HEAT SOURCE AND MASS TRANSFER EFFECTS ON MHD FLOW OF AN ELASTO-VISCOUS FLUID THROUGH A POROUS MEDIUM

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1 1. V. RAJESH HEAT SOURCE AND MASS TRANSFER EFFECTS ON MHD FLOW OF AN ELASTO-VISCOUS FLUID THROUGH A POROUS MEDIUM 1. DEPARTMENT OF MATHEMATICS, NARAYANA ENGINEERING COLLEGE, NELLORE, ANDHRA PRADESH, INDIA ABSTRACT: An analial sud is pefomed o examine he effes of empeaue dependen hea soue on he unsead fee onveion and mass ansfe flow of an elaso-visous fluid pas an exponeniall aeleaed infinie veial plae in he pesene of magnei field hough poous medium. The plae empeaue is aised lineal wih ime and he onenaion level nea he plae is aised oc w. The Laplae ansfom mehod is used o obain he expession fo veloi. The effe of vaious paamees, ouing ino he poblem, on veloi field is disussed wih he help of gaphs. KEYWORDS: MHD, Fee onveion, Mass diffusion, Viso-elasi fluid, Poous medium, hea soue INTRODUCTION The poblem of fee-onveion and mass ansfe flow of an eleiall-onduing fluid pas an infinie plae unde he influene of a magnei field has aaed inees in view of is appliaion o geophsis, asophsis, engineeing, and o he bounda lae onol in he field of aeodnamis. Soundalgeka (1979) has deived an exa soluion fo he flow pas an impulsivel saed infinie veial plae in is own plane when he onveion uen is se up due o empeaue as well as onenaion gadien. MHD effes on impulsivel saed veial infinie plae wih vaiable empeaue in he pesene of ansvese magnei field wee sudied b Soundalgeka e al. (191). Bu in hese papes he effes of empeaue-dependen hea soues have no been aking ino aoun. Suh a siuaion exiss in man indusial o ehnologial appliaions, sola eneg poblems, o in poblem of spae sienes. Fom his poin of view, Rapis and Tzivanidis (191) have sudied he effes of mass ansfe, fee onveion uens, and hea soues on he Soke s poblem fo an infinie veial plae. Basan kuma Jha and Ravinda Pasad (1991) have sudied he effes of hea soue on MHD fee-onveion and mass ansfe flow hough a poous medium. In man indusial appliaions, he flow pas an exponeniall aeleaed infinie veial plae plas an impoan ole. This is paiulal impoan in he design of spaeship, sola eneg olleos, e. Fom his poin of view, fee onveion effes on flow pas an exponeniall aeleaed veial plae was sudied b Singh and Naveen kuma (19). The skin fiion fo aeleaed veial plae has been sudied analiall b Hossain and Shao (19). Basan kuma Jha e al. (1991) analzed mass ansfe effes on exponeniall aeleaed infinie veial plae wih onsan hea flux and unifom mass diffusion. Reenl Muhuumaaswam e al. () sudied hea ansfe effes on flow pas an exponeniall aeleaed veial plae wih vaiable empeaue. On he ohe hand, onsideable inees has been developed in he sud of he ineaion beween magnei fields and he flow of eleiall-onduing inompessible elaso-visous fluid due o is wide appliaions in moden ehnolog. The sud of an elaso-visous pulsaile flow helps o undesand he mehanisms of dialsis of blood hough an aifiial kidne. One suh model of a visoelasi fluid has been poposed b wale (19) fo a liquid of small elaxaion ime. Soundalgeka (197) sudied he flow of an elaso-visous fluid (wale s liquid B 1 ) pas an infinie plae. Singh (193) sudied he MHD flow of an elaso-visous fluid (wale s liquid B 1 ) pas an infinie hoizonal plae fo boh he lasses of impulsivel as well as unifoml aeleaed moion. Again Singh (19) sudied he fee-onveion flow of viso-elasi fluid pas an aeleaed veial plae. Samia e al. (199) sudied he lamina flow of an eleial-onduing wale s liquid B 1, pas an infinie non-onduing veial plae fo impulsive as well as unifoml aeleaed moion of he plae, in he pesene of a ansvese magnei field. Chowdhu and Islam () sudied MHD fee onveion flow of a viso- opigh FACULTY of ENGINEERING - HUNEDOARA, ROMANIA

2 ANNALS OF FACULTY ENGINEERING HUNEDOARA Inenaional Jounal Of Engineeing elasi fluid pas a veial poous plae. Reenl Rajesh and Vama (1) sudied he effes of hemal adiaion on unsead fee onveion flow of an elaso-visous fluid ove a moving veial plae wih vaiable empeaue in he pesene of magnei field hough poous medium This pape deals wih he analial sud of Hea soue and mass ansfe effes on MHD fee onveion flow of a viso-elasi fluid pas an exponeniall aeleaed infinie veial plae wih vaiable empeaue hough poous medium. The dimensionless govening equaions ae solved using he Laplae ansfom ehnique. MATHEMATICAL ANALYSIS The unsead fee onveion and mass ansfe flow of an eleiall onduing inompessible elaso-visous fluid pas an infinie veial plae hough poous medium in he pesene of hea soue has been onsideed. A magnei field of unifom sengh B is applied ansvesel o he plae. The indued magnei field is negleed as he magnei Renolds numbe of he flow is aken o be ve small. The flow is assumed o be in x -dieion whih is aken along he veial plae in he upwad dieion.the -axis is aken o be nomal o he plae. Iniiall he plae and he fluid ae a he same empeaue T wih onenaion level C a all poins. A ime >, he plae is exponeniall aeleaed wih a veloi u = u exp( a ) in is own plane and he plae empeaue is aised lineal wih ime and he level of onenaion nea he plae is aised o C w. The effe of visous dissipaion is assumed o be negligible. Then b usual Boussinesq s appoximaion, he unsead flow is govened b he following equaions. 3 u u K u σ Bu νu = ν + gβ ( T T ) + gβ ( C C ) ρ ρ K (1) T T ρcp = κ + Q () C C = D (3) Wih he iniial and bounda ondiions:, u =, T = T, C = C fo all >,, T = T + ( T w T ) A, C = C w a = () u =, T T, C C, as. u whee A =. ν On inoduing he following non-dimensional quaniies: u u u =, u T T gβν ( T w T ) =, =, θ =, G u ν ν Tw T =, 3 u C C gβν ( C w C ) μc p ν σ Bν C =, G C w C m =, P 3 =, S =, M =, u κ D ρu Qν Ku uk aν F =, S =, K =, a = κu ρν ν u () in equaions (1) o (), leads o 3 u u u u = S Mu + G θ + GmC K () θ θ P = Fθ (7) C C S = () Wih he iniial and bounda ondiions: : u =, θ =, C = fo all > : u = exp( a), θ =, C = 1 a = u =, θ, C as (9) Tome IX (Yea 11). Fasiule. ISSN 1

3 ANNALS OF FACULTY ENGINEERING HUNEDOARA Inenaional Jounal Of Engineeing In he pesen analsis we have onsideed he hea geneaion (absopion) of he pe Q = Q T T (1) whee Q is he volumei ae of hea geneaion (absopion). All he phsial paamees ae ρcp defined in he nomenlaue. Fo solving he poblem, we ake, Bead and Wales (19), u in he fom u = U + SU1 The soluion of equaions () o () unde iniial and bounda ondiion (9) and b use of equaion (11) b he Laplae Tansfom ehnique is given b S C ef = (1) whee P P F θ = + exp( F) ef + F P P P F + exp( F) ef F P G G u = B13 + ( B1 B1 ) + ( B3 + B1 B11 B ) d d G G m S ( F P) + ( B + B B B ) + B B M d 1 P opigh FACULTY of ENGINEERING - HUNEDOARA, ROMANIA FG S SbG S S B d P M S m B B ( B B ) SG SG b SG F ( B B ) B B d M d P 1 ( B ) m ( ) exp( ) exp B1 M ef + M + M ( ) exp( ) exp M ef M M ( ) ( ) 1 B = exp M ef M exp M ef M + + exp( ) B3 ( M ) ef ( M ) + exp M ef + M exp( b) b exp( b) b exp( b) B ( b+ M ) ef ( b+ M ) + b+ M ( ) ( ) ( b) b ( b) b ( b) exp exp exp b+ M ef + b+ M + + b+ M exp M + 1 B = a + π aexp a a + M ( a + M ) ef ( a M ) + ( ) ( ) exp a + M ef + a + M (13) (1)

4 ANNALS OF FACULTY ENGINEERING HUNEDOARA Inenaional Jounal Of Engineeing exp M + M B = ( M ) ef π M exp M ef + M exp M + exp( ) M B7 = ( M ) ef ( M ) π exp M ef + M exp M + exp ( b) M + b B = ( M + b) ef ( M b) π + exp M + b ef + M + b P F P exp( ) P exp( ) B9 ( F P ) ef + + P F P P F P exp( ) P exp( ) ( F P ) ef P F P 1 P 1 exp F P exp F B = F ef + ( F ) ef + P P ( ) exp( ) P F P B11 ( F P ) ef P ( ) P F P + exp( F P ) ef + P S exp( b) b exp( b) bs exp( b) B1 ( bs ) ef b + bs S exp( b) b exp( b) bs exp( b) ( bs ) ef + b + + bs exp ( a) B13 = exp ( a + M ) ef ( a M ) + ( ) ( ) + exp a + M ef + a + M B 1 ( M ) ef + M + M ( M ) ef M M exp ( b) B1 ( b + M ) ef ( b M ) + + exp( b + M ) ef + ( b + M ) P F P B1 ( F ) ef + P + F P F P exp ( F ) + ef P F Tome IX (Yea 11). Fasiule. ISSN 1

5 ANNALS OF FACULTY ENGINEERING HUNEDOARA Inenaional Jounal Of Engineeing B 17 S ef = exp( b) S 1 exp S exp B = bs ef b + ( bs ) ef + b 1 M F M M = M +, b=, =, d = ( F M ) K S 1 P 1 DISCUSSION, RESULTS AND CONCLUSIONS In ode o ge he phsial insigh ino he poblem, we have ploed veloi pofiles fo diffeen paamees M (Magnei paamee), K (pemeabili paamee), S (viso-elasi paamee), Gm (Mass gashof numbe), (ime), a (aeleaing paamee), F (Hea soue paamee) and S (Shmid numbe) in Figues (1) o (1) fo he ases of heaing (G<) and ooling (G>) of he plae. The heaing and ooling ake plae b seing up fee onveion uen due o empeaue and onenaion gadien. Figues (1) and () illusae he influenes of M (Magnei paamee) on he veloi field in ases of ooling and heaing of he plae a =. espeivel. Fom hese figues he veloi is found o deease wih an inease in M fo he ase of heaing of he plae. I is beause ha he appliaion of ansvese magnei field will esul a esisiive pe foe (Loenz foe) simila o dag foe, whih ends o esis he fluid flow and hus eduing is veloi. Bu he evese effe is found in he ase of ooling of he plae. I is also found ha in he ase of ooling, he veloi ineases nea he sufae of he plae and beomes maximum and hen deeases awa fom he plae. The evese phenomenon is found in he ase of heaing of he plae M=.1,G=1,Gm= M=.3,G=1,Gm= M=.,G=1,Gm= M=1, G=1,Gm= - Veloi Veloi - - M=.1,G=-1,Gm=- M=.3,G=-1,Gm=- M=.,G=-1,Gm=- M=1, G=-1,Gm= Figue 1. Veloi pofiles when P=.1, S=.7, k=., Figue. Veloi pofiles when P=.1, S=.7, k=., a=., S=., F=, =. a=., S=., F=, =. Figues (3) and () epesen he veloi pofiles due o he vaiaions in K(pemeabili paamee) in ases of ooling and heaing of he plae a =. espeivel. Fom hese figues he veloi is obseved o inease wih an inease in pemeabili paamee K fo he ase of heaing of he plae. This is due o he fa ha he pesene of a poous medium ineases he esisane o flow. Bu he evese effe is obseved in he ase of ooling of he plae. Figues () and () displa he effes of S (viso-elasi paamee) on he veloi field fo he ases ooling and heaing of he plae a =. espeivel. In he ase of ooling of he plae, i is obseved ha he veloi is less fo Newonian fluid (S is equal o zeo) han he Non-Newonian fluid (S is no equal o zeo) and also he veloi ineases wih an inease in S. Bu he opposie phenomenon is obseved in he ase of heaing of he plae. Figues (7) and () eveal veloi vaiaions wih Gm (mass gashof numbe) in he ases of ooling and heaing of he plae a =. espeivel. Fom he figues i is obseved ha he veloi ineases wih an inease in mass gashof numbe Gm in he ase of ooling of he plae. I is due o he fa inease in he values of mass Gashof numbe has he enden o inease he mass buoan effe. This gives ise o an inease in he indued flow. The evese effe is obseved in he ase of heaing of he plae. Figues (9) and (1) epesens he veloi pofiles fo diffeen values of (ime) in ases of ooling and heaing of he plae espeivel. Fom he figues, in he ase of ooling of he plae, he veloi is found o inease wih an inease in ime. Bu he evese effe is obseved in he ase of heaing of he plae. opigh FACULTY of ENGINEERING - HUNEDOARA, ROMANIA 9

6 ANNALS OF FACULTY ENGINEERING HUNEDOARA Inenaional Jounal Of Engineeing K=.,G=1,Gm= K=1, G=1,Gm= K=1.,G=1,Gm= K=, G=1,Gm= - Veloi Veloi - - K=.,G=-1,Gm=- K=1, G=-1,Gm=- K=1.,G=-1,Gm=- K=, G=-1,Gm= Figue 3. Veloi pofiles when P=.1, S=.7, k=., a=., S=., M=1, F=, = Figue. Veloi pofiles when P=.1, S=.7, k=., a=., S=., M=1, F=, =. Veloi S=, G=1,Gm= S=., G=1,Gm= S=.1, G=1,Gm= S=.1, G=1,Gm= Veloi S=, G=-1,Gm=- S=.,G=-1,Gm=- S=.1, G=-1,Gm=- S=.1, G=-1,Gm= Figue. Veloi pofiles when P=.1, S=.7, k=., a=., M=1, F=, = Figue. Veloi pofiles when P=.1, S=.7, k=., a=., M=1, F=, =. Veloi 3 1 G=1,Gm= G=1,Gm= G=1,Gm=1 G=1,Gm=1 Veloi G=-1,Gm=- G=-1,Gm=- G=-1,Gm=-1 G=-1,Gm= Figue 7. Veloi pofiles when P=.1, S=.7, k=., a=., M=1, F=, = Figue. Veloi pofiles when P=.1, S=.7, k=., a=., S=., M=1, F=, = =.,G=1,Gm= =.3,G=1,Gm= =.,G=1,Gm= =.,G=1,Gm= 1 - Veloi 1 Veloi -1-1 =.,G=-1,Gm=- =.3,G=-1,Gm=- =.,G=-1,Gm=- =.,G=-1,Gm= Figue 9. Veloi pofiles when P=.1, S=.7, k=., a=., S=., M=1, F= Figue 1. Veloi pofiles when P=.1, S=.7, k=., a=., S=., M=1, F= 1 Tome IX (Yea 11). Fasiule. ISSN 1

7 ANNALS OF FACULTY ENGINEERING HUNEDOARA Inenaional Jounal Of Engineeing Figues (11) and (1) epesen he veloi pofiles fo diffeen values of a (aeleaing paamee) in ases of ooling and heaing of he plae a =. espeivel. Fom he figues he veloi is found o inease wih an inease in a (aeleaing paamee) in ases of boh ooling and heaing of he plae. I is also found ha he fluid veloi due o he impulsive sa of he plae (a is equal o zeo) is less han due o he exponeniall aeleaed sa (a is no equal o zeo) in ases of boh ooling and heaing of he plae. 1 1 a=,g=1,gm= a=3,g=1,gm= a=,g=1,gm= a=9,g=1,gm= a=,g=-1,gm=- a=3,g=-1,gm=- a=,g=-1,gm=- a=9,g=-1,gm=- Veloi Veloi Figue 11. Veloi pofiles when P=.1, S=.7, k=., Figue 1. Veloi pofiles when P=.1, S=.7, S=., F=, M=1, =. k=., F=, S=., M=1, =. To obseve he effe of F, he veloi pofiles fo diffeen F (Hea soue paamee) ae pesened in figues (13) and (1) in ases of ooling and heaing of he plae a =. espeivel. I is obseved ha hee is negligible effe of F on he veloi in ases of boh ooling and heaing of he plae. I is also obseved ha, in he ase of ooling of he plae, he veloi ineases nea he sufae of he plae and beomes maximum and hen deeases awa fom he plae. Bu he opposie effe is obseved in he ase of heaing of he plae F=,G=1,Gm= F=,G=1,Gm= F=,G=1,Gm= F=,G=1,Gm= - Veloi Veloi - - F=,G=-1,Gm=- F=,G=-1,Gm=- F=,G=-1,Gm=- F=,G=-1,Gm= Figue 13. Veloi pofiles when P=.1, S=.7, k=., a=., S=., M=1, = Figue 1. Veloi pofiles when P=.1, S=.7, k=., a=., S=., M=1, =. 1 1 S=., G=1,Gm= S=.,G=1,Gm= S=.7, G=1,Gm= S=.7,G=1,Gm= - Veloi Veloi S=., G=-1,Gm=- S=.,G=-1,Gm=- S=.7, G=-1,Gm=- S=.7,G=-1,Gm= Figue 1. Veloi pofiles when P=.1, k=., a=., Figue 1. Veloi pofiles when P=.1, k=., a=., S=., F=, M=1, =. S=., F=, M=1, =. Figues (1) and (1) displa he effes of S (Shmid numbe) on he veloi field fo he ases of ooling and heaing of he plae a =. espeivel. Fom he figues, in he ase of ooling of he plae, i is found ha he veloi ineases wih an inease in S. Bu he evese effe is found in he ase of heaing of he plae. opigh FACULTY of ENGINEERING - HUNEDOARA, ROMANIA 11

ANNALS OF FACULTY ENGINEERING HUNEDOARA Inenaional Jounal Of Engineeing ACKNOWLEDGEMENT I would like o aknowledge D. S. Vijaa Kuma Vama, Pofesso of Mahemais, S.V. Univesi, Tiupai (A.

8 ANNALS OF FACULTY ENGINEERING HUNEDOARA Inenaional Jounal Of Engineeing ACKNOWLEDGEMENT I would like o aknowledge D. S. Vijaa Kuma Vama, Pofesso of Mahemais, S.V. Univesi, Tiupai (A.P), India fo fuiful disussion on he subje of his pape. APPENDIX - NOMENCLATURE C Conenaion in he fluid fa awa fom he plae C w Conenaion of he plae A Consan ' Coodinae axis nomal o he plae C Dimensionless onenaion Dimensionless oodinae axis nomal o he plae u Dimensionless veloi B Exenal magnei field G m Mass Gashof numbe P Pandl numbe S C Shmid numbe C Speies onenaion in he fluid C p Speifi hea a onsan pessue T Tempeaue of he fluid fa awa fom he plae T Tempeaue of he fluid nea he plae T w Tempeaue of he plae k Themal onduivi of he fluid G Themal Gashof numbe ' Time u Veloi of he fluid in he x -dieion u Veloi of he plae a Aeleaing paamee D Chemial Moleula diffusivi g Aeleaion due o gavi K pemeabili paamee M Magnei field paamee F Hea Soue paamee S Viso-elasi paamee Dimensionless ime m Coeffiien of visosi ef Complemena eo funion Densi of he fluid q Dimensionless empeaue s Elei onduivi ef Eo funion u Kinemai visosi a Themal diffusivi b* Volumei oeffiien of expansion wih onenaion b Volumei oeffiien of hemal expansion w Condiions on he wall Fee seam ondiions REFERENCES [1.] Bead DM and Wales K (19) Elasio-visous bounda lae flows, Two dimensional flow nea a sagnaion poin. Po. Camb. Phil. So. : 77. [.] Chowdhu MK and Islam MN () MHD fee onveion flow of viso-elasi fluid pas an infinie veial poous plae. Hea and Mass Tansfe 3: 39. [3.] Hossain MA and Shao LK (19). The skin fiion in he unsead fee onveion flow pas an aeleaed plae, Asophsis and Spae Siene, 1, pp [.] Jha BK and Pasad R (1991). MHD fee onveion and mass ansfe flow hough a poous medium wih hea soue, Asophsis and Spae Siene, 11, pp [.] Jha BK, Pasad R and Rai S (1991). Mass Tansfe Effes on he flow pas an exponeniall aeleaed veial plae wih onsan hea flux, Asophsis and Spae Siene, 11, pp [.] Muhuumaaswam R, Sahappan KE and Naaajan R (). Hea Tansfe Effes on flow pas an exponeniall aeleaed veial plae wih vaiable empeaue, Theoe. Appl. Meh., Vol.3, No., pp [7.] Rajesh V and Vama SVK (1) Radiaion effes on MHD flow hough a poous medium wih vaiable empeaue o vaiable mass diffusion. In. J. of Appl. Mah and Meh. (1): [.] Rapis AA and Tzivanidis GJ (191). Effes of mass ansfe, fee onveion uens and hea soues on he Sokes' poblemfo an infinie veial plae. Asophs. Spae Siene, 7(), 31. [9.] Samia NK, Pasad R and Redd MUS (199) MHD fee onveion flow of an elaso-visous fluid pas an infinie veial plae. Aso Phsis and Spae Siene 11:1-13. [1.] Singh AK (19) Vijnana Paishad Anusandhan Paika 7 (No. ):193. [11.] Singh AK and Kuma N (19). Fee onveion flow pas an exponeniall aeleaed veial plae, Asophsis and Spae Siene, 9, pp.-. [1.] Singh AK and Singh J (193) MHD flow of an elasio-visous fluid pas an aeleaed plae. Na. Aad. Si. Lees (No. 7): [13.] Soundalgeka VM (197) Rheol. Aa 13: 177. [1.] Soundalgeka VM (1979) J. Appl. Meh. (Tans. ASME) : 77. [1.] Soundalgeka VM, Pail MR and Jahagida MD (191) MHD sokes poblem fo a veial plae wih vaiable empeaue. Nulea Engg. Des. Vol.: 39-. [1.] Wales K (19) J. Meanique 1:7. ANNALS OF FACULTY ENGINEERING HUNEDOARA INTERNATIONAL JOURNAL OF ENGINEERING opigh Univesi Poliehnia Timisoaa, Faul of Engineeing Hunedoaa,, Revoluiei, 3311, Hunedoaa, ROMANIA hp://annals.fih.up.o 1 Tome IX (Yea 11). Fasiule. ISSN 1

Engineering Accreditation. Heat Transfer Basics. Assessment Results II. Assessment Results. Review Definitions. Outline

Engineering Accreditation. Heat Transfer Basics. Assessment Results II. Assessment Results. Review Definitions. Outline Hea ansfe asis Febua 7, 7 Hea ansfe asis a Caeo Mehanial Engineeing 375 Hea ansfe Febua 7, 7 Engineeing ediaion CSUN has aedied pogams in Civil, Eleial, Manufauing and Mehanial Engineeing Naional aediing