Physics Letters A 374 (2010) 4607–4613

Contents lists available at ScienceDirect

Physics Letters A

www.elsevier.com/locate/pla

Stability of thermal convection of an Oldroyd-B fluid in a porous mediumwith Newtonian heating

Jun Niu, Ceji Fu ∗, Wenchang Tan ∗

State Key Laboratory for Turbulence and Complex Systems and Department of Mechanics and Aerospace Engineering, College of Engineering, Peking University, Beijing 100871, China

a r t i c l e i n f o a b s t r a c t

Article history:Received 5 August 2010Received in revised form 10 September2010Accepted 17 September 2010Available online 21 September 2010Communicated by F. Porcelli

Keywords:StabilityOldroyd-B fluidPorous mediumConvection

A linear stability analysis determining the onset of oscillatory convection of an Oldroyd-B fluid in abounded two-dimensional rectangular porous medium generated by Newtonian heating is conducted.Influences of viscoelastic parameters and Biot number on the onset of oscillatory convection, preferredmodes and patterns of disturbed temperature contours are discussed.

© 2010 Elsevier B.V. All rights reserved.

1. Introduction

Study of the stability of thermal convection in a fluid-saturatedporous medium is very important for exploiting the dynamicsof thermal convection and potential applications in many engi-neering fields, such as geothermal reservoirs, energy conservation,and cooling of electrical machines. In the past several decades,linear stability analysis of thermal convection in a Newtonianfluid-saturated porous medium has been conducted by many re-searchers based on the Darcy’s law [1–5]. In contrast, publicationson study of thermal convection in a viscoelastic fluid-saturatedporous medium are far less than fruitful though interest in theflow and heat transfer characteristics for viscoelastic fluids in-creases rapidly in the past several years [6–9]. This is due to thelack of suitable models, as Darcy’s law for Newtonian fluids, to de-scribe the motion of viscoelastic fluids in porous media.

A linear stability analysis of a layer of pure viscoelastic fluidheated from below was first performed by Sokolov and Tanner[10] based on a simple constitutive relation for the viscoelas-tic fluid. They identified two possible instability mechanisms: ex-change of stabilities and overstability. The former is similar tothat for a Newtonian fluid, but the latter is unique to the vis-coelastic fluid. Recently, Kim et al. [11] performed linear and non-linear stability analyses of thermal convection for an Oldroyd-B

* Corresponding authors. Tel.: +8610 6275 3675 (C.J. Fu), +8610 6275 4244 (W.C.Tan); fax: +8610 6275 1812 (C.J. Fu, W.C. Tan).

E-mail addresses: cjfu@pku.edu.cn (C. Fu), tanwch@mech.pku.edu.cn (W. Tan).

0375-9601/$ – see front matter © 2010 Elsevier B.V. All rights reserved.doi:10.1016/j.physleta.2010.09.028

fluid saturated in a horizontal porous layer with isothermal topand bottom boundaries based on a modified Darcy’s law for de-scription of the viscoelastic fluid motion in the porous layer. Thismodel was first proposed by Alishaev and Mitzadajanzade [12],which is a macroscopic phenomenological model mimicking theDarcy’s law but incorporating the viscoelastic effect. Subsequently,this model has been employed by several other researchers forstudying thermal convection of viscoelastic fluids saturated inporous media [13–16]. Zhang et al. [17] investigated the stabil-ity of thermal convection in a porous cylinder saturated witha viscoelastic fluid; two different boundary heating conditionswere considered: constant temperature heating and constant heatflux heating. They obtained the critical Rayleigh number for on-set of thermal convection and the corresponding preferred modeof flow pattern for different combinations of the elastic param-eters. Nevertheless, the heating conditions in these works are inthe form of isothermal bottom boundary or uniform heat fluxfrom below. No work has been conducted for the stability ofa viscoelastic fluid in a porous medium subject to Newtonianheating, which is more commonly met in engineering applica-tions.

Carslaw and Jaeger [18] were the first to introduce at the bot-tom wall a variable heat flux boundary condition expressed interms of the Biot number, which represents a Robin boundary con-dition. This boundary condition is then referred to as Newtonianheating or forced convection heat transfer [19,20]. Kubitschek andWeidman [21] performed a linear stability analysis of a Newtonianfluid-saturated porous medium heated from below by Newtonianheating. They proved that when Bi → 0, the thermal convection

4608 J. Niu et al. / Physics Letters A 374 (2010) 4607–4613

phenomenon approaches to that subject to uniform heat flux heat-ing. On the other hand, when Bi → ∞, it coincides with that underisothermal bottom heating. Therefore, thermal convection subjectto different bottom heating conditions (i.e., Dirichlet, Neumann andRobin boundary conditions) can be achieved by tuning properly thevalue of the Biot number.

In this work, a two-dimensional linear stability analysis of anOldroyd-B fluid saturated thin rectangular porous medium is per-formed under the Newtonian heating boundary condition. The crit-ical Biot-modified Rayleigh numbers marking the onset of oscilla-tory convection are obtained numerically. The effects of the elasticparameters and the Biot number on the onset of oscillatory con-vection and preferred cellular modes are investigated, as well asthe patterns of disturbed temperature distributions upon onset ofoscillatory convection.

2. Model formulation

A bounded three-dimensional thin porous layer with height H∗and rectangular dimensions a∗ and b∗ is considered, in which b∗ ismuch smaller than H∗ . The four vertical boundaries of the modelare adiabatic and impermeable. The impermeable horizontal topboundary is isothermal with a constant temperature T ∗

1 , while theimpermeable horizontal bottom boundary is heated by Newtonianheating with external ambient temperature T ∗∞ and convectiveheat transfer coefficient h. This porous medium has a permeabilityK and is saturated with an incompressible viscoelastic fluid whichhas a constant dynamic viscosity μ, a coefficient of thermal expan-sion β and a density ρ . The fluid-saturated porous medium hasa thermal diffusivity κ . Here, flow of the Oldroyd-B fluid in theporous medium is described with the modified Darcy’s law [12].Therefore, the governing equations of the current problem, underthe Oberbeck–Boussinesq approximation, are given as [11,13,14]

∇∗ · v∗ = 0 (1)(1 + λ̄

∂

∂t∗

)(−∇∗ p∗ + ρ∗gk̂) = μ

K

(1 + ε̄

∂

∂t∗

)v∗ (2)

∂T ∗

∂t∗ + (v∗ · ∇∗)T ∗ = κ∇∗2T ∗ (3)

ρ∗ = ρ∗1

[1 − β

(T ∗ − T ∗

1

)](4)

where v∗ is the Darcy velocity with v∗ = (u∗, v∗, w∗) in Cartesiancoordinates, p∗ the pressure, g the gravitational acceleration, ε̄ andλ̄ respectively the strain retardation time and the stress relax-ation time, k̂ a unit vector along the z-direction which is verticallyupward, and ρ∗

1 the density at temperature T ∗1 . The boundary con-

ditions are given as

u∗ = 0 at x∗ = 0,a∗ (5a)

v∗ = 0 at y∗ = 0,b∗ (5b)

w∗ = 0 at z∗ = −H∗,0 (5c)

∂T ∗

∂x∗ = 0 at x∗ = 0,a∗ (5d)

∂T ∗

∂ y∗ = 0 at y∗ = 0,b∗ (5e)

T ∗ = T ∗1 at z∗ = 0 (5f)

h(T ∗∞ − T ∗) = −k

∂T ∗

∂z∗ at z∗ = −H (5g)

In Eq. (5g), k is the effective thermal conductivity of the viscoelas-tic fluid-saturated porous layer.

3. Linear stability analysis

3.1. Linear stability equations

For convenience, a heat flux q can be introduced into the cur-rent work as [21]

q = Bi

Bi + 1· k�T ∗

H∗ (6)

Here, �T ∗ = T ∗∞ − T ∗1 > 0 and Bi is the Biot number defined as

Bi = hH∗

k(7)

It should be noticed that q is not a constant heat flux in this prob-lem, but a function of Bi and �T ∗ .

In order to perform the linear stability analysis, Eqs. (1)–(4) arefirst non-dimensionalized by scaling lengths with H∗ , time withH∗ 2/κ , velocities with κ/H∗ , pressure with κμ/K and tempera-ture with qH∗/k. Then θ and w are introduced to represent re-spectively the infinitesimal disturbances of the dimensionless tem-perature and vertical velocity over the pure conduction solution.As a consequence, Eqs. (1)–(4) can be, after eliminating the pres-sure term, rewritten as(

1 + ε∂

∂t

)∇2 w = Ra

(1 + λ

∂

∂t

)∇2

1θ (8)

∂θ

∂t− w = ∇2θ (9)

where ε and λ are respectively the dimensionless relaxation andretardation characteristic times. ∇2 = ∂2

∂x2 + ∂2

∂ y2 + ∂2

∂z2 and ∇21 =

∂2

∂x2 + ∂2

∂ y2 . Ra in Eq. (8) is the Biot-modified Rayleigh number de-

fined as

Ra = qKρ1βg H∗2

μkκ= Bi

Bi + 1· K gβ�T ∗H∗

νκ= Bi

Bi + 1· Ra (10)

In Eq. (10), ν is the kinematic viscosity of the viscoelastic fluid, Rais the typical Rayleigh number in the case of a porous mediumwith constant temperature difference �T ∗ between isothermalbottom and top boundaries. The corresponding dimensionlessboundary conditions of the disturbances are

w = 0 at z = −1,0 (11a)∂θ

∂x= 0 at x = 0,a (11b)

∂θ

∂ y= 0 at y = 0,b (11c)

θ = 0 at z = 0 (11d)

Bi θ − θz = 0 at z = −1 (11e)

where a = a∗H∗ and b = b∗

H∗ are dimensionless side lengths. In thecase when b∗ is much smaller than H∗ , i.e., b � 1, the problemcan be simplified into a two-dimensional one.

3.2. The characteristic equation and solution procedure

Under the normal mode analysis, temperature and velocity dis-turbances are supposed to be horizontally periodic. Taking theadiabatic vertical boundary conditions into account, the solutionshould be of the form:

θ = cos(αx) cos(β y)Θ(z)eσ t , α = mπ

a, β = nπ

b(12)

w = ∂θ − ∇2θ = (σ + L2 − D2)θ, L2 = α2 + β2 (13)

∂t

J. Niu et al. / Physics Letters A 374 (2010) 4607–4613 4609

where σ is the temporal growth rate, m and n arbitrary integers,and Θ(z) a function only related with z. D is a differential op-erator as D = ∂

∂z . In general, the problem can be considered forany values of m and n. However, solving the three-dimensionalproblem with numerical search is a daunting task. Therefore, weassume n equals zero in the case of b � 1, then the whole prob-lem can be simplified into a two-dimensional one as Wang [22]did in his work.

By substituting Eqs. (12) and (13) into Eq. (8), one gets the fol-lowing normal differential equation

(1 + εσ )[

D4 − (2L2 + σ

)D2 + (

L4 + σ L2)]Θ= L2Ra(1 + λσ )Θ (14)

Obviously, the corresponding solution has the form

Θ(z) = A1ek1 z + A2ek2z + A3ek3z + A4ek4z (15)

where A j ( j = 1,2,3,4) are unknown constants and k j are theroots of the following equation:

k4 − (2L2 + σ

)k2 + L4 + σ L2 − L2Ra

1 + λσ

1 + εσ= 0 (16)

According to the boundary conditions (11a)–(11e) and Eq. (15),A2 = −A1, A4 = −A3 and A1, A3, k1, k3 satisfy

A1(σ + L2 − k2

1

)sinh(k1) + A3

(σ + L2 − k2

3

)sinh(k3) = 0 (17a)

A1[Bi sinh(k1) + k1 cosh(k1)

] + A3[Bi sinh(k3) + k3 cosh(k3)

]= 0 (17b)

In order to have nontrivial solution to the above equation, the de-terminant of its coefficient matrix should be zero, viz.

sinh(k1)(σ + L2 − k2

1

)[Bi sinh(k3) + k3 cosh(k3)

]− sinh(k3)

(σ + L2 − k2

3

)[Bi sinh(k1) + k1 cosh(k1)

] = 0 (18)

In the steady convection case σ = 0, Eq. (18) degenerates to

sinh(k1)[Bi sinh(k3) + k3 cosh(k3)

]+ sinh(k3)

[Bi sinh(k1) + k1 cosh(k1)

] = 0 (19)

This is exactly the same as Kubitschek and Weidman [21] ob-tained for the case of a Newtonian fluid under the same conditions.However, for oscillatory convection as considered in this work, σshould be a purely imaginary number iω in which ω is definedas the oscillation frequency. In this case, k j ( j = 1,2,3,4) cou-ples with σ ; a numerical algorithm will be employed for solvingEq. (18) for the critical value of Ra.

3.3. The limiting case of Bi number

As mentioned above, we consider the problem under differentheating conditions, depending on the value of the Biot number. AsBi approaches infinity which corresponds to the temperature at thebottom boundary equal to T ∗∞ , one gets

limBi→∞

Ra = limBi→∞

(Bi

Bi + 1

)Ra = Ra (20)

Therefore, the Biot-modified Rayleigh number in this case degen-erates to the classical Rayleigh number defined under isothermaltop and bottom boundaries with a temperature difference �T ∗ .The whole problem is then similar to that in the work by Kim etal. [11] in which the required solution for Θ(z) can be easily ob-tained as

Θ = An sin(nπ z) (n = 1,2,3, . . .) (21)

On the other hand, in the limit of vanishingly small Bi, one has

limBi→0

Ra = limBi→0

(Bi

Bi + 1

)Ra = Bi Ra (22)

The quantity Bi Ra equals the Rayleigh number defined for theproblem subject to bottom heating with a constant heat flux. Inother words, as Bi → 0, the problem coincides with that underuniform heat flux heating. Accordingly, Eq. (18) in this case is sim-plified to

k3(σ + L2 − k2

1

)sinh(k1) cosh(k3)

− k1(σ + L2 − k2

3

)sinh(k3) cosh(k1) = 0 (23)

For steady convection (σ = 0), Eq. (23) can be further simplifiedinto the form as that obtained for a Newtonian fluid in a porousmedium [23].

4. Test case

A numerical algorithm is employed in the current work to solveEq. (18) for the critical Rayleigh number that marks onset of ther-mal convection for the Oldroyd-B fluid saturated in the porousmedium. In order to verify that the numerical algorithm is appli-cable and correct, we compare the results of the critical Rayleighnumber for onset of steady convection under a limiting case ofBi with published results for a Newtonian fluid in the porousmedium. As has been discussed above, when the value of Bi isinfinitely large, it represents an isothermal bottom heating condi-tion. It has been shown that in the steady convection case withσ = 0, the critical Rayleigh number for onset of steady convec-tion is independent of the elastic parameters and is the same asthat for a Newtonian fluid [11,17]. Therefore, we set the tempo-ral growth rate σ equal to zero and Bi to be as large as 104, andsolve Eq. (18) numerically for the critical Rayleigh number Rac,s .The results are illustrated in Fig. 1 as a function of a, in which theresults for a Newtonian fluid under the same boundary conditions[21] are also plotted for comparison. It can be seen that our nu-merical results agree well with those in the previous work. It canalso be identified that the global minimum Rayleigh number forBi = 104 is equal to 39.474 at a = 1, which is in close agreementwith the result for a Newtonian fluid under an isothermal bottomboundary [3]. Therefore, our numerical algorithm is applicable tothe present problem.

5. Results and discussion

The critical value Rac which marks the onset of oscillatory con-vection in the porous medium and the corresponding oscillationfrequency ω are determined from Eq. (18) over all modes m byassuming a, λ, ε and Bi to be constants. As a typical viscoelas-tic fluid, the dilute polymeric solution has values of λ and ε inthe range between 0.1 and 2.0 [24]. However, bio-viscoelastic flu-ids such as blood [25] may have values of λ and ε beyond thisrange. Therefore, without loss of generality, we select in this workthe values of λ and ε of viscoelastic fluids in wider ranges thanthose of dilute polymeric solutions [13]. First of all, we studiedthe following three cases: (1) λ = 0.003, ε = 0.001, (2) λ = 0.3,ε = 0.1 and (3) λ = 15, ε = 5. Notice that the values of λ andε are at different magnitudes for these three cases, but the ratioof λ over ε is the same and equal to 3. Secondly, for revealingthe effect of ε only, the value of λ is fixed and the followingthree cases are investigated: (4) λ = 5, ε = 0.1, (5) λ = 5, ε = 0.5and (6) λ = 5, ε = 3. Finally, the values of λ and ε for Case (7)are set as λ = 0.5, ε = 0.1 such that Cases (2), (4) and (7) havethe same value of ε and the corresponding results can be used

4610 J. Niu et al. / Physics Letters A 374 (2010) 4607–4613

Fig. 1. Comparison of the critical Rayleigh numbers for the onset of thermal con-vection of a Newtonian fluid obtained with our numerical algorithm (solid curve) atBi = 104 with those previously published (markers).

to reveal the effect of λ on the heat transfer and flow patterns.In order to cover all the three kinds of boundary conditions atthe bottom boundary, our calculations are performed for Bi in therange from 10−4 to 104. Based on our numerical results, bottomboundary conditions with Bi = 10−4, 10 and 104 can be suitablyselected to represent constant heat flux heating, Newtonian heat-ing and isothermal heating conditions, respectively. Especially, ournumerical results for Bi = 10 are discussed in detail since no workhas been found on the stability of thermal convection in a non-Newtonian fluid-saturated porous medium subject to Newtonianheating.

5.1. Influence of the viscoelastic parameters on the onset of oscillatoryconvection

The influence of the values of λ and ε on the critical Rayleighnumber Rac for onset of oscillatory convection is studied withBi = 10, namely, the bottom boundary under Newtonian heat-ing condition. In our discussion, Ramin denotes the global mini-mum value of Rac . The variation of Rac as a function of a forCases (1)–(3) is respectively shown in Fig. 2(a)–(c), along with thepreferred mode m for thermal convection. The corresponding Raminfor these three cases is identified as 36.538, 18.502 and 12.322 re-spectively. Therefore, the value of Ramin decreases with increase ofthe magnitude of λ and ε, though the ratio λ/ε is the same inCases (1)–(3). However, when comparing our results of Ramin withthat for a Newtonian fluid obtained by Kubitschek and Weidman[21], we find that the value of Ramin for Case (1) is exactly thesame as that for a Newtonian fluid under the same boundary con-ditions. Since the value of Ramin for a Newtonian fluid marks onsetof steady convection, we conclude that there exists no oscillatoryconvection but only steady one upon onset of thermal convectionfor Case (1) with very small values of λ and ε. In other words, theeffect of the viscoelastic parameters is negligible in this case. Butoscillatory convection exists when the viscoelastic parameters arelarge enough and emerges earlier than steady convection for theOldroyd-B fluid, as in Cases (2) and (3). It can be easily seen thatfor any fixed a, the critical Rayleigh number Rac is always smallerfor the case with larger viscoelastic parameters. Therefore, withfixed ratio λ/ε, onset of oscillatory convection gets earlier whenthe magnitude of λ and ε increases. Furthermore, the variation ofthe preferred mode m with a illustrated in Fig. 2 indicates that a

Fig. 2. (a)–(c) Variation of the critical Rayleigh number with rectangular dimensiona at Bi = 10 for Cases (1), (2) and (3), respectively.

Fig. 3. (a)–(c) Variation of the critical Rayleigh number with rectangular dimensiona at Bi = 10 for Cases (4), (5) and (6), respectively.

longer horizontal dimension will result in a larger preferred modein the temperature and velocity distributions.

Plotted in Fig. 3(a)–(c) are the curves of Rac varying with afor Cases (4)–(6), along with the preferred mode m for thermalconvection. In these cases, the value of λ is the same and equalto 5, but the value of ε is different. The corresponding Ramin isdetermined to be 1.110, 4.066 and 22.344, respectively, each ofwhich is smaller than Ramin = 36.538 for onset of steady convec-tion and is thus the global minimum for oscillatory convection to

J. Niu et al. / Physics Letters A 374 (2010) 4607–4613 4611

Fig. 4. Variation of the critical Rayleigh number with rectangular dimension a atBi = 10 for Case (7).

occur. One can easily find that for any fixed a, a larger value ofRac corresponds to a larger value of ε. Therefore, onset of oscil-latory convection can be postponed as the value of ε increases. Itcan also be found that in all the three cases the preferred mode mincreases with an increasing a, but the interval between two adja-cent extrema of Rac decreases with a decrease of ε.

In order to reveal the effect of the value of λ on the onset ofoscillatory convection, we can compare the results for Cases (2),(4) and (7), in which ε = 0.1 but the value of λ is different. Sincethe Rac vs. a curves for Cases (2) and (4) have been shown inFigs. 2(b) and 3(a), plotted in Fig. 4 is only the result for Case (7)with the corresponding Ramin equal to 11.102. It can be found fromFigs. 2(b), 4(a) and 7 that for any constant a, the critical Rayleighnumber Rac is larger for the case with smaller value of λ, whichindicates the relaxation time λ tends to facilitate instability of theOldroyd-B fluid in the porous medium. In addition, except for thedifferent values of Rac , the varying tendency of these three curvesappear similar and the length of interval between two adjacentextrema of Rac is the same, which is different from those shownin Fig. 3 and indicates that the value of λ has no effect on thevariation of preferred mode m.

At other fixed values of Bi, the effects of the viscoelastic param-eters on the onset of thermal convection in the Oldroyd-B fluid-saturated porous medium is similar to those shown in Figs. 2–4and will not be repeated here. Instead, the influence of Bi on theonset of thermal convection will be discussed in detail in the fol-lowing section.

5.2. The effect of the Biot number

Fig. 5 shows the global minimum Rayleigh number Ramin asfunctions of the Biot number for Cases (1)–(7). In all the sevencases, we find that onset of thermal convection marked by Raminis continuously delayed with increasing Bi, especially in the range10−2 � Bi � 103. As Bi → 0 and Bi → ∞, Ramin will saturate atthe value corresponding to bottom heating by a uniform heat fluxand by an isothermal bottom boundary, respectively. Therefore, forgiven values of λ and ε, onset of thermal convection occurs at thelowest Ramin with bottom subject to uniform heat flux heating;while the highest Ramin corresponds to onset of thermal convec-tion under an isothermal bottom heating condition. Especially, thevalue of Ramin for Case (1) at Bi = 10−4 is determined to be 27.098while that at Bi = 104 is identified equal to 39.474. These twovalues are exactly the same as those obtained by Kubitschek andWeidman [21] for a Newtonian fluid, which further confirm thatthe value of Ramin in Case (1) marks onset of steady convectionand the values of λ and ε are too small to induce oscillatory con-vection in prior to steady convection. From the curves in Fig. 5,it can be seen that the value of Ramin increases with increasing εbut decreases with increase of λ, which holds for any value of Bi.Furthermore, it is interesting to find that the difference betweenthe value of Ramin corresponding to Bi → 0 and that correspond-ing to Bi → ∞ is the largest for steady convection. For oscillatoryconvection, the difference decreases with an increase of λ and a

Fig. 5. Variation of the global minimum Rayleigh number with the Biot number forCases (1)–(7), respectively.

decrease of ε. When the ratio of λ to ε gets large enough as thatin Case (4), the Ramin vs. Bi curve is almost horizontal, indicatingthat the onset of oscillatory convection is insensitive to the bottomheating condition.

5.3. Temperature disturbance

The most illustrative way to distinguish the effects of differ-ent bottom heating conditions is to analyze the transition of thetemperature field as a function of Bi with a constant horizontal di-mension a. The analytical solution of the temperature disturbanceθ can be obtained from Eqs. (15) and (17) as:

θ = A1 cos(ax)

[sinh(k1z) − (iω + a2 − k2

1) sinh(k1)

(iω + a2 − k23) sinh(k3)

sinh(k3z)

]

× eiωt (24)

Fig. 6(a)–(c) shows the patterns of the disturbed temperaturecontours for Case (5) with dimension a = 3.25 and time t = 2pπ

ω

(p is a positive integer) at Bi = 10−4, 10 and 104, respectively.The corresponding critical Rayleigh number Rac and oscillation fre-quency ω are adopted in the calculations in order to obtain theresults at the onset point. The cold and hot fluid regions are dis-tinguished by light and dark shaded areas.

For Bi = 10−4, the preferred mode is m = 3, isotherms at thebottom in Fig. 6(a) are perpendicular to the horizontal bound-ary. For Bi = 10, Fig. 6(b) shows the preferred mode maintainsm = 3, all the cold and hot spots are located higher than those inFig. 6(a); isotherms at the bottom now intersect obliquely the hor-izontal boundary. As Bi increases to 104, however, the isotherms inFig. 6(c) make a profound change; all contours at the bottom haveno intersection but become parallel to the horizontal boundary.Furthermore, the isotherms in this case appear to be symmetricin the z-direction. Although not shown here, the disturbed tem-perature distribution patterns become more complicated as a in-creases. Therefore, the pattern of isothermal contours upon onset

4612 J. Niu et al. / Physics Letters A 374 (2010) 4607–4613

Fig. 6. (a)–(c) The patterns of disturbed temperature contours for Case (5) for a = 3.25 and t = 2nπω (n is a positive integer) at Bi = 10−4, Bi = 10, Bi = 104 respectively.

Fig. 7. (a)–(c) The patterns of disturbed temperature contours for Cases (1)–(3) for a = 3.25 and t = 2nπω (n is a positive integer) at Bi = 10−4.

of oscillatory convection is strongly dependent on the rectangulardimension and the Biot number.

In order to investigate the effects of the viscoelastic parame-ters on the temperature distribution, the patterns of the disturbedtemperature contours for Cases (1)–(3) are plotted in Fig. 7(a)–(c)respectively with dimension a = 3.25 and time t = 2pπ

ω (p is apositive integer) at Bi = 10−4. It can be seen that the isothermpattern in Fig. 7(b) is more complicated than that in Fig. 7(c)because the smaller value of ε corresponds to smaller distancebetween adjacent extrema in Fig. 7(b) than in Fig. 7(c), resultingin larger preferred mode in Fig. 7(b). However, the isotherm pat-tern in Fig. 7(a) is less complex than those in Fig. 7(b) and (c)though the value of ε in Case (1) is the smallest. As has alreadybeen discussed above, there exists no oscillatory convection upononset for Case (1) because the values of λ and ε are too smallto induce oscillatory convection. Therefore, the isotherm patternin Fig. 7(a) corresponds to steady convection while those appearin Fig. 7(b) and (c) are for oscillatory convection. We also inves-tigated the effect of ε on the disturbed temperature distributionwith fixed values of λ, Bi and a, it is found that the isotherm pat-tern is more complicated for the case with smaller ε due to thelarger preferred mode. Although not shown here, we have checkedthat the pattern of the disturbed temperature contours is indepen-dent of relaxation time λ. Therefore, the variations of disturbedtemperature distribution further confirm our conclusions in Sec-tion 5.1.

6. Conclusion

The onset of oscillatory convection in a thin porous rectangu-lar box saturated with an Oldroyd-B fluid heated from below byforced convection is studied in this work. A modified Darcy’s lawis employed to describe the flow of the viscoelastic fluid in theporous medium. A two-dimensional linear stability analysis wasfirst performed in terms of the disturbed temperature and flow

velocity. Then the critical Biot-modified Rayleigh numbers mark-ing the onset of thermal convection were obtained, along with thepreferred modes using a numerical algorithm. The Biot numberBi is introduced to describe the characteristics of bottom heatingconditions and seven different set values of the viscoelastic pa-rameters are chosen in our calculations. The results reveal someimportant properties for the fluid upon onset of thermal convec-tion. First, with fixed Biot number and rectangular dimension, theform of thermal convection depends on the magnitude of the vis-coelastic parameters. With very small values of the parameters,there exists only steady convection, but when the magnitude ofthe viscoelastic parameters is large enough, oscillatory convectioncan be induced and is in prior to the onset of steady convection.The critical Biot-modified Rayleigh number for oscillatory convec-tion strongly depends on λ and ε; the value of Rac increases withincreasing ε and with decrease of λ. Second, with fixed λ and ε,the global minimum Rayleigh number becomes larger as the Biotnumber increases. The global minimum Rayleigh number saturateswhen Bi → 0 and Bi → ∞, which coincides respectively with Raminunder bottom heating by a uniform heat flux and by an isother-mal bottom heating. The difference between Ramin for Bi → 0 andRamin for Bi → ∞ becomes smaller with an increase of λ and adecrease of ε. Third, for fixed relaxation and retardation times,the pattern of disturbed temperature contours upon onset of os-cillatory convection and the preferred mode depend on the Biotnumber; the pattern of disturbed temperature contours dependson the retardation time and the aspect ratio of the porous layer,but is independent of the relaxation time, which holds at any valueof Bi.

Acknowledgements

This work is supported by the National Natural Science Foun-dation of China (Grant Nos. 10825208 and 10632010) and theNational Key Basic Research Program (Grant No. 2007CB814800).

J. Niu et al. / Physics Letters A 374 (2010) 4607–4613 4613

References

[1] D.A. Nield, A. Bejan, Convection in Porous Media, Springer-Verlag, New York,2006.

[2] D.B. Ingham, I. Pop, Transport Phenomena in Porous Media, Pergamon, Oxford,1998.

[3] E.R. Lapwood, Proc. Cambridge Philos. Soc. 44 (1948) 508.[4] E.M. Sparrow, R.J. Goldstein, V.K. Jonsson, J. Fluid Mech. 18 (1964) 513.[5] C.W. Horton, F.T. Rogers, J. Appl. Phys. 6 (1945) 367.[6] T. Hayat, S.B. Khan, M. Khan, Nonlinear Dyn. 47 (2007) 353.[7] M.S. Malashetty, W.C. Tan, M. Swamy, Phys. Fluids 21 (2009) 084101.[8] S.C. Hirata, M.N. Ouarzazi, Phys. Lett. A 374 (2010) 2661.[9] A.M. Salem, Phys. Lett. A 369 (2007) 315.

[10] M. Sokolov, R.I. Tanner, Phys. Fluids 15 (1972) 534.[11] M.C. Kim, S.B. Lee, S. Kim, B.J. Chung, Int. J. Heat Mass Transfer 46 (2003) 5065.

[12] M.G. Alishaev, A.K. Mitzadajanzade, Izvestiia VUZov Neft i Gaz 6 (1975) 71.[13] J. Niu, C.J. Fu, W.C. Tan, J. Non-Newtonian Fluid Mech. 165 (2010) 203.[14] D.Y. Yoon, M.C. Kim, C.K. Choi, Trans. Porous Media 55 (2004) 275.[15] B. Bertola, E. Cafaro, Int. J. Heat Mass Transfer 49 (2006) 4003.[16] C.J. Fu, Z.Y. Zhang, W.C. Tan, Phys. Fluids 19 (2007) 104107.[17] Z.Y. Zhang, C.J. Fu, W.C. Tan, Phys. Fluids 19 (2007) 098104.[18] H.S. Carslaw, J.C. Jaeger, Conduction of Heat in Solids, Clarendon Press, Oxford,

1947.[19] D. Lesnic, D.B. Ingham, I. Pop, Int. J. Heat Mass Transfer 42 (1999) 2621.[20] D. Lesnic, D.B. Ingham, I. Pop, C. Storr, Heat Mass Transfer 40 (2004) 665.[21] J.P. Kubitschek, P.D. Weidman, Int. J. Heat Mass Transfer 46 (2003) 3697.[22] C.Y. Wang, Trans. Porous Media 46 (2002) 37.[23] C.Y. Wang, Phys. Fluids 11 (1999) 1673.[24] S. Rosenblat, J. Non-Newtonian Fluid Mech. 21 (1986) 201.[25] K.K. Yeleswarapu, M.V. Kameneva, K.R. Rajagopal, J.F. Antaki, Mech. Res. Com-

mun. 25 (1998) 257.