an autonomous system for thermal convection of viscoelastic fluids in a porous layer using a thermal...

RESEARCH ARTICLE

Qi WEI, Xiaohui ZHANG

An autonomous system for thermal convection of viscoelasticfluids in a porous layer using a thermal nonequilibrium model

© Higher Education Press and Springer-Verlag Berlin Heidelberg 2010

Abstract Thermal convection of viscoelastic fluidssaturating a horizontal porous layer heated from below isanalyzed using a thermal nonequilibrium model to takeaccount of the interphase heat transfer between the fluidand the solid. The viscoelastic character of the flow isconsidered by a modified Darcy’s law. An autonomoussystem with five differential equations is deduced byapplying the truncated Galerkin expansion to the momen-tum and heat transfer equations. The effects of interphaseheat transfer H on the thermal convection of viscoelasticfluids in a porous medium are analyzed and discussed. Theresults show that the weak interphase heat transfer tends tostabilize the steady convection.

Keywords thermal convection, porous media, viscoelas-tic fluid, thermal nonequilibrium model

1 Introduction

The thermal convection in a saturated porous media is asubject of practical interest for its applications inengineering, such as solar energy storage systems,geothermal reservoirs, passive cooling of nuclear reactors,pollutant transport in underground waters, soil decontami-nation, storage of chemical or agricultural products, etc.More detailed discussions of its theory and applications arepresented by Nield and Bejan [1].One interesting case of convection in porous media

arises when the fluid is viscoelastic. Recently, Kim et al.[2] have studied thermal instability driven by buoyancyforces in an initially quiescent and horizontal porous layersaturated by viscoelastic fluids and found that theoverstability is a preferred mode of instability for a certainrange of elastic parameters. Yoon et al. [3] have considered

the onset of oscillatory convection in a horizontal porouslayer saturated with viscoelastic liquid. Bertola and Cafaro[4] have studied theoretically the instability of a viscoe-lastic fluid saturating a horizontal porous layer heated frombelow with a dynamical system approach.However, the work of Bertola and Cafaro has been

conducted under the assumption that the fluid and porousmedium are in local thermodynamic equilibrium every-where. However, for many practical applications involvinghigh-speed flows or large temperature differences betweenthe solid and fluid phases, the assumption of local thermalequilibrium is inadequate, and it is important to takeaccount of the thermal nonequilibrium effects. Nield andBejan [1] have stated that the thermal nonequilibriummodel utilizes two equations to separately model the fluidand solid phases instead of using a single energy equationthat describes the common temperature of the saturatedmedium (the one-field model). In the thermal none-quilibrium model, the two energy equations are coupledtogether by terms that account for the heat lost to or gainedfrom the other phase. As a matter of fact, the thermalnonequilibrium effects on the convection of Newtonianfluids in porous media have been considered already. Forexample, Banu and Rees [5] have studied the effect ofthermal nonequilibrium on the onset of convection in aporous layer and found that the critical Rayleigh numberand wave number are modified by the presence of thermalnonequilibrium. Sheu [6] has studied the transition tochaos in a porous layer using thermal nonequilibrium, andthe results show that the interphase heat transfer stabilizessteady convection and alters the routes to chaos.The aim of the present paper is to study the thermal

convection of a viscoelastic fluid saturating a horizontalporous layer using a thermal nonequilibrium model to takeaccount of the interphase heat transfer between the fluidand the solid. The viscoelastic character of the flow isconsidered by a modified Darcy’s law. The truncatedGalerkin expansion is applied to the governing equationsof the thermal convection in a porous medium to deduce anautonomous system with five ordinary differential equa-tions. The system is used to investigate the dynamic

Received March 27, 2009; accepted August 29, 2009

Qi WEI (✉), Xiaohui ZHANGSchool of Physics Science and Technology, Soochow University,Suzhou 215006, ChinaE-mail: [email protected]

Front. Energy Power Eng. China 2010, 4(4): 507–516DOI 10.1007/s11708-010-0017-x

behavior of the thermal convection of viscoelastic fluids inthe porous medium in order to study the effect ofinterphase heat transfer on the convection.

2 Formulations

Let us consider a fluid-saturated porous layer of width land height d, which is heated from below and cooled fromabove. The bottom surface at y ¼ 0 is held at constanttemperature Th, the top one at y ¼ d is held at constanttemperature Tc, and the vertical walls are adiabatic.The momentum equation on unsteady flows of a

Newtonian fluid in porous media is usually expressed inthe form of Darcy’s law, which states that

�fK

ε�∂u∂t

þ u ¼ –K

�ðrp – �fgÞ, (1)

where u ¼ ui þ vj is the fluid velocity vector, g is thegravity, p is the pressure of the fluid, �f is the density of thefluid, � is the dynamic viscosity of the fluid, K is a quantitycalled permeability, and ε is the porosity.However, because the pressure drop in viscoelastic

flows is nonlinearly related to the filtration velocity, thepermeability would also depend on the relaxation time ofthe fluid. Following a well-established approach [7], themodified Darcy’s law can be obtained by including arelaxation term about the pressure gradient, with acharacteristic time τ depending on viscoelasticity:

�fK

ε�∂u∂t

þ u ¼ –K

�1þ τ

∂∂t

� �ðrp – �fgÞ: (2)

This equation implicitly assumes that the fluid has aconstant viscosity and a single relaxation time that is thecase, for instance, of an upper convected Maxwell fluid [8].It is assumed in this paper that the convective fluid and

the porous medium are not in local thermodynamicequilibrium, and therefore, a two-temperature model ofmicroscopic heat transfer applies. The governing equationsfor the fluid and solid temperatures are [1]

εð�cÞf∂Tf∂t

þ ð�cÞfu$rTf ¼ εkfr2Tf þ hðTs – Tf Þ, (3)

ð1 – εÞð�cÞs∂Ts∂t

¼ ð1 – εÞksr2Ts – hðTs – Tf Þ, (4)

and

� ¼ �0½1 – βðTf – TcÞ�, (5)

where T is the temperature. The subscripts f and s denotethe fluid and solid phases, respectively. The properties ofthe fluid and porous media include the specific heat ðcÞ, thecoefficient of cubical expansion ðβÞ, the thermal con-ductivity ðkÞ, and the interphase heat transfer coefficientbetween the fluid and the porous medium ðhÞ.

Equations (2)–(5) and the continuity equation r⋅u ¼ 0constitute the basic equations on convection of aviscoelastic fluid saturating a porous layer. They can benondimensionalized by using the following transforma-tions:

x,yð Þ ¼ dðx,yÞ,

u,vð Þ ¼ εkfð�cÞfd

ðu,vÞ,Tf ¼ ðTh – TcÞ�þ Tc,

Ts ¼ ðTh – TcÞfþ Tc,

t ¼ ð�cÞfkf

d2t:

8>>>>>>>>>>><>>>>>>>>>>>:

(6)

The pressure terms can be eliminated by introducing thestream function, u ¼ –ψy and v ¼ ψx. In the streamfunction formulation, the continuity equation is automati-cally satisfied. Eqs. (2)–(5) then become

τ1∂∂t

þ 1

� �ðψxx þ ψyyÞ ¼ Ra τ2

∂∂t

þ 1

� ��x, (7)

�t –ψy�x þ ψx�y ¼ �xx þ �yy þ hðf – �Þ, (8)

and

αft ¼ fxx þ fyy þ lhð� –fÞ, (9)

where

Ra ¼ �fgβðTh – TcÞKdε�kf

,

l ¼ εkfð1 – εÞks

, h ¼ hd2

εkf, α ¼ ð�cÞskf

ð�cÞfks:

8>><>>:

(10)

are Darcy-Rayleigh number ðRaÞ, a porosity-modifiedconductivity ratio ðlÞ, a scaled interphase heat transfercoefficient ðhÞ, and a diffusivity ratio ðαÞ, respectively. Inparticular, τ1 is the dimensionless retardation time due tothe action of the porous matrix, while τ2 is the dimension-less relaxation time depending on viscoelasticity.

3 Truncated Galerkin expansion

The finite amplitude analysis was carried out using atruncated representation of the Galerkin expansion byconsidering only one term for stream function and twoterms for temperature distributions in the respective forms of

ψ ¼ A11sinðaxÞsinðπyÞ, (11)

� ¼ 1 – yþ B11cosðaxÞsinðπyÞ þ B02sinð2πyÞ, (12)

and

508 Front. Energy Power Eng. China 2010, 4(4): 507–516

φ ¼ 1 – yþ C11cosðaxÞsinðπyÞ þ C02sinð2πyÞ: (13)

Substituting Eqs. (11)–(13) into Eqs. (7)–(9), multiplyingthe equations by orthogonal eigenfunctions correspondingto (11)–(13), integrating them over the spatial domain, andrescaling the amplitudes yield the following system ofnonlinear ordinary differential equations:

dX

dτ¼ –

1 –RD2

D1X þ 1 –D2

D1Y – ðR – 1ÞD2

D1XZ

þ D2

D1

H

g2ðU –Y Þ,dY

dτ¼ RX – Y – ðR – 1ÞXZ þ HðU –Y Þ,

dZ

dτ¼ 4gðXY –ZÞ –HðW þ ZÞ,

dU

dτ¼ 1

α½ –U þ lHðY –UÞ�,

dW

dτ¼ 1

α½ – 4gW – lHðZ þW Þ�,

8>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>:

(14)

where the time was rescaled, and the following notationwas introduced:

τ ¼ δ2t, D1 ¼ δ2τ1, D2 ¼ δ2τ2,

g ¼ π2

δ2, R ¼ Ra

δ2, H ¼ h

δ2,

8><>: (15)

and the amplitudes were rescaled in the form of

X ¼ A11ag

2πffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2gðR – 1Þp , Y ¼ πRB11

2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2gðR – 1Þp ,

Z ¼ –πRB02

ðR – 1Þ, U ¼ πRC11

2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2gðR – 1Þp ,

W ¼ πRC02

ðR – 1Þ:

8>>>>>><>>>>>>:

(16)

System (14) provides a set of nonlinear ordinarydifferential equations with seven parameters. ParameterD1 is a modified dimensionless retardation time, D2 is amodified dimensionless relaxation time, R is the modifiedDarcy-Rayleigh number, and H is the modified interphaseheat transfer coefficient; parameter α is the diffusivity ratio,and l is the porosity-modified conductivity ratio. The valueof g has to be consistent with the wave number at theconvection threshold, which is required so that theconvection cells fit into the domain and fulfill the boundaryconditions. System (14) is similar to that given by Eqs. (13)in Sheu [6], the only differences being in the first equationthat has two additional terms and different coefficients.System (14) atD2 ¼ 0 are reduced to the system of Sheu [6].The steady state solutions are useful because they

predict that a finite amplitude solution to the system ispossible for subcritical values of the Rayleigh number andthat the minimum values of Ra for which a steady solutionis possible lies below the critical values for instability toeither a marginal state or an overstable infinitesimalperturbation.

The critical (or equilibrium) points correspond to thesteady state solutions of the dynamical system (14), fromwhich they are obtained by setting time derivatives at zero.Set t ing X ¼ ½X ,Y ,Z,U ,W �, and X1 ¼ ½0,0,0,0,0� i sobviously an equilibrium point. Let

a ¼ ðlH þ 1ÞðR – 1Þ – ½1þ ð1=g2 – 1ÞRD2�H½ðlH þ 1Þ – ð1=g2 – 1ÞD2H �ðR – 1Þ ,

b ¼ lH þ H þ 4g

lH þ 4g, c ¼ lH

lH þ 1,

d ¼ –lH

lH þ 4g, e ¼ ðlH þ 1Þ – ð1=g2 – 1ÞD2H

lH þ 1:

8>>>>>><>>>>>>:

(17)

Thus, two nonzero equilibrium points are given by

X2 ¼ffiffiffiffiffiffiffiabe

p,

ffiffiffiffiffiffiab

e

r,a,c


e

r, – da

� �,

X3 ¼ –ffiffiffiffiffiffiffiabe

p, –


e

r,a, – c


e

r, – da

� �:

8>><>>:

(18)

The solution X1 ¼ ½0,0,0,0,0� corresponds to pure con-duction, which is known to be a possible solution though itis unstable when RðRaÞ is sufficiently large. The remainingsolutions X2 and X3 characterize the onset of finiteamplitude steady motions.

4 Numerical results

By considering the thermal nonequilibrium model of heattransfer between the fluid and solid phase, system (14) wasdeduced to describe the dynamics of thermal convection ofviscoelastic fluids in porous media. The objective of thisstudy was to analyze the thermal nonequilibrium effect onthe dynamic behavior of thermal convection of viscoelasticfluids in a porous medium as the Rayleigh numberchanges. The important parameters representing this effectare H , the interphase heat transfer coefficient, and k, theporosity-modified conductivity ratio. In order to simplifythe analysis, the other parameters were kept constant. Thevalue of g used in all computations was 0.5, which isconsistent with the critical wave number at marginalstability in porous medium convection. All solutions wereobtained using the same initial conditions that wereselected to be in the neighborhood of the positiveconvection fixed point, i.e., at τ ¼ 0: X ¼ Y ¼ Z ¼ 0:9,and U ¼ W ¼ 0. System (14) was solved by applyingRunge-Kutta method. All computations were carried out toa value of a maximum time, τmax, of 100 with a constanttime step Δτ ¼ 0:001.In the following discussions, three values of H (0.01,

0.05, 5) were chosen to investigate the effects of interphaseheat transfer parameter H on the dynamic behaviors ofsystem (14).By carefully examining Figs. 1–3, several interesting

Qi WEI et al. Thermal convection of viscoelastic fluids in a porous layer 509

and important physical phenomena can be found. For smallvalues, e.g., H = 0.05, which indicates weak interphaseheat transfer between the fluid and solid. As the Rayleighnumber increases, a cascade of a period-2 (R = 2), period-4

(R = 5) route to multiperiods (R = 40) ensue. Furthermore,it can be predicted that chaos will occur as R increases.Figures 4–6 show the phase portraits at H = 0.01 for

various values of the Rayleigh number (R = 2, 5 and 40).

Fig. 1 Time history of X and phase portraits at R ¼ 2, H = 0.05(a) Time history of X ; (b) phase portrait of X-Z; (c) phase portrait of X-W; (d) phase portrait of Y-U


Fig. 2 Time history of X and phase portraits at R = 5, H = 0.05(a) Time history of X ; (b) phase portrait of X-Z; (c) phase portrait of X-W; (d) phase portrait of Y-U



Steady-state solutions that are different from the initialstate can be gained at R = 40, as shown in Fig. 6. Thewaveform of X reaches a stable value as the nondimen-sional time approaches 40, which means that the numericalsolution is independent on the time. The phase trajectories

of X-Z, X-W, and Y-U tend to be steady points. It is,however, interesting to observe in Figs. 4(a) and 5(a), thatthe waveform of X is periodically oscillatory.The equilibrium model corresponding to temperature

equilibrium between the fluid and solid suggests that the



solution should lie on the plane Y = U and Z = –W. InFig. 4, it can be observed that the solution departs from thisplane. Z is 1 order of magnitude smaller than –W. In

Fig. 5, it can be seen that W and U are four or three ordersof magnitude smaller than –Z and Y.The steady state solution is obtained at R = 2 asH further




increases to 5, as shown in Fig. 7. Figure 7(a) may bereviewed as the overdamped oscillation with the dampingeffect being weaker at lower Rayleigh number, and the

flows tend to be stable pure conduction state:X1 ¼ ½0,0,0,0,0�, which means that the numerical solutionis independent on initial state X ¼ ½0:9,0:9,0:9,0,0�.

Fig. 7 Time history of X and phase portraits at R ¼ 2, H = 5(a) Time history of X ; (b) phase portrait of X-Z; (c) phase portrait of X-W; (d) phase portrait of Y-U


As the Rayleigh number continues to increase to 5,Fig. 8(a) illustrates the oscillations of the time history of X.It is found that a periodic oscillation appears. Moreover,

when a period-doubling route to chaos is found at R = 40,system (14) becomes chaotic. The destabilization occursearlier when R increases. Figures 9(b), (c), and (d) give the

Fig. 8 Time history of X and phase portraits at R = 5, H = 5(a) Time history of X ; (b) phase portrait of X-Z; (c) phase portrait of X-W; (d) phase portrait of Y-U

Fig. 9 Time history of X and phase portraits at R = 40, H = 5(a) Time history of X ; (b) phase portrait of X-Z; (c) phase portrait of X-W; (d) phase portrait of Y-U


chaotic trajectory of the equation. The temporal oscillatingflows are quite periodic at lower R number, as shown inFig. 8(a), but aperiodic at higher R number, as shown inFig. 9(a).It is noted in Fig. 8 that U and W are one order of

magnitude smaller than Y and Z. In Fig. 9, it can be foundthat U and W are two or one order of magnitude smallerthan Y and Z.

5 Conclusions

A five-dimensional autonomous dynamic system wasdeduced to analyze the thermal convection of viscoelasticfluids in a porous medium by applying the thermalnonequilibrium model. The problem was examined bymeans of stability of equilibria, time history, and phaseportraits. It was found that the interphase heat transferbetween the fluid and the solid alters the routes to chaos.Also, the very weak interphase heat transfer (H = 0.01)tends to stabilize steady convection. With weak interphaseheat transfer (H = 0.05), as the Rayleigh number increases,a cascade of a period-2, period-4 route to multiperiodsensue. Furthermore, it can be predicted that chaos willoccur as R increases. As interphase heat transfer H furtherincreases to 5, the destabilization and chaos occurs earlierwhen R increases.

Acknowledgements This work was supported by the National NaturalScience Foundation of China (Grant No. 50576061).

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an autonomous system for thermal convection of viscoelastic fluids in a porous layer using a thermal...

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