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Convection in a porous layer with a surface reaction
International Journal of Heat and Mass Transfer 54 (2011) 5653–5657
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Technical Note
Convection in a porous layer with a surface reaction q Nicola L. Scott, B. Straughan ⇑ Department of Mathematics, University of Durham, DH1 3LE, UK
a r t i c l e
i n f o
Article history: Received 12 August 2011 Accepted 12 August 2011 Available online 14 September 2011 Keywords: Porous convection Chemical reactions Oscillatory convection
a b s t r a c t We develop a problem recently studied by Postelnicu in which a horizontal layer of porous material is subject to a chemical reaction at the base of the layer. The porous medium is of Darcy type and the layer is saturated with a non-isothermal liquid containing a concentration of the reacting chemical. The equations are developed and then it is shown that if the reaction parameter B of Postelnicu exceeds a critical value convection commences but as oscillatory convection, not stationary convection, as previously thought. Ó 2011 Elsevier Ltd. All rights reserved.
1. Introduction The study of thermal convection problems driven by chemical reactions is dominated by applications and has recently been witnessing much activity. For convection in a pure fluid McTaggart and Straughan [8] developed an energy method to yield global nonlinear stability thresholds and further references may be found therein. Within the field of convection in porous media the area of chemical reactions is perhaps newer and richer. McKay [7] studies the effect of chemical reaction upon convection in a porous layer overlain by a viscous fluid. Rahman and AlLawatia [12] investigate the effect of high order chemical reactions while Mahdy [5] continues the analysis in double diffusive convection. Andres and Cardoso [1] study instability in the presence of a chemical reaction in a finite layer and link this to the important environmental topic of carbon dioxide storage. Phase change effects may also play a prominent role, for example in compositional convection where material is deposited at the Earth’s solid inner core, see e.g. Eltayeb et al. [3,4]. Nguyen et al. [9] study flow driven by a chemical reaction on an anisotropic porous cylinder, Malashetty and Biradar [6] analyse the onset of double diffusive convection with a chemical reaction present in an anisotropc porous layer. Of particular interest to the present article is the paper by Postelnicu [11] who models the situation where a chemical reaction at the base of a horizontal layer of porous material gives rise to a convective instability. Our major goal here is to revisit the situation studied by Postelnicu [11] where we have a layer of Darcy porous material saturated with a non-isothermal, incompressible viscous fluid, in which is
q
This work was supported in part by a grant from the Leverhulme Trust, ‘‘Tipping points: mathematics, metaphors and meanings’’, and in part by a DTA award from EPSRC. ⇑ Corresponding author. E-mail addresses: [email protected] (N.L. Scott), [email protected] (B. Straughan). 0017-9310/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijheatmasstransfer.2011.08.032
dissolved a chemical of concentration C. As there are several misprints in Postelnicu [11] we rederive the basic equations at the outset. Firstly we assume Darcy’s Law and then
rp ¼
l K
v qgk;
ð1Þ
where v, p are velocity and pressure, q, g are the density and gravity (acting in the negative z-direction), k = (0, 0, 1), l is the dynamic viscosity of the fluid and K is the permeability of the porous medium. The Boussinesq approximation is adopted whereby the density is assumed constant everywhere except in the body force term in (1). In line with Postelnicu [11], the density in (1) is taken to depend only on the temperature, T, and not on the concentration C. This dependence is assumed linear so that
q ¼ q0 ð1 aðT T 0 ÞÞ;
ð2Þ
q0 being the density at reference temperature T0, and a is the coefficient of thermal expansion. Insertion of (2) in (1) leads to the momentum equation p;i ¼
l K
v i q0 gð1 aðT T 0 ÞÞki ;
ð3Þ
where standard indicial notation is employed. The fluid is incompressible so that
v i;i ¼ 0:
ð4Þ
The temperature field and the concentration field satisfy the equations, see Straughan [13],
1 T ;t þ v i T ;i ¼ jDT; M /C ;t þ v i C ;i ¼ /kc DC;
ð5Þ
where M = (q0cp)f/(q0c)m with (q0c)m = /(q0cp)f + (1 /)(qc)s and j = km/(q0cp)f is the thermal diffusivity of the porous medium, km
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being given by km = js(1 /) + jf/. The terms cp and cs are the specific heat at constant pressure of the fluid and the specific heat of the solid, respectively, and / is the porosity of the solid. The saturated porous medium occupies the infinite layer fðx; yÞ 2 R2 g fz 2 ð0; hÞg and on the upper wall the temperature and reactant concentration are fixed and constant with zero mass flux there. Thus the upper boundary conditions are
h;t þ b1 w ¼ Dh; 1 Dc; Le Dw RD h ¼ 0;
ð12Þ
M/c;t þ b3 w ¼
On the lower wall (z = 0) there is an exothermic reaction where the reactant is converted into an inert product at a rate, r, where
where D⁄ = @ 2/@x2 + @ 2/@y2 is the horizontal Laplacian, Le = j/(/kc) is the Lewis number and the Rayleigh number, R, is defined by R = Khgq0aTU/(jl). By standard linear analysis we wish to find time dependent horizontally periodic solutions, therefore w, h and / take the Fourier form
E r ¼ k0 C exp ; RT
w¼
T ¼ TU;
v i ni ¼ w ¼ 0;
C ¼ CU ;
on z ¼ h:
ð6Þ
ð7Þ
k0 is a rate constant, R⁄ is the universal gas constant and E is the activation energy of the reaction. The rate of change of temperature is proportional to the heat of the reaction, Q, the rate at which it occurs, and inversely proportional to the rate at which the heat is conducted away from the surface, kT. The rate of change of the reactant concentration is proportional to the rate of the reaction and inversely proportional to the porosity of the solid and the diffusivity of the reactant, kc. There is zero mass flux across the lower wall, hence the boundary lower boundary conditions are
@T E ¼ Qk0 C exp ; @z RT @C E ¼ k0 C exp ; /kc @z RT kT
v i ni ¼ w ¼ 0
ð8Þ
Þ, the fluid velocity is zero and the ; T; C; p At the steady state, ðv temperature and reactant concentration are independent of time; v i ¼ 0; T ;t ¼ 0; C ;t ¼ 0. If we now consider the temperature and reactant concentration fields to be functions of z only (5) immediately imply that
ð9Þ
ðzÞ, is found from (3). The steady pressure, p We now introduce small perturbations ui, h, c, p to the steady state such that
v i ¼ vi þ ui ;
T ¼ T þ h;
C ¼ C þ c;
þ p: p¼p
h¼
1 X
c¼
1 X
rH þ b1 W ¼ ðD2 k2 ÞH;
3. Perturbation boundary conditions After non-dimensionalising using (11) the upper boundary conditions are
w ¼ 0;
l p;i ¼ ui þ q0 g ahki ; K
where w = u3. Eq. (10) are linearised by considering the terms uih,i and uic,i to be small and we may therefore neglect them. The pressure term is removed by taking curl curl of (10)4 and retaining the third component (i = 3). The non-dimensionalisations 2
t ¼ jhM t ; C ¼ CU C ;
T ¼ T U T ; p ¼ jKl p
are employed to find
T ¼ 1 on z ¼ 1
@T n ¼ AC exp ; @z T @C n ; ¼ BC exp @z T
A¼
Qhk0 C u ; kT T U
ð15Þ
B¼
hk0 ; /kc
ð16Þ
n¼
E R T U
are non-dimensional coefficients. Eqs. (15), (16) and (19) imply that the linearised perturbation boundary conditions are
W ¼ H ¼ U ¼ 0;
on z ¼ 1
ð17Þ
and
W ¼ 0; ð11Þ
C ¼ 1;
and the lower boundary conditions (on z = 0) are
where
ð10Þ
ð14Þ
where D = d/dz.
1 h;t þ b1 w þ ui h;i ¼ jDh; M /c;t þ b3 w þ ui c;i ¼ /kc Dc;
erj t fj ðx; yÞUj ðzÞ;
where f is some function such that D⁄f = k2f for a wave number k, and rj is an eigenvalue (growth rate). Since we are interested in the first instance of instability, we choose one mode of the Fourier expansions to find a typical eigenvalue which satisfies the system of equations
w ¼ 0;
xi ¼ hxi ; v i ¼ jh v i ;
ð13Þ
j¼1
Eqs. (3)–(5) then become
ui;i ¼ 0;
erj t fj ðx; yÞHj ðzÞ;
j¼1
M/rU þ b3 W ¼
2. Basic solution and perturbation equations
CðzÞ ¼ b3 z þ b4 :
erj t fj ðx; yÞW j ðzÞ;
j¼1
1 2 2 ðD k ÞU; Le 2 2 ðD2 k ÞW þ Rk H ¼ 0;
on z ¼ 0:
TðzÞ ¼ b1 z þ b2 ;
1 X
dH ¼ AU; dz
dU ¼ BU on z ¼ 0; dz
ð18Þ
where w is now the third component of u. Evaluating the steady state, (9), on the boundary walls one obtains the relations
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1 ¼ b1 þ b2 ;
Eq. (20) may be solved for b1 and b3 for specific values of A, B and n.
1 ¼ b3 þ b4 ;
n ; b1 ¼ Ab4 exp b2 n : b3 ¼ Bb4 exp b2
ð19Þ
We may eliminate b2 and b4 using (19)1 and (19)2, then use (19)3 and (19)4 to give two coupled equations for b1 and b3;
n ; b1 ¼ Að1 b3 Þ exp 1 b1 n : b3 ¼ Bð1 b3 Þ exp 1 b1
ð20Þ
4. Numerical results We employed both the Natural D and D2 Chebyshev-Tau methods to numerically compute the critical Rayleigh number for various values of the boundary parameters, A, B, n, for (14) with boundary conditions (17) and (18). Details of these methods may be found in [10,2], respectively, and complete agreement was found between them. We set M/ = 1, 0.5, 0.25 and obtained the values given in Tables 1–3, respectively. Postelnicu [11] claims that exchange of stability holds so that one may select r = 0 in Eq. (14). This is seen by our results to be
Table 1 Values of the critical Rayleigh number Rc, critical wave number kc and Im (r1) for M/ = 1. Le = 0.1
Le = 1
Le = 10
Le = 100
A
B
n
b1
b3
Rc
kc
Im (r1)
Rc
kc
Im (r1)
Rc
kc
Im (r1)
Rc
kc
Im (r1)
0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 1 1 1 1 1 1 1 1 1 5 5 5 5 5 5
0.5 0.5 0.5 1 1 1 5 5 5 0.5 0.5 0.5 1 1 1 5 5 5 1 1 1 5 5 5
0 0.15 0.5 0 0.15 0.5 0 0.15 0.5 0 0.15 0.5 0 0.15 0.5 0 0.15 0.5 0 0.15 0.5 0 0.15 0.5
0.33333 0.30836 0.25109 0.25000 0.23483 0.19860 0.08333 0.08132 0.07585 0.66667 0.62632 0.53008 0.50000 0.47459 0.41241 0.16667 0.16293 0.15284 2.50000 2.44559 2.31163 0.83333 0.82158 0.79088
0.33333 0.30836 0.25109 0.50000 0.46967 0.39720 0.83333 0.81317 0.75854 0.33333 0.31316 0.26504 0.50000 0.47459 0.41241 0.83333 0.81464 0.76418 0.50000 0.48912 0.46233 0.83333 0.82158 0.79088
81.847 88.475 108.655 109.651 116.732 138.030 333.906 342.184 366.828 40.923 43.559 51.468 54.825 57.761 66.469 166.952 170.783 182.060 10.965 11.209 11.859 33.390 33.868 35.183
2.342 2.342 2.342 2.351 2.351 2.351 2.372 2.372 2.372 2.342 2.342 2.342 2.352 2.352 2.352 2.372 2.372 2.372 2.351 2.352 2.352 2.372 2.372 2.372
0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
86.836 93.868 115.277 121.075 128.895 152.412 417.313 427.658 458.458 43.418 46.214 54.605 60.538 63.779 73.395 208.655 213.443 227.537 12.108 12.377 13.094 41.731 42.328 43.971
2.490 2.490 2.490 2.598 2.598 2.598 2.908 2.908 2.908 2.490 2.490 2.490 2.598 2.598 2.598 2.908 2.908 2.908 2.598 2.598 2.598 2.908 2.908 2.908
0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
83.160 89.894 110.397 108.047 115.025 136.011 322.772 330.774 354.596 41.580 44.258 52.293 54.023 56.916 65.497 161.385 165.089 175.990 10.805 11.045 11.685 32.277 32.739 34.010
2.248 2.248 2.248 2.216 2.216 2.216 2.200 2.200 2.200 2.248 2.248 2.248 2.216 2.216 2.216 2.200 2.200 2.200 2.216 2.216 2.216 2.200 2.200 2.200
1. 982 1.982 1.982 2.616 2.616 2.616 4.502 4.502 4.502 1.982 1.982 1.982 2.616 2.616 2.616 4.502 4.502 4.502 2.616 2.616 2.616 4.502 4.502 4.502
73.484 79.434 97.552 94.488 100.591 118.944 261.307 267.786 287.072 36.742 39.108 46.209 47.244 49.774 57.278 130.653 133.651 142.477 9.449 9.659 10.219 26.131 26.504 27.533
2.181 2.181 2.181 2.135 2.135 2.135 2.040 2.040 2.040 2.181 2.181 2.181 2.135 2.135 2.135 2.040 2.040 2.040 2.135 2.135 2.135 2.040 2.040 2.040
1.339 1.339 1.339 1.723 1.723 1.723 3.036 3.036 3.036 1.339 1.339 1.339 1.723 1.723 1.723 3.036 3.036 3.036 1.723 1.723 1.723 3.036 3.036 3.036
Table 2 Values of the critical Rayleigh number Rc, critical wave number kc and Im (r1) for M/ = 0.5. Le = 0.1
Le = 1
Le = 10
Le = 100
A
B
n
b1
b3
Rc
kc
Im (r1)
Rc
kc
Im (r1)
Rc
kc
Im (r1)
Rc
kc
Im (r1)
0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 1 1 1 1 1 1 1 1 1 5 5 5 5 5 5
0.5 0.5 0.5 1 1 1 5 5 5 0.5 0.5 0.5 1 1 1 5 5 5 1 1 1 5 5 5
0 0.15 0.5 0 0.15 0.5 0 0.15 0.5 0 0.15 0.5 0 0.15 0.5 0 0.15 0.5 0 0.15 0.5 0 0.15 0.5
0.33333 0.30836 0.25109 0.25000 0.23483 0.19860 0.08333 0.08132 0.07585 0.66667 0.62632 0.53008 0.50000 0.47459 0.41241 0.16667 0.16293 0.15284 2.50000 2.44559 2.31163 0.83333 0.82158 0.79088
0.33333 0.30836 0.25109 0.50000 0.46970 0.39720 0.83333 0.81317 0.75854 0.33333 0.31316 0.26504 0.50000 0.47459 0.41241 0.83333 0.81464 0.76418 0.50000 0.48912 0.46233 0.83333 0.82158 0.79088
81.847 88.475 108.655 109.651 116.732 138.030 333.906 342.184 366.828 40.923 43.559 51.468 54.825 57.761 66.469 166.952 170.783 182.060 10.965 11.209 11.859 33.390 33.868 35.183
2.342 2.342 2.342 2.352 2.352 2.352 2.372 2.372 2.372 2.342 2.342 2.342 2.352 2.352 2.352 2.372 2.372 2.372 2.352 2.352 2.352 2.372 2.372 2.372
0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
86.836 93.868 115.277 121.075 128.895 152.412 417.313 427.658 458.458 43.418 46.214 54.605 60.538 63.779 73.395 208.655 213.443 227.537 12.108 12.377 13.094 41.731 42.328 43.971
2.490 2.490 2.490 2.598 2.598 2.598 2.908 2.908 2.908 2.490 2.490 2.490 2.598 2.598 2.598 2.908 2.908 2.908 2.598 2.598 2.598 2.908 2.908 2.908
0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
89.462 96.707 118.764 115.723 123.197 145.674 350.703 359.398 385.281 44.731 47.612 56.257 57.862 60.959 70.150 175.350 179.374 191.219 11.572 11.830 12.515 35.070 35.572 36.953
2.244 2.244 2.244 2.209 2.209 2.209 2.187 2.187 2.187 2.244 2.244 2.244 2.209 2.209 2.209 2.187 2.187 2.187 2.209 2.209 2.209 2.187 2.187 2.187
2.816 2.816 2.816 3.804 3.804 3.804 6.548 6.548 6.548 2.816 2.816 2.816 3.804 3.804 3.804 6.548 6.548 6.548 3.804 3.804 3.804 6.548 6.548 6.548
70.954 76.700 94.194 89.896 95.701 113.162 244.142 250.195 268.214 35.477 37.762 44.618 44.948 47.354 54.494 122.071 124.872 133.117 8.990 9.190 9.722 24.414 24.763 25.725
2.122 2.122 2.122 2.059 2.059 2.059 1.946 1.946 1.946 2.122 2.122 2.122 2.059 2.059 2.059 1.946 1.946 1.946 2.059 2.059 2.059 1.946 1.946 1.946
1.964 1.964 1.964 2.499 2.499 2.499 4.300 4.300 4.300 1.964 1.964 1.964 2.499 2.499 2.499 4.300 4.300 4.300 2.499 2.499 2.499 4.300 4.300 4.300
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Table 3 Values of the critical Rayleigh number Rc, critical wave number kc and Im (r1) for M/ = 0.25. Le = 0.1
Le = 1
Le = 10
Le = 100
A
B
n
b1
b3
Rc
kc
Im (r1)
Rc
kc
Im (r1)
Rc
kc
Im (r1)
Rc
kc
Im (r1)
0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 1 1 1 1 1 1 1 1 1 5 5 5 5 5 5
0.5 0.5 0.5 1 1 1 5 5 5 0.5 0.5 0.5 1 1 1 5 5 5 1 1 1 5 5 5
0 0.15 0.5 0 0.15 0.5 0 0.15 0.5 0 0.15 0.5 0 0.15 0.5 0 0.15 0.5 0 0.15 0.5 0 0.15 0.5
0.33333 0.30836 0.25109 0.25000 0.23483 0.19860 0.08333 0.08132 0.07585 0.66667 0.62632 0.53008 0.50000 0.47459 0.41241 0.1667 0.16293 0.15284 2.50000 2.44559 2.31163 0.83333 0.82158 0.79088
0.33333 0.30836 0.25109 0.50000 0.46967 0.39720 0.83333 0.81317 0.75854 0.33333 0.31316 0.26504 0.50000 0.47459 0.41241 0.83333 0.81464 0.76418 0.50000 0.48912 0.46233 0.83333 0.82158 0.79088
81.847 88.475 108.655 109.651 116.732 138.030 333.906 342.184 366.828 40.923 43.559 51.468 54.825 57.761 66.469 166.952 170.783 182.060 10.965 11.209 11.859 33.390 33.868 35.183
2.342 2.342 2.342 2.352 2.352 2.352 2.372 2.372 2.372 2.342 2.342 2.342 2.352 2.352 2.352 2.372 2.372 2.372 2.352 2.352 2.352 2.372 2.372 2.372
0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
86.836 93.868 115.277 121.075 128.895 152.412 417.313 427.658 458.458 43.418 46.214 54.605 60.538 63.779 73.395 208.655 213.443 227.537 0.000 0.000 0.000 0.000 0.000 0.000
2.490 2.490 2.490 2.598 2.598 2.598 2.908 2.908 2.908 2.490 2.490 2.490 2.598 2.598 2.598 2.908 2.908 2.908 12.108 12.377 13.094 41.731 42.328 43.971
0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 2.598 2.598 2.598 2.908 2.908 2.908
102.436 110.731 135.987 132.082 140.612 166.267 408.139 418.258 448.381 51.218 54.517 64.415 66.041 69.577 80.067 204.068 208.751 222.536 13.208 13.502 14.285 40.814 41.398 43.005
2.241 2.241 2.241 2.203 2.203 2.203 2.166 2.166 2.166 2.241 2.241 2.241 2.203 2.203 2.203 2.166 2.166 2.166 2.203 2.203 2.203 2.166 2.166 2.166
3.863 3.863 3.863 5.481 5.481 5.481 9.558 9.558 9.558 3.863 3.863 3.863 5.481 5.481 5.481 9.558 9.558 9.558 5.481 5.481 5.481 9.558 9.558 9.558
68.136 73.654 90.453 84.653 90.120 106.563 227.249 232.883 249.655 34.068 36.262 42.846 42.326 44.593 51.316 113.624 116.231 123.906 8.465 8.654 9.155 22.725 23.050 23.945
2.046 2.046 2.046 1.965 1.965 1.965 1.841 1.841 1.841 2.046 2.046 2.046 1.965 1.965 1.965 1.841 1.841 1.841 1.965 1.965 1.965 1.841 1.841 1.841
2.846 2.846 2.846 3.565 3.565 3.565 5.989 5.989 5.989 2.846 2.846 2.846 3.565 3.565 3.565 5.989 5.989 5.989 3.565 3.565 3.565 5.989 5.989 5.989
Fig. 1. Variation of Rc with A, for B = 1, n = 0, M/ = 1.
Fig. 2. Variation of Rc with B, for A = 0.5, n = 0, M/ = 1.
Fig. 3. Variation of Rc with n, for A = 0.5, B = 0.5, M/ = 1.
Fig. 4. Variation of Rc with M/, for A = 0.5, B = 0.5, n = 0.
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N.L. Scott, B. Straughan / International Journal of Heat and Mass Transfer 54 (2011) 5653–5657 Table 4 Values of the critical Rayleigh number, Rc, critical wave number, kc and Im (r1) for M/ = 1. Le = 0.1
Le = 1
Le = 10
Le = 100
A
B
n
b1
b3
Rc
kc
Im (r1)
R_c
k_c
Im (r1)
Rc
kc
Im (r1)
Rc
kc
Im (r1)
0.5 0.5 0.5 0.5 0.5 0.5 0.5 1 5
0.5 0.5 0.5 0.5 1 5 1 1 1
0 0.15 0.5 0 0 0 0 0 0
0.33333 0.30836 0.25109 0.33333 0.25000 0.08333 0.25000 0.50000 2.50000
0.33333 0.30836 0.25109 0.33333 0.50000 0.83333 0.50000 0.50000 0.50000
81.847 88.475 108.655 81.847 109.651 333.906 109.651 54.825 10.965
2.342 2.342 2.342 2.342 2.351 2.372 2.351 2.352 2.351
0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
86.836 93.868 115.277 86.836 121.075 417.313 121.075 60.538 12.108
2.490 2.490 2.490 2.490 2.598 2.908 2.598 2.598 2.598
0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
83.160 89.894 110.397 83.160 108.047 322.772 108.047 54.023 10.805
2.248 2.248 2.248 2.248 2.216 2.200 2.216 2.216 2.216
1.982 1.982 1.982 1.982 2.616 4.502 2.616 2.616 2.616
73.484 79.434 97.552 73.484 94.488 261.307 94.488 47.244 9.449
2.181 2.181 2.181 2.181 2.135 2.040 2.135 2.135 2.135
1.339 1.339 1.339 1.339 1.723 3.036 1.723 1.723 1.723
false and for certain parameter ranges oscillatory convection (Im (r) – 0) is dominant. In Figs. 1–4 for fixed parameters A, B, n, M/ we see that as the Lewis number increases from 0 the instability curve increases. This is due to stationary convection. However, at a critical value of the Lewis number the stationary convection eigenvalue ceases to be the dominant one and a second eigenvalue which has Im (r2) – 0 is the most destabilizing (although in Tables 1–4 we denote the leading eigenvalue by r1). This is why, for example in Fig. 1, the instability curve increases, but appears to then kink and decrease with further increase of Le. On the right of the curve after the kink, the instability is oscillatory convection. Figs. 1–3 show that increasing A decreases the critical Rayleigh number for a particular Lewis number, whereas increasing B or n increases the value of Rc. It appears from close examination of the values in Tables 1–3 that, when n = 0, increasing A by a factor of n exactly decreases Rc by a factor of n. Fig. 4 shows that when stationary convection is dominant altering the value of M/ had no impact on Rc. One would expect this as when r is set equal to zero M/ does not appear in (14), (17) or (18). The critical Lewis number, the one at which oscillatory convection begins to dominate stationary convection, is dependent on B and M/, this is shown in Figs. 2 and 4, but is not dependent on A or n, shown in Figs. 1 and 3. This is further demonstrated in Table 4, where it is seen that changing A or n has no effect on the imaginary part of r. However, increasing B increases Im (r1) and therefore oscillatory convection becomes dominant at a lower value of Le. Tables 1–3 show that decreasing M/ increases the imaginary part of r and therefore oscillatory convection becomes dominant at a lower Lewis number.
References [1] J.T.H. Andres, S.S.S. Cardoso, Onset of convection in a porous medium in the presence of chemical reaction, Phys. Rev. E 83 (2011) 046312. [2] J.J. Dongarra, B. Straughan, D.W. Walker, Chebyshev-tau-QZ algorithm methods for calculating spectra of hydrodynamic stability problems, Appl. Numer. Math. 22 (1996) 399–434. [3] I.A. Eltayeb, E.A. Hamza, J.A. Jervase, E.V. Krishan, D.E. Loper, Compositional convection in the presence of a magnetic field. I. a single interface, Proc. Roy. Soc. London A 460 (2004) 3505–3528. [4] I.A. Eltayeb, E.A. Hamza, J.A. Jervase, E.V. Krishan, D.E. Loper, Compositional convection in the presence of a magnetic field. II. cartesian plume, Proc. Roy. Soc. London A 461 (2005) 2605–2633. [5] A. Mahdy, Effect of chemical reaction and heat generation or absorption on double diffusive convection from a vertical truncated cone in porous media with variable viscosity, Int. Commun. Heat and Mass Transfer 37 (2010) 548– 554. [6] M.S. Malashetty, B.S. Biradar, The onset of double diffusive reaction– convection in an anisotropic porous layer, Phys. Fluids 23 (2011) 064102. [7] G. McKay, Onset of buoyancy-driven convection in superposed reacting fluid and porous layers, J. Eng. Math. 33 (1998) 31–46. [8] C.L. McTaggart, B. Straughan, Chemical surface reactions and nonlinear stability by the method of energy, SIAM J. Math. Anal. 17 (1986) 342–351. [9] H.D. Nguyen, S. Paik, R.W. Douglass, I. Pop, Unsteady non-darcy reactiondriven flow from an anisotropic cylinder in porous media, Chem. Eng. Sci. 51 (1996) 4963–4977. [10] L.E. Payne, B. Straughan, A naturally efficient technique for porous convection stability with non-trivial boundary condition, Int. J. Numer. Anal. Geomech. 24 (2000) 815–836. [11] A. Postelnicu, Onset of convection in a horizontal porous layer driven by catalytic surface reaction on the lower wall, Int. J. Heat and Mass Transfer 52 (2009) 2466–2470. [12] M.M. Rahman, M. Al-Lawatia, Effects of higher order chemical reaction on micropolar fluid flow on a power law permeable stretched sheet with variable concentration in a porous medium, Can. J. Chem. Eng. 88 (2010) 23–32. [13] B. Straughan, Stability and wave motion in porous media, Applied Mathematical Sciences, Springer, New York, 2008.
Contents lists available at SciVerse ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Technical Note
Convection in a porous layer with a surface reaction q Nicola L. Scott, B. Straughan ⇑ Department of Mathematics, University of Durham, DH1 3LE, UK
a r t i c l e
i n f o
Article history: Received 12 August 2011 Accepted 12 August 2011 Available online 14 September 2011 Keywords: Porous convection Chemical reactions Oscillatory convection
a b s t r a c t We develop a problem recently studied by Postelnicu in which a horizontal layer of porous material is subject to a chemical reaction at the base of the layer. The porous medium is of Darcy type and the layer is saturated with a non-isothermal liquid containing a concentration of the reacting chemical. The equations are developed and then it is shown that if the reaction parameter B of Postelnicu exceeds a critical value convection commences but as oscillatory convection, not stationary convection, as previously thought. Ó 2011 Elsevier Ltd. All rights reserved.
1. Introduction The study of thermal convection problems driven by chemical reactions is dominated by applications and has recently been witnessing much activity. For convection in a pure fluid McTaggart and Straughan [8] developed an energy method to yield global nonlinear stability thresholds and further references may be found therein. Within the field of convection in porous media the area of chemical reactions is perhaps newer and richer. McKay [7] studies the effect of chemical reaction upon convection in a porous layer overlain by a viscous fluid. Rahman and AlLawatia [12] investigate the effect of high order chemical reactions while Mahdy [5] continues the analysis in double diffusive convection. Andres and Cardoso [1] study instability in the presence of a chemical reaction in a finite layer and link this to the important environmental topic of carbon dioxide storage. Phase change effects may also play a prominent role, for example in compositional convection where material is deposited at the Earth’s solid inner core, see e.g. Eltayeb et al. [3,4]. Nguyen et al. [9] study flow driven by a chemical reaction on an anisotropic porous cylinder, Malashetty and Biradar [6] analyse the onset of double diffusive convection with a chemical reaction present in an anisotropc porous layer. Of particular interest to the present article is the paper by Postelnicu [11] who models the situation where a chemical reaction at the base of a horizontal layer of porous material gives rise to a convective instability. Our major goal here is to revisit the situation studied by Postelnicu [11] where we have a layer of Darcy porous material saturated with a non-isothermal, incompressible viscous fluid, in which is
q
This work was supported in part by a grant from the Leverhulme Trust, ‘‘Tipping points: mathematics, metaphors and meanings’’, and in part by a DTA award from EPSRC. ⇑ Corresponding author. E-mail addresses: [email protected] (N.L. Scott), [email protected] (B. Straughan). 0017-9310/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijheatmasstransfer.2011.08.032
dissolved a chemical of concentration C. As there are several misprints in Postelnicu [11] we rederive the basic equations at the outset. Firstly we assume Darcy’s Law and then
rp ¼
l K
v qgk;
ð1Þ
where v, p are velocity and pressure, q, g are the density and gravity (acting in the negative z-direction), k = (0, 0, 1), l is the dynamic viscosity of the fluid and K is the permeability of the porous medium. The Boussinesq approximation is adopted whereby the density is assumed constant everywhere except in the body force term in (1). In line with Postelnicu [11], the density in (1) is taken to depend only on the temperature, T, and not on the concentration C. This dependence is assumed linear so that
q ¼ q0 ð1 aðT T 0 ÞÞ;
ð2Þ
q0 being the density at reference temperature T0, and a is the coefficient of thermal expansion. Insertion of (2) in (1) leads to the momentum equation p;i ¼
l K
v i q0 gð1 aðT T 0 ÞÞki ;
ð3Þ
where standard indicial notation is employed. The fluid is incompressible so that
v i;i ¼ 0:
ð4Þ
The temperature field and the concentration field satisfy the equations, see Straughan [13],
1 T ;t þ v i T ;i ¼ jDT; M /C ;t þ v i C ;i ¼ /kc DC;
ð5Þ
where M = (q0cp)f/(q0c)m with (q0c)m = /(q0cp)f + (1 /)(qc)s and j = km/(q0cp)f is the thermal diffusivity of the porous medium, km
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being given by km = js(1 /) + jf/. The terms cp and cs are the specific heat at constant pressure of the fluid and the specific heat of the solid, respectively, and / is the porosity of the solid. The saturated porous medium occupies the infinite layer fðx; yÞ 2 R2 g fz 2 ð0; hÞg and on the upper wall the temperature and reactant concentration are fixed and constant with zero mass flux there. Thus the upper boundary conditions are
h;t þ b1 w ¼ Dh; 1 Dc; Le Dw RD h ¼ 0;
ð12Þ
M/c;t þ b3 w ¼
On the lower wall (z = 0) there is an exothermic reaction where the reactant is converted into an inert product at a rate, r, where
where D⁄ = @ 2/@x2 + @ 2/@y2 is the horizontal Laplacian, Le = j/(/kc) is the Lewis number and the Rayleigh number, R, is defined by R = Khgq0aTU/(jl). By standard linear analysis we wish to find time dependent horizontally periodic solutions, therefore w, h and / take the Fourier form
E r ¼ k0 C exp ; RT
w¼
T ¼ TU;
v i ni ¼ w ¼ 0;
C ¼ CU ;
on z ¼ h:
ð6Þ
ð7Þ
k0 is a rate constant, R⁄ is the universal gas constant and E is the activation energy of the reaction. The rate of change of temperature is proportional to the heat of the reaction, Q, the rate at which it occurs, and inversely proportional to the rate at which the heat is conducted away from the surface, kT. The rate of change of the reactant concentration is proportional to the rate of the reaction and inversely proportional to the porosity of the solid and the diffusivity of the reactant, kc. There is zero mass flux across the lower wall, hence the boundary lower boundary conditions are
@T E ¼ Qk0 C exp ; @z RT @C E ¼ k0 C exp ; /kc @z RT kT
v i ni ¼ w ¼ 0
ð8Þ
Þ, the fluid velocity is zero and the ; T; C; p At the steady state, ðv temperature and reactant concentration are independent of time; v i ¼ 0; T ;t ¼ 0; C ;t ¼ 0. If we now consider the temperature and reactant concentration fields to be functions of z only (5) immediately imply that
ð9Þ
ðzÞ, is found from (3). The steady pressure, p We now introduce small perturbations ui, h, c, p to the steady state such that
v i ¼ vi þ ui ;
T ¼ T þ h;
C ¼ C þ c;
þ p: p¼p
h¼
1 X
c¼
1 X
rH þ b1 W ¼ ðD2 k2 ÞH;
3. Perturbation boundary conditions After non-dimensionalising using (11) the upper boundary conditions are
w ¼ 0;
l p;i ¼ ui þ q0 g ahki ; K
where w = u3. Eq. (10) are linearised by considering the terms uih,i and uic,i to be small and we may therefore neglect them. The pressure term is removed by taking curl curl of (10)4 and retaining the third component (i = 3). The non-dimensionalisations 2
t ¼ jhM t ; C ¼ CU C ;
T ¼ T U T ; p ¼ jKl p
are employed to find
T ¼ 1 on z ¼ 1
@T n ¼ AC exp ; @z T @C n ; ¼ BC exp @z T
A¼
Qhk0 C u ; kT T U
ð15Þ
B¼
hk0 ; /kc
ð16Þ
n¼
E R T U
are non-dimensional coefficients. Eqs. (15), (16) and (19) imply that the linearised perturbation boundary conditions are
W ¼ H ¼ U ¼ 0;
on z ¼ 1
ð17Þ
and
W ¼ 0; ð11Þ
C ¼ 1;
and the lower boundary conditions (on z = 0) are
where
ð10Þ
ð14Þ
where D = d/dz.
1 h;t þ b1 w þ ui h;i ¼ jDh; M /c;t þ b3 w þ ui c;i ¼ /kc Dc;
erj t fj ðx; yÞUj ðzÞ;
where f is some function such that D⁄f = k2f for a wave number k, and rj is an eigenvalue (growth rate). Since we are interested in the first instance of instability, we choose one mode of the Fourier expansions to find a typical eigenvalue which satisfies the system of equations
w ¼ 0;
xi ¼ hxi ; v i ¼ jh v i ;
ð13Þ
j¼1
Eqs. (3)–(5) then become
ui;i ¼ 0;
erj t fj ðx; yÞHj ðzÞ;
j¼1
M/rU þ b3 W ¼
2. Basic solution and perturbation equations
CðzÞ ¼ b3 z þ b4 :
erj t fj ðx; yÞW j ðzÞ;
j¼1
1 2 2 ðD k ÞU; Le 2 2 ðD2 k ÞW þ Rk H ¼ 0;
on z ¼ 0:
TðzÞ ¼ b1 z þ b2 ;
1 X
dH ¼ AU; dz
dU ¼ BU on z ¼ 0; dz
ð18Þ
where w is now the third component of u. Evaluating the steady state, (9), on the boundary walls one obtains the relations
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1 ¼ b1 þ b2 ;
Eq. (20) may be solved for b1 and b3 for specific values of A, B and n.
1 ¼ b3 þ b4 ;
n ; b1 ¼ Ab4 exp b2 n : b3 ¼ Bb4 exp b2
ð19Þ
We may eliminate b2 and b4 using (19)1 and (19)2, then use (19)3 and (19)4 to give two coupled equations for b1 and b3;
n ; b1 ¼ Að1 b3 Þ exp 1 b1 n : b3 ¼ Bð1 b3 Þ exp 1 b1
ð20Þ
4. Numerical results We employed both the Natural D and D2 Chebyshev-Tau methods to numerically compute the critical Rayleigh number for various values of the boundary parameters, A, B, n, for (14) with boundary conditions (17) and (18). Details of these methods may be found in [10,2], respectively, and complete agreement was found between them. We set M/ = 1, 0.5, 0.25 and obtained the values given in Tables 1–3, respectively. Postelnicu [11] claims that exchange of stability holds so that one may select r = 0 in Eq. (14). This is seen by our results to be
Table 1 Values of the critical Rayleigh number Rc, critical wave number kc and Im (r1) for M/ = 1. Le = 0.1
Le = 1
Le = 10
Le = 100
A
B
n
b1
b3
Rc
kc
Im (r1)
Rc
kc
Im (r1)
Rc
kc
Im (r1)
Rc
kc
Im (r1)
0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 1 1 1 1 1 1 1 1 1 5 5 5 5 5 5
0.5 0.5 0.5 1 1 1 5 5 5 0.5 0.5 0.5 1 1 1 5 5 5 1 1 1 5 5 5
0 0.15 0.5 0 0.15 0.5 0 0.15 0.5 0 0.15 0.5 0 0.15 0.5 0 0.15 0.5 0 0.15 0.5 0 0.15 0.5
0.33333 0.30836 0.25109 0.25000 0.23483 0.19860 0.08333 0.08132 0.07585 0.66667 0.62632 0.53008 0.50000 0.47459 0.41241 0.16667 0.16293 0.15284 2.50000 2.44559 2.31163 0.83333 0.82158 0.79088
0.33333 0.30836 0.25109 0.50000 0.46967 0.39720 0.83333 0.81317 0.75854 0.33333 0.31316 0.26504 0.50000 0.47459 0.41241 0.83333 0.81464 0.76418 0.50000 0.48912 0.46233 0.83333 0.82158 0.79088
81.847 88.475 108.655 109.651 116.732 138.030 333.906 342.184 366.828 40.923 43.559 51.468 54.825 57.761 66.469 166.952 170.783 182.060 10.965 11.209 11.859 33.390 33.868 35.183
2.342 2.342 2.342 2.351 2.351 2.351 2.372 2.372 2.372 2.342 2.342 2.342 2.352 2.352 2.352 2.372 2.372 2.372 2.351 2.352 2.352 2.372 2.372 2.372
0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
86.836 93.868 115.277 121.075 128.895 152.412 417.313 427.658 458.458 43.418 46.214 54.605 60.538 63.779 73.395 208.655 213.443 227.537 12.108 12.377 13.094 41.731 42.328 43.971
2.490 2.490 2.490 2.598 2.598 2.598 2.908 2.908 2.908 2.490 2.490 2.490 2.598 2.598 2.598 2.908 2.908 2.908 2.598 2.598 2.598 2.908 2.908 2.908
0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
83.160 89.894 110.397 108.047 115.025 136.011 322.772 330.774 354.596 41.580 44.258 52.293 54.023 56.916 65.497 161.385 165.089 175.990 10.805 11.045 11.685 32.277 32.739 34.010
2.248 2.248 2.248 2.216 2.216 2.216 2.200 2.200 2.200 2.248 2.248 2.248 2.216 2.216 2.216 2.200 2.200 2.200 2.216 2.216 2.216 2.200 2.200 2.200
1. 982 1.982 1.982 2.616 2.616 2.616 4.502 4.502 4.502 1.982 1.982 1.982 2.616 2.616 2.616 4.502 4.502 4.502 2.616 2.616 2.616 4.502 4.502 4.502
73.484 79.434 97.552 94.488 100.591 118.944 261.307 267.786 287.072 36.742 39.108 46.209 47.244 49.774 57.278 130.653 133.651 142.477 9.449 9.659 10.219 26.131 26.504 27.533
2.181 2.181 2.181 2.135 2.135 2.135 2.040 2.040 2.040 2.181 2.181 2.181 2.135 2.135 2.135 2.040 2.040 2.040 2.135 2.135 2.135 2.040 2.040 2.040
1.339 1.339 1.339 1.723 1.723 1.723 3.036 3.036 3.036 1.339 1.339 1.339 1.723 1.723 1.723 3.036 3.036 3.036 1.723 1.723 1.723 3.036 3.036 3.036
Table 2 Values of the critical Rayleigh number Rc, critical wave number kc and Im (r1) for M/ = 0.5. Le = 0.1
Le = 1
Le = 10
Le = 100
A
B
n
b1
b3
Rc
kc
Im (r1)
Rc
kc
Im (r1)
Rc
kc
Im (r1)
Rc
kc
Im (r1)
0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 1 1 1 1 1 1 1 1 1 5 5 5 5 5 5
0.5 0.5 0.5 1 1 1 5 5 5 0.5 0.5 0.5 1 1 1 5 5 5 1 1 1 5 5 5
0 0.15 0.5 0 0.15 0.5 0 0.15 0.5 0 0.15 0.5 0 0.15 0.5 0 0.15 0.5 0 0.15 0.5 0 0.15 0.5
0.33333 0.30836 0.25109 0.25000 0.23483 0.19860 0.08333 0.08132 0.07585 0.66667 0.62632 0.53008 0.50000 0.47459 0.41241 0.16667 0.16293 0.15284 2.50000 2.44559 2.31163 0.83333 0.82158 0.79088
0.33333 0.30836 0.25109 0.50000 0.46970 0.39720 0.83333 0.81317 0.75854 0.33333 0.31316 0.26504 0.50000 0.47459 0.41241 0.83333 0.81464 0.76418 0.50000 0.48912 0.46233 0.83333 0.82158 0.79088
81.847 88.475 108.655 109.651 116.732 138.030 333.906 342.184 366.828 40.923 43.559 51.468 54.825 57.761 66.469 166.952 170.783 182.060 10.965 11.209 11.859 33.390 33.868 35.183
2.342 2.342 2.342 2.352 2.352 2.352 2.372 2.372 2.372 2.342 2.342 2.342 2.352 2.352 2.352 2.372 2.372 2.372 2.352 2.352 2.352 2.372 2.372 2.372
0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
86.836 93.868 115.277 121.075 128.895 152.412 417.313 427.658 458.458 43.418 46.214 54.605 60.538 63.779 73.395 208.655 213.443 227.537 12.108 12.377 13.094 41.731 42.328 43.971
2.490 2.490 2.490 2.598 2.598 2.598 2.908 2.908 2.908 2.490 2.490 2.490 2.598 2.598 2.598 2.908 2.908 2.908 2.598 2.598 2.598 2.908 2.908 2.908
0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
89.462 96.707 118.764 115.723 123.197 145.674 350.703 359.398 385.281 44.731 47.612 56.257 57.862 60.959 70.150 175.350 179.374 191.219 11.572 11.830 12.515 35.070 35.572 36.953
2.244 2.244 2.244 2.209 2.209 2.209 2.187 2.187 2.187 2.244 2.244 2.244 2.209 2.209 2.209 2.187 2.187 2.187 2.209 2.209 2.209 2.187 2.187 2.187
2.816 2.816 2.816 3.804 3.804 3.804 6.548 6.548 6.548 2.816 2.816 2.816 3.804 3.804 3.804 6.548 6.548 6.548 3.804 3.804 3.804 6.548 6.548 6.548
70.954 76.700 94.194 89.896 95.701 113.162 244.142 250.195 268.214 35.477 37.762 44.618 44.948 47.354 54.494 122.071 124.872 133.117 8.990 9.190 9.722 24.414 24.763 25.725
2.122 2.122 2.122 2.059 2.059 2.059 1.946 1.946 1.946 2.122 2.122 2.122 2.059 2.059 2.059 1.946 1.946 1.946 2.059 2.059 2.059 1.946 1.946 1.946
1.964 1.964 1.964 2.499 2.499 2.499 4.300 4.300 4.300 1.964 1.964 1.964 2.499 2.499 2.499 4.300 4.300 4.300 2.499 2.499 2.499 4.300 4.300 4.300
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Table 3 Values of the critical Rayleigh number Rc, critical wave number kc and Im (r1) for M/ = 0.25. Le = 0.1
Le = 1
Le = 10
Le = 100
A
B
n
b1
b3
Rc
kc
Im (r1)
Rc
kc
Im (r1)
Rc
kc
Im (r1)
Rc
kc
Im (r1)
0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 1 1 1 1 1 1 1 1 1 5 5 5 5 5 5
0.5 0.5 0.5 1 1 1 5 5 5 0.5 0.5 0.5 1 1 1 5 5 5 1 1 1 5 5 5
0 0.15 0.5 0 0.15 0.5 0 0.15 0.5 0 0.15 0.5 0 0.15 0.5 0 0.15 0.5 0 0.15 0.5 0 0.15 0.5
0.33333 0.30836 0.25109 0.25000 0.23483 0.19860 0.08333 0.08132 0.07585 0.66667 0.62632 0.53008 0.50000 0.47459 0.41241 0.1667 0.16293 0.15284 2.50000 2.44559 2.31163 0.83333 0.82158 0.79088
0.33333 0.30836 0.25109 0.50000 0.46967 0.39720 0.83333 0.81317 0.75854 0.33333 0.31316 0.26504 0.50000 0.47459 0.41241 0.83333 0.81464 0.76418 0.50000 0.48912 0.46233 0.83333 0.82158 0.79088
81.847 88.475 108.655 109.651 116.732 138.030 333.906 342.184 366.828 40.923 43.559 51.468 54.825 57.761 66.469 166.952 170.783 182.060 10.965 11.209 11.859 33.390 33.868 35.183
2.342 2.342 2.342 2.352 2.352 2.352 2.372 2.372 2.372 2.342 2.342 2.342 2.352 2.352 2.352 2.372 2.372 2.372 2.352 2.352 2.352 2.372 2.372 2.372
0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
86.836 93.868 115.277 121.075 128.895 152.412 417.313 427.658 458.458 43.418 46.214 54.605 60.538 63.779 73.395 208.655 213.443 227.537 0.000 0.000 0.000 0.000 0.000 0.000
2.490 2.490 2.490 2.598 2.598 2.598 2.908 2.908 2.908 2.490 2.490 2.490 2.598 2.598 2.598 2.908 2.908 2.908 12.108 12.377 13.094 41.731 42.328 43.971
0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 2.598 2.598 2.598 2.908 2.908 2.908
102.436 110.731 135.987 132.082 140.612 166.267 408.139 418.258 448.381 51.218 54.517 64.415 66.041 69.577 80.067 204.068 208.751 222.536 13.208 13.502 14.285 40.814 41.398 43.005
2.241 2.241 2.241 2.203 2.203 2.203 2.166 2.166 2.166 2.241 2.241 2.241 2.203 2.203 2.203 2.166 2.166 2.166 2.203 2.203 2.203 2.166 2.166 2.166
3.863 3.863 3.863 5.481 5.481 5.481 9.558 9.558 9.558 3.863 3.863 3.863 5.481 5.481 5.481 9.558 9.558 9.558 5.481 5.481 5.481 9.558 9.558 9.558
68.136 73.654 90.453 84.653 90.120 106.563 227.249 232.883 249.655 34.068 36.262 42.846 42.326 44.593 51.316 113.624 116.231 123.906 8.465 8.654 9.155 22.725 23.050 23.945
2.046 2.046 2.046 1.965 1.965 1.965 1.841 1.841 1.841 2.046 2.046 2.046 1.965 1.965 1.965 1.841 1.841 1.841 1.965 1.965 1.965 1.841 1.841 1.841
2.846 2.846 2.846 3.565 3.565 3.565 5.989 5.989 5.989 2.846 2.846 2.846 3.565 3.565 3.565 5.989 5.989 5.989 3.565 3.565 3.565 5.989 5.989 5.989
Fig. 1. Variation of Rc with A, for B = 1, n = 0, M/ = 1.
Fig. 2. Variation of Rc with B, for A = 0.5, n = 0, M/ = 1.
Fig. 3. Variation of Rc with n, for A = 0.5, B = 0.5, M/ = 1.
Fig. 4. Variation of Rc with M/, for A = 0.5, B = 0.5, n = 0.
5657
N.L. Scott, B. Straughan / International Journal of Heat and Mass Transfer 54 (2011) 5653–5657 Table 4 Values of the critical Rayleigh number, Rc, critical wave number, kc and Im (r1) for M/ = 1. Le = 0.1
Le = 1
Le = 10
Le = 100
A
B
n
b1
b3
Rc
kc
Im (r1)
R_c
k_c
Im (r1)
Rc
kc
Im (r1)
Rc
kc
Im (r1)
0.5 0.5 0.5 0.5 0.5 0.5 0.5 1 5
0.5 0.5 0.5 0.5 1 5 1 1 1
0 0.15 0.5 0 0 0 0 0 0
0.33333 0.30836 0.25109 0.33333 0.25000 0.08333 0.25000 0.50000 2.50000
0.33333 0.30836 0.25109 0.33333 0.50000 0.83333 0.50000 0.50000 0.50000
81.847 88.475 108.655 81.847 109.651 333.906 109.651 54.825 10.965
2.342 2.342 2.342 2.342 2.351 2.372 2.351 2.352 2.351
0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
86.836 93.868 115.277 86.836 121.075 417.313 121.075 60.538 12.108
2.490 2.490 2.490 2.490 2.598 2.908 2.598 2.598 2.598
0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
83.160 89.894 110.397 83.160 108.047 322.772 108.047 54.023 10.805
2.248 2.248 2.248 2.248 2.216 2.200 2.216 2.216 2.216
1.982 1.982 1.982 1.982 2.616 4.502 2.616 2.616 2.616
73.484 79.434 97.552 73.484 94.488 261.307 94.488 47.244 9.449
2.181 2.181 2.181 2.181 2.135 2.040 2.135 2.135 2.135
1.339 1.339 1.339 1.339 1.723 3.036 1.723 1.723 1.723
false and for certain parameter ranges oscillatory convection (Im (r) – 0) is dominant. In Figs. 1–4 for fixed parameters A, B, n, M/ we see that as the Lewis number increases from 0 the instability curve increases. This is due to stationary convection. However, at a critical value of the Lewis number the stationary convection eigenvalue ceases to be the dominant one and a second eigenvalue which has Im (r2) – 0 is the most destabilizing (although in Tables 1–4 we denote the leading eigenvalue by r1). This is why, for example in Fig. 1, the instability curve increases, but appears to then kink and decrease with further increase of Le. On the right of the curve after the kink, the instability is oscillatory convection. Figs. 1–3 show that increasing A decreases the critical Rayleigh number for a particular Lewis number, whereas increasing B or n increases the value of Rc. It appears from close examination of the values in Tables 1–3 that, when n = 0, increasing A by a factor of n exactly decreases Rc by a factor of n. Fig. 4 shows that when stationary convection is dominant altering the value of M/ had no impact on Rc. One would expect this as when r is set equal to zero M/ does not appear in (14), (17) or (18). The critical Lewis number, the one at which oscillatory convection begins to dominate stationary convection, is dependent on B and M/, this is shown in Figs. 2 and 4, but is not dependent on A or n, shown in Figs. 1 and 3. This is further demonstrated in Table 4, where it is seen that changing A or n has no effect on the imaginary part of r. However, increasing B increases Im (r1) and therefore oscillatory convection becomes dominant at a lower value of Le. Tables 1–3 show that decreasing M/ increases the imaginary part of r and therefore oscillatory convection becomes dominant at a lower Lewis number.
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