Free convection over a non-isothermal axisymmetric body immersed in a porous medium saturated with an electrically conducting non-newtonian fluid

Transcription

1 Free covectio over a o-isothermal aisymmetric body immersed i a porous medium saturated ith a electrically coductig o-netoia fluid Abstract Shobha Bagai, Chadrashekhar Nishad Departmet of Mathematics Uiversity of Delhi Ne Delhi, Idia The study ivestigates the problem of magetohydrodyamic (MHD) free covectio over a o-isothermal aisymmetric body uder the actio of trasverse magetic field. The body is embedded i a porous medium saturated ith electrically coductig o-netoia poer la fluid. I order to obtai similarity solutio, it is assumed that the viscosity of the fluid decays epoetially ith temperature. The qualitative results are illustrated for a vertical flat plate, horizotal cylider ad sphere. Keyords Aisymmetric body, Electrically coductig o-netoia fluid, Free covectio, Magetohydrodyamics (MHD), No-isothermal, Porous medium.. Itroductio Magetohydrodyamic (MHD) free covectio flo of electrically coductig fluids through differet geometries is of cosiderable iterest to the techical field due to its frequet occurrece i geothermal operatios, petroleum idustries, thermal isulatio ad desig of solid-matri heat echagers chemical catalytic reactor. As a eample, the geothermal regio gases are electrically coductig ad udergo the ifluece of magetic field. It also has applicatios i uclear egieerig i coectio ith reactors coolig. Colig [] provides a etesive study of applicatio of magetohydrodyamics to geophysical ad astroomical problems. Refereces of comprehesive surveys regardig the subject of porous media ca be had i the recet books [ 5]. Most of the studies cocered ith the problem of MHD free covectio flo i oporous ad porous media have bee published durig the past several decades. Although, MHD free covectio flo of a electrically coductig fluid across a vertical plate i a oporous medium have bee cosidered by various authors [6 ], very fe authors have cosidered the problem for aisymmetric bodies. The problem of MHD free covectio flo of a electrically coductig fluid flo i porous medium across vertical plate ithout iteral heat geeratio has bee studied by Raptis ad Perdikis []. Al-Nimr ad Hader [] have preseted aalytical solutios for fully developed MHD atural covectio flo i

2 ope eded vertical porous chaels. A aalysis for o-darcy free covectio flo of a electrically coductig fluid over a impermeable vertical plate embedded i a thermally stratified, fluid saturated porous medium for the case of poer-la surface temperature as preseted by Afify [3]. A hadful of orks are available i the literature for MHD free covectio flo i porous medium for differet geometries ith iteral heat geeratio. Yih [4] umerically aalyses the effects of viscous dissipatio, Joule heatig ad heat source/ sik o o-darcy MHD atural covectio flo over a isoflu permeable sphere i a porous medium. The MHD free covectio from a sphere embedded i a electrically coductig fluid saturated porous regime ith heat geeratio as eamied theoretically ad umerically by Beg et al [5]. I the preset paper e study the effect of variable temperature depedet viscosity o MHD free covective heat trasfer rates i the presece of iteral heat geeratio for a o-isothermal to dimesioal or aisymmetric bodies embedded i porous medium saturated ith electrically coductig o-netoia fluid. Similarity solutios are obtaied for epoetially decayig viscosity of the fluid. For the eistece of the similarity solutio, the all temperature distributio is take as a arbitrary poer of the all profile parameter. Although the equatios are derived for a arbitrary aisymmetric shape, umerical results are illustrated oly for the three geometries amely, vertical flat plate, horizotal cylider ad sphere.. Mathematical Formulatio Figure Physical model Cosider the problem of free covectio boudary- layer flo over a o-isothermal aisymmetric body embedded i a o-netoia poer la fluid saturated porous medium as sho i Figure. A strog magetic field of uiform stregth B is applied i the y-

3 directio ormal to the body surface. The fluid is assumed to be electrically coductig. The iduced magetic field due to motio of electrically coductig fluid is eglectig. This assumptio is valid because the velocity of the fluid is very small i porous medium. The body is maitaied at a all temperature T everyhere. T hich is higher tha the ambiet temperature Uder the Boussiesq approimatio, the equatios for MHD free covectio i a curviliear co-ordiate system ca be ritte as [] y ( r u) ( r v) () u K ( ) g T K ( ) B u () u T T T q ''' v m y y cpf (3) here the fuctio r ( ) describes the geometric cofiguratio of the body ad is give as : plae flo 4(a) r ( ) r ( ) : aisymmetric flo 4(b) The acceleratio due to gravity g alog the -compoet is related to the body shape fuctio r( ) as g dr g (5) d The poer la fluid ide is less tha for pseudoplastic fluids, equal to for Netoia fluids ad greater tha for dilatat fluids. The o-netoia poer las proposed by Christopher ad Middlema [6] ad Dharmadhikari ad Kale [7] have a commo form amely dp u g (6) d K ( ) here is dyamic viscosity parameter hile a fuctio of the poer la ide. The appropriate boudary coditios associated ith the problem are K ( ) is the modified permeability give as At y : v, T T ( ) (7)

4 As y : u, T T (8) We itroduce the geeral trasformatios A f (9) B y () T T T () b T T e, here is the viscosity at T T () here is the stream fuctio, hich satisfies the equatio of cotiuity () u, v (3) r y r O usig equatios (9) (), equatios () ad (3) reduce to r mra K B r f ' f ' e A B A B B q''' P f ' M f ' mr '' A T Ram cpf (5) here T T T (4) Ra K gt m P d l T dl (6) M b T d l A dl is the dimesioless viscosity parameter. The fuctios A() ad B() are chose so as to satisfy

5 r mra A B (7) B mr A I (8) Simplifyig (7) ad (8) yields I s s T r g d T r g A r Ra I (9) B m Ra I O usig equatio (9), equatios (4) ad (5) ca be ritte as f ' Nf ' e () s s '' s P I f ' P I f ' e () The o-negative om-dimesioal MHD parameter N is give as m B N Ra () gt hereas the iteral heat geeratio per uit volume q ''' is ritte as [8] k mt Ra s ''' q e (3) s I The primes i equatio () ad () deote differetiatio ith respect to the similarity variable. The appropriate boudary coditios o associated ith the problem i terms of the similarity variable are

6 At : f, As : f, (4) The trasformed Darcia velocities alog the body ad i the ormal directio are give as mra u f ' (5) m Ra v s P I f I s I d lg dl r P I I f ' d l d l (6) 3. Solutio Procedure Equatios () ad () suggest that eact solutios are possible he the lumped parameter P I remais costat i the stream ise directio. The lumped parameter P I ca be ritte as P I here s is a o-egative iteger. O itroducig a e trasformed variable s T r g d d l T (7) dl T r g s r g d (8) equatio (7) ca be ritte as P I s T d d l T (9) d l s T The variable is the distace measured from the loer stagatio poit for the case of plae flo ad the volume segmet cut by a horizotal plae for the case of aisymmetric flo. Equatio (9) reveals that the similarity solutios eists he the all temperature varies accordig to,

7 (3) T Hece, the lumped parameter ad o-egative iteger s by the relatio P I is related to the costat epoet, poer ide P I s (3) Equatio () therefore takes the form '' f ' f ' s e s s s O usig equatios (9), (8) ad (3), the fuctio I i equatio (9) ca be ritte as (3) I s g r (33) The trasformed goverig equatios are valid for a aisymmetric body of arbitrary shape. The geometric cofiguratio is itrisic due to the trasformed variable. Three geometric shapes are cosidered amely vertical flat plate, horizotal cylider ad sphere. For these geometries the variable, from equatio (8) is give by, g : vertical flat plate si g r d :horizotal cylider (34) 3 g r si d :sphere here si r ad r is the radius of the horizotal cylider or sphere. The local surface heat flu is give by q k m T y y (35) hereas the o dimesioal local heat flu defied by

8 q ql r L K r gt r Trk m m (36) T r is the all ambiet temperature differece at a trailig edge or rear stagatio poit ad L r is the referece legth such as the legth of the vertical plate or radius of the cylider or sphere. The local Nusselt umber defied by Nu q T k m (37) ca be ritte usig equatio (35) as Nu Ra ' (38) I Equatio (36) for three geometries uder cosideratio ca be ritte as q s ' vertical flat plate (39) Lr si d s q ' si horizotal cylider (4) si d si d s q ' si sphere (4) si d 4. Result ad Discussio Table gives the heat trasfer rate for differet values of ad the dimesioless MHD parameter N for the vertical flat plate at L r. The results are preseted for isothermal ad o-isothermal plate ith costat viscosity. For the

9 pseudoplastic fluid it is observed that heat trasfer rate i the absece of magetic effect N is less tha the heat trasfer rate i the presece of magetic effect N. Whereas i the case of Netoia ad dilatat fluid the heat trasfer rate decreases as the value of N icrease from to. This ca be attributed to the fact that pseudoplastic fluids are shear-thiig fluids that have a loer apparet viscosity at higher shear rates. Table gives the heat trasfer rate for differet values of ad the dimesioless MHD parameter N for the vertical flat plate at Lr for.5. The results are preseted for isothermal ad o-isothermal. It is observed that the heat trasfer rate decreases as the value of N icreases from to for all the three cases, amely, preudoplastic, Netoia ad dilatat fluid for varyig viscosity. Figures 4 illustrate the effect of o-dimesioal MHD parameter N o the heat trasfer rate for a isothermal cylider ad sphere. The qualitative behavior of the heat trasfer rate is same as that obtaied by Bagai ad Nishad [9] i the absece of the magetic field. As the value of N icreases the heat trasfer rate decreases. Moreover, for the problem uder cosideratio, the fluid heats the isothermal body immersed i it. Figures 5 7 depict the effect of the parameter N for a o-isothermal body. It is observed that for higher values of N the body is still heated by the fluid hereas for loer values of N the body heats the fluid. There ill be a critical value of N, betee. ad 5., for hich there ill be egligible heat trasfer eve though the temperature of the body is varyig. For a oisothermal body, it is observed that the local heat flu has a poit of etrema betee the loer ad the upper stagatio poits. But as the value of icreases the poit of maima shifts toards the loer stagatio poit.

10 Table values of ' for differet values of MHD parameter N ad fluid ide for isothermal body ad o-isothermal body. he viscosity parameter. N Pseudoplastic fluid Isothermal body.5 Netoia fluid. No-isothermal body. Isothermal body Dilatat fluid. No-isothermal body. Isothermal body No-isothermal body Table values of for isothermal body.5. ' for differet values of MHD parameter N ad fluid ide ad o-isothermal body. he viscosity parameter N Pseudoplastic fluid Isothermal body.5 Netoia fluid. No-isothermal body. Isothermal body Dilatat fluid. No-isothermal body. Isothermal body No-isothermal body

11 (a) (b) Figure Effect of MHD parameter N o local heat flu for Pseudoplastic fluid.5 here, for (a) cylider ad (b) sphere. (a) (b) Figure 3 Effect of MHD parameter N o local heat flu for Netoia fluid. here, for (a) cylider ad (b) sphere. (a) (b) Figure 4 Effect of MHD parameter N o local heat flu for dilatat fluid. here, for (a) cylider ad (b) sphere.

12 (a) (b) Figure 5 Effect of MHD parameter N o local heat flu for Pseudoplastic fluid.5 here.,.5 for (a) cylider ad (b) sphere. (a) (b) Figure 6 Effect of MHD parameter N o local heat flu for Netoia fluid. here.,.5 for (a) cylider ad (b) sphere. (a) (b) Figure 7 Effect of MHD parameter N o local heat flu for dilatat fluid. here.,.5 for (a) cylider ad (b) sphere.

13 List of symbols A Fuctio adjustig the geeral trasformatio mr RaI B Fuctio adjustig the geeral trasformatio Ra I B Magetic field stregth c pf Specific heat at costat pressure of the fluid f Dimesioless stream fuctio gg, Acceleratio due to gravity ad its stream ise compoet I Fuctio adjustig the boudary layer legth scale k m Effective thermal coductivity of porous medium K Modified permeability L r Referece legth N Nu Dimesioless MHD parameter Poer la ide for the fluid Local Nusselt umber m B Ra gt P Fuctio associated ith all temperature p Pressure q ''' Iteral heat geeratio per uit volume q Local surface heat flu q Dimesioless local heat flu

14 Ra Modified local Rayleigh umber K gt m r, r Geometric cofiguratio of the body T Temperature uv, Velocity compoet i ad y-directio y, Boudary layer co-ordiates Greek symbols m Thermal diffusivity of the porous medium Coefficiet of thermal epasio Similarity variable Dimesioless temperature Epoet associated ith all temperature icrease Fluid desity Electric coductivity Kiematic viscosity Dyamic viscosity parameter Stream fuctio Viscosity parameter Boudary-layer thickess Porosity Trasformed variable Peripheral agle measured from the loer stagatio poit

15 Subscripts Wall coditio Ambiet coditio The leadig edge coditio Refereces [] T.G. Colig, Magetohydrodyamics, Joh Wiley & Sos, Ne York (957) [] R.W. Johso, The Hadbook of Fluid Dyamics, Spriger-verlag Germay (998) [3] D.A. Nield, A. Beja, Covectio i Porous Media, third ed. Spriger, Ne York (6) [4] D.B. Igham, I. Pop, Trasport i porous media, Pergamo, Oford () [5] K. (ed.) Vafai, Hadbook of Porous Media, secod ed. Mercel Dekker, Ne York (5) [6] E. M. Sparro, R.D. Cess, The effect of magetic field o free covectio heat trasfer, It. J. Heat Mass Trasfer. 3 (96) [7] N. Riley, Magetohydrodyamic free covectio, J. Fluid Mech. 8 (964) [8] A. Rapits, A.K. Sigh, MHD free covectio flo past a accelerated vertical plate. It. Comm. Heat Mass Trasfer, (983) [9] M.A. Hossai, Viscous ad joule heatig effect o MHD-free covectio ith variable plate temperature. It. J. Heat Mass Trasfer.35 (99) [] M.A. El-Hakiem, A.A. Mohammadei, S.M.M. El-Kabeir, R.S.R. Gorla, Joule heatig effects o magetohydrodyamic free covectio flo of a micropolar fluid, It. Comm. Heat Mass Trasfer, 6 (999) 9-7. [] A. Raptis, C. Perdikis, Magetohydrodyamics effects o mass trasfer flo through porous medium. Astrophysics ad Space Sciece 3 (985) [] M.A. Al-Nimr, M.A. Harder, MHD free covectio flo i ope-eded vertical porous chaels. Chem. Eg. Sc. 54 (999) [3] A.A. Afify, Effect of variable viscosity o o-darcy MHD free covectio alog a o-isothermal vertical surface i a thermally stratified porous medium, Applied Mathematical Modellig 3 (7) [4] K.A. Yih, Viscous ad joule heatig effects o o-darcy MHD atural covectio flo over a permeable sphere i porous media ith iteral heat geeratio. It. Comm. Heat Trasfer, 7 () 59-6.

16 [5] O.A. Beg, J. Zueco, R. Bhargava, H.S. Takhar, Magetohydrodyamic covectio flo from a sphere to a o-darcia porous medim ith heat geeratio or absorptio effects: etork simulatio, It. J. Thermal Sc. 48 (9) [6] R.V. Christopher, Middlema S.: Poer La flo through a packed tube, Id. Egg. Chem. Fudametals 4 (965) [7] R.V. Dharmadhikari, D.D. Kale, Flo of o-netoia fluids through porous media, Chem. Egg. Sc. 4 (985) [8] A. Hadim, L.C. Burmeister, Oset ofcovectio i a porous medium ith iteral heat geeratio ad doard flo, AIAA J. Thermophys. Heat Trasfer (976) [9] S. Bagai, C. Nishad, Effect of variable viscosity o free covective heat trasfer over a o-isothermal body of arbitrary shape i a o-netoia fluid saturated porous medium ith iteral heat geeratio, Trasp. Porous Med 94 ()