The effect of a horizontal pressure gradient on the onset of a Darcy–Bénard convection in thermal non-equilibrium conditions

International Journal of Heat and Mass Transfer 53 (2010) 68–75

Contents lists available at ScienceDirect

International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt

The effect of a horizontal pressure gradient on the onset of a Darcy–Bénard convection in thermal non-equilibrium conditions Adrian Postelnicu Department of Thermal Engineering and Fluid Mechanics, Transilvania University of Brasov, Bd Eroilor 29, 500036 Brasov, Romania

a r t i c l e

i n f o

Article history: Received 14 July 2009 Accepted 5 October 2009 Available online 2 November 2009 Keywords: Porous layer Thermal non-equilibrium Convection onset

a b s t r a c t In this paper there is studied the effect of a horizontal pressure gradient on the onset of Darcy–Bénard convection in a fluid-saturated porous layer heated from below, when the fluid and solid phases are not in local equilibrium. In the context of a linearized stability analysis, the problem is transformed into an eigenvalue equation. The problem, when cast in dimensionless form, contains three parameters (the pressure gradient, the porosity-scaled conductivity ratio and the scaled inter-phase heat transfer coefficient). This problem is solved numerically by using two methods: Galerkin approach and the numerical solver dsolve from Maple and comparisons between these methods are performed. Critical values of Rayleigh number, wave number and frequency are obtained for various values of the problem parameters. Ó 2009 Elsevier Ltd. All rights reserved.

1. Introduction The problem of convection onset in a porous layer heated from below dates back since the classical works by Horton and Rogers [1] and Lapwood [2]. A critical review of the state-of-the art in this area of research was done by Rees [3]. We restrict our literature survey only to those papers dealing with the onset of convection in horizontal porous layers saturated with Newtonian fluids. Various themes and combined effects were studied in the literature pertaining to the above mentioned problem. The effect of a moving thermal wave on Bénard convection in a horizontal saturated porous layer was analyzed by Mamou et al. [4]. The same configuration was analyzed by Banu and Rees [5], but in the frame of a weak nonlinear stability analysis.. On the other hand, in a porous medium the volume averaged temperatures of the solid and fluid phases are generally different from one another and this is termed as local thermal non-equilibrium (LTNE). Basic information on the LTNE in porous medium convection can be found in an excellent review by Rees and Pop [6]. Banu and Rees [7] were able to find how the onset criterion for convection in a horizontal fluid layer is affected by thermal nonequilibrium conditions, by adopting a two-equation model for the separate modelling of the solid and fluid phase temperature fields in the fluid-saturated porous medium. Postelnicu and Rees [8] extended the work by Banu and Rees [7], by including the boundary effects as modelled by the Brinkman terms. The form-drag was included also, but these authors shown

E-mail address: [email protected] 0017-9310/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijheatmasstransfer.2009.10.006

that these terms have no effect on stability criteria, since the basic state whose stability is being analyzed was one of no flow. Postelnicu [9] continued this study, by taking into account isothermal rigid boundaries instead of stress-free boundaries. Because an analytical approach was no more possible, he used a Galerkin approach in order to find the critical Darcy–Rayleigh number and wave number at which the convection occurs. Malashetty et al. [10] performed an analytical study on the stability of a horizontal anisotropic porous layer heated from below and cooled from above, in thermal non-equilibrium conditions, using a Darcy model, in conjunction with a two-field model for the energy equation (for the fluid and solid phases). In a subsequent paper, Malashetty et al. [11] used the Lapwood–Brinkmann model for the momentum equation, also in LTNE conditions. Malashetty and Heera [12] examined very recently the effect of LTNE on double diffusion convection in a fluid-saturated sparsely packed porous layer, heated from below and cooled from above. A Brinkman model was again employed and all the computations were performed analytically, both in linear and nonlinear analysis, due to the typical stress-free boundary conditions. Postelnicu [13] dealt with the effect of inertia on the onset of mixed convection in a porous layer using a thermal non-equilibrium model. The steady mixed convection in in a vertical porous layer using a thermal non-equilibrium model was analyzed by Saeid [14]. Prats [15] investigated the effect of horizontal fluid motion on thermally induced convection currents in porous medium. When thermal equilibrium conditions hold, he shown that the convection pattern not only moves in the direction of the basic flow, but it also travels at exactly the same speed. Further information on the effect of a horizontal pressure gradient on the onset of Darcy–Brinkman

69

A. Postelnicu / International Journal of Heat and Mass Transfer 53 (2010) 68–75

Nomenclature c d f g

specific heat depth of the porous layer y-dependent part of the perturbed stream function y-dependent part of the perturbed fluid phase temperature. Also, acceleration due to gravity y-dependent part of the perturbed solid phase temperature. Also the inter-phase heat transfer coefficient scaled inter-phase heat transfer coefficient horizontal wave number. With subscript, thermal conductivity permeability Nield number pressure Darcy–Rayleigh number time temperature Darcian velocities in the x- and y-direction dimensionless Cartesian co-ordinates taken along the lower surface of the porous layer and normal to it, respectively

h H k K Ni p R t T u, v x, y

Greek letters diffusivity ratio b coefficient of cubical expansion e porosity

convection in a fluid-saturated porous layer heated from below, in local thermal equilibrium (LTE), can be found in [3]. The present study is aimed at examining the effect of a horizontal pressure gradient on the onset of convection in conditions of thermal local non-equilibrium. A Darcy formulation is adopted and a linear stability analysis is carried out. 2. Analysis We consider the classical Darcy–Bénard problem, consisting of an infinite horizontal fluid-saturate porous layer sandwiched between the planes y = 0 and y = d, which are heated isothermally at the temperatures Th and Tc respectively, where Th > Tc. The solid and fluid phases are not in thermal equilibrium and a pressure gradient acts in the horizontal direction. The governing equations are

 K @p ¼ ; u l @ y @T

ð1Þ  q gbK K @p v ¼   þ f ðT f  T c Þ l @x l 

@T

@T

eðqcÞf f þ ðqcÞf u f þ v f @x @y @t ¼ ekf

@2Tf @2Tf þ 2 @ x2 @y

q u U h

H

c

Subscripts c cold f fluid phase h hot s solid phase Superscripts ‘ differentiation with respect to y

2

ðqcÞf d 1 1 t; u ¼ ekf u ; t ¼ ; x ¼ x; y ¼ y d d kf ðqcÞf d p¼

ð2Þ

lkf  Tf  Tc Ts  Tc ; u¼ p; h ¼ ðqcÞf K Th  Tc Th  Tc

ð3Þ !  hðT s  T f Þ

ð4Þ

see Nield and Bejan [16], where the LTNE is taken into account by a two-field model that describes the fluid and solid phase energy equations separately. The bars denote dimensional quantities and the notations are usual and are listed in the Nomenclature. Eqs. (1)–(4) are non-dimensionalised using the transformations

ð5Þ

ð6Þ

@h @h @h @ 2 h @ 2 h þ þ Hðu  hÞ þu þv ¼ @t @x @y @x2 @y2

ð7Þ

a

@u @2u @2u ¼ 2 þ 2 þ cHðh  uÞ @t @x @y

ð8Þ

where the stream function w was introduced, according to the relationships

@w ; @y

v¼

@w @x

ð9Þ

In Eqs. (6)–(8), the following dimensionless constants were introduced 2

þ hðT s  T f Þ

ekf  v; ðqcÞf d

@2w @2w @h þ ¼R @x2 @y2 @x

H¼

!

v¼

to give

u



@T s @2T s @2T s þ 2 ð1  eÞðqcÞs  ¼ ð1  eÞks  @ x2 @y @t

stream function perturbation of the stream function dimensionless pressure gradient density dimensionless temperature of the fluid phase in the basic conduction state perturbation of the solid phase temperature dimensionless temperature of the fluid phase in the basic conduction state perturbation of the fluid phase temperature porosity-scaled conductivity ratio

W P

Acronyms LTE local thermal equilibrium LTNE local thermal non-equilibrium

a

 @ v @u þ ¼0  @ x @ y

w

hd ; ekf

c¼

ekf ; ð1  eÞks

a¼

ðqcÞs kf  ; ðqcÞf ks

R¼

qf gbðT h  T c ÞKd elf kf ð10Þ

which are the scaled inter-phase heat transfer coefficient, a porosity-modified conductivity ratio, the diffusivity ratio and the Darcy–Rayleigh number based on the fluid properties. It should be noted that Vadasz [17] defined the Nield number as eÞks , so that we may define Ni ¼ ð1 hd2  H ¼ 1=Nif , where Nif is the fluid related number, defined as ek Nif ¼ hdf2 , se also Malashetty and Hera [12],  or alternatively, Hc ¼ 1=Ni , with Ni given as above.

70

A. Postelnicu / International Journal of Heat and Mass Transfer 53 (2010) 68–75

Given the form of (5), the boundary conditions for (6)–(8) are

w ¼ h ¼ u ¼ 1 on y ¼ 0 and y ¼ 1

ð11Þ

If the non-dimensional horizontal pressure gradient has the magnitude P, the basic state is characterized by u ¼ P. So, the basic conduction profile, whose stability is the subject of this study, is given by

hb ¼ ub ¼ 1  y

Multiplying (19)–(21) by sinpy, and integrating from 0 to 1, we obtain a homogeneous algebraic system which has non-zero solutions if

 2  k þ p2    ik   0

ikR 2

ðk þ p2 þ H þ ipk þ kÞ2

cH

    ¼0 H  2 ðk þ p2 þ cH þ akÞ  0

ð12Þ

ð24Þ

where we have chosen the constant of integration for the stream function to be such that w is antisymmetric about y = 1/2, see [3]. The basic state given by (12) is perturbed by setting:

It is readily seen that the stationary convection is not possible, so only oscillatory convection must be studied in this problem. Looking for neutrally stable disturbances, i.e. Re(k) = 0, we set k = ix in (24). Equating to zero the real and imaginary parts of (24), we obtain the following equations

wb ¼ Pðy  1=2Þ;

w ¼ Pðy  1=2Þ þ W;

h ¼ 1  y þ H; u ¼ 1  y þ U

ð13Þ

h i 2 2 2 ðk þ p2 Þ ðk þ p2 þ HÞðk þ p2 þ cHÞ  cH2  ðPk þ xÞx

and after linearization, we obtain

@2W @2W @H þ 2 ¼R @x2 @y @x

ð14Þ

@H @2H @2H @W @H þ 2 þ ¼ P þ HðU  HÞ @t @x2 @y @x @x

ð15Þ

2

a

ð16Þ

The boundary conditions are

on y ¼ 0 and y ¼ 1

ð17Þ

The principle of exchange of stabilities does not hold here, so that we assume that Eqs. (14)–(16) admit solutions in the form









H ¼ Re gðyÞekt eikx ;



U ¼ Re hðyÞekt eikx



ð18Þ where k is the horizontal wave number. By substituting (18) into Eqs. (14)–(16), we obtain 2

f 00  k f ¼ Rkg

ð19Þ

2

ð20Þ

2

ð21Þ

g 00  ðk þ H þ iPk þ kÞg þ ikf þ Hh ¼ 0 00

h  ðk þ cH þ akÞh þ cHg ¼ 0 The boundary conditions for the perturbed problem are

f ¼ g ¼ h ¼ 0;

on y ¼ 0 and y ¼ 1

ð22Þ

Now we have to solve the eigenvalue problem formed by Eqs. (19)–(21) together with the boundary conditions (22). This is a complex eigenvalue problem which must be solved for the Rayleigh number R and growth rate k as functions of k, P, a, H and c. We note that the growth rate k is in general a complex quantity: when Re(k) < 0 the system is stable, for Re(k) > 0 it is unstable, while Re(k) = 0 states for neutral stability stands. No analytical solution exist for this problem, so that numerical methods must be employed. 3. Numerical analysis Two methods are used in this paper: a one-term-Galerkin technique and the numerical solver dsolve from Maple. We provide below a short description of these approaches. 3.1. One-term-Galerkin approach The perturbed quantities are expressed as

f ¼ A sin py;

g ¼ B sin py;

h ¼ C sin py

ð23Þ

ð25Þ

h i 2 2 2 ðk þ p2 Þ ðk þ p2 þ cHÞðPk þ xÞ  ðk þ p2 þ HÞax 2

@U @ U @ U ¼ 2 þ 2 þ cHðH  UÞ @t @x @y

W ¼ Re f ðyÞekt eikx ;

2

 k axR ¼ 0

2

W ¼ H ¼ U ¼ 0;

2

 k xðk þ p2 þ cHÞR ¼ 0

ð26Þ

Eliminating the Rayleigh number between (25) and (26), we get the following cubic algebraic equation in x

ax3 þ Pakx2 þ ½ðk2 þ p2 þ cHÞ2 þ acH2 x 2 þ Pkðk þ p2 þ cHÞ2 ¼ 0

ð27Þ

Once, x is obtained from (27), the Rayleigh number is derived on using either (25) or (26). A minimization of R over k is performed numerically, producing finally the critical values kc, Rc and xc. 3.2. Second approach For this approach, we need to rewrite Eqs. (19)–(21) by separating them into real and imaginary parts. So, we express f ¼ fR þ ifI and similarly g and h. Looking for neutrally stable disturbances, i.e. Re(k) = 0, we set k = ix in (20) and (21), where x is a real number. Imposing, as usual, a normalization condition: g 0 ð0Þ ¼ 1, i.e. g 0R ð0Þ ¼ 1 and g 0I ð0Þ ¼ 0, we have 6 ODEs for the 6 unknowns 0 0 fR0 ð0Þ, fI0 ð0Þ, hR ð0Þ, hI ð0Þ, R and x. 4. Results Numerical values for a are chosen in the range (0.01 ... 1). When applying the Galerkin method, we found in all the cases analyzed in this paper that the algebraic Eq. (27) has a real root and two complex conjugated roots In Table 1 there are reported the results obtained with the two numerical methods described previously. 4.1. Effect of diffusivity ratio a The first investigation in the parameters space was performed on the influence of a on the critical values of wave number, Rayleigh number and frequency. It is found that this influence is weak for small values of the pressure gradient: for instance, kc is practically insensitive to the variation of a till the third digit for P 6 1. Several results are reported in Table 2 for c = 1 and in Table 3 for c = 10. In both tables, the Galerkin method was used to find the critical values of wave number, Rayleigh number and frequency. For the subsequent figures, we will consider a = 0.5 and we point out that irrespective of the method used to solve the eigenvalue problem, the curves are the same.

71

A. Postelnicu / International Journal of Heat and Mass Transfer 53 (2010) 68–75 Table 1 Critical wave number, Rayleigh number and frequency, when a = 0.5. One-term Galerkin

P = 0.1, H = 10, c = 1 P = 1, H = 1, c = 1 P = 10, H = 10, c = 1 P = 10, H = 1000, c = 1 P = 10, H = 1000, c = 10 P = 100, H = 100, c = 0.01 P = 100, H = 1000, c = 10

Maple

kc

Rc

xc

kc

Rc

xc

3.436 3.211 3.529 3.148 3.142 5.730 3.111

52.360 41.363 54.387 78.617 43.427 185.564 44.304

0.327 3.207 34.240 21.260 29.929 573.628 296.347

3.436 3.211 3.504 3.127 3.142 5.737 3.123

52.360 41.363 53.739 79.313 43.424 185.531 46.960

0.362 3.215 36.373 60.192 33.067 574.040 328.665

Table 2 Critical wave number, Rayleigh number and frequency, when c = 1.

a

kc

Rc

xc

P = 1, H = 1 0.01 0.5 1

3.211 3.211 3.211

3.437 3.437 3.438

41.362 41.363 41.364

3.211 3.207 3.204

3.271 3.270 3.270 3.270 3.270 3.270 3.270 3.270 3.270 3.270

52.366 52.390 52.416

3.403 3.275 3.129

72.350 72.358 72.364 72.368 72.372 72.375 72.377 72.379 72.380 72.381

3.059 2.8745 2.711 2.564 2.433 2.315 2.207 2.109 2.020 1.937

3.157 3.157 3.157 3.157 3.157 3.157 3.157 3.157 3.157 3.157

3.211 3.215 3.216

3.461 3.529 3.577

78.191 78.191 78.191 78.191 78.191 78.191 78.191 78.191 78.191 78.191

3.257 3.246 3.236 3.227 3.220 3.213 3.208 3.203 3.199 3.196

41.364 41.411 41.426

32.109 32.133 32.138

3.154 3.152 3.150 3.149 3.148 3.148 3.148 3.147 3.147 3.147

Rc

xc

3.216 3.218 3.218

41.426 41.454 41.454

321.598 321.797 321.797

57.362 57.665 57.706

371.451 373.436 373.635

136.607 158.036 166.690 171.219 173.999 175.878 177.232 178.256 179.056 179.699

336.569 419.197 463.262 486.370 500.205 509.501 516.232 521.177 525.036 528.102

P = 100, H = 10 52.958 54.387 55.277

34.307 34.240 34.241

73.360 74.134 74.727 75.184 75.539 75.813 76.025 76.187 76.310 76.401

30.483 28.563 26.875 25.377 24.047 22.844 21.765 20.780 19.883 19.064

3.717 3.737 3.739

P = 100, H = 100

P = 10, H = 1000 0.288 0.265 0.245 0.228 0.213 0.200 0.189 0.178 0.169 0.161

kc

P = 100, H = 1

P = 10, H = 100

P = 0.1, H = 1000 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

xc

P = 10, H = 10

P = 1, H = 100 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Rc

P = 10, H = 1

P = 1, H = 10 0.1 0.5 1

kc

3.494 4.354 4.802 5.036 5.176 5.270 5.338 5.388 5.427 5.458

P = 100, H = 1000

78.346 78.454 78.528 78.581 78.617 78.641 78.658 78.668 78.674 78.676

28.774 26.437 24.449 22.744 21.260 19.964 18.817 17.788 16.872 16.045

2.913 2.789 2.715 2.667 2.636 2.614 2.600 2.590 2.584 2.581

Rc

xc

kc

93.070 102.746 109.290 113.786 116.885 118.998 120.402 121.289 121.792 122.010

265.839 234.153 211.084 193.085 178.560 166.362 156.034 147.044 139.188 132.248

Table 3 Critical wave number, Rayleigh number and frequency, when c = 10.

a

kc

Rc

xc

P = 1, H = 1 0.1 0.5 1

3.176 3.176 3.176

P = 10, H = 1 40.802 40.805 40.808

3.172 3.158 3.141

42.775 42.779 42.784

3.140 3.056 2.956

43.350 43.351 43.351

3.114 3.000 2.868

43.418 43.419 43.419

3.111 2.992 2.857

P = 1, H = 10 0.1 0.5 1

3.162 3.162 3.162

3.144 3.144 3.144

3.142 3.142 3.142

40.868 41.037 41.148

31.758 31.757 31.774

3.160 3.153 3.145

42.887 43.296 43.730

31.383 30.501 29.510

3.144 3.141 3.139

43.368 43.435 43.505

31.141 29.970 28.638

43.420 43.427 43.435

31.110 29.929 28.565

3.214 3.217 3.218

41.402 41.443 41.449

321.371 321.669 321.769

3.206 3.519 3.619

49.535 55.070 56.315

319.304 349.602 359.409

45.145 51.252 57.211

305.685 279.994 257.255

P = 100, H = 100

P = 10, H = 1000 3.142 3.142 3.141

xc

P = 100, H = 10

P = 10, H = 100

P = 1, H = 1000 0.1 0.5 1

3.179 3.187 3.194

Rc

P = 100, H = 1

P = 10, H = 10

P = 1, H = 100 0.1 0.5 1

kc

3.086 2.930 2.806

P = 100, H = 1000 3.135 3.111 3.086

43.611 44.304 45.026

310.408 296.347 280.664

72

A. Postelnicu / International Journal of Heat and Mass Transfer 53 (2010) 68–75

4.2. Variation with the pressure gradient Another set of numerical investigation was carried-out with the aim to study the effect of the pressure gradient on the critical values of the wave number, Rayleigh number and frequency. The variation of the critical wave number with the pressure gradient is shown in Fig. 1, for different values of the parameters H and c.  The curves corresponding to (H = 1, c = 10) and (H = 10, c = 0.01) are almost horizontal, indicating a weak effect of the pressure gradient in these cases.  The curve for (H = 10, c = 1) raises for P 6 50 and then remains horizontal. A similar pattern is observed for (H = 100, c = 0.01) and P 6 25.  A different behavior is observed for (H = 100, c = 10), when the curve is continously decreasing, falling for higher values of the pressure gradient below the critical value p.  Variation of the curve corresponding to (H = 100, c = 10) is different, its general trend is to increase with the increase of the pressure gradient.  Fig. 2 shows the dependence of the critical Rayleigh number on the pressure gradient parameter.  Largest values of the Rayleigh number are seen to occur when H = 100 and c = 0.01, but they are almost constant, between 180 and 190, being almost insensitive to the effect of the pressure gradient.  A somewhat similar situation occurs for (H = 10, c = 1) , (H = 1, c = 10) and (H = 10, c = 0.01), but at smaller values of the Rayleigh number. The last two cuurves are basically almost horizontal.  The curve corresponding to (H = 100, c = 10) is the single one which increases, especially for large P, and one may interpret that as a sensitivity of the critical Rayleigh number to the effect of the pressure gradient in conditions of transportation of heat through both solid and fluid phases (large .) and approach to local thermal equilibrium (large H).

Fig. 2. Critical Rayleigh number vs the pressure gradient P, when a = 0.5.

All curves in Fig. 3, where the critical frequency is represented vs the pressure gradient, decrease linearly, the sharpest variation being for H = 100 and c = 0.01, while the highest values are obtained also for (H = 100, c = 10).

Fig. 3. Critical frequency vs the pressure gradient P, when a = 0.5.

4.3. Effects of the inter-phase heat transfer coefficient and porosityscaled conductivity ratio The following graphs, Figs. 4–10 are plotted on a logarithmic scale in the abscissa, see also Banu and Rees [7], Malashetty et al. [11], Postelnicu and Rees [8], Postelnicu [13]. We chosen to illustrate the above mentioned effects at three values of P: 0.1, 10 and 100.

Fig. 1. Critical wave number vs the pressure gradient P, when a = 0.5.

a) The variation of the critical wave number with log H for various values of c is depicted in Figs. 4 and 5. It is typical to get a maximum of the critical wave number (see for instance the above mentioned references) at medium values of H, while in the large H-limit, all curves tend to the value of p, characteristic for LTE conditions. However, since we are interested here in the role played by the pressure

A. Postelnicu / International Journal of Heat and Mass Transfer 53 (2010) 68–75

Fig. 4. Variation of the critical wave number with H for varying c, when P = 0.1, a = 0.5.

Fig. 5. Critical wave number as a function of H for varying c, when P = 10, a = 0.5.

gradient, we remark that the peaks are slightly increased when the pressure gradient is intensified. We mention that when the critical wave number attains large values, this corresponds to tall thin convection cells, see for instance Banu and Rees [7]. b) The variation of the critical Rayleigh number, shown in Figs. 6–8. Several general features, found also by previous researchers, see above, are listed below: (b1) For very small values of c (0.001 and 0.01) the Rayleigh number increases indefinitely. (b2) At small values of H, there is a small rate of heat transfer between the solid and fluid phases. Consequently the solid phase has a small influence on the onset of convection and Rayleigh number is almost insensitive to H in this domain.

73

Fig. 6. Critical Rayleigh number as a function of H for varying c, when P = 0.1, a = 0.5.

Fig. 7. Critical Rayleigh number vs H for varying c, when P = 10, a = 0.5.

(b3) But, as H acquires large values, the critical Rayleigh number is more and more sensitive to c. It decreases with increasing c and becomes constant in the large H-limit, when LTE limit is approached. (b4) Now, concerning the effect of the pressure gradient, we observe that its intensification alters the shape of the curves by producing a maximum of the Rayleigh number in the medium range of H, see the curves corresponding to c = 1 and 10 in Fig. 8, where P = 100. c) The variation of the critical frequency is represented in Figs. 9 and 10. It is seen that only negative values are obtained and the general features of Rayleigh number variation with H are retrieved, in the small and large H-limits. Otherwise, a minimum is obtained, more pronounced and placed towards larger values of H, as c decreases. As regard to the

74

A. Postelnicu / International Journal of Heat and Mass Transfer 53 (2010) 68–75

Fig. 8. Variation of Rc with H for varying c, when P = 100, a = 0.5.

Fig. 10. Variation of xc with H for varying c, when P = 10, a = 0.5.

ium acts independently of the solid phase. In this case, the limiting behavior is Rc = 4p2 and kc = p.  Between these two extremes, Rc increases with H, while kc rises to a maximum and then decays back to p. The question of how accurate may be the Galekin approach when solving eigenvalue problems in thermal and hydrodynamic stability is frequently raised. In this respect, the Galerkin approximation was discussed in a recent paper by Nield and Kuznetsov [18], where it has been used for a Rayleigh-Bénard problem. These authors found that this method lead to an overestimate of the Rayleigh number by no more than 3% (see page 1214 in their paper). For a comparison between the one-term and N-terms Galerkin approach, the interested reader may also consult [9]. In the present paper, we have used and compared the one-termGalerkin approach with a rigorous numerical technique. The first one has obvious advantages related to an analytical approach and it is proved that it produces results with minimal losses of accuracy. Fig. 9. Critical frequency number vs H for varying c, when P = 0.1, a = 0.5.

pressure gradient influence, it is easily to remark the larger absolute values acquired by the critical frequency as P increases. 5. Conclusion This study has been undertaken with the aim to quantify the effect of a horizontal pressure gradient on the onset of Darcy–Benard, when the fluid and solid phases are not in thermal local equilibrium. Known conclusions reported in related studies, such as Banu and Rees [7], Malashetty et al. [11], Postelnicu and Rees [8], Postelnicu [9] and [13] are obtained in the present paper at a constant value of the pressure gradient  The LTE situation is recovered in the large H-limit.  For small H, the perturbation in the solid phase temperature is much smaller than that in the fluid phase; now the porous med-

References [1] C.W. Horton, F.T. Rogers, Convective currents in a porous medium, J. Appl. Phys. 16 (1945) 367–370. [2] E.R. Lapwood, Convection of a fluid in a porous medium, Pro. Camb. Philos. Soc. 44 (1948) 508–521. [3] D.A.S. Rees, The stability of Darcy–Bénard convection, in: K. Vafai (Ed.), Handbook of Porous Media, Marcel Dekker, 2000, pp. 521–558. [4] M. Mamou, L. Robillard, E. Bilgen, P. Vasseur, Effect of a moving thermal wave on Bénard convection in a horizontal saturated porous layer, Int. J. Heat Mass Transfer 39 (1996) 347–354. [5] N. Banu, D.A.S. Rees, Effect of a traveling thermal wave on weakly nonlinear convection in a porous layer heated from below, J. Porous Med. 4 (2001) 225– 239. [6] D.A.S. Rees, I. Pop, Local thermal nonequilibrium in porous medium convection, in: D.B. Ingham, I. Pop (Eds.), Transport in Porous Media, vol. 3, Pergamon, 2005, pp. 147–173. [7] N. Banu, D.A.S. Rees, Onset of Darcy–Bénard convection using a thermal nonequilibrium model, Int. J. Heat Mass Transfer 45 (2002) 2221–2228. [8] A. Postelnicu, D.A.S. Rees, The onset of Darcy–Brinkman convection in a porous medium using a thermal nonequilibrium model. Part 1. Stress-free boundaries, Int. J. Energy Res. 27 (2003) 961–973. [9] A. Postelnicu, The onset of a Darcy–Brinkman convection using a thermal nonequilibrium model. Part II, Int. J. Therm. Sci. 47 (2008) 1587–1594. [10] M.S. Malashetty, I.S. Shivakumara, Sridhar Kulkarni, The onset of convection in a anisotropic porous layer using a thermal non-equilibrium model, Transport Porous Med. 27 (2005) 199–215.

A. Postelnicu / International Journal of Heat and Mass Transfer 53 (2010) 68–75 [11] M.S. Malashetty, I.S. Shivakumara, Sridhar Kulkarni, The onset of Lapwood– Brinkman convection using a thermal non-equilibrium model, Int. J. Heat Mass Transfer 48 (2005) 1155–1163. [12] M.S. Malashetty, R. Heera, The onset of double diffusive convection in sparsely packed porous layer using a non-equilibrium model, Acta Mech. 204 (2009) 1– 20. [13] A. Postelnicu, The effect of inertia on the onset of mixed convection in a porous layer using a thermal non-equilibrium model, J. Porous Med. 10 (2007) 515– 524. [14] N.H. Saeid, Analysis of mixed convection in a vertical porous layer using nonequilibrium model, Int. J. Heat Mass Transfer 47 (2004) 5619–5627.

75

[15] M. Prats, The effect of horizontal fluid motion on thermally induced convection currents in porous medium, J. Geophys. Res. 71 (1967) 4835–4848. [16] D.A. Nield, A. Bejan, Convection in Porous Media, third ed., Springer, New York, 2006. [17] P. Vadasz, Explicit conditions on the local thermal equilibrium model in porous media heat conduction, Transport Porous Med. 59 (2005) 341–355. [18] D.A. Nield, A.V. Kuznetsov, The effects of combined horizontal and vertical heterogeneity and anisotropy on the onset of convection in a porous medium, Int. J. Thermal Sci. 46 (2007) 1211–1218.