Axial Magnetic Field Effect on Taylor-Couette Flow

Transcription

1 Jounal of Applied Fluid Mechanics, Vol. 8, No., pp , 05. Available online at ISSN , EISSN DOI: /acadpub.jafm Axial Magnetic Field Effect on Taylo-Couette Flow S. Abekane, M. Ihdene, M. Modees 3 and A. Ghezal 3 Enegetic depatment, faculty of engineeing Sciences, Boumedes-35000, Algeia Univesity of Yahia Faès, Médéa- 6000, Algeia 3 Laboatoy of theoetical and applied fluid mechanics, univesity of sciences and technology Houai Boumediene Bab Ezzoua, Algies-6, Algeia Coesponding Autho abekane.sofian@gmail.com (Received Des, 03; accepted May 9, 04) ABSTRACT This study is inteested in the effect of an axial magnetic field imposed on incompessible flow of electically conductive fluid between two hoizontal coaxial cylindes. The imposed magnetic field is assumed unifom and constant. The effect of heat geneation due to viscous dissipation is also taken into account. The inne and oute cylindes ae maintained at diffeent unifom tempeatues. The movement of the fluid is due to otation of the cylinde with a constant speed. An exact solution of the equations govening the flow was obtained in the fom of Bessel functions. A finite diffeence implicit scheme was used in the numeical solution. The velocity and tempeatue distibutions wee obtained with and without the magnetic field. The esults show that fo diffeent values of the tmann numbe, the velocity between the two cylindes deceases as the tmann numbe inceases. Also, it is found that by inceasing the tmann numbe, the aveage Nusselt numbe deceases. On the othe hand, the tmann numbe does not affect the tempeatue. Keywods: Rotating cylindes, viscous dissipation, heat tansfe, magnetic field, Bessel function, finite diffeence. NOMENCLATURE a B 0 Cp d Ec P t T k themal diffusivity extenal magnetic field density specific heat width of the annula space Ecket numbe tmann numbe Pandtl numbe adius time tempeatue themal conductivity u v w μ υ θ ρ σ Ω η adial velocity angula velocity axial velocity dynamic viscosity fluid kinematic viscosity non-dimensional tempeatue fluid density electic conductivity otational speed adius atio viscous dissipation function. INTRODUCTION The study of flow of electically conductive fluids, called magnetohydodynamic (MHD) has attacted much attention due to its vaious applications. In astophysics and geophysics, it is applied to the study of stella stuctues, teestial coes and sola plasma. In industial pocesses, it finds its application in MHD pumps, nuclea eactos, the extaction of geothemal enegy, metallugical and cystal gowth in the field of semiconductos, the contol of the behavio of fluid flow and heat and mass tansfe and the stability of convective flows. Seveal studies have been conducted to evaluate the effect of magnetic field on the convective flows fo diffeent conditions. Chandasekha (96) has made the pediction of the linea stability of hydodynamics and hydomagnetic Taylo-Couette flow. Tatsuo et al (993) caied out expeimental investigations about the natual convection of a magnetic fluid between two concentic cylindes and hoizontal isothems. Ben did, and Heny (996) investigated numeically the effect of a constant magnetic field on a thee-dimensional buoyancyinduced flow in a cylindical cavity, they put in light the stuctual changes of the flow induced by the magnetic field fo each field oientation. Singh et al (997) pesented exact solutions fo fully developed natual convection in open-ended vetical concentic annuli unde a adial magnetic field. Bessaih et al

2 S. Abekane et al. / JAFM, Vol. 8, No., pp , 05. (999) studied numeically MHD lamina flow of a liquid metal contained in a cylindical enclosue having an aspect atio equal to, and whose uppe wall is in otation. The assembly is subjected to a vetical extenal magnetic field. El Amin (003) studied the effects of both fist- and second-ode esistance due to the solid matix on foced convective flow fom a hoizontal cicula cylinde in the pesence of a magnetic field and viscous dissipation, with a vaiable suface tempeatue bounday condition. The study of the effects of the azimuthal magnetic field of an electically conducting fluid in a otating annulus has also been pesented by Kut et al (004). yat and Kaa (006) investigated the Couette time-dependent flow of an incompessible thid-gade fluid subjected to a magnetic field of vaiable stength analytically. Goup theoetic methods wee employed to analyze the nonlinea poblem and a solution fo the velocity field was obtained analytically. Sanka et al (006) studied numeically a natual convection of a low Pandtl numbe electically conducting fluid unde the influence of eithe axial o adial magnetic field in a vetical cylindical annulus. They showed that the magnetic field can be suppess the flow and heat tansfe. Bessaih et al (009) studied the MHD stability of an axisymmetic otating flow in a cylindical enclosue containing liquid metal (P = 0.05), with an aspect atio equal to, and subjected to a vetical tempeatue gadient and an axial magnetic field. Wobel et al (00) pesented an expeimental and numeical analysis of a themo-magnetic convective flow of paamagnetic fluid in an annula enclosue with a ound od coe and a cylindical oute wall unde gavitational and magnetic envionments. Azim et al (00) studied numeically the effect of magnetic field and Joule heating on the coupling of convection flow along and conduction inside a vetical flat plate in the pesence viscous dissipation and heat geneation. Ellahi et al (00) detemined analytic solutions fo a nonlinea poblem govening the MHD flow of a thid gade fluid in the annulus of otating concentic cylindes. Makinde and Onyejekwe (0) investigated a steady flow and heat tansfe of an electically conducting fluid with vaiable viscosity and electical conductivity between two paallel plates in the pesence of a tansvese magnetic field. Venkatachalappa et al (0) caied out numeical computations to investigate the effect of axial o adial magnetic field on the double-diffusive natual convection in a vetical cylindical annula cavity. Kakaantzas et al (0) studied numeically the combined effect of a hoizontal magnetic field and volumetic heating on the natual convection flow and heat tansfe of a low Pandtl numbe fluid in a vetical annulus. Seth et al (0) studied the effects of otation and magnetic field on unsteady Couette flow of a viscous incompessible electically conducting fluid between two hoizontal paallel poous plates in a otating medium. Mozayyeni and Rahimi (0) investigated numeically the poblem of mixed convection of a fluid in the fully developed egion between two hoizontally concentic cylindes with infinite lengths, in the pesence of a constant magnetic field with a adial MHD foce diection, consideing the effects of viscous heat dissipation in the fluid in both steady and unsteady states. Seth et al (0) investigated the effects of ll cuent on unsteady hydomagnetic Couette flow of a viscous incompessible electically conducting fluid in a otating system in the pesence of an inclined magnetic field. Seth and Singh (03) studied theoetically the effect of ll cuent and a unifom tansvese magnetic field on unsteady MHD Couette flow of class-ii in a otating system. J. Pakash (04) poved analytically that the pinciple of the exchange of stabilities in convection in a Rotating Feofluid Satuated Poous is not, in geneal, valid fo the case of fee boundaies but the study shown that a sufficient condition fo the validity of this pinciple can be deived. Bhuiyan et al (04) studied numeically the effects of joule heating on magneto-hydodynamic natual convection flow in pesence of viscous dissipation and pessue stess wok fom a hoizontal cicula cylinde. Although the exact solutions fo the tmann flow and the MHD Couette flow have been achieved fo moe than seventy yeas, the solutions fo a heat tansfe in flow between concentic otating cylindes, also known as Taylo Couette flows, unde extenal magnetic field have been esticted to high tmann numbes. The aim of the pesent study is to examine analytically and numeically the effects of an extenal axial magnetic field applied to the foced convection flow of an electically conducting fluid between two hoizontal concentic cylindes, consideing the effects of viscous heat dissipation in the fluid. It should be noted that the natual convection is supposed negligible in this wok, which is not always the case of the vetical cylinde. The foced flow is induced by the otating inne cylinde, in slow constant angula velocity and the othe is fixed.. FORMULATION OF THE PROBLEM Conside a lamina flow of a viscous incompessible electically conductive fluid between two coaxial cylindes. The inne cylinde of adius is otated at a constant speed Ω and the oute cylinde of adius is kept fixed. The inne and oute walls ae maintained at a constant and diffeent tempeatues T and T espectively, while the top and bottom walls ae insulated. The two cylindes ae electically isolated. The flow is subjected to a constant unifom and axially magnetic field B 0. Geomety of the poblem is pesented in Fig.. We assume that the magnetic Reynolds numbe is neglected. When the magnetic field is unifom and extenally applied, its time vaiations can be neglected and the set of flow equations futhe simplified to involve only the Navie-Stokes equations and the consevation of the electic cuent. Also we assume that the electic field is zeo. In this study the viscous dissipation tem in the enegy equation is consideed. 56

3 S. Abekane et al. / JAFM, Vol. 8, No., pp , 05. M C and C ae the constants of integation, which ae detemined fom the bounday conditions on the velocity. K M bk M C I M K M K M I M b I M I M C I M K M K M I M Fig.. Geomety of the poblem. ANALYTICAL STUDY The flow is assumed to be steady, lamina and unidiectional, theefoe the adial and axial components of the velocity and the deivatives of the velocity with espect to θ and z ae zeo. Unde these assumptions and in cylindical coodinates, the govening equations fo the flow following the azimuthal diection can be witten as follows: v v v vb0 0 k T v v () () : v( ), T T (3) : v( ) 0, T T (4) The govening equation and bounday conditions, eq.() to (4), witch ae in non-adimensional fom, become: * * v v * v 0 (5) * * * * * * * v v Ec P * * * * * (6) * * * : v ( ), (7) * * * : v ( ) 0, 0 (8) Whee * * v T T, v,, B0d,, T T P, Ec a Cp T Whee, the stas ae dopped fo convenience. The velocity pofile in the annula space is obtained by solving the Eq. (5) as follows: v C I C K Whee: 0 (9) I is the modified Bessel function of the fist kind of ode, and K is the modified Bessel function of the second kind of ode. To obtain the tempeatue field fom Eq. (6), we pefomed calculations by using the fist, second and thid tem used by Omid M. et al (0) of the expansions of the modified Bessel functions I M and K M fo small values of. 3. Expansion with one tem of Bessel modified functions I( M) M (0) ( ) K M () M By substituting the values of I M and K M fom the above expansions in the velocity equation, Eq. (9), and using the new velocity distibution in Eq. (6) to find the tempeatue field. The tempeatue gadient is given then by the following equation: C C B 3 3 M () The tempeatue pofile is given by: Ec PC C 3ln( ) C4 (3) M Whee C 3 and C 4 ae the constants of integation with espect to θ: BC C4 M C3 ln( ) BC C 4 M 3. Expansion with two tems of modified Bessel functions M 3 I( M) M 6 (4) M K( M) ln( ) ( ) ( M) M 4 Whee: γ is Eule's constant defined by: (5) 57

4 S. Abekane et al. / JAFM, Vol. 8, No., pp , 05. lim... ln( m) x 3 4 m 0, The tempeatue gadient is theefoe expessed as follows: C M CC M CC M C B C C ln( ) C M 3 M The tempeatue pofile is given by: 8 (6) C M CC M CC M C3ln( R) C4 B C C M C (ln( )) M 6 (7) Whee C 3 and C 4 ae the constants of integation, which ae detemined fom the bounday conditions on the tempeatue C M CC M CC M C3 C4 B 6 6 C C M C (ln( )) M C M CC M CC M C C M C4 B M 6 (8) 3. Expansion with thee tems of modified Bessel functions 3 5 M M (9) I( M) M M K( M) ln( ) ( ) ( M) M 4 M 5 3 ln( ) ( ) ( M) 6 3 (0) The tempeatue gadient is given by the following equation: M C 6 M C ln C5M ln M 6 6 M C6M C M ln 384 C3 B 4 4 M C7 M C8 M C M ln M C9 M CC M ln C M 960 () Whee The constants C 5 to C 9 ae given in tems of C and C as follows: C5 C CC C C6 C CC C C C CC C7 C C CC C8 C C CC C9 CC CC C The solution of the enegy equation is: C0 M C M C M C7 M C M 9600 M M C 6 M C ln C ln C5 M ln M 6 C4 C3ln( ) B 6 6 M 6 M C6 M C M ln C M ln M C M ln 8 M CC M ln () Whee The Constants C 0, C and C ae given as follows: C0 C5 C6 C C C9 CC C C8 C NUMERICAL STUDY In this numeical study, we conside a two-dimensional and axisymmetic unsteady flow. We opted fo the velocity - pessue fomulation due to its apidity of pediction, its lowe cost, and its ability to simulate eal conditions. The finite diffeence scheme adopted fo the esolution is vey simila to that used by R.Peyet (976), A.Ghezal et al (99) and (0), this is a semi implicit scheme of Cank-Nicholson type. The spatial discetization using the Make And Cell (MAC) is shown in fig.. The iteative pocedue is assumed conveged when the following test is veified max( Lu, Lv, Lw, L D ) whee L u, L v, L w, L θ and D epesents opeatos diffeences elating to system equations coesponding to the poblem vaiables u, v, w,θ and Π espectively, ε is of the ode of 0-4 depending on the consideed case. We then poceeded to a study of the mesh sensitivity of the field of study. This study led us to etain a mesh of 336 nodes along the diection and 48 nodes in the z diection. 58

5 S. Abekane et al. / JAFM, Vol. 8, No., pp , 05. V n i-/,j+/ T n i-/,j+/ W n i-/,j V n i-/,j-/ T n i-/,j-/ U n i,j+/ Π i,j U n i,j-/ V n i+/,j+/ T n i+/,j+/ W n i+/,j V n i+/,j-/ T n i+/,j-/ With: The aveage Nusselt numbe on the inne and oute cylindes is given by: z Nui Nui() z dz L 0 z Nue Nue() z dz L 0 Fig.. M.A.C cell 4. Mathematical equations Based on these dimensionless vaiables, the consevation equations of mass, momentum and enegy ae witten in non otating fame cylindical coodinates as follows (whee the stas ae dopped fo convenience): u u w 0 (3) z u u v u u u w u u u u t z Ta z Ta v v vu v v v v v v u w t z Ta z Ta w w w w w w u w t z z Ta z u w t z PTa z Ec Ta Whee: Bd is the tmann numbe, Ta d is the Taylo numbe, d is the width of the annula space, u u w u w z z v v v z (4) (5) (6) (7) is the viscous dissipation function The ate of heat tansfe in non dimensional fo the inne and oute cylinde is given by: Nui ( z) e ( ) Nu z 4. Initial and bounday conditions At the time t=0: u(, z,0) v(, z,0) w(, z,0) (, z,0) (, z,0) 0 (8) The bounday conditions ae as follows: z 0 : u(, z ) v (, z ) w (, z ) 0, (, z ) = z 0 : u(, z ) = v(, z ) = w(, z) (, z ) =0 (9) (30) z 0 : u v = w=0, 0 z u v z L : w 0, 0 z z z 4. RESULTS AND DISCUSSION (3) (3) In ode to undestand the physical situation of the poblem and the effects of the tmann and Ecket numbes, we have found the numeical and analytical values of the velocity, tempeatue, and the Nusselt numbe. The esults obtained though the numeical code wee pesented in figs. 3 and 4 wee compaed with those calculated using the thee analytical appoach fo small value of tmann numbe. It is noticed fom fig. 3 that the analytical esults fo the thee cases of the expansions with one, two and thee tem of modified Bessel functions coincide well with the numeical esults fo small tmann numbe.5. It can also be obseved that the influence of the ode of development on tempeatue is negligible As can be seen fom fig. 4 that whethe fo the aveage Nusselt numbe on inne and oute sufaces vesus tman numbes. The analytic appoach coesponding to the expansion of thee tems is close to the numeical appoach. Fo the tmann numbe values less than. the influence of the ode of development in the analytical appoach is insignificant. So the next analytical esults in this wok ae done by the expansions with thee tems of modified Bessel functions. The velocity and tempeatue ae evaluated analytically and numeically fo diffeent values of tmann numbe in figs. 5 and 6. 59

6 S. Abekane et al. / JAFM, Vol. 8, No., pp , 05. st ode nd ode 3d ode Numeical 0, 0, Fig. 3. Effect of the development of Bessel functions on tempeatue distibution, fo = 0.5, η = 0.5, P = 0.0, Ec=0.5.,430,45 Nu i v Analytical (3 d ode) = =4 Numeical = =4 0, 0, Fig. 5. Compaison of analytical and numeical esults of velocity pofile, fo η = 0.5, Ta=0,t * =0. Analytical (3 d ode).4.8 Numeical.4.8,40,45 st ode nd ode 3d ode Numeical 0, Nu e,40 0,73 0,730 0,78 0,76 0, st ode nd ode 3d ode Numeical 0, Fig. 6. Compaison of analytical and numeical esults Tempeatue pofile, fo η = 0.5, Ta=0, P = 0.0, Ec=0.000, t * =0. which causes a eduction of the velocity in the annula space because the centifugal foce is countepoductive and the Loentz electomagnetic foce acts as a flow dampe. 0,74 0,7 0, Fig. 4. Effect of the development of Bessel functions on aveage Nusselt numbe on inne and oute sufaces of the cylinde against the tmann numbe, fo η = 0.5, P = 0.0, Ec=0.5 Obviously, the velocity and tempeatue pofiles, fo vaious obtained via these two diffeent methods, agee with each othe easonably well. We can notice that the velocity pofile without magnetic field is quasi-linea, and an incease in tman numbe, It is obseved fom fig. 6 that the effect of weak magnetic field on the adial pofile of tempeatue is insignificant. It It should be noted that the effect of magnetic field on the tempeatue distibution is insignificant, wheeas the changes induced by the magnetic field on the tempeatue gadient and theefoe on the Nusselt numbe is consideable. Fig. 7 displays the effect of tmann numbe on the tempeatue, as shown in this figue, the tempeatue pofile is simila to those shown in Figs. 6. It is evident that the effect of weak and stong magnetic field on the adial pofile of tempeatue is insignificant. It It is valid in the case of low and high values of tmann 60

7 S. Abekane et al. / JAFM, Vol. 8, No., pp , 05. = =4 =6 =8 =0 =30 =40 =50 Nu i,44,4,40,38 Ec=0 Ec=0.0 Ec=0. Ec=0.3 Ec=0.5 0,,36 0, Fig. 7. Tempeatue pofile as a function of tmann numbe, fo η = 0.5, Ta=0, P = 0.0, Ec=0.5, t * =0.,44,4 = =5 =0 =30 =50 Nu e 0,78 0,76 0, z/d Ec=0 Ec=0.0 Ec=0. Ec=0.3 Ec=0.5,40 0,7 Nu i,38 0,70, z/d, z/d Fig. 9. Effect of Ecket numbe on local Nusselet numbe distibution on inne and oute cylindes, fo η = 0.5, P = 0.0,, t * =0. Nu e 0,78 0,77 0,76 0,75 0,74 0,73 = =5 =0 =30 =50 In fact when the Ecket numbe is consideable. The heat geneation in the fluid inceases due to viscous dissipation. Thus the tempeatue of the fluid in the annula space inceases causing a decease in the tempeatue gadient in the vicinity of the inne cylinde and an incease of the gadient in the vicinity of the oute cylinde. A significant incease in the tmann numbe, causes a eduction of the centifugal foce, which esults in a gadual decease in the Nusselt numbe. The analysis of the vaiation of local Nusselt numbe on the inne and oute cylinde shows that this numbe tends to a limit value. 0, z/d Fig. 8. Effect of tman numbe on local Nusslet numbe distibution on inne and oute cylindes, fo η = 0.5, P = 0.0, Ec=0.5, t * =0 numbe. Also we can notice that the tempeatue pofiles don t change fo Ec=0.5 and Ec=0.000, so the Ecket numbe don t affect the tempeatue. Fig. 8 shows the effect of tmann numbe on the local Nusselt numbe on the inne and oute sufaces, fo an Ecket numbe Ec = 0.5. It is found that fo high values of tmann numbe, the local Nusselt numbe on the inne and oute sufaces deceases. Effect of Ecket numbe on the distibution of local Nusselt numbe on the inne and oute cylindes is displayed in Fig. 9, fo = 0. As can be seen, with incease of Ecket numbe, the influence of heat tansfe due to the viscous dissipation in the annula space is impoved, which leads to the incease in the aveage tempeatue of the fluid at this egion. The dimensionless tempeatues of inne and oute cylindes ae maintained at.0 and 0.0, espectively. It is evident that by inceasing the aveage tempeatue of fluid in annula space, the ate of heat tansfe between the fluid and inne cylinde deceases due to the eduction of the tempeatue diffeence between them. Secondly, the convective heat tansfe between the fluid and the oute cylinde is impoved because of the incease in the tempeatue. 6

8 v v S. Abekane et al. / JAFM, Vol. 8, No., pp , 05.,44 Nu i,4,40,38 t=, Nu e,36 0,77 0,76 0,75 0,74 0,73 Ec=0 Ec=0.0 Ec=0. Ec=0.3 Ec=0.5 Ec=0 Ec=0.0 Ec=0. Ec=0.3 Ec=0.5 0, 0, 0, t= 0,7 0, Fig. 0. Effect of Ecket numbe on aveage Nusselt numbe on inne and oute sufaces of the cylinde against the tmann numbe, η = 0.5, P = 0.0, t * =0. It is obseved fom fig. 0 that the effect of inceasing tmann numbe is the decease the aveage Nusselt numbe on both sufaces of the cylinde. So a consideably inceasing tmann numbe, which leads to a eduction of the centifugal foce, esults in a pogessive decease in the Nusselet numbe. 0, Fig.. Velocity distibution at diffeent times at z/d=7 fo = and = 50 fo Ta=0. t= Fom this figue, it can also be noticed that the aveage Nusselt numbe on the oute cylinde is lowe than on the inne cylinde, because the velocity and tempeatue gadient ae highe fo the cold inne cylinde than fo the oute cylinde. Also the esults show the effects of viscous dissipation tems on the ate of heat tansfe, the aveage Nusselt numbe inceases with an incease in the Ecket numbe on the oute cylinde, but it deceases on the inne cylinde. In fact, as the Ecket numbe is lage the heat geneated in the annulus inceases due to viscous dissipation, and thus the tempeatue of the fluid inceases. This causes a decease in the tempeatue gadient close to the inne cylinde, and an incease in the gadient in the vicinity of the oute cylinde. 0, 0, 0, t= In this pat, some esults ae pesented in diffeent nondimensional time values fo the distibution of velocity and tempeatue in the annulus Figs (,, 3 and 4). 0, Fig.. Tempeatue distibution at diffeent times at z/d=7 fo = 0.0 and = 50 fo Ta=0, Ec=0,P= 6

9 S. Abekane et al. / JAFM, Vol. 8, No., pp , 05. t= t= 0, 0, 0, t= 0, t= 0, 0, 0, 0, Fig. 3. Tempeatue distibution at diffeent times at z/d=7 fo = and = 50 fo Ta=0, Ec=0, P=. Thee is not much diffeence in velocity at t= 0 compaed to t = 0, but compaing tempeatue distibution at with values geate than 0, it indicates that much moe time is still needed to each steady-state. Fom fig., we can notice that fo a small value of Pandtl numbe (P=), The effect of the time vaiation is found to be not significant on the tempeatue, it eaches faste a steady-state to the point that we can t notice the diffeence between the steady and unsteady states flows. As we know, fo lage fluid Pandtl numbe, the momentum flow tansfe is faste than heat tansfe. This can be seen clealy in Fig. 4 (fo a fluid with P = 7) and the distibution of the azimuthal component of velocity eaching a steady-state quicke than the tempeatue at the mid-length. CONCLUSION In this study, the foced convection flow of an electically conducting fluid between two hoizontal concentic cylindes in the pesence of an axial magnetic field and a tempeatue gadient consideing the effects of viscous heat dissipation in the fluid has been investigated numeically and analytically. The velocity distibution in the annulus is obtained analytically in tems of the modified Bessel functions Fig. 4. Tempeatue distibution at diffeent times at z/d=7 fo = and = 50 fo Ta=0, Ec=0, P=7. whose agument contains tmann numbe and adial coodinate. To obtain the tempeatue, the expansions of the modified Bessel functions, with thee tems which coincides bette with the numeical esults, ae used in the enegy equation. It is found that the velocity deceases in the annulus with incease of tmann numbe. Howeve an incease in tmann numbe does not affect the tempeatue. The effects of magnetic field stength and Ecket numbe on local and aveage Nusselt numbe have been examined. The esults show that an incease in tmann numbe educes the Nusselt numbe on both sufaces of the cylindes. Also it was noticed that as the Ecket numbe inceases aveage Nusselt numbe inceases on the oute cylinde, but opposite tend is obseved on the inne cylinde. In addition, some esults of the unsteady state have been discussed in this wok. REFERENCES Azim M.A., Mamun A.A., Rahman M.M. (00), Viscous Joule heating MHD conjugate heat tansfe fo a vetical flat plate in the pesence of heat geneation, Intenational Communications in Heat and Mass Tansfe, 37, Ben did H., and Heny D. (996). Numeical simulation of convective thee-dimensional flows in a hoizontal cylinde unde the action of a 63

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