Porous convection with Cattaneo heat flux

International Journal of Heat and Mass Transfer 53 (2010) 2808–2812

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Porous convection with Cattaneo heat flux B. Straughan Department of Mathematical Sciences, Durham University, DH1 3LE, UK

a r t i c l e

i n f o

Article history: Received 12 August 2009 Accepted 13 February 2010 Available online 11 March 2010 Keywords: Thermal convection Convection in porous media Cattaneo theory of heat propagation Christov heat flux equation Cattaneo–Fox heat flux equation Oscillatory convection

a b s t r a c t We study the problem of thermal convection in a horizontal layer of Darcy porous material saturated with an incompressible Newtonian fluid, with gravity acting downward. The constitutive equation for the heat flux is taken to be one of Cattaneo type. Care must be taken with the choice of objective derivative for the rate of change of the heat flux. Here we employ a recent model due to Professor C. Christov as well as one suggested many years ago by Professor N. Fox. The thermal relaxation effect in both classes of heat flux law is found to be significant if the Cattaneo number is sufficiently large, and the convection mechanism switches from stationary convection to oscillatory convection with narrower cells. The transition point is calculated and the convection thresholds are derived analytically. Ó 2010 Elsevier Ltd. All rights reserved.

1. Introduction Heat propagation via a wave mechanism instead of simply by diffusion is one of great current interest. Recent studies confirm this is not simply a low temperature phenomenon, but one which has potentially important real applications. Mundane applications of hyperbolic heat propagation are in fields such as skin burns, Dai et al. [6], chemotaxis, Dolak and Hillen [8], virus spread, Barbera et al. [1], heat transfer in one of Saturn’s moons, Bargmann et al. [2], traffic flow, Jordan [11], heat propagation in biological tissues, Vedavarz et al. [31], Mitra et al. [16], phase changes, Liu et al. [14], Miranville and Quintanilla [15], food technology, Saidane et al. [25], and in nanofluids, Vadasz et al. [30]. Other related applications are discussed in Quintanilla and Racke [22,23], Jordan [12], Reverberi et al. [24], Straughan [28], chapters 7 and 8, Vadasz [29], Vadasz et al. [30]. A key way of introducing finite temperature wave motion has been to use the Cattaneo [3] law for the heat flux. In thermal convection in a viscous fluid analysis of the Cattaneo law was initiated by Straughan and Franchi [26], with further work by Lebon and Cloot [13]. The higher derivative Guyer–Krumhansl effects were analysed by Franchi and Straughan [10] and by Dauby et al. [7]. Further work in the area of fluid mechanics employing the Cattaneo law for the heat flux may be found in the interesting papers of Puri and Jordan [19,18], and Puri and Kythe [20,21]. Vadasz [29] investigates whether oscillatory heat motion will be possible in a block of porous material by employing a dual phase lag theory for heat conduction. We here continue with such a study by inves-

E-mail address: [email protected] 0017-9310/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijheatmasstransfer.2010.02.017

tigating whether oscillatory motion is possible in a layer of porous material saturated with a viscous fluid which is heated from below. The heat flux is a vector field and so the equation governing its evolutionary behaviour must involve an objective time derivative. Straughan and Franchi [26] employed an objective time derivative due to N. Fox and worked with what might be described as a Cattaneo–Fox theory. Recently, Christov [5] has written a very interesting paper which revisits the question of which objective derivative one should employ when dealing with a Cattaneo type theory for a fluid. He proposes an alternative frame-indifferent generalization of Fourier’s law with relaxation of the heat flux. The objective of the current paper is to investigate thermal convection in a layer of saturated porous material employing Cattaneolike heat flux laws. We employ both the Cattaneo–Fox and Cattaneo–Christov models and compare the results. 2. Equations for porous convection The equations for thermal convection in a porous medium may be found in the books by Nield and Bejan [17], or by Straughan [27,28]. They consist of the balances of linear momentum, mass, and energy, and these may be written as

v i;t ¼ 

1

q

p;i þ agki T 

l v; qK i

ð1Þ

v i;i ¼ 0;

ð2Þ

1 T t þ v i T ;i ¼ Q i;i ; M

ð3Þ

where v i ; p; T are the velocity, pressure and temperature fields, q; a; g; l and K are density, thermal expansion coefficient, gravity,

B. Straughan / International Journal of Heat and Mass Transfer 53 (2010) 2808–2812

dynamic viscosity and permeability, respectively. The quantity Q i is the heat flux vector, k ¼ ð0; 0; 1Þ and standard indicial notation is used throughout. The parameter M is defined below where we derive equation (3). We now write energy balances for the solid and fluid parts of the porous medium separately, and additionally write a Cattaneo heat flux law for the solid and a Cattaneo–Fox heat flux law for the fluid. This is analogous to the development of the non-Cattaneo case by e.g. [28, pp. 14–15]. Let V i be the actual velocity of the fluid in the pores, let / be the porosity, and set v i ¼ /V i , v i being the pore averaged velocity. Denote by subscript s; f the solid and fluid parts. Then we have the energy balance and Cattaneo law for the solid,

e i;i ; ðq0 cÞs T ;t ¼  Q ss Qe i;t ¼  Qe i  ks T ;i ;

ð4Þ ð5Þ

where c is the specific heat, ss is the relaxation time, ks is the there i is the heat flux. The energy balance in the mal conductivity, and Q fluid and the Cattaneo–Fox heat flux laws may be written, cf. Straughan and Franchi [26],

e i;i ; ðq0 cp Þf ðT ;t þ V i T ;i Þ ¼  Q   1 1 sf Qe i;t þ V j Qe i;j  Qe j V i;j þ Qe j V j;i ¼  Qe i  kf T ;i ; 2 2

ð6Þ ð7Þ

where cp is the specific heat at constant pressure, kf is the thermal conductivity, and sf is a relaxation time. To derive an averaged equation governing the (averaged) porous media properties, we multiply equation (4) by ð1  /Þ, Eq. (6) by /, and add, and likewise multiply equation (5) by ð1  /Þ and Eq. (7) by /, and add. We now define the quantities Q i ; ðq0 cÞm ; km ; j; M and s by

Qi ¼

ei Q ; ðq0 cp Þf

ðq0 cÞm ¼ /ðq0 cp Þf þ ð1  /Þðq0 cÞs ;

j¼

km ; ðq0 cp Þf

M¼

ðq0 cp Þf ; ðq0 cÞm

1 2

1 2

ð8Þ

ð9Þ

The complete system of equations for movement and heat propagation in the fluid in a porous medium therefore consist of (1)–(3) coupled with either (8) or (9). The saturated porous medium occupies the horizontal layer fðx; yÞ 2 R2 ; z 2 ð0; dÞg and Eqs. (1)–(3) with (8) or (9) hold in the domain R2  ð0; dÞ  ft > 0g. The boundary conditions are

w  v 3 ¼ 0 on z ¼ 0; d; T ¼ TU;

z ¼ d;

 ¼ ð0; 0; jbÞ; Q

where b is the temperature gradient given by

b¼

TL  TU : d

2

2d

;

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi agd2 bK q : R¼

lj

The linearized, non-dimensional equations which follow from (1)–(3) with (8) are

ui;t ¼ p;i þ Rki h  ui ; ui;i ¼ 0; Pr ht ¼ Rw  qi;i ; MDa Pr ^ðui;z  w;i Þ  qi  h;i ; 2C q ¼ CRs Da i;t

ð12Þ

whereas from (1)–(3) with (9) we derive

ui;t ¼ p;i þ Rki h  ui ; ui;i ¼ 0; Pr ht ¼ Rw  qi;i ; MDa Pr ^ui;z  qi  h;i : 2C q ¼ 2CRs Da i;t

ð13Þ

To study linear instability of the conduction solution (11) we write the variables ui ; h; qi and p by explicitly separating the time dependent parts as

hðx; tÞ ¼ ert hðxÞ;

pðx; tÞert pðxÞ:

r

Pr h ¼ Rw  Q; MDa

2rC

ð14Þ

Pr ^Dw  Q  Dh; Q ¼ kCRs Da

where D ¼ @ 2 =@x2 þ @ 2 =@y2 is the horizontal Laplacian, and where k ¼ 0 for the Cattaneo–Christov theory, whereas k ¼ 1 when the Cattaneo–Fox model is employed. The boundary conditions to be used in conjunction with (14) are

w ¼ 0;

h ¼ 0;

z ¼ 0; 1;

ð15Þ

and w and h satisfy a plane tiling periodicity in the horizontal variables x and y.

ð10Þ 3. Linear instability, Cattaneo–Fox theory

with T L > T U , both constants. The steady solution to (1)–(3) with either (8) or (9) which satisfies the boundary conditions (10) and whose stability we are interested in is

T ¼ bz þ T L ;

C¼

sj

rDw ¼ RD h  Dw;



sQ i;t þ sf ðv j Q i;j  Q j v i;j Þ ¼ qi  jT ;i :

v i  0;

number, C, and Rayleigh number, Ra ¼ R2 , introduced as

Then p is eliminated and we put Q ¼ qi;i to reduce (12) or (13) to studying the system

s ¼ /sf þ ð1  /Þss :

An equivalent derivation to that of Eq. (8), but for the Cattaneo– Christov heat flux law, Christov [5], yields

z ¼ 0;

U is a velocity scale. The non-dimensional numbers Pr and Da are the Prandtl and Darcy numbers. Key in this work are the Cattaneo

qi ðx; tÞ ¼ e qi ðxÞ;

sQ i;t þ sf v j Q i;j  Q j v i;j þ Q j v j;i ¼ Q i  jT ;i :

T ¼ TL;

Q i ¼ Q i þ qi . Then, we linearize and non-dimensionalize with the scalings xi ¼ xi d; t ¼ t K q=l; p ¼ p dlU=K; Pr ¼ l=jq; qi ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 ^ ¼ sf =s, where qi Q ] ; Q ] ¼ jT ] =d; T ] ¼ U bd l=jqK ag ; Da ¼ K=d ; s

rt

Then one shows Eqs. (4)–(7) lead to Eq. (3) and



To analyse the instability of solution (11) we introduce pertur i þ ui ; T ¼ T þ h; p ¼ p  þ p; bations ðui ; h; p; qi Þ such that v i ¼ v

ui ðx; tÞ ¼ ert ui ðxÞ;

km ¼ /kf þ ð1  /Þks ;

2809

ð11Þ

In this section, we consider equations (14) when k ¼ 1. In the first instance we consider stationary convection, i.e. when r ¼ 0. Then one finds from (14),

RD h  Dw ¼ 0; Rw ¼ Q ; s^CRDw ¼ Q  Dh:

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B. Straughan / International Journal of Heat and Mass Transfer 53 (2010) 2808–2812

This leads to

^Dw þ wÞ: D2 w ¼ R2 D ðC s

ð16Þ

Due to boundary conditions (15) we may write W as a series of terms of form w ¼ WðzÞf ðx; yÞ where f is a function satisfying D f ¼ a2 f ; a being a wave number. Further, WðzÞ may be written in the form

WðzÞ ¼ sin ðnpzÞ;

n ¼ 1; 2; . . .

Then equation (16) reduces to

R2 ¼

a2 ð1

K2 ; ^ KÞ  Cs

ð17Þ

where K ¼ n2 p2 þ a2 . Eq. (17) leads to some interesting possibilities including R2 switching to negative values. This does not contradict physics, we replace R2 by Ra and interpret it as heating from above. This unpleasant feature of the Cattaneo–Fox model was witnessed for a fluid by Straughan and Franchi [26] and is one reason to perhaps consider Christov’s [5] replacement. However, we now proceed to analyse (17) with n ¼ 1. In this case we find by taking 2 2 dR =da that the critical value of a2 ; a2c , at which R2 achieves a minimum is when

a2c ¼

p2 ð1  C s^p2 Þ : ^Þ ð1 þ p2 C s 4p2 : ^p2 Þ ð1  C s

a4 ðk3 c1  c2 k2 Þ  2c2 k1 a2  c1 k1 : The coefficient k3 c1  c2 k2 ¼ 1=2  p2 =2P  PC. This is zero when pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P ¼ ð1  1  8C p2 Þ=4C. So k3 c1  c2 k2 < 0 whenever C < 1=8p2 . We generally expect C to be small and P to be large. For example, for water a typical value of Pr ¼ 6, whereas for sand K takes a value in the range 2  107 to 1:8  106 cm2 , see Nield and Bejan [17]. For a 3cm layer d ¼ 3 cm and this yields a value of P ¼ Pr=Da ¼ 2

Prd =K in the range 3  105 to 2:7  106 . Thus, for practical values 2

2

dR =da < 0. This means that we find the critical value of R2 by allowing a2 ! 1. Hence, from (24),

Raosc ¼

ð18Þ

To study oscillatory convection we return to equations (14) with k ¼ 1. Then replacing Pr=Da ¼ P we must solve

     ¼ 0:  2PC r þ 1 

  ðr þ 1ÞK Ra2    R r MP   C s ^ RK K

0 1

C > CT ;

4p 2 1  C p2

Ra ¼

1 2PC 2

:

The transition is given by

ð19Þ

ð20Þ

Ra ¼

whereas for

CT ¼

Due to the boundary conditions (15) we replace D by K and then from (19) we have to solve the determinant equation

ð25Þ

:

Thus, with the Cattaneo–Fox model in porous convection we find the Rayleigh number threshold is the smaller of (18) and (25). By equating these quantities we find



ðr þ 1ÞDw  RD h ¼ 0; P Q  Rw þ r h ¼ 0; M ^RDw þ Dh þ ð2PC r þ 1ÞQ ¼ 0: Cs

1 2PC 2

C < CT ;

This yields a critical Rayleigh number of the form

Ra ¼ R2 ¼

^ we take s ^ ¼ 1 and also In the absence of accurate values for s restrict attention to the case of M ¼ 1. The analysis below is not 2 2 qualitatively altered by these values. By taking dR =da we find the denominator is positive while the numerator is

p

2

1 pffiffiffiffiffiffi : þ 8P p

ð26Þ

Thus, for C small we have stationary convection, but once C exceeds C T the convection mechanism switches to one of oscillatory convection and the cell structure breaks down. For example, for P ¼ 6; C T ¼ 3:1610361  102 and Ra at the transition has value Ra ¼ 83:3988. 4. Linear instability, Cattaneo–Christov theory

This leads to a quadratic in r. We follow the method of Chandrasekhar [4] and write r ¼ rr þ ir1 and since we are interested in critical conditions we put rr ¼ 0. Taking the real and imaginary parts of the equation (20) we find

We now consider equations (14) with k ¼ 0. For the stationary convection case, r ¼ 0, we see that (14) reduce to

P 2 r ð2PC þ 1Þ ¼ 0; M 1 2 P C P R2 a2 2PC  K2 þ 2 Kr21  K ¼ 0: M M

Rw ¼ Q ;

^ Þ  K2 þ K R2 a2 ð1  KC s

Eliminating

Q ¼ Dh; ð22Þ which results in

r21 leads us to the equation

 K

2

ð21Þ

 1

K þ M P þ 2C R2 ¼ 2PC : ^Þ a2 ð2PC þ KC s 2

D2 w ¼ R2 D w ð23Þ

2

One may show dR =da P 0 and so we may take n ¼ 1. Then (23) may be written as

R2 ¼

k1 þ k2 a2 þ k3 a4 ; c1 a2 þ c2 a4

ð24Þ

  p p 1 ; k1 ¼ þ Pþ 2C 2PC M 2 p P 1 k2 ¼ ; þ þ PC M 2CM 1 ^; k3 ¼ ; c1 ¼ 2PC þ C p2 s 2PC

ð27Þ

and using the boundary conditions we have w is formed from a series of terms of the form WðzÞf ðx; yÞ with WðzÞ having representation by terms like sin npz so that (27) leads to

R2 ¼

K2 : a2

Taking the derivative we find @R2 =@n2 P 0 and so we take n ¼ 1. 2 2 Then dR =da ¼ 0 yields

where 4

Dw ¼ RD h;

2

ac ¼ p

ð28Þ

and

^: c2 ¼ C s

R2stat ¼ 4p2 :

ð29Þ

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B. Straughan / International Journal of Heat and Mass Transfer 53 (2010) 2808–2812

To investigate oscillatory convection we solve

Table 2 Linear instability values for the Cattaneo–Christov model.



ðr þ 1ÞDw ¼ RD h; ð2PC r þ 1ÞQ ¼ Dh; P Q ¼ Rw  r h; M

ð30Þ

which lead to

ð2PC r þ 1ÞR2 D w ¼ r

C

Ra

P

a

r1

0.10 0.11 0.12 0.15 0.20

39.4784 36.9066 31.5843 21.2933 12.9714

6 6 6 6 6

3.14 3.90 3.90 4.00 4.20

0 ±0.4401632 ±0.6361547 ±0.8694543 ±0.9699395

P ð2PC r þ 1Þðr þ 1ÞDw  ðr þ 1ÞD2 w: M ð31Þ

Put r ¼ rr þ ir1 with this equation to find

rr ¼ 0 and take real and imaginary parts of

K  a2 R2 ¼ 0; M P2 C 3 P  2PC r1 a2 R2  2 r K þ r1 K þ r1 K2 ¼ 0: M 1 M

K2  ð2P2 C þ PÞr21

Recall

1

2

R ¼

r1 – 0 and then we derive

2PCa2

"

 # K2 K 1 : þ Pþ 2C 2PC M

ð32Þ 2

2

One shows n ¼ 1 yields R2 smallest and then dR =da ¼ 0 gives

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2c ¼ p p2 þ ð2P 2 C þ PÞ=M

ð33Þ R2osc

from which one may calculate using (32). However, for direct comparison with the results of Section 3 we now take M ¼ 1. Employing equation (33) in Eq. (32) with M ¼ 1 leads to

2

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2

2 2 2 1 6 p þ p p þ 2P C þ P pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 R2osc ¼ 2PC 2PC p p2 þ 2P2 C þ P

3  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  1 7 2 þðp2 þ p p2 þ 2P C þ PÞ P þ 5: 2C

ð34Þ

5. Conclusions We have investigated linearized instability for thermal convection in a porous medium when the heat flux vector satisfies either a Cattaneo–Fox law or a Cattaneo–Christov one. For both theories we find that stationary convection is preferred for small Cattaneo numbers, but there is a transition after which oscillatory convection is witnessed with a much different convection cell aspect ratio. However, the behaviour before and at the transition is very different for both models. In fact, the two objective derivatives proposed by Fox and by Christov lead to very different thermal con^¼1 vection results. The discussion below is based on taking s and M ¼ 1, but the qualitative behaviour remains the same even without these restrictions. With the Cattaneo–Fox model pffiffiffiffiffiffithe transition is found for Cattaneo number C T ¼ 1=ðp2 þ 8PpÞ. For C < C T the critical Rayleigh and wave numbers are Ra ¼ 4p2 =ð1  C p2 Þ and a2 ¼ p2 ð1  C p2 Þ=ð1 þ C p2 Þ, whereas for C > C T ; Ra ¼ 1=2PC 2 with ac ! 1. Alternatively, for the Cattaneo–Christov model the transition Cattaneo number is found numerically. In this case for C < C T ; Ra ¼ 4p2 with a ¼ p and when C > C T ,

R2 ¼

p2 2P2 C

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

þ 2

p p2 þ Pð1 þ 2PCÞ 2P2 C 2

þ

1 4PC 2

þ

1 2C

with Eq. (34) yields analytically the minimum values of R2 as a function of P and C. Eq. (34) may be rearranged as

R2osc ¼

p2 2P2 C

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

þ 2

p p2 þ Pð1 þ 2PCÞ 2P2 C 2

þ

1 4PC

þ 2

1 : 2C

ð35Þ

From this expression we see that as P ! 1; R2osc ! 1=2C. One may show from (35) that @R2 =@C < 0. Then using Eqs. (29) and (35) we deduce that for C < C T we find R2 ¼ 4p2 with a2 ¼ p2 , but once C > C T then R2 has the value given by (35) with a2 given by (33). Thus, at C T the convection changes from stationary convection to oscillatory convection and the wave number increases discontinuously which means the convection cells switch to a narrower hexagonal shape. The transition depends on P and for example we show some results in Table 1. From Table 1 we see that for P ¼ 6 the transition is in the (small) interval C T 2 ½0:1059; 0:1060 whereas when P ¼ 3  105 the transition is for C T in the interval C T 2 ½0:1266; 0:1267. Note how the wave number increases strongly at the transition from stationary to oscillatory convection, a transition associated with much narrower cells. Table 1 Linear instability results at transition. P

R2stat

astat

R2osc

aosc

C

6 6 3  105 3  105

39.4784 39.4784 39.4784 39.4784

3.14159 3.14159 3.14159 3.14159

39.5182 39.4510 39.4966 39.4655

3.90226 3.90256 688.66 688.79

0.1059 0.1060 0.1266 0.1267

a2 ¼ p

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p2 þ 2P2 C þ P:

In the Cattaneo–Fox case the critical Rayleigh number increases from 4p2 to the transition point. For example, with P ¼ 6; C T ¼ 3:161  102 ; RaT ¼ 83:3988 and a2 ¼ 5:1758. Beyond the transition, in C, Ra decreases while ac becomes infinite. With P ¼ 3  105 ; C T ¼ 2:0505  104 ; RaT ¼ 39:5585. When the Cattaneo– Christov law is used Ra ¼ 4p2 until C reaches the transition C T after which it decreases. The critical wave number increases, but the convection cell remains hexagonal, albeit much narrower. With the Cattaneo–Christov model when P ¼ 6; C T ¼ 1:059  101 ; RaT ¼ 39:4784, and a2 changes from a2 ¼ 9:8696 to a2 ¼ 15:2276. For P ¼ 3  105 ; C T ¼ 1:266  101 and RaT ¼ 39:4784. For completeness we include some numerical results in Table 2 for the Cattaneo–Christov model. These results were obtained by solving (30) numerically using a D2 Chebyshev tau method, cf. see Dongarra et al. [9], Straughan [28], chapter 9. In this way we display relevant values of r1 at criticality. Acknowledgments I am indebted to a anonymous referee who spotted errors in an earlier version of the manuscript and for his/her trenchant remarks. References [1] E. Barbera, C. Curro, G. Valenti, A hyperbolic reaction–diffusion model for the hantavirus infection, Math. Methods Appl. Sci. 31 (2008) 481–499.

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