layer Using a Thermal Non-equilibrium Model CHEN Xi, WANG Shaowei, TAN Wenchang Department of Mechanics and Aerospace Engineering, Peking University, Beijing 100871, P.R.China [email protected] ABSTRACT: We use a linear stability analysis to investigate the onset of thermosolutal convection in a horizontal porous layer subject to fixed temperatures and chemical equilibrium on the boundaries’ surfaces. The critical Rayleigh number and the corresponding wave number for the exchange of stability and overstability are obtained. The critical value of the parameters for oscillatory convection occurs are obtained. Besides, asymptotic analysis for both small and large values of inter-phase heat transfer coefficient H. KEYWORDS: Double diffusive, Porous media, Non-equilibrium model, Critical rayleigh number 1 INTRODUCTION In many geological systems, with the fluid convect in a porous medium, the amount of solute in the fluid will be dissolved or precipitated because of temperature, pressure or chemical reactions, and it will also affect the convection of the fluid. Then double-diffusive convection (thermosolutal convection) occurs. The first linear stability analysis of a double-diffusive system in a porous layer was investigated by Nield [7], the critical Rayleigh numbers for the onset of stationary and overstable convection are obtained for different thermal and solutal boundary conditions using a linear stability analysis. Recently, Pritchard and Richardson [2] discussed how the dissolution or precipitation of the solute affect the onset of convection. The other emphasis of this paper is the thermal non-equilibrium model. To be convenient, most of the former works on convective heat transfer in porous medium are under the assumption that the solid and fluid phases are in local thermal equilibrium (LTE). But in fact, in many applications, they are not in local thermal equilibrium. Then Nield and Bejan [1] give us the equations of the thermal non-equilibrium model. Recently, Malashetty et al.[11] present the effect of thermal non-equilibrium on the onset of convection in a porous layer using the Lapwood-Brinkman model. Nevertheless, the former studies focused on either double-diffusive convection or non-equilibrium model, none of them investigated the double-diffusive convection using a non-equilibrium model in spite of it has many important applications. Thus, the aim of this paper is to make the linear stability analysis of double-diffusive convection in a horizontal porous layer using a thermal non-equilibrium model. temperatures are T0 and T1 at the bottom and top, respectively, and solutal mass concentrations are C0 and C1 . Taking a horizontal coordinate x and a vertical coordinate z , assuming the fluid satisfies Darcy Law, neglecting the small quantity of the second order, we get the linear non-dimensional equations: xx zz C C x T x (1) t x xx zz H ( ) (2) t xx zz H ( ) (3) Ct x (4) Here, is stream function, 、 are perturbations of temperature of fluid and solid, C is perturbations of solute. T and C are related to the Rayleigh numbers that emerge naturally from the pure thermal and pure solutal problems. H is scaled inter-phase heat transfer coefficient. is porosity-modified conductivity ratio. is diffusivity ratio. k is the thermal conductivity. Le is Lewis number. 3 LINEAR STABILITY ANALYSIS We use Fourier-mode solutions to discuss the stability (see Pritchard & Richardson [2]). 0 et e imx sin(nz) , 0 et e imx sin( nz) (5) t imx 0 e e sin(nz) , C 0 et e imx sin(nz) (6) Substituting (5) and (6) into (1)-(4), and eliminate 0 , 0 , 2 DOUBLEDIFFUSIVE CONVECTION MODEL 1 2 C k ( C ) Le 0 and 0 , we obtain: A 3 B 2 C D 0 REACTION- (7) Here, A, B, C and D are functions with respect to M, N, 2 2 T and C , where M m 2 , N n . For simplicity, we In a horizontal porous layer of depth d, we assume that won’t give the expressions of the coefficients A, B, C and D © ST. PLUM-BLOSSOM PRESS PTY LTD Stability Analysis of Thermosolutal Convection in a Horizontal Porous Layer Using a Thermal Non-equilibrium Model DOI: 10.5503/J.PNSGE.2013.10.007 in the paper. It is easy to find that D=0 is the critical condition for the exchange of stability and the critical condition for oscillatory convection occurs is BC AD 0 and C 0 . M T( c ) N 2) C (M 2 ) 2 M 2 C ( ) H O( H 2 ) M M M 2 k By setting T( c ) / M 0 , then we obtain 4 M T( c ) 2 H O( H 2 ) 2(2 2 k ) 2 ( 2 1) Hence, C C (1 ) H O( H 2 ) 2 2 k T( c ) 4 2 M 1 ( C 4 2 ) /[ 2(1 ) / 2 2 C k /( 2 2 k ) 3 ] and C must satisfy (14) (13) C k 0 (14) 5 DISCUSSION (9) Obviously, when N 2 , T take the lowest value. So we assume that N 2 , then paint the plot of variation of T with M for specific values of k and H. The values (10) of the other parameters are H=5, Le 2 , 1 , C 30 . (M 2 k ) C (M ) (M ) 1 1 ) O( 2 ) H 2 (M 2 k ) M 2 H 2 ( (12) (8) (ii)For large values of H, we obtain (assumed that Le 1 and N 2 ) (M 2 ) 2 1 M 2 T( c ) C (1 ) (11) M 1 1 O( ) H H2 Therefore, we assume that k=0, than our result recovers the result of non-reaction. Thus, 1 1 1 (15) T( c ) (4 2 C ) 1 O( 2 ) H H Here, 2 2 C 8 4 (16) 1 2 2 Now we discuss the asymptotic analysis for both small and large values of H. For large H, LTE is recovered. Following, we will find the asymptotic solutions, and in section 5 we will compare it with numerical solutions. (i)When H is small, we get (assumed that Le 1 T( c ) M1 and substitute it into T( c ) / M 0 , we get 4 RESULTS AND ASYMPTOTIC ANALYSIS and 2 2 2 3 Assuming that, Figure 1 Variation of T with values of k M Figure 2 Variation of T with values of H for specific M for specific compare O(c ) and T(c) , and their corresponding wave numbers. We take five virtual numbers. When k = 1.8708 , O( c ) T( c ) . So we obtain the critical value of k (k=1.8708) From Figure 1, we can see that the critical Rayleigh number T decreases with k increases. So when k becomes smaller, the system will be more stable. Thus, the existent of reaction will make the system instable. Form Figure 2 ( Le 2 , 1 , k 2 , C 30 ), we know that (c ) for the exchange of oscillatory convection and stationary convection. So when k 1.8708 , over-stability will be considered, when k 1.8708 , oscillatory convection won't occur. From Table 1, we can see when 0 k 2 , the variation of O with M for k is very small. So the Figure can't give a obviously variation. But from the numerical solutions, it can be seen that the critical Rayleigh number O(c ) decreases with k increases. So when k becomes the critical Rayleigh number T(c) increases with H increases. So when H becomes bigger, the system will be more stable. Thus, the existent of non-equilibrium between fluid and solid will make the system instable. When O( c ) T( c ) , oscillatory convection can occur. In Table 1 (( Le 2 , 1 , H 5 , 0.2 , C 30 )), we 30 Physical and Numerical Simulation of Geotechnical Engineering 10th Issue, Mar. 2013 smaller, the system will be more stable. Table 1 Comparison of k Table 114.8983 113.4234 112.3849 112.0339 2 ( Le 2 , 1 M O(c ) 12.859 12.796 12.070 11.973 112.5922 112.4721 112.3849 112.3548 15.293 15.181 15.098 15.070 will be considered, when H 4.8535 , oscillatory , convection won't occur. From Figure3, we can see, O( c ) T( c ) . So we obtain the critical value of H with O(c ) M T(c ) and with M O(c ) for variation of H T(c) M T(c ) O(c ) 4.0000 4.5000 4.8535 5.0000 109.1689 110.6322 111.6293 112.0339 11.553 11.773 11.916 11.973 107.3666 109.8712 111.6293 112.3548 Figure 3 Variation of O O(c ) increases with H increases. Hence when H becomes larger, the system will be more stable. ( H = 4.8535 ) for the exchange of oscillatory convection and stationary convection. When H 4.8535 , over-stability T(c) k O(c ) k 2 , 0.2 , C 30 ) we get, when H = 4.8535 , Table 2 Comparison of M O(c ) for variation of M T(c ) (c ) T 1.0000 1.2000 1.8708 2.0000 From T(c) with O(c ) and M T(c) with H M O(c ) 14.092 14.582 14.927 15.070 with M for specific values of H In Table 3, we compare the exact and asymptotic values of the wave number and the critical Rayleigh number when H is small. The other parameters are Le 1 , 1 , k 10 , C 30 . We can see that when H is small, exact values are almost the same as the asymptotic values, but when H grows bigger, the contrast of them are more obviously. Table 3 Comparison of exact and asymptotic values of M T(c ) and T(c) for small H. (E denotes the exact solution and A denotes the asymptotic solution) log10 H M T(c ) ( E ) M T(c ) ( A) T(c ) ( E) T(c ) ( A) -2.0 -1.5 -1.0 -0.5 0.0 0.5 9.86627 9.88063 9.92556 10.0630 10.4549 11.3640 9.85965 9.85976 9.86011 9.86120 9.86467 9.87562 69.4685 69.5334 69.7374 70.3708 72.2674 77.4214 69.4685 65.5019 69.6393 70.0738 71.4478 75.7923 H is large. The other parameters are Le 1 , 1 , In Table (4), we compare the exact and asymptotic values of the wave number and the critical Rayleigh number when 31 Stability Analysis of Thermosolutal Convection in a Horizontal Porous Layer Using a Thermal Non-equilibrium Model DOI: 10.5503/J.PNSGE.2013.10.007 k 0 , C 30 . We can see when H is large, exact values are almost the same as the asymptotic values. Table 4 Comparison of exact and asymptotic values of and T(c) for large H. log10 H M T(c ) ( E ) M T(c ) ( A) T(c ) ( E) T(c ) ( A) 2.5 3.0 3.5 4.0 4.5 5.0 10.1763 9.98205 9.91777 9.89715 9.89061 9.88853 9.56052 9.76503 9.82969 9.85014 9.85661 9.85865 74.9185 76.4605 76.9723 77.1367 77.1889 77.2055 74.7985 76.4497 76.9718 77.1369 77.1891 77.2056 [2]. 6 CONCLUSIONS Linear stability of double-diffusive convection in a horizontal porous layer using a thermal non-equilibrium model has been studied. The critical Rayleigh number and the corresponding wave number for the exchange of stability and over-stability are obtained. We find that the critical Rayleigh number for stability increases when H increases or k decreases. Differ from stability, the critical Rayleigh number for over-stability increases when H increases or k increases. We also find the critical values of the parameters for exchange of stationary convection and oscillatory convection. Then asymptotic solutions for both small and large values of H are studied. We present (c ) comparison of numerical and asymptotic values of M T (c ) and T and for both small and large values of H . We (c) 2 can see when H or 0 , M T and (c ) T has a certain limit. For convenience, we assumed that Le 1 , the errors of numerical and asymptotic values are produced by the other parameters as k and C , and we take the appropriate values to decreased the errors. Although the variation of C is very small, it makes a (c ) considerable affection of T , that's the affection of double-diffusive convection. [3]. [4]. [5]. [6]. [7]. D.Pritchard, C.N.Richardson, The effect of temperature-dependent solubility on the onset of thermosolutal convection in a horizontal poros layer. J.Fluid Mech 571 (2007): 59-95. N.Banu, D.A.S.Rees, Onset of Darcy-Benard convection using a thermal non-equilibrium model. International Journal of Heat and Mass Transfer 45, 2002: 2221-2228. J.H.Jeans, Astronomy and Cosmogony. Cambridge University Press. Wenchang Tan, Takashi Masuoka, Stability analysis of a Maxwell fluid in a porous medium heated from below. Physics Letters A 360 (2007): 454-460. T.E.Jupp, A.W,Woods. Thermally-driven reaction fronts in porous media. J. Appl. Phys 16 (2003): 367-370. D.A.Nield, Onset of thermohaline convection in a porous medium. Water Resour. Res. 4 (3): 553-560. D.A.S.Rees, I.Pop, Free convective stagnation point flow in a porous medium using thermal non-equilibrium model. Int.Common. Heat Mass Transfer, 26 (1999): 945-954. [9]. D.A.S.Rees, I.Pop, Vertical free convective boundary layer flow in a porous medium using a thermal non-equilibrium model. J. Porous Media 3 (2000): 31-44. [10]. A.V.Kuznetsov, Thermal non-equilibrium forced convection in porous media, in: Derek B.Ingham and I.Pop (Eds.). [8]. Transport Phenomena in Porous Media, Pergamon Press, Oxford, (1998): 103-130. [11]. M.S.Malashetty, I.S.Shivakumara, S.Kulkarni, The onset of Lapwood-Brinkman convection using a thermal non-equilibrium model. International Journal of Heat and Mass Transfer, 48 (2005): 1155-1163. REFERENCES [1]. M T(c ) D. A. Nield and A. Bejan, Convection in Porous Media, 3rd ed., Springer, New York, 2006. 32