IJMET Issue 2, May July (2011), pp

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1 Inernaional Journal of of Mechanical Engineering Engineering and echnology (IJME ISSN (Prin and ISSN echnology (Online (IJME Volume ISSN Issue 976 May- 634(Prin July ( IAEME ISSN (Online Volume IJME Issue May July ( pp. 99- I A E M E IAEME hp:// NSEADY MHD FREE ONVEIVE FLOW IN A ROAING POROS MEDIM WIH MASS RANSFER Dr. Sundarammal Kesavan Supervisor SRM niversiy hennai M. Vidhya Research Scholar SRM niversiy hennai Dr. A. Govindarajan 3 sundarammalkesavan@kr.srmuniv.ac.in mvidhya_978@yahoo.co.in govindarajana@kr.srmuniv.ac.in ABSRA An exac analysis of unseady MHD free convecive flow and mass ransfer during he moion of a viscous incompressible fluid hrough a porous medium bounded by an infinie verical porous surface in a roaing sysem is presened. he porous plane surface and he porous medium are assumed o roae in a solid body roaion. he verical surface is subjeced o uniform consan sucion perpendicular o i and he emperaure a his surface flucuaes in ime abou a non-ero consan mean. Analyical expressions for he velociy emperaure and concenraion fields are obained using he perurbaion echnique. he effecs of R (roaion parameer k (permeabiliy parameer M (Harmann number and (frequency parameer on he flow characerisics are discussed. I is observed ha he primary velociy componen decreases wih he increase in eiher of he roaion parameer R he permeabiliy parameer k or he Harmann number M. I is also noed ha he primary skin fricion increases whenever here is an increase in he Grashof number Gr or he modified Grashof number Gm. I is clear ha he hea ransfer coefficien in erms of he Nussel number decreases in he case of boh air and waer when here is an increase in he Harmann number M. Keywords: Mass ransfer free convecion porous medium MHD roaion.. INRODION Free convecion flows are of grea ineres in a number of indusrial applicaions such as fiber and granular insulaion geohermal sysems ec. Buoyancy is also of imporance in an environmen where differences beween land and air emperaures can give rise o complicaed flow paerns. Magneohydrodynamic (MHD flows have araced he aenion of a large number of scholars due o heir diverse applicaions. In asrophysics and geophysics hey are applied o sudy he sellar and solar srucures inersellar maer radio propagaion hrough he ionosphere ec. In engineering MHD flows find heir applicaion in MHD pumps Professor in Deparmen of Mahemaics SRM niversiy Kaankulahur amil Nadu 63 3 India Senior Lecurer Dep of mahemaics Sahyabama niversiy Rajiv Gandhi Road hennai Professor Deparmen of Mahemaics SRM niversiy Kaankulahur amil Nadu 63 3 India 99

2 Inernaional Journal of Mechanical Engineering and echnology (IJME ISSN (Prin ISSN (Online Volume Issue May- July ( IAEME MHD bearings ec. onvecion in porous media has applicaions in geohermal energy recovery oil exracion hermal energy sorage and flow hrough filering devices. he phenomenon of mass ransfer is also very common in he heory of sellar srucure and observable effecs are deecable a leas on he solar surface. he sudy of effecs of magneic field on free convecion flow is imporan in liquidmeals elecrolyes and ionied gases. he hermal physics of hydromagneic flow problems wih mass ransfer is of ineres in power engineering and meallurgy. Malalhy and Srinivas [] invesigaed he pulsaing flow of a hydromagneic fluid beween wo permeable beds. Singh [] analyed he influence of a moving magneic field on hree dimensional couee flow. Das e al. [3] discussed mass ransfer effecs on MHD flow and hea ransfer pas a verical porous plae hrough a porous medium under oscillaory sucion and hea source.muhucumaraswamy [4] sudied unseady flow of an incompressible fluid pas an impulsively sared verical plae wih hea and mass ransfer. Acharya e al. [5] discussed magneic field effecs on free convecion and mass ransfer flow hrough porous medium wih consan sucion and consan hea flux. haudhary and Jain [6] analyed combined hea and mass ransfer effecs on MHD free convecion flow pas an oscillaing plae embedded in porous medium. Agrawal and Kishor [7] sudied he effecs of hermal and mass diffusion on MHD naural convecion flow beween wo infinie verical moving and oscillaing porous parallel plaes. Muhuraj and Srinivas [8] discussed hea ransfer effecs on MHD oscillaory flow in an asymmeric wavy channel. Muhucumaraswamy e al. [9] analyed chemical reacion effecs on infinie verical plae wih uniform hea flux and variable mass diffusion. Singh e al. [] discussed hea ransfer effecs in a hree dimensional flow hrough a porous medium wih a periodic permeabiliy. Gersen e al. [] analyed flow and hea ransfer effecs along a plane wall wih periodic sucion. Singh [] discussed he effec of injecion/sucion parameer on hree dimensional couee flow wih ranspiraion cooling. Gupa and Johari [3] sudied he effec of magneohydrodynamic incompressible flow pas a highly porous medium which was bounded by a verical infinie porous plae. Singh e al. [4] analyed he hea ransfer effecs on hree dimensional flucuaing flow hrough a porous medium wih a variable permeabiliy. Ahmed and Ahmed [5] discussed wo dimensional MHD oscillaory flow along a uniformly moving infinie verical porous plae bounded by a porous medium. Sharma and Yadav [6] sudied hea ransfer effecs on hree dimensional flow hrough porous medium bounded by a porous verical surface wih a variable permeabiliy and a hea source. Jain and Gupa [7] discussed free convecion effecs on hree dimensional couee flow wih ranspiraion cooling. Singh and Sharma [8] analyed he magneohydrodynamic effecs on hree dimensional couee flow wih ranspiraion cooling. Singh e al. [9] sudied he effecs of permeabiliy and roaion parameers on oscillaory couee flow hrough a porous medium in a roaing sysem. Rapis and Perdikis [] discussed he effec of permeabiliy on oscillaory and free convecion flow hrough a porous medium. In he sudies menioned above unseady free convecive flow wih hea and mass ransfer effecs in a roaing porous medium have no been discussed while such flows are very imporan in geophysical and asrophysical problems. herefore he objecive of he presen paper is o analye he effecs of permeabiliy variaion and mass ransfer on flow of a viscous incompressible fluid pas an infinie verical porous surface in a roaing sysem when he emperaure of he surface varies wih ime abou a non-ero consan mean and he emperaure a he free sream is consan.

3 Inernaional Journal of Mechanical Engineering and echnology (IJME ISSN (Prin ISSN (Online Volume Issue May- July ( IAEME. FORMLAION OF HE PROBLEM onsider unseady flow of a viscous incompressible fluid hrough a porous medium occupying a semi-infinie region of he space bounded by a verical infinie porous surface in a roaing sysem under he acion of a uniform magneic field applied normal o he direcion of flow. he emperaure of he surface varies wih ime abou a non-ero consan mean and he emperaure a he free sream is consan. he porous medium is in fac a non-homogenous medium which may be replaced by a homogenous fluid having dynamical properies equal o hose of a nonhomogenous coninuum. Also we assume ha he fluid properies are no affeced by he emperaure and concenraion differences excep by he densiy ρ in he body force erm; he influence of he densiy variaions in he momenum and energy equaions is negligible. We consider ha he verical infinie porous plae roaes in unison wih a viscous fluid occupying he porous region wih he consan angular velociy Ω abou an axis which is perpendicular o he verical plane surface. he aresian coordinae sysem is chosen such ha x y axes respecively are in he verical upward and perpendicular direcions on he plane of he verical porous surface while -axis is normal o i as shown in Fig. wih he above frame of reference and assumpions he physical variables excep he pressure p are funcions of and ime only. onsequenly he equaions expressing he conservaion of mass momenum and energy and he equaion of mass ransfer neglecing he hea due o viscous dissipaion which is valid for small velociies are given by W ( u u u σbu Ωv gβ( gβ ( u ( K ρ v v Ωu v K v σbv ρ (3

4 Inernaional Journal of Mechanical Engineering and echnology (IJME ISSN (Prin ISSN (Online Volume Issue May- July ( IAEME K p ρ (4 p ρ k (5 D (6 wih he boundary condiions i e ε( v u a as v u (7 where all he symbols are defined in he Nomenclaure secion. In a physically realisic siuaion we canno ensure perfec insulaion in any experimenal seup. here will always be some flucuaions in he emperaure. he plae emperaure is assumed o vary harmonically wih ime. I varies from ε( ± as varies from o π/. Since ε is small he plae emperaure varies only slighly from he mean value. For consan sucion we have from Eq. ( in view of (7 (8 onsidering iv u and aking ino accoun Eq. (8 hen Eqs. ( and (3 can be wrien as K ( gβ gβ( Ωi (9 We inroduce he following non-dimensional quaniies: 3 3 p Ω R ( gβ Gm gβ( Gr K k K ρ P D Sc Z Z In view of he above non-dimensional quaniies Eqs. (9 (5 and (6 reduce respecively o M k Gm Gr ir ( Pr ( Sc ( and he boundary condiions (7 become as a ε e i (3 3. MEHOD OF SOLION

5 Inernaional Journal of Mechanical Engineering and echnology (IJME ISSN (Prin ISSN (Online Volume Issue May- July ( IAEME In order o reduce he sysem of parial differenial Eqs. (( under heir boundary condiions (3 o a sysem of ordinary differenial equaions in he nondimensional form we assume he following for velociy emperaure and concenraion of he flow field as he ampliude ε (<< of he permeabiliy variaions is very small. ( ( εe ( i ( ( ε e ( i ( ( ε e ( (4 Subsiuing (4 ino he sysem (( and equaing harmonic and nonharmonic erms we ge ir M (Gr Gm k (5 M i( R (Gr Gm k (6 Pr (7 Pr ipr (8 Sc (9 Sc isc ( he appropriae boundary condiions reduce o ( ( ( ( ( ( ( ( ( ( ( ( ( ( hus he soluion of he problem is M5 M Pr Sc M i (e e 3 Le Le L3e ε e Gr (M M (M M ( ( ε e ( Pr i M e e 4 5 (3 Sc e (4 Now i is convenien o wrie he primary and secondary velociy fields in erms of he flucuaing pars separaing he real and imaginary par from Eqs. ( and (3 and aking only he real pars as hey have physical significance he velociy and emperaure disribuion of he flow field can be expressed in flucuaing pars as given below. u u ε(n rcos Nisin (5 w v v ε(n rsin Nicos (6 w where u iv and N r ini. π Hence he expressions for he ransien velociy profiles for are given by 3

6 Inernaional Journal of Mechanical Engineering and echnology (IJME ISSN (Prin ISSN (Online Volume Issue May- July ( IAEME u w v w π u ( εn i ( and π v ( εn r (. 3.. Skin Fricion he skin fricion a he plae in erms of ampliude and phase is given by d d i w d ε e dz dz dz ( PrL e Pr L Sce Sc L M e du ( PrL LSc L3M3 εe dz a 3 M3 3 i εe i M5 M Gr(M5e Me (M M (M M Gr(M5 M (M M (M M i Gr ( PrL LSc L3M 3 εe (7 (M M 4 he skin fricion coefficien for various values of Gr Gm k R M are given in able afer separaing he real and imaginary pars of he equaion ( Rae of Hea ransfer he hea ransfer coefficien in erms of he Nussel number a he plae in erms of ampliude and phase is given by d d i w d ε e dz dz dz ( d dz ( Pr εe i w M he consans L L L 3 M M M 3 M 4 M 5 N i N r are given in he Appendix secion. 4. RESLS AND DISSSION he problem of unseady MHD free convecive flow wih hea and mass ransfer effecs in a roaing porous medium has been considered. he soluions for primary and secondary velociy field emperaure field and concenraion profiles are obained using he perurbaion echnique. he effecs of flow parameers such as he magneic parameer M Grashof numbers for hea and mass ransfer Gr and Gm porosiy parameer k Prandl number Pr and he roaion parameer R on he velociy field have been sudied analyically and presened wih he help of Figs. and 3. he effecs of flow parameer on concenraion profiles have been presened wih he help of Fig. 4. he effecs of flow parameers on he ransien velociy profiles u/w and v/w have been presened in able. Furher he effecs of flow parameers on he skin fricion coefficien and rae of hea ransfer have been discussed wih he help of ables and (8 4

7 Inernaional Journal of Mechanical Engineering and echnology (IJME ISSN (Prin ISSN (Online Volume Issue May- July ( IAEME 4. Primary Velociy Profile (u/w From Eqs. ( (3 and (4 i is observed ha he seady par of he velociy field has a hree layer characer. hese layers may be idenified as he hermal layer arising due o ineracion of he hermal field and he velociy field and is conrolled by he Prandl number; he concenraion layer arising due o he ineracion of he concenraion field and he velociy field and he sucion layer as modified by he roaion and he porosiy of he medium. On he oher hand he oscillaory par of he velociy field exhibis a wo layer characer. hese layers may be idenified as he modified sucion layers arising as a resul of he riangular ineracion of he coriolis force and he unseady convecive forces wih he porosiy of medium. he dimensionless primary and secondary velociy componens for differen values of Gr Gm k and R are shown in Figs. and 3 considering Pr.7 (air π 5 ε. and Sc.6. he value of Sc.6 is chosen in such a way o represen waer vapor a approximaely 5 and am. I is clear from Fig. ha he primary velociy profiles increases whenever here is eiher an increase in he Grashof number or he modified Grashof number for mass ransfer whereas he profiles show he reverse rend whenever here is an increase in eiher of he roaion parameer he permeabiliy of he porous medium or he Harmann number. his shows ha he roaion permeabiliy of he porous medium and he magneic field exer rearding influence on he primary flow. Fig.. Effecs of Gm Gr k R M on primary velociy profiles wih Pr.7 5 π ε. and Sc.6. From Fig. i is noed ha all he velociy profiles increase seadily near he lower plae and hereafer hey show a consan decrease and reach he value ero a he oher plae bu he profiles show a reverse rend in Fig. 3. he magneic parameer is found o decelerae he velociy of he flow field o a significan amoun due o he magneic pull of he Loren force acing on he flow field. In he case of Singh [] he magneic parameer shows he reverse effec. 5

8 Inernaional Journal of Mechanical Engineering and echnology (IJME ISSN (Prin ISSN (Online Volume Issue May- July ( IAEME 4. Secondary Velociy Profile (v/w he Secondary velociy profiles are shown in Fig. 3 for various values of he modified Grashof number permeabiliy of he porous medium roaion parameer and he Harmann number or magneic parameer. I is observed ha he magniude of he secondary velociy profiles increases whenever here is an increase in eiher of he Grashof number or he modified Grashof number for mass ransfer or he permeabiliy of he porous medium. On he oher hand he velociy profiles show he opposie rend whenever here is an increase in he roaion parameer or he Harmann number. Fig. 3. Effecs of Gm Gr k R and M on secondary velociy profiles wih Pr.7 π 5 ε. and Sc ransien Velociy Profiles able : Variaions of velociies u/ and v/ when Gm. Gr. Pr.7 Sc.6 ε. k. R.. Z 5 v/ u/ u/ v/ he numerical values of he ransien velociy componen u/ and v/ for differen values of he frequency are given in able. I is seen ha for fixed values of k and R he componens of he primary velociy and he magniude of he secondary velociy decrease as he frequency parameer increases. his is in keeping wih he view ha he frequency of he oscillaion of he plae emperaure 6

9 Inernaional Journal of Mechanical Engineering and echnology (IJME ISSN (Prin ISSN (Online Volume Issue May- July ( IAEME has an acceleraing effec on he flow field. I is also remarked ha since he permeabiliy parameer k involves he sucion velociy he resuls discussed above and displayed in Figs. and 3 for he variaions of he parameer k correspond also o he variaions in he sucion velociy a he porous surface in he manner k. 4.4 oncenraion Profiles he concenraion profiles for various values of he Schmid number are ploed in Fig. 4. I is noed from Fig. 4 ha he concenraion profiles decreases wih an increase in he Schmid number. he values of he Schmid number Sc are chosen in such a way ha hey represen he diffusing chemical species of mos common ineres in air. For example he values of Sc for H H NH 3 propyl benene and helium in air are and.3 respecively as repored by Perry []. I is noed ha for heavier diffusing foreign speices i.e. increasing he Schmid number reduces he velociy in boh magniude and exen and hining of hermal boundary layer occurs. Subsanial increase in he velociy profiles is observed near he plae wih decreasing values of he Schmid number (ligher diffusing paricle. his shows ha he heavier diffusing species have greaer rearding effecs on he concenraion profiles of he flow field. he concenraion profiles agrees well wih he resuls of Das e al. [3]. Fig. 4. Effecs of Sc on concenraion disribuion wih ε. 5 and π 7

10 Inernaional Journal of Mechanical Engineering and echnology (IJME ISSN (Prin ISSN (Online Volume Issue May- July ( IAEME 4.5 Skin Fricion able : Variaions of Skin Fricion (primary and magniude of secondary when ε. Pr.7 Sc.6 and various values of Gm Gr R k M. Gm Gr k R M Primary skin fricion Secondary skin fricion Magniude of secondary skin fricion d u d v d d I is noed from able ha he primary skin-fricion componen increases due o an increase in eiher of he Grashof number or he modified Grashof number for mass ransfer. On he oher hand i decreases due o an increase in eiher of he roaion parameer R permeabiliy of he porous medium k or he Harmann number M. I is also noed from he above able ha he magniude of he secondary skin fricion increases due o an increase in eiher of he roaion parameer R modified Grashof number Gm permeabiliy of he porous medium k or he Grashof number Gr. However i decreases wih an increase in he Harmann number. he effecs of all parameers excep Gm closely agrees wih he resuls of Das e al. [3] Rae of Hea ransfer able 3: Variaions of hea ransfer when ε. in he case air and waer. Pr.7 (Air Pr 7. (Waer M Nu M Nu he magniude of he hea ransfer coefficien for various values of and M are given in able 3. I is clear from able 3 ha he magniude of he hea ransfer coefficien in he case of boh air and waer decrease whenever here is an increase in eiher he Harmann number M or ime. Bu hey increase due o an increase in he frequency parameer. I is also observed from able (3 ha he hea ransfer coefficien in he case of waer for any paricular values of and M is significanly higher when compared wih ha of air. Our resuls are in good agreemen wih he resuls of Mahao and Maii []. 8

11 Inernaional Journal of Mechanical Engineering and echnology (IJME ISSN (Prin ISSN (Online Volume Issue May- July ( IAEME 5. ONLSION he above analysis brings ou he following resuls of physical ineres on he velociy (primary and secondary emperaure and concenraion profiles of he flow field.. he modified Grashof number for mass ransfer and Grashof number have he effec of acceleraing he primary velociy profiles he magniude of he secondary velociy profiles and he skin fricion whereas he Harmann number has he effec of decreasing he flow field a all he poins due o he magneic pull of he Loren force acing on he flow field.. he roaion parameer and he frequency parameer have he effec of decreasing he primary velociy profiles as well as he magniude of he secondary velociy profiles whereas hey have he effec of increasing he skin fricion and he rae of hea ransfer. 3. he permeabiliy parameer has he influence of decreasing he primary velociy and he primary skin fricion whereas i has he influence of increasing he magniude of he secondary velociy profiles and he secondary skin fricion. 4. he presence of foreign species reduces he velociy as well as he hermal boundary layer and furher reducion occurs wih increasing values of he Schmid number. 5. he velociy of he fluid layer decreases and he hickness of he hermal boundary layer increases wih increasing values of he Schmid number. 6. An increase in he magneic parameer causes decreases in boh he primary velociy profiles and he magniude of he secondary velociy profiles. 7. When he magneic parameer is negleced. i.e. (M and he frequency of oscillaion is kep consan he resuls obained in his paper coincide wih he resul obained by Mahao and Maii [8]. In he absence of magneic field he primary velociy and he magniude of he secondary velociy obained by he above researchers increase as he frequency parameer increases. However when he frequency parameer increases he componens of he primary velociy profiles and he magniude of he secondary velociy profiles decrease as could be seen from able in his paper. his is due o he presence of he magneic field.[8] 6. REFERENES [] Malahy and Srinivas s 8 Pulsaing flow of a hydro magneic fluid beween permeable beds Inernaional ommunicaions in Hea and Mass ransfer 35: [] Singh K. D. 4 Influence of moving magneic field on hree dimensional couee flow ZAMP 55: [3] Das S.S. Saapahy A. Das J.K. and Panda J.P 9 Mass ransfer effecs on MHD flow and hea ransfer pas a verical porous plae hrough a porous medium under oscillaory sucion and hea source Inernaional Journal of Hea and Mass ransfer 5: [4] Muhucumaraswamy R. and Ganesh P. 3 nseady flow pas an impulsively sared verical plae wih hea and mass ransfer Inernaional Journal of hea and mass ransfer 3:

12 Inernaional Journal of Mechanical Engineering and echnology (IJME ISSN (Prin ISSN (Online Volume Issue May- July ( IAEME [5] Acharya M. Dash G.. and Singh L.P. Magneic field effecs on he free convecion and mass ransfer flow hrough porous medium wih consan sucion and consan hea flux Indian J. Pure Appl. Mah. 3(: -8. [6] haudhary and Arpia Jain 7 ombined hea and mass ransfer effecs on MHD free convecion flow pas an oscillaing plae embedded in porous medium Rom. Journ. Phys. 5: Buchares. [7] Agrawal and Kishor 988 hermal and mass diffusion on MHD naural convecion flow beween wo infinie verical moving and oscillaing porous parallel plaes Indian Journal of echnology 6: 4-. [8] Muhuraj and Srinivas S. A noe on hea ransfer o MHD oscillaory flow in an asymmeric wavy channel Inernaional ommunicaions in Hea and Mass ransfer 37: [9] Muhucumaraswamy R. Manivannan and hangaraj 7 hemical reacion effecs on infinie verical plae wih uniform hea flux and variable mass diffusion Inernaional Review of Pure and Applied Mahemaics 3: [] Singh K.D. and Verma 995 hree dimensional oscillaory flow hrough a porous medium wih periodic permeabiliy ZAMM Z. Angew. Mah. Mech. 75: [] Gersen and Gross 974 Flow and hea ransfer along a plane wall wih periodic soluion Journal of Applied Mahemaics and Physics (ZAMP 5: - 5. [] Singh K.D. 999 hree dimensional couee flow wih ranspiraion cooling Z. Angew. Mahs. Phys. (ZAMP 5: [3] Gupa G.D. and Rajesh Johari MHD hree dimensional flow pas a porous plae Indian J. Pure Appl. Mah. 3(3: [4] Singh K.D. Rakesh Sharma and Khem hand hree dimensional flucuaing flow and hea ransfer hrough a porous medium wih variable permeabiliy (ZAMM Z. Angew. Mech. 8 : [5] Ahmed. S. and Ahmed N. 4 wo-dimensional MHD oscillaory flow along a uniformly moving infinie verical porous plae bounded by porous medium Indian J. Pure Appl. Mah. 35 (: [6] Sharma and Yadav 6 hree-dimensional flow and hea ransfer hrough porous medium bounded by a porous verical surface wih variable permeabiliy and hea source Bull. al. Mah. Soc. 98(3: [7] Jain and Gupa 6 hree dimensional free convecion couee flow wih ranspiraion cooling Journal of Zhejiang niv. Science A 7(3: [8] Singh and Rakesh Sharma MHD hree dimensional couee flow wih ranspiraion cooling (ZAMM Z. Angew. Mah. Mech. 8(: [9] Singh K.D. Gorla M.G. and Hans Raj 5 Periodic soluion of oscillaory couee flow hrough porous medium in roaing sysem Indian J. Pure Appl. Mah. 36(3: [] Rapis and Perdikis 985 Oscillaory flow hrough a porous medium by he presence of free convecive flow In. J. Engng. Sci. 3(: [] Mahao and Maii 988 nseady free convecive flow and mass ransfer in a roaing porous medium Indian Journal of echnology 6: [] Perry E.D. 963 hemical Engineers Handbook (4 h Edn. McGraw-Hill Book ompany New York.