Physics Letters A 375 (2011) 2683–2688

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Physics Letters A

www.elsevier.com/locate/pla

From cat’s eyes to disjoint multicellular natural convection flow in tall tiltedcavities

Alfredo Nicolás a,∗, Elsa Báez b, Blanca Bermúdez c

a Depto. Matemáticas, 3er Piso Ed. AT-Diego Bricio, UAM-I, 09340 México D.F., Mexicob Depto. Matemáticas Aplicadas y Sistemas, UAM-C, 01120 México D.F., Mexicoc Facultad de C. de la Computación, BUAP, 72570 Puebla, Pue., Mexico

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

Article history:Received 5 June 2010Received in revised form 23 March 2011Accepted 4 May 2011Available online 18 May 2011Communicated by A.P. Fordy

Keywords:Unsteady Boussinesq approximationRayleigh numberInclination angleAspect ratioCat’s eyesDisjoint multicellular flow

Numerical results of two-dimensional natural convection problems, in air-filled tall cavities, are reportedto study the change of the cat’s eyes flow as some parameters vary, the aspect ratio A and the angleof inclination φ of the cavity, with the Rayleigh number Ra mostly fixed; explicitly, the range of thevariation is given by 12 � A � 20 and 0◦ � φ � 270◦; about Ra = 1.1 × 104. A novelty contribution ofthis work is the transition from the cat’s eyes changes, as A varies, to a disjoint multicellular flow, as φ

varies. These flows may be modeled by the unsteady Boussinesq approximation in stream function andvorticity variables which is solved with a fixed point iterative process applied to the nonlinear ellipticsystem that results after time discretization. The validation of the results relies on mesh size and time-step independence studies.

© 2011 Elsevier B.V. All rights reserved.

The physics involved on natural convection flows in enclosuresmodels many engineering applications: nuclear reactor insulation,ventilation of buildings, and cooling of electronic devices; for tallcavities the applications rely mostly on solar energy and buildingservices engineering systems [1]. These flows, thermally coupledto viscous fluids in a gravitational system, may be modeled bythe unsteady Boussinesq approximation which is based on the factthat temperature variations are small enough to imply that densityvariations are negligible except for the buoyancy force in the mo-mentum equation, leading to an incompressible structure; on thisregard, the 2D formulation in stream function and vorticity vari-ables is considered.

The results are obtained with a numerical method which, af-ter time discretization, leads to the solution of a nonlinear systemof elliptic equations which in turn is solved through an iterativefixed point process leading to the solution of uncoupled, well-conditioned, symmetric linear elliptic problems for which efficientsolvers exist regardless of the space discretization. This numericalmethod, reported in [2] for moderate tilted cavities has turned outto be robust enough to undertake a study of the effects on thecharacteristics of flows in tall cavities, filled with air and heated

* Corresponding author. Tel.: +52 55 5804 4656; fax: +52 55 5804 4660.E-mail address: anc@xanum.uam.mx (A. Nicolás).

0375-9601/$ – see front matter © 2011 Elsevier B.V. All rights reserved.doi:10.1016/j.physleta.2011.05.022

from the side, as the aspect ratio A of the cavity (A = ratio of theheight to the width) and the angle of inclination of the cavity φ,vary; with the Rayleigh number Ra mostly fixed.

In this Letter, the variation of the parameters is given by 12 �A � 20 and 0◦ � φ � 270◦; about Ra = 1.1 × 104. Interest thereexists not only on the dynamics and evolution of the fluid flowbut also on the heat transfer; then, the study is complementedwith the Nusselt numbers, local Nu and global Nu; the time Tss

when the steady state of the flow is reached is also reported, thenthe time length of the transient stage is determined. Mesh size andtime-step independence studies are made to validate the results;then, the validation is not depending on the comparison with otherworks using different dimensionless forms. However, for φ = 0◦ acomparison is done with [1] and [3,4] on the cat’s eyes flow as φ

and A vary: some agreements and disagreements are shown.To get natural convection flows for large A is not possible for

arbitrary high Ra unless an appropriate dimensionless form beused, smoother than the one used here; something like this is re-ported in [2] and [5], for Ra = 3.5 × 105 with A = 8 and untilRa = 1011 with A = 4 respectively.

To the best of our knowledge various findings from this studyare being reported for the first time: i) with φ = 0◦ fixed: unlike[1] and [3,4], where the appearing and disappearing of the cat’seyes occurs with Ra variations about 1.1 × 104 we show that thisphenomenon occurs also when A varies about 16, and Ra fixed;

2684 A. Nicolás et al. / Physics Letters A 375 (2011) 2683–2688

ii) for φ > 0◦ , and A = 16 fixed: the change of cat’s eyes flow todisjoint multicellular flow for φ � 60◦; iii) if Ra changes about anangle φ, say φ = 75◦ , the number of disjoint cells holds constantas Ra � 1.1 × 104 and when Ra > 1.1 × 104 but with a smallerconstant; iv) when the cavity is cooled from below there are notcat’s eyes nor disjoint multicellular cells: there is not convection atall but conduction. From hereafter, A∗ = 16 and Ra∗ = 1.1 × 104.

1. Mathematical model and numerical method

Let Ω ⊂ RN (N = 2,3) be the region of the flow of an unsteadythermal viscous incompressible fluid flow, and let Γ its boundary.Under the Boussinesq approximation these flows may be modeledby the dimensionless system

ut − ∇2u + ∇p + (u · ∇)u = Ra

Prθe, (1a)

∇ · u = 0, (1b)

θt − 1

Pr∇2θ + u · ∇θ = 0, (1c)

in Ω , t > 0; where u, p and θ are the velocity, pressure, and tem-perature of the flow respectively, e is the unitary vector in thegravitational direction. The dimensionless parameters Ra and Pr arethe Rayleigh and Prandtl numbers. The dimensionless temperatureθ is given by θ = T −T0

T1−T0, where T0 and T1 are reference temper-

atures, T0 < T1. The system must be supplemented with initialconditions u(x,0) = u0(x) and θ(x,0) = θ0(x) in Ω; and bound-ary conditions, say u = f and Bθ = 0 on Γ , t � 0, where B is a θ

boundary operator.Restricting Eqs. (1) to a bi-dimensional region Ω , taking the

curl in both sides of Eq. (1a) and taking into account

u1 = ∂ψ

∂ y, u2 = −∂ψ

∂x, (2)

which follow from (1b), with ψ the stream function and(u1, u2) = u; the component in the direction k = (0,0,1) givesthe scalar system, in Ω , t > 0,

∇2ψ = −ω, (3a)

ωt − ∇2ω + u · ∇ω = Ra

Pr

(∂θ

∂xcosφ − ∂θ

∂ ysinφ

), (3b)

θt − γ ∇2θ + u · ∇θ = 0, (3c)

where the vector e in (1a) has been replaced by the angle of in-clination φ of the region Ω: e = (sin φ, cosφ); γ = 1/Pr, ω isthe vorticity which, from ωk = ∇ × u = −∇2ψk, gives (3a) andω = ∂u2

∂x − ∂u1∂ y as well. Then, system (3) gives the Boussinesq ap-

proximation in stream function and vorticity variables.The natural convection flows here are set in rectangular cavities

Ω = (0,a) × (0,b); a > 0, b > 0. To construct the boundary condi-tion ωbc for ω, which is not a trivial task to deal with, variousalternatives have been proposed. Here the second order approx-imation in [6] extended to natural convection problems is used:Taylor expansion of ψ on Γ combined with (3a); [2]. However,ωbc , [2], depends on values still unknown of ψ in Ω , t > 0; thisproblem is solved within a fixed point iterative process.

The local Nusselt number Nu measures the heat transfer at eachpoint on the hot wall where θ is specified and the global Nusseltnumber Nu the average of such heat transfer. They are defined by:Nu(x) = − ∂θ

∂ y |y=0,b or Nu(y) = − ∂θ∂x |x=0,a; Nu|y=0,b = 1

A

∫ a0 Nu(x)dx

or Nu|x=0,a = 1A

∫ b0 Nu(y)dy.

Once the time derivatives of ω and θ in (3) are approximatedby

ft(x, (n + 1)t

) = 3 f n+1 − 4 f n + f n−1

2t+ O

(t2), (4)

n � 1, x ∈ Ω , t > 0 the time step, and f r = f (x, rt); at eacht = (n + 1)t a semi-discrete system with unknowns (ψn+1,ωn+1,

θn+1) is obtained. After renaming these unknowns by (ψ,ω, θ),a nonlinear elliptic system results

∇2ψ = −ω, ψ |Γ = 0, (5a)

αω − ∇2ω + u · ∇ω = Ra

Pr

(∂θ

∂xcosφ − ∂θ

∂ ysinφ

)+ fω,

ω|Γ = ωbc, (5b)

αθ − γ ∇2θ + u · ∇θ = fθ , Bθ |Γ = 0, (5c)

where α = 32t , fω = 4ωn−ωn−1

2t , and fθ = 4θn−θn−1

2t ; ωbc denotesthe boundary condition of ω, as stated before, B the boundary op-erator for θ , and the components u1 and u2 of u, in terms of ψ ,are given by (2). To initiate (4), (ω1, θ1,ψ1) need to be computed:a first order approximation may be applied on a subsequence withsmaller time step; a system of the form (5) is also obtained.

Denoting

Rω(ω,ψ)

≡ αω − ∇2ω + u · ∇ω − Ra

Pr

(∂θ

∂xcosφ − ∂θ

∂ ysinφ

)− fω,

and similarly Rθ (θ,ψ). Then, system (5) is equivalent, in Ω , to

∇2ψ = −ω, ψ |Γ = 0; Rθ (θ,ψ) = 0, Bθ |Γ = 0;Rω(ω,ψ) = 0, ω|Γ = ωbc. (6)

To solve (6), the following fixed point iterative process is applied,in Ω:

With {θ0,ω0} = {θn,ωn} given, solve until convergence on θ and ω

∇2ψm+1 = −ωm, ψm+1|Γ = 0,

θm+1 = θm − ρθ

(α I − γ ∇2)−1

Rθ

(θm,ψm+1),

Bθm+1|Γ = 0, ρθ > 0,

ωm+1 = ωm − ρω

(α I − ∇2)−1

Rω

(ωm,ψm+1),

ωm+1|Γ = ωm+1bc , ρω > 0 (7)

and take (ψn+1, θn+1,ωn+1) = (ψm+1, θm+1,ωm+1).It should be noted that the ω boundary condition ωbc , given

by unknown values of ψ in Ω , [2], is performed as part of theiterative process in (7). Moreover, (7) is equivalent to

∇2ψm+1 = −ωm, ψm+1|Γ = 0,(α I − γ ∇2)θm+1 = (

α I − γ ∇2)θm − ρθ Rθ

(θm,ψm+1),

Bθm+1|Γ = 0, ρθ > 0, (8)

similarly for ωm+1. Then, at each iteration, three uncoupled linearelliptic problems, associated with the operators ∇2, α I −γ∇2, andα I − ∇2 must be solved. For the space discretization of problemslike those in (8), either finite differences or finite elements maybe applied, if rectangular domains are used; in either case efficientsolvers exist. For finite elements, variational formulations have tobe chosen, [7]. For the results in the next section, it is enough touse the second order finite differences approximations of the Fish-pack solver, [8]. Then, such second order approximation combinedwith the second order approximation in (4) for the first derivativesin time, the second order approximation for ωbc , the approxima-tion with second order central differences at the interior points,and with (4) on the boundary, for all the first space derivatives,

A. Nicolás et al. / Physics Letters A 375 (2011) 2683–2688 2685

including those in Nu, and de second order trapezoidal rule to cal-culate Nu imply that the whole discrete problem relies on secondorder approximations only.

2. Results and discussion

For natural convection, all the walls of the cavity are solid andfixed then, by viscosity, u is 0 everywhere on Γ , by (2) ψ is con-stant and it can be chosen to be zero. The boundary condition forω, ωbc is the one given in [2] whereas the one for θ , given in theoperator B , is

θ = 1 on Γ |x=0,

θ = 0 on Γ |x=a; ∂θ

∂n= 0 on Γ |y=0,b,

then the horizontal walls are insulated and on the vertical wallsθ is constant, and heating occurs on the left wall. The cavities aresupposed to be filled with air, then the Prandtl number Pr = 0.71;hence, from hereafter the flows will depend on A and φ only: 12 �A � 20 and 0◦ � φ � 270◦; and about Ra∗ .

Denoting by Tss the time where the flow reaches its asymp-totic steady state, Tss is computed by the point-wise discrete L∞absolute criterion on Ω , [9],

ψ : ‖ψn+1hx,hy − ψn

hx,hy‖∞, θ : ‖θn+1hx,hy − θn

hx,hy‖∞,

since by its physical definition, [10], Tss is the time when the so-lution does not change any more at any spatial point occupied bythe fluid.

The results, converged flows to a steady state, are reportedthrough the streamlines and the isotherms which, unless otherwisestated, correspond to ten values obtained by default using Mathe-matica. For θ such values start from 0.09 close to the cold rightwall, which is 0.9 divided by 10, and then they increase a multipleinteger until reach 0.9 close to the hot left wall; if some clari-fication were needed the streamlines values would be indicated.(hx,hy) and t denote the mesh size and the time step, they willbe specified in each case under study.

To support that the flows are correct, mesh size and time-stepindependence studies are made in terms of the point-wise discreteL∞ relative error on Ω , [9],

t fixed: ‖ fhx1,hy1;t − fhx2,hy2;t‖∞‖ fhx1,hy1;t‖∞

;

{hx,hy} fixed: ‖ fhx,hy;t1 − fhx,hy;t2‖∞‖ fhx,hy;t1‖∞

.

To this end, for Ra∗ and A∗ a mesh size independence studywas performed for φ = 0◦ considering the uniform meshes, untilthe boundary, 1) (hx,hy) = ( 1

16 , 16256 ), 2) (hx,hy) = ( 1

24 , 16384 ), and

3) (hx,hy) = ( 132 , 16

512 ); with t = 0.0001 fixed. Table 1 shows thediscrepancies (Dis.) for ψ and θ . Then, the correct flows are ob-tained with 2); about time step independence study, with hx = 1

24fixed, a calculation with a smaller time step, t = 0.00001, givesdiscrepancies: 0.06% for ψ and 0.0001% for θ ; a smaller timestep is not necessary since for more complicated problems, likemixed convection, [9], the discrepancies are even smaller. Then,t = 0.0001 is enough to get the right flows; with t bigger themethod blows up. The description of the results follows.

For φ = 0◦ , 40◦ , and 60◦ , Fig. 1 shows streamlines andisotherms for Ra∗ and A∗ , that are obtained with t = 0.0001and the optimal uniform mesh in 2) above. Fig. 2a) displays therespective heat transfer with the Nu(y) graphs. The result in Fig. 1with φ = 0◦ , which is the motivation of this study, has three cat’seyes: the three inner cells in the central part of the streamlines;

Table 1Mesh independence: Ra = 1.1 × 104, A = 16; h = 1/24, dt = 0.0001.

Mesh Dis. ψ Dis. θ

1 vs. 2 2.02% 0.64%1 vs. 3 2.68% 0.87%2 vs. 3 0.66% 0.22%

Table 2Various quantities: Ra = 1.1 × 104 with A = 16; h = 1/24, dt = 0.0001.

φ ψmin ψmax Nu Tss Cells

0◦16 −38.229 0 1.532 4.586 3 (cey)

30◦16 −34.773 0 1.499 8.754 3 (cey)

40◦16 −30.689 0 1.477 5.306 4 (cey)

60◦16 −20.329 7.805 1.967 72.288 5

70◦16 −16.699 11.744 2.54 40.003 11

75◦16 −15.803 12.316 2.653 53.106 13

80◦16 −14.153 11.812 2.772 69.889 17

120◦16 −7.805 20.329 1.967 72.296 5

10475◦ −14.932 11.525 2.577 54.352 13

1.4 × 10475◦ −18.7 14.768 2.737 44.34 11

1.8 × 10475◦ −21.546 17.286 2.906 27.696 11

104270◦ −8.65 × 10−10 6.93 × 10−10 1.022 0.322 0

this number agrees with [1] but not with [3], who has four; in [4]there appear also three cat’s eyes for Ra = 1 × 104 and 1.2 × 104

which are the closest to Ra∗ .About the contour values of ψ : For φ = 60◦ the contour val-

ues of the three big cells at the center and extremes of the cavityrange from −17.5 to −2.5: −17.5 for the innermost contour and−2.5 the closest to the boundary; the innermost contour of thesmall cells has value 5 and 2.5 the next; the streamline separatingthem has value 0; then, they are disconnected; because of positiveand negative contour values, the small cells move counterclockwiseand the big cells clockwise; and a disjointed multicellular patternappears.

To know more on cat’s eyes and disjointed multicellular flow,for Ra∗ and A∗ , Fig. 3 pictures the flow pattern for φ = 30◦ ,75◦ , and 80◦; the heat transfer is displayed in Fig. 2b). Then, thethree cat’s eyes persist up to 30◦ and the disjointed multicellu-lar on 60◦ � φ � 120◦ . The contour values for φ = 30◦ run from−30 to −5, from the innermost contour, out of six, toward theboundary; for φ = 75◦ , the big cells, moving clockwise, have val-ues {−15,−10,−5}, and {10,5} for the small ones, moving coun-terclockwise, separated by the one of contour value 0; and forφ = 80◦ , {−10,−5} for the big ones, of elliptic shape, and {10,5}for the small ones; the nine isotherms contour vary from 0.9 closeto hot left wall to 0.1 close to the cold right wall.

Table 2 shows characteristics of the flow: the minimum andmaximum values of ψ , ψmin and ψmax, Nu, Tss , and the numberof cells, where (cey) means cat’s eyes and disjointed multicellularflow the others. We shall describe and discuss Table 2 in threeparts:

I) Ra∗ and A∗ fixed, varying φ: A) |ψmin| decreases as φ in-creases; B) for 60◦ � φ � 80◦ , unlike the cat’s eyes flow, ψmax > 0and this value increases till 75◦ and then it decreases; C) Nudecreases till 40◦ and then it increases; and D) Tss is oscilla-tory. About the congruence of Nu(y) with respect to Nu in C):for φ = 60◦ , disjointed multicellular flow, the graph of Nu(y) inFig. 2a) displays several (big) maximum and (small) minimum val-ues, a fact that does not occur for cat’s eyes flow, which forcesin some way that Nu increases from φ = 60◦ on, after it has

2686 A. Nicolás et al. / Physics Letters A 375 (2011) 2683–2688

Fig. 1. Ra = 1.1 × 104: A = 16 and various φ ’s.

decreased at φ = 40◦; for φ = 0◦ and 40◦ there exist various max-ima and minima, in both cases they are significantly smaller (big-ger) than those for φ = 60◦ . Fig. 2b) shows that for φ = 30◦ , Nu(y)

looks very close to the one for 0◦ in Fig. 2a) (both have three cat’seyes); for φ > 60◦ the number of maxima and minima increases asφ increases being implied by the increasing of number of cells inthe streamlines and the thermal effects.

i) The streamlines in Fig. 1 show that for φ = 40◦ the num-ber of cat’s eyes increases to four and for φ = 60◦ they disappearand a very striking fact occurs: a disjointed multicellular flow pat-tern appears which may be strongly associated to what happensin B), C), and D). In Fig. 1 is also observed that the cat’s eyesand the disjointed multicellular flow cause a noticeable pattern onthe isotherms: the isotherms wave along with them, the numberof waves being bigger for the disjointed multicellular pattern. InFig. 3 the three cat’s eyes hold until 30◦; for φ � 60◦ , the numberof cells in the disjointed multicellular flow increases as φ does, butit can decrease as is shown for φ = 120◦ in last row of Table 2; thewaving effect on the isotherms increases also. ii) For φ = 0◦ , 30◦ ,and 40◦ the values of the streamlines do not change sign: the flow

moves clockwise throughout the cavity but for φ � 60◦ there ischange in sign: the big cells move clockwise and the small coun-terclockwise. iii) Tss increases significantly at φ = 60◦ , being oscil-latory before and after, and increases from 70◦ , all this togetherwith ψmax > 0 makes the flow more unstable and the disjointedmulticellular flow appears. iv) Tss = 4.586 for Ra∗ , A∗ , and φ = 0◦is significantly smaller than t = 200 claimed to be the time whenthe 3 cat’s eyes appear in [1] and much smaller than t = 500 in[4]. v) For φ = 120◦ , all the characteristics of the flow with respectto φ = 60◦ , row 4, are the same but the motion is reversed be-cause of the interchange, magnitude and sign, of ψmin and ψmax:for φ = 0◦ the cavity is heated on the left wall, for φ = 180◦ isheated on the right one (the position of the bottom and top is re-versed), and for φ = 90◦ is heated from below; thus, the positionsof the cavity for 90◦ < φ < 180◦ are changed to the other side,symmetrically, with respect to those φ’s in 0◦ < φ < 90◦ , then theflows rotate in opposite direction depending on whether heatingoccurs on the tilted left wall or on the right one.

II) The three rows before the last one display the characteristicsfor φ = 75◦ about Ra∗: Ra = 104, 1.4 × 104, 1.8 × 104. It is seen

A. Nicolás et al. / Physics Letters A 375 (2011) 2683–2688 2687

Fig. 2. a) Local Nusselt number for Ra = 1.1 × 104: A = 16 and various φ ’s. b) Local Nusselt number for Ra = 1.1 × 104: A = 16, φ = 30◦ and φ > 60◦ .

Fig. 3. Ra = 1.1 × 104: A = 16, φ = 30◦ and φ > 60◦ .

2688 A. Nicolás et al. / Physics Letters A 375 (2011) 2683–2688

that ψmin, ψmax, and Nu are about the same with 70◦ � φ � 75◦for Ra∗ , given rising about the same number of cells; for Ra = 104,the bigger number of cells (13) agrees with the bigger value ofTss = 54.352 which is the same it happens for Ra∗ with φ = 80◦(17 cells and Tss = 69.889), that is, the flows are more unstable.

III) The last row shows the characteristics for Ra = 104 at 270◦which means heating occurs on top, and cooling below. Surpris-ingly, the number of cells is 0, Tss is very short. And, ψmin andψmax imply an almost motionless flow; the heat transfer is alsoweak because of N and because Nu(y) is constant, slightly above 1;that is, it behaves like heat transfer by conduction. Almost thesame occurs for Ra∗ and Ra = 1.4 × 104. Then, there is not convec-tion at all but conduction. In other words, there is an oppositionto buoyancy: the hot fluid goes up and the cold goes down.

About some variations of A: Calculations for φ = 0◦ fixed, vary-ing A = 12, 14, and 20, show: with A = 12 the cat’s eyes disappear,with A = 14 the number of cat’s eyes reduces to two whereas withA = 20 this number increases to five; the waves in the isothermsincrease also as the number of cat’s eyes increases. For A = 12, nocat’s eyes, Tss = 1.827 which is much smaller than those Tss ’s withcat’s eyes, meaning that the flow is more stable.

3. Conclusions

Numerical results on natural convection in large tilted cavities,filled with air, have been presented to study the effects on theflows as the aspect ratio A and the angle of inclination of the cav-ity φ vary, with Ra mostly fixed. The study shows several aspectson motion and thermal activity of the flows. For φ > 0◦ and A = 16fixed, the results show a transition to hold or to increase the num-ber of cat’s eyes until some value of φ and then, for higher valuesof φ a disjoint multicellular flow appears which increases its num-ber of cells as φ increases but this number can decrease for someangle. The isotherms, since the appearing of the cat’s eyes, are ac-

companied with a waving effect which increases as the number ofcells increases. When Ra varies about Ra = 1.1 × 104, with an an-gle fixed, the number of cells holds constant. When cooling is frombelow there are no cells; there is not convection at all but con-duction. For φ = 120◦ the motion is reversed, the rest of the flowcharacteristics are kept constant, with respect to φ = 60◦ . Aboutthe physics of the fluid, the results show that a more vigorous mo-tion is obtained when heating is from the side, that is, for φ = 0◦;however, this does not contribute to the heat transfer at all. The re-sults also suggest that the maximum heat transfer will be reachedwhen heating is from below. In general, as the number of cells in-creases Tss increases too, meaning that the flow is more unstableto arrive at its steady state.

Acknowledgements

We would like to thank the unknown reviewer for the carefulreading of the manuscript and his remarks which have improvedthe presentation of this work.

References

[1] Z.J. Zhu, H.X. Yang, Heat Mass Transfer 39 (2003) 579.[2] E. Báez, A. Nicolás, Int. J. Heat Mass Transfer 49 (2006) 4773.[3] P. Le Quéré, J. Heat Transfer 112 (1990) 965.[4] S. Wakitani, J. Heat Transfer 119 (1997) 97.[5] F.X. Trias, A. Gorobets, M. Soria, A. Oliva, Int. J. Heat Mass Transfer 53 (2010)

665.[6] A. Nicolás, B. Bermúdez, Comput. Methods Eng. Sci. 16 (5) (2004) 441.[7] R. Glowinski, Hanbook of Numerical Analysis: Numerical Methods for Fluids

(Part 3), North-Holland, 2003.[8] J. Adams, P. Swarztrauber, R. Sweet, FISHPACK: A Package of Fortran Subpro-

grams for the Solution of Separable Elliptic PDE’s, The National Center forAtmospheric Research, Boulder, CO, USA, 1980.

[9] A. Nicolás, B. Bermúdez, Int. J. Numer. Methods Fluids 48 (5) (2005) 349.[10] L.D. Landau, E.M. Lifshitz, Fluid Mechanics, second ed., Pergamon Press, 1989.