Combined Effects of Hall Current and Radiation on MHD Free Convective Flow in a Vertical Channel with an Oscillatory Wall Temperature

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1 Opn Journal of Fluid Dnamics, 03, 3, 9- ttp://d.doi.org/0.436/ojfd Publisd Onlin Marc 03 (ttp:// Combind Effcts of Hall Currnt and Radiation on MHD Fr Convctiv Flow in a Vrtical Cannl wit an Oscillator Wall Tmpratur Sankar Kumar Gucait, Sanatan Das, Rabindra Nat Jana Dpartmnt of Applid Matmatics, Vidasagar Univrsit, Midnapor, India Dpartmnt of Matmatics, Univrsit of Gour Banga, Englis Bazar, India jana67@aoo.co.in Rcivd Dcmbr, 0; rvisd Januar 5, 03; accptd Januar 6, 03 ABSTRACT T combind ffcts of Hall currnt and radiation on an unstad MHD fr convctiv flow of a viscous incomprssibl lctricall conducting fluid in a vrtical cannl wit an oscillator wall tmpratur av bn studid. W av considrd two diffrnt cass ) flow du to t impulsiv motion of on of t cannl walls and ) flow du to t acclratd motion of on of t cannl walls. T govrning quations ar solvd analticall using t Laplac transform tcniqu. It is found tat t primar vlocit and t magnitud of t scondar vlocit incrasd wit an incras in Hall paramtr for t impulsiv as wll as t acclratd motions of on of t cannl walls. An incras in itr radiation paramtr or frqunc paramtr lads to dcras in t primar vlocit and t magnitud of t scondar vlocit for t impulsiv as wll as acclratd motions of on of t cannl walls. T fluid tmpratur dcrass wit an incras in radiation paramtr. Furtr, t sar strsss at t lft wall rduc wit an incras in itr radiation paramtr or frqunc paramtr for t impulsiv as wll as t acclratd motions of on of t cannl wall. Kwords: Hall Currnt; MHD Fr Convction; Radiation; Prandtl Numbr; Grasof Numbr; Frqunc Paramtr; Impulsiv Motion; Acclratd Motion. Introduction T mcanism of conduction in ionizd gass in t prsnc of a strong magntic fild is diffrnt from tat in mtallic substanc. T lctric currnt in ionizd gass is gnrall carrid b lctrons, wic undrgos succssiv collisions wit otr cargd or nutral particls. In t ionizd gass, t currnt is not proportional to t applid potntial cpt wn t fild is vr wak in an ionizd gas wr t dnsit is low and t magntic fild is vr strong, t conductivit normal to t magntic fild is rducd du to t fr spiraling of lctrons and ions about t magntic lins of forc bfor suffring collisions and a currnt is inducd in a dirction normal to bot lctric and magntic filds. Tis pnomnon, wll known in t litratur, is calld t Hall ffct. T stud of dromagntic flows wit Hall currnts as important nginring applications in problms of magntodrodnamic gnrators and of Hall acclrators as wll as in fligt magntodrodnamics. It is wll known tat a numbr of astronomical bodis posss fluid intriors and magntic filds. It is also important in t solar psics involvd in t sunspot dvlopmnt, t solar ccl and t structur of magntic stars. In spac tcnolog applications and at igr oprating tmpraturs, radiation ffcts can b quit significant. T radiativ convctiv flows ar frquntl ncountrd in man scintific and nvironmntal procsss, suc as astropsical flows, watr vaporation from opn rsrvoirs, ating and cooling of cambrs, and solar powr tcnolog. T unstad dromagntic flow of a viscous incomprssibl lctricall conducting fluid troug a vrtical cannl is of considrabl intrst in t tcnical fild du to its frqunt occurrnc in industrial and tcnological applications. T Hall ffcts on t flow of ionizd gas btwn paralll plats undr transvrs magntic fild av bn studid b Sato []. Miatak and Fujii [] av discussd t fr convction flow btwn vrtical plats on plat isotrmall atd and otr trmall insulatd. Natural convction flow btwn vrtical paralll plats on plat wit a uniform at flu and t otr trmall insulatd as bn invstigatd b Tanaka t al. [3]. Gupta and Gupta [4] av studid t radiation ffct on dromagntic convction in a vrtical cannl. Hall ffcts on t dromagntic convctiv

2 0 flow troug a vrtical cannl wit conducting walls av bn invstigatd b Dutta and Jana [5]. T unstad dromagntic fr convctiv flow wit radiativ at transfr in a rotating fluid as bn dscribd b Bstman and Adjpong [6]. Josi [7] as studid t transint ffcts in natural convction cooling of vrtical paralll plats. Sing [8] av dscribd t natural convction in unstad Coutt motion. Sing t al. [9] av studid t unstad fr convctiv flow btwn two vrtical paralll plats. T natural convction in unstad MHD Coutt flow wit at and mass transfrs as bn analzd b Ja [0]. Naraari t al. [] av studid t unstad fr convctiv flow btwn long vrtical paralll plats wit constant at flu at on boundar. T unstad fr convctiv flow in a vrtical cannl du to smmtric ating av bn dscribd b Ja t al. []. Sing and Paul [3] av studid t unstad natural convctiv btwn two vrtical walls atd/coold asmmtricall. Sanal and Adikari [4] av studid t ffcts of radiation on MHD vrtical cannl flow. T radiation ffcts on MHD Coutt flow wit at transfr btwn two paralll plats av bn amind b Mbin [5]. Grosan [6] as studid t trmal radiation ffct on t full dvlopd mid convctiv flow in a vrtical cannl. Guria and Jana [7] av discussd Hall ffcts on t dromagntic convctiv flow troug a rotating cannl undr gnral wall conditions. Ja and Ajibad [8] av studid t unstad fr convctiv Coutt flow of at gnrating/absorbing fluid. Effcts of trmal radiation and fr convction currnts on t unstad Coutt flow btwn two vrtical paralll plats wit constant at flu at on boundar av bn studid b Naraari [9]. Rajput and Sau [0] av studid t unstad fr convction MHD flow btwn two long vrtical paralll plats wit constant tmpratur and variabl mass diffusion. Das t al. [] av studid t radiation ffcts on fr convction MHD Coutt flow startd ponntiall wit variabl wall tmpratur in t prsnc of at gnration. T ffct of radiation on transint natural convction flow btwn two vrtical walls av bn dscribd b Mandal t al. []. Das t al. [3] av studid t radiation ffcts on unstad MHD fr convctiv Coutt flow of at gnration/absorbing fluid. T ffcts of radiation on MHD fr convctiv Coutt flow in a rotating sstm av bn discussd b Sarkar t al. [4]. Sarkar t al. [5] av studid an oscillator MHD fr convctiv flow btwn two vrtical walls in a rotating sstm. T aim of t prsnt papr is to stud t combind ffcts of Hall currnt and radiation on t unstad MHD fr convctiv flow of a viscous incomprssibl lctricall conducting fluid in a vrtical cannl wit an oscillator wall tmpratur of on of t cannl walls. It is found tat t primar vlocit u and t magnitud of t scondar vlocit dcras wit an v incras in itr radiation paramtr R or frqunc paramtr n or Prandtl numbr Pr for t impulsiv as wll as t acclratd motions of on of t cannl walls. It is also obsrvd tat t primar vlocit u and t magnitud of t scondar vlocit v incras wit an incras in itr Hall paramtr m or Grasof numbr Gr or tim for t impulsiv as wll as acclratd motions. An incras in Grasof numbr Gr lads to fall t fluid vlocit componnts. An incras in t radiation paramtr R lads to incras t fluid tmpratur. Furtr, t sar strss at t wall 0 du to t primar flow and t absolut valu of t sar strss at t wall 0 du to t scondar flow dcras for t impulsiv as wll as acclratd motions of on of t cannl walls wit an incras in radiation paramtr R. T rat of at transfr 0, at t wall 0 dcrass wil t rat of at transfr 0, at t wall incrass wit an incras in Prandtl numbr Pr.. Formulation of t Problm and Its Solution Considr t unstad MHD flow of a viscous incomprssibl lctricall conducting radiativ fluid btwn two infinitl long vrtical paralll walls sparatd b a distanc. T flow is st up b t buoanc forc arising from t tmpratur gradint. Coos a Cartsian co-ordinats sstm wit t -ais along t cannl wall at 0 in t vrticall upward dirction, t -ais prpndicular to t cannl walls and z-ais is normal to t -plan (s Figur ). Initiall, at tim t 0, t two walls and t fluid ar assumd to b at t sam tmpratur T and stationar. At tim t >0, t wall at 0 starts to mov in its own plan wit a vlocit Ut and its tmpratur is raisd to T T0 Tcost wras t wall at is stationar and maintaind at a constant tmpratur T, wr is t frqunc of t tmpratur oscillations. A uniform transvrs magntic fild B 0 is applid prpndicular to t cannl walls. W assum tat t flow is laminar and t prssur gradint trm in t momntum quation can b nglctd. It is assumd tat t ffct of viscous and Joul dissipations ar ngligibl. It is also assumd tat t radiativ at flu in t -dirction is ngligibl as compard to tat in t -dirction. As t cannl walls ar infinitl long, t vlocit fild and tmpratur distribution ar functions of and t onl. Undr t usual Boussinsq approimation, t flow is govrnd b t following Navir-Stoks quations u u B0 g T T j, () t

3 Figur. Gomtr of t problm. v v t B 0 j, wr is t fluid dnsit, t kinmatic viscosit, u and ar fluid vlocit componnts and g t acclration du to gravit. T nrg quation is c p T k t T q r, T t fluid tmpratur, k t trmal conductivit, c p t spcific at at constant prssur and q t r radiativ at flu. T initial and boundar conditions for t vlocit and tmpratur distributions ar t 0: u v 0, T T for0, u U t, v 0, t 0: T T T0 Tcost at 0, t 0: u v 0, T T at. () (3) It as bn sown b Cogl t al. [6] tat in t opticall tin limit for a non-gra gas nar quilibrium, t following rlation olds q r p 4T T K d, 0 T (5) wr K is t absorption cofficint, is t wav lngt, p is t Planck s function and subscript ' ' indicats tat all quantitis av bn valuatd at t tmpratur T wic is t tmpratur of t walls at tim t 0. Tus, our stud is limitd to small diffrnc of wall tmpraturs to t fluid tmpratur. On t us of t Equation (5), t Equation (3) bcoms T T cp k 4 T TI, (6) t wr (4) p I K d. 0 T (7) T gnralizd Om s law, on taking Hall currnts into account and nglcting ion-slip and trmo-lctric ffct, is (s Cowling [7]) j jb EqB, (8) B 0 wr j is t currnt dnsit vctor, B t magntic fild vctor, E t lctric fild vctor, t cclotron frqunc, t lctrical conductivit of t fluid and t collision tim of lctron. W sall assum tat t magntic Rnolds numbr for t flow is small so tat t inducd magntic fild can b nglctd. Tis assumption is justifid sinc t magntic Rnolds numbr is gnrall vr small for partiall ionizd gass. T solnoidal rlation B 0 for t magntic fild givs Bz B0 constant vrwr in t fluid wr B 0,0, B 0. Furtr, if j, j, j z b t componnts of t currnt dnsit j, tn t quation of t consrvation of t currnt dnsit j 0 givs j z constant. Tis constant is zro sinc j z 0 at t walls wic ar lctricall non- conducting. Tus j z 0 vrwr in t flow. Sinc t inducd magntic fild is nglctd, t B Mawll s quation E bcoms E 0 t E wic givs E 0 and 0. Tis implis tat z z E constant and E constant vrwr in t flow. W coos tis constants qual to zro, i.. E E 0. In viw of t abov assumption, t Equation (8) givs j mj vb (9) 0, j mj ub (0) wr m is t Hall paramtr. Solving (9) and (0) for j and j, w av B0 j vmu, () m B0 j mv u. () m On t us of () and (), t momntum Equations () and () along - and -dirctions bcom u u B0 g T T umv, t m 0, (3) v v B0 vmu. (4) t m Introducing non-dimnsional variabls

4 uv, t u, v,,, U0 T T, U t U 0 f, T T 0 Equations (6), (3) and (4) bcom (5) u u Gr M u mv, (6) m v v M v mu, (7) m wr M B0 Gr g T T Pr R, (8) wr d d R spr 0, (4) M im a. m T initial and boundar conditions for F, s, s F0, s f s, F, s 0, ar 0, s,, s0. sin sin (5) and (6) Solutions of Equations (3) and (4) subjct to t boundar conditions (6) ar givn b is t magntic paramtr, s sin RsPr 0 t Grasof numbr,,for Pr s n sin R spr Pr cp k t Prandtl numbr and R 4I k, s (7) s sin Rs t radiation paramtr., for Pr s n sin R s T initial and boundar conditions (4) bcom 0: u F, s v 0, 0for0, 0: u f, v 0, cosn at 0, (9) sin as sgr f s 0: u v 0, 0at, sin a s Prs s n sin as sin RsPr wr n is t frqunc paramtr. sin as sin RsPr Combining Equations (6) and (7), w gt for Pr F F im M Gr F, (0) sin as sgr m f s sin a s Ras n wr sin as sin Rs F uiv andi. () sin as sin Rs T initial and boundar conditions for F, ar for Pr, 0: F 0for0, (8) 0: F f at 0, () Now, w sall considrd t following cass. 0: F 0at. ) Wn t wall at 0 startd impulsivl: Taking t Laplac transform of Equations (0) and In tis cas f, i.. f s. Tn t invrs (8) and on t us of (9) and (), w av d F d a s F Gr, (3) s Laplac transforms of Equations (7) and (8) giv t solution for t tmpratur distribution and t vlocit fild as s in sin RinPr in sin R inpr π ks sinkπ for Pr sin R inpr sin R inpr Pr k s n, sin Rin sin Rin ks s3 in in π 3 sin k π for Pr, sin Rin sin Rin k s3 n (9)

5 3 s in sin a k Gr sin ain π sin kπ sin a k s Pr in sin a in in sin RinPr sin ain sin RinPr sin RinPr in sin ain sin RinPr s s s s F, 4π k sin kπ for Pr (30) k s s n Pr s s n s sin a k Gr in sin ain sin Rin π sin kπ sin a k s R a sin ain sin Rin in sin ain sin Rin s s3 s s3 4π ksin kπ for Pr, sin ain sin Rin k s n s3 n wr M im s Rk π, s k π and s 3 Rk π. (3) Pr m ) Wn t wall at 0 startd acclratdl:, i.. In tis cas f f s s. Tn t invrs Laplac transforms of Equations (7) and (8) giv t solution for t tmpratur distribution and t vlocit fild as s in sin RinPr in sin R inpr π ks sinkπ for Pr sin R inpr sin R inpr Pr k s n, s3 i sin i Pr i sin i Pr 3 n R n n R n ks π sin k π for Pr, sin RinPr sin RinPr k s3 n s sin a k π sin kπ sin acos a sin a k s asin a i Gr n sin a in sin R inpr cos asin a Pr in sin ain sin RinPr in sin ain sin RinPr in sin a in sin R inpr s s s s 4π k sin π for Pr k F, k s s n Pr s s n s sin a k π sin kπ sin acos a sin a k s asin a Gr in sin ain cos asin a sin Rin R a sin a in sin R in s s3 in sin a in sin R in s s3 4π ksin kπ sin ain sin Rin k s n s3 n for Pr, (3) (33)

6 4 wr s, s and s 3 ar givn b (3). 3. Rsults and Discussion W av prsntd t non-dimnsional vlocit componnts and tmpratur distribution for svral valus of Hall paramtr m, radiation paramtr R, Prandtl numbr Pr, frqunc paramtr n, Grasof numbr Gr and tim against wn M 5 and π n in Figurs -7. It is sn from Figurs and 3 4 tat t primar vlocit u and t magnitud of t scondar vlocit v incras wit an incras of Hall paramtr m for t impulsiv as wll as acclratd motions of on of t cannl walls. Figurs 4 and 5 sow tat t primar vlocit u and t magnitud of t scondar vlocit v dcras wit an incras in radiation paramtr R for bot t impulsiv and acclratd motions of on of t cannl Figur 4. Primar vlocit u for diffrnt R wn m 0.5, Pr 0.7, n, Gr 5 and 0.. Figur. Primar vlocit for diffrnt m wn R, Pr 0. 7, n, Gr u 5 and 0.5. Figur 5. Scondar vlocit v for diffrnt R wn m 0.5, Pr 0.7, n, Gr 5 and 0.. Figur 3. Scondar vlocit v for diffrnt m R, Pr 0.7, n, Gr 5 and 0.5. wn Figur 6. Primar vlocit u for diffrnt Pr wn m 0.5, R, n, Gr 5 and 0..

7 5 Figur 7. Scondar vlocit v for diffrnt Pr w n m05., R, ngr, 5 and 0.. walls. It is illustratd from Fi gurs 6 and 7 tat t primar vlocit u and t magnitud of t scondar vlocit v dcras wit an incras in Prandtl numbr Pr for t impulsiv as wll as acclratd motions of on of t cannl walls. Figurs 8 and 9 sow tat bot t primar vlocit u and t magnitud of t scondar vlocit v dcras wit an incras in frqunc paramtr n for bot t impulsiv and acclratd motions of on of t cannl walls. An incras in Grasof numbr Gr lads to incras t primar vlocit u and t magnitud of t scondar vlocit v for bot t impulsiv and acclratd motions of on of t cannl walls sown in Figurs 0 and. It is sn form Figurs and 3 tat t primar vlocit u and t magnitud of t scondar vlocit v incras wit an incras in tim for bot t impulsiv and acclratd motions of on of t cannl wa lls. It is sn from Figur 4 tat t fluid tmpratur dcrass wit an incras in radiation paramtr R. Tis rsult qualitativl agrs wit pctations, sinc t ffct of radiation is to dcras t rat of nrg transport to t fluid, trb dcrasing t tmpratur of t fluid. It is obsrvd from Figur 5 tat t fluid tmpratur incrass wit an incras in Prandtl numbr Pr. Tis is in agrmnt wit t psical fact tat t trmal boundar lar ticknss dcrass wit incrasing Pr. Figur 6 sows tat t fluid tmpratur dcrass wit an incras of frqunc paramtr n. Figur 7 sows tat t fluid tmpratur incrass wn tim progrsss. It is sn from Figurs -3 tat t fluid vlocitis for t impulsiv motion of on of t cannl walls is alwas gratr tan t acclratd motion. T rat of at transfr at t cannl walls 0 ar rspctivl 0, and =0 and, and ar givn b (s t Equa- = tions (34) and (35) blow). wr s, s and s 3 ar givn b (3). Numrical rsults of t rat of at transfr at t cannl walls 0 and ar rspctivl 0, and, wic ar prsntd in Tabls -3 for svral valus of Prandtl numbr Pr, tim, π frqunc paramtr n wn n. Tabl sows 4 tat t rat of at transfr 0, dcrass wil t rat of at transfr, incrass wit an incras in Prandtl numbr Pr. Tabl sows tat t rat of at transfr 0, at t wall 0 dcrass wras t rat of at transfr, at t wall incrass wn tim progrsss. It is sn from Tabl 3 tat t rat of at transfrs 0, and, dcras wit an incras in frqunc paramtr n. For t impulsiv motion, t non-dimnsional sar strss at t wall 0 is givn b (s t Equations (36) and (37) blow). s in in π ks R inpr cot R inpr R inpr cot R inpr for Pr Pr k s n 0, ks s3 in i Prcot i in i cot i π 3 R n R n R n R n for Pr k s3 n (34) s in in π k ks R inpr cosc R inpr R inpr cosc R inpr for Pr Pr k s n, s3 i i k 3 n i Pr cosc i n ks R n R n R i n cosc R i n π f or Pr k s3 n (35)

8 6 s k acot a π k s Pr i in Gr a incot a in R inpr cot RinPr n in a incot a in R inpr cot R inpr in s s s s 4k π k s s n Pr ss n F i for Pr 0 s k acot a π k s Gr in aincot ain Rincot Rin R a in aincot ain Rincot Rin s s3 s s3 4k π k s n s3 n for Pr, wr s, s and s 3 ar givn b (3). For t acclratd motion, t non-dimnsional sar strss at t wall 0 is as s k sin acos a a acot a π k s asin a in Gr aincot ain RinPr cot RinPr Pr in in aincot ain RinPr cot RinPr in wr s s s s 4k π k s s n Pr ss n F i for Pr 0 s k sin acos a a acot a π k s asin a s, s and s 3 ar givn b (3). Gr in aincot ain Rincot Rin R a in aincot ain Rincot Rin s s3 s s3 4k π k s n s3 n for Pr, (36) (37)

9 7 Figur 8. Primar vlocit for diffrnt n wn m0.5, R, Pr 0.7, Gr 5 and 0.. u Figur. Scondar vlocit v for diffrnt Gr wn m 0.5, R, Pr 0.7, n and 0.. Figur 9. Scondar vlocit for diffrnt n wn m0.5, R, Pr 0.7, Gr 5 and 0.. v Figur. Primar vlocit u for diffrnt m 0.5, R, Pr 0.7, n and M 5. wn Figur 0. Primar vlocit u for diffrnt Gr wn Figur 3. Scondar vlocit v for diffrnt wn m0.5, R, Pr 0.7, n and 0.. m 0.5, R, Pr 0.7, n and M 5.

10 8 Tabl. Rat of at transfr 0, and, wn n and ,, R \Pr R \ Tabl. Rat of at transfr 0, and, wn n and Pr. 0,, Tabl 3. Rat of at transfr 0, and, wn 0.5 and Pr. 0,, R\ n Figur 4. Tmpratur for diffrnt R wn Pr 0.7, n and 0.. Figur 5. Tmpratur for diffrnt R, n and 0.. Pr wn Numrical rsults of t non-dimnsional sar strsss and at t wall 0 du to t primar and t scondar flows ar plottd in Figurs 8-7 against Hall paramtr m for svral valus of radiation param tr R, Pran dtl num br Pr, frqunc param tr n, Gr asof numbr Gr and tim wn π M 5 and n. Figurs 8 and 9 sow tat t 4 sar strss du t o t primar flow and t magnitud of t sar strss du to t scondar flow at t wall 0 dcras for t impulsiv as wll as acclratd motions of on of t cannl

11 9 Figur 6. Tmpratur for diffrnt n for Pr 0.7, R and 0.. Figur 9. Sar strss Pr = 0.7, n =, Gr = 5 and 0.. for diffrnt R wn Figur 7. Tmpratur for diffrnt tim for Pr 0.7, n and R. Figur 0. Sar strss for diffrnt Pr wn R=, n =,Gr=5 and 0.. Figur 8. Sar strss Gr 5, Pr 0.7, n and 0.. for diffrnt R wn Figur. Sar strss for diffrnt Pr wn R, n, 0. and Gr 5.

12 0 Figur. Sar strss R, Pr 0.7, Gr 5 and 0.. for diffrnt n wn Figur 5. Sar strss for diffrnt Gr wn 0., Pr 0.7, n and R. Figur 3. Sar strss R, 0., Pr 0.7 and Gr 5. for diffrnt n wn Figur 6. Sar strss R, Pr 0.7, n and Gr 5. for diffrnt tim w n Figur 4. Sar strss for diffrnt Gr wn R, Pr 0.7, n and 0.. Figur 7. Sar strss for diffrnt tim R, Pr 0.7, n and Gr 5. wn

13 walls wit an incras in radiation paramtr R. It is sn from Figurs 0 and tat for t impulsiv and acclratd motions of on of t cannl walls t sar strss and t magnitud of t sar strss incras wit an incras of Prandtl numbr Pr. Figurs and 3 sow tat t sar strss and t magni tud of t sar strss dcras wit an incras in frqunc paramtr n for t impulsiv as wll as acclratd motions of on of t cannl walls. An incras of Grasof numbr Gr lads to incras in t sar strss wil t magn itud of t sar strss dcrass for m 0.3 and incrass for m 0.3 for bot t impulsiv and acclratd motions of on of t cannl walls sow in Figurs 4 and 5. Figurs 6 and 7 sow tat for bot t impulsiv and acclratd motions of on of t cannl walls, t magnitud of t sar strss dcrass wras t magnitud of t sar strss incras s wit an incras in tim. 4. Conclusion T combind ffcts of Hall currnt and radiation on t unstad MHD fr convctiv flow in a vrtical cannl wit an oscillator wall tmpratur av bn studid. Radiation as a rtrding influnc on t fluid vlocit componnts for bot t impulsiv as wll as acclratd motions of on of t cannl walls. Hall currnts acclrats t fluid vlcit componnts for t impulsiv as wll as acclratd motions of on of t cannl walls. In t prnc of radiation t fluid tmpratur dcrass. Furtr, t sar strss and t absolut valu of t sar strss at t wall 0 dcr as wit an incras in radiation paramtr R for t impulsiv as wll as acclratd motions of on of t cannl walls. T rat of at transfrs 0, and, incras wit an incras in radiation paramtr R. REFERENCES [] H. Sato, T Hall Effcts in t Viscous Flow of Ionizd Gas btwn Paralll Plats undr Transvrs Magntic Fild, Journal of Psical Socit of Japan, Vol. 6, 96, pp doi:0.43/jpsj.6.47 [] O. Miatak and T. Fujii, Fr Convction Hat Transfr btwn Vrtical Plats On Plat Isotrmall Hatd and Otr Trmall Insulatd, Hat Transfr Japans Rsarc, Vol., 97, pp [3] H. Tanaka, O. Miatak, T. Fujii and M. Fujii, Natural Convction Hat Transfr btwn Vrtical Paralll Plats On Plat wit a Uniform Hat Flu and t Otr Trmall Insulatd, Hat Transfr Japans Rsarc, Vol., 973, pp [4] P. S. Gupta and A. S. Gupta, Radiation Effct on Hdromantic Convction in a Vrtical Cannl, Intrna- tional Journal of Hat Mass Transfr, Vol. 7, No., 974, pp doi:0.06/ (74) [5] N. Datta and R. N. Jana, Hall Effcts on Hdromagntic Convctiv Flow troug a Cannl wit Conducting Walls, Intrnationa Journal of Enginring Scinc, Vol. 5, No. 9-0, 977, pp doi:0.06/000-75(77) [6] A. R. Bstman and S. A. Adjpong, Unstad Hdromagntic Fr Convction Flow wit Radiativ Hat Transfr in a Rotating Fluid, Spac Scinc, Vol. 43, No., 988, pp doi:0.007/bf [7] H. M. Josi, Transint Effcts in Natural Convction Cooling of Vrtical Paralll Plats, Intrnational Communication Hat and Mass Transfr, Vol. 5, No., 988, pp doi:0.06/ (88) [8] A. K. Sing, Natural Convction in Unstad Coutt Motion, Dfns Scinc Journal, Vol. 38, No., 988, pp [9] A. K. Sing, H. K. Golami and V. M. Soundalgkar, Transint Fr Convction F low btwn Two Vrtical Paralll Plats, Hat and Mass Transfr, Vol. 3, No. 5, 996, pp doi:0.007/bf [0] B. K. Ja, Natural Convction in Unstad MHD Coutt Flow, Hat and Mass Transfr, Vol. 37, No. 4-5, 00, pp doi:0.007/pl [] M. Naraari, S. Srnad and V. M. Soundalgkar, Transint Fr Convction Flow btwn Long Vrtical Paralll Plats wit Constant Hat Flu at On Boundar, Journal of Trmopsics and Aromcanics, Vol. 9, No., 00, pp [] B. K. Ja, A. K. Sing and H. S. Takar, Transint Fr Convction Flow in a Vrtical Cannl Du to Smmtric Hating, Intrnational Journal of Applid Mcical Enginring, Vol. 8, No. 3, 003, pp [3] A. K. Sing and T. Paul, Transint Natural Convction btwn Two Vrtical Walls Hatd/Coold Asmtricall, Intrnational Journal of Applid Mcanical En- ginring, Vol., No., 006, pp [4] D. C. Sanal and A. Adikari, Effcts of Rad iation on MHD Vrtical C annl Flow, Bulltin of Cal cutta Matmatical Socit, Vol. 98, No. 5, 006, pp [5] P. Mbin, Radiation Effcts on MHD Coutt Flow wit Hat Transfr btwn Two Paralll Plats, Global Journal of Pur and Applid Matmatics, Vol. 3, No., 007, pp. -. [6] T. Grosan and I. Pop, Trmal Radiation Effct on Full Dvlopd Mid Convction Flow in a Vrtical Cannl, Tcnisc Mcanik, Vol. 7, No., 007, pp [7] M. Guria and R. N. Jana, Hall Effcts on t Hdromagntic Convctiv Flow troug a Rotating Cannl undr Gnral Wall Conditions, Magntodrodnamics, Vol. 43, No. 3, 007, pp [8] B. K. Ja and A. O. Ajibad, Unstad Fr Convctiv Coutt Flow Of Hat Gnrating/Absorbing Fluid, Intrnational Journal of Enrg and Tcnolog, Vol., No., 00, pp. -9. [9] M. Naraari, Effcts of Trmal Radiation and Fr

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