II. FORMULATION OF THE PROBLEM

Transcription

1 Intnational Jounal of Engining Scinc Invntion ISSN (Onlin): ISSN (Pint): Volum 6 Issu 9 Sptmb 7 PP. - Study of Unstady Magntohydodynamic Flow of n Incompssibl Viscous Elctically Conducting Fluid Boundd By Two Non-Conducting Vtical Plats in Psnc of Inclind Magntic Fild D.majyoti Goswami* Minmoy Goswami* D.Kishna Gopal Singha # *Kazianga Univsity ssam India #Kaanga Gils H.S.S. ssam India. Cosponding utho: D.majyoti Goswami bstact: Th psnt aticl addsss th ffct of unstady MHD flow of an incompssibl viscous fluid boundd by two no-conducting paalll plats placd vtically in psnc of unifom inclind magntic fild. On of th plats is considd to b in motion with constant vlocity whas th oth plat is adiabatic. Using tansfomation associatd with dcay facto w hav dducd a st of odinay diffntial quations which a solvd analytically fo th flow fild tmpatu fild and inducd magntic fild fo diffnt valus of MHD flow paamts. Th sults obtaind fo vlocity tmpatu and inducd magntic fild a discussd and analyzd gaphically. Kywods: MHD fluid flow Hat tansf unstady Pandtl numb adiabatic analytical solution Dat of Submission: Dat of accptanc: I. INTRODUCTION Th study of MHD flow und th action of a unifom tansvs magntic fild has gnatd much intst in cnt yas in viw of its numous industial applications such as th MHD gnatos and plasma MHD acclatos pumps flowmts ptolum industy puification of cud oil lctostatic pcipitation polym tchnology tc. Th consqunt ffct of th psnc of solid paticls on th pfomanc of such dvics has ld to th studis of paticulat suspnsions in conducting fluids in th psnc of xtnally applid magntic fild. Sth and Ghosh [] considd th unstady hydomagntic flow of a viscous incompssibl lctically conducting fluid in a otating channl und th influnc of a piodic pssu gadint and of unifom magntic fild which was inclind with th axis of otation. n analytical solution to th poblm of stady and unstady hydomagntic flow of viscous incompssibl lctically conducting fluid und th influnc of constant and piodic pssu gadint in psnc of inclind magntic fild has bn obtaind xactly by Ghosh [] to study th ffct of slowly otating systms with low fquncy of oscillation whn th conductivity of th fluid is low and th applid magntic fild is wak. Yang and Yu [3] hav invstigatd th ntanc poblm of convctiv magntohydodynamic channl flow btwn two paalll plats subjctd simultanously to an axial tmpatu gadint and a pssu gadint. Thy hav also considd both th cass of constant hat flux and constant wall tmpatu. Th unstady magntohydodynamic flow of an lctically conducting viscous incompssibl non- Nwtonian Bingham fluid boundd by two paalll non-conducting poous plats with hat tansf considing th Hall Effct has bn studid by ttia and hmd [4]. Bokakati and Bhaali [5] hav studid th poblm of flow and hat tansf btwn two infinit hoizontal paalll poous plats wh th low plat is a sttching sht and th upp on is a poous solid plat in psnc of a tansvs magntic fild. Th hat tansf in an axisymmtic flow btwn two paalll poous disks und th ffct of a tansvs magntic fild was studid by Bhaali and Bokakati [6]. Shih-I-Pai [7] studid an unstady motion of an infinit flat insulatd plat st impulsivly into th unifom motion with vlocity in its own plan in th psnc of a tansvs unifom magntic fild. Th poblm of combind f and focd convctiv magntohydodynamic flow in a vtical channl has bn studid by Umavathi and Malashtty [8]. Thy had also considd th ffct of viscous and ohmic dissipations. It has bn obsvd that th viscous dissipation nhancs th flow vsal in th cas of downwad flow whil it countd th flow in th cas of upwad flow. Singha and Dka[9] considd th poblm of two phas MHD flow and hat tansf poblm in a hoizontal channl. Jodán [] has invstigatd th tansint f convction MHD flow of a dissipativ fluid along a smi-infinit vtical plat with mass tansf th sufac of which is xposd to a constant hat flux. In his pap h also studid th influncs of th viscous dissipation buoyancy atio paamt Schmidt numb and magntic paamt on hat and mass tansf and on th tim ndd to ach th stady-stat. Pag

2 Study of Unstady Magntohy dodynamic Flow of n Incompssibl Th ffcts of hat tansf on unstady hydomagntic flow in a paalll-plat channl of an lctically conducting viscous incompssibl fluid hav bn invstigatd by Singha []. H found that vlocity distibution incass na th plats and thn dcass vy slowly at th cntal potion btwn th two plats. Th pincipal numical sults psntd in his wok showd that th flow fild is appciably influncd by th applid magntic fild. Singha[] invstigatd th ffct of tansvsly applid xtnal magntic fild on th unstady lamina flow of an incompssibl viscous lctically conducting fluid in a channl of two hoizontal hatd plats. Hnc th psnt pap invstigats th ffct of th unstady MHD flow of an incompssibl viscous lctically conducting fluid boundd by two non-conducting paalll plats vtically in psnc of unifom inclind magntic fild. on of which is at st oth moving in its own plan with a vlocity u. Th analytical solutions fo th fluid vlocitis magntic fild and tmpatu distibutions a obtaind. Th ffcts of vaious paamts on th flow and hat tansf a shown gaphically. II. FORMULTION OF THE PROBLEM Th unstady lamina flow of an incompssibl viscous lctically conducting fluid btwn two nonconducting paalll plats placd vtically at a distanc h apat is considd in psnc of unifom inclind magntic fild. Th flow is assumd to b along th X -axis paalll to vtical diction though th cntal lin of th channl and Y -axis is nomal to it. Figu: Gomtical configuation Th plats of th channl a at y h and that th lativ vlocity btwn th two plats is u and also th is no pssu gadint in th flow fild. n xtnal unifom magntic fild of stngth B maks an angl with th positiv diction of X -axis which inducs a magntic fild B ( y ) maks also an angl to th f stam vlocity. Th plat at y h is maintaind at tmpatu T whil th oth plat y h is kpt at tmpatu T ( T T ) and th plats a lctically non-conducting. Th componnts of th vlocitis and th magntic fild a givn as follows: V u v w u y t B ( ) = x y z B B B = co s B ( y t ) co s (9 ) B = co s B y t sin B { ( ) } = { B ( y t ) B } wh p = constant pssu gadint in th flow diction cos and ' t ' is th tim. 3 Pag

3 Study of Unstady Magntohy dodynamic Flow of n Incompssibl III. SSUMPTIONS In od to div th govning quations of th poblm th following assumptions a mad. (i) Th fluid is finitly conducting and th viscous dissipation and th Joul hat a nglctd. (ii) Hall ffct and polaization ffct a ngligibl. 4 Govning quations Th govning quations of th poblm und th abov conditions a as follows: V. () V t B t c p V V p V J B Z ( V B ) B. (3) T d d T. (4) t d y d y H th thid tm in th ight hand sid of Eq. () is th magntic body foc and dnsity du to th magntic fild dfind by () J is th lctic cunt J ( E V B ) (5) Z is th foc du to buoyancy Z g ( T T ) (6) It has bn takn that E. That is in th absnc of convction outsid th bounday lay B B to E. and C u l B J thn (5) lads Th fluid motion stats fom st at t and th no-slip condition at th plats implis that th fluid vlocity has nith a z-no an x-componnt at y h. Using th vlocity and magntic fild distibution as statd abov th Eq. ()- (4) a as followd: u u B ( ) u g ( T T ) t y (7) B u B B t y y (8) T T t c y p T T (9) t y Lt us assum that th oth plat is adiabatic i.. thmally insulatd walls thn th bounday conditions a 4 Pag

4 Study of Unstady Magntohy dodynamic Flow of n Incompssibl t : u B B T T t : u u B B T T a t y h T t : u u B B a t y h. y Intoducing th following non-dimnsional quantitis: * u u * y y * t t u b B T T T. u h h B T T In tms of th abov non-dimnsional vaiabls and paamts th basic quations (7)-(9) tak th fom u G ( ) a u H R u T t R y R b u b t y R R P y m T T t P y Th astisks hav bn doppd with th undstanding that all th quantitis a now dimnsionlss. Fo lation () th bounday conditions () bcoms t : u b T t : u b T a t y= + T t : u b a t y=. y In od to solv Eqs. () - (4) w consid n t n t n t u f ( y ) b g ( y ) a n d T F ( y ) wh n is th dcay constant. Substituting (6) in Eqs. () - (4) w gt G f ''( y ) R H R ( ) n f ( y ) F ( y ) R and (7) a g ''( y ) ( n R R P ) g ( y ) R R P f '( y ) m m F ''( y ) n P F ( y ) (9) Fo lation (6) th bounday conditions (5) again bcoms t : f g F t : f n t n t n t g F a t y n t n t F t : f g a t y. y () () () (3) (4) (5) (6) (8) () Th solutions of Eqs. (7) - (9) with th hlp of th bounday conditions () and substituting in th lations (6) a 5 Pag

5 Study of Unstady Magntohy dodynamic Flow of n Incompssibl y y u ( y ) co s[( y ) ] C C () y y y b ( y ) ( sin[( y ) ] ( y ( sin[ y ] C co s[ y ] C ))) () T c o s[( y ) ] 5 (3) c o s[ ] 5 wh a R H R ( ) n G R n R R P 3 m R R P 4 m n P C o s c [ ] 5 ( )( ) ( ) C 4 C S c [ ] ( ) C C Pag

6 Study of Unstady Magntohy dodynamic Flow of n Incompssibl C C ( sin[ ] ) ( sin[ ] ) c o s c [ ] sin[ ] ( ) C 3 9 c [ ] [ ] s c o s ( ) C ( c o s[ ] ) C 5 4 ( c o s[ ] ) C 6 4 C C 8 7 c o s c [ ] c o s[ ] s in[ ] s in h[ ]( ) sin[ ] s c [ ] [ ]( ) 3 c o s IV. RESULTS ND DISCUSSION Numical solutions fo th Eqs. () - (3) a obtaind fo diffnt valus of wh = cos which vais as = ll plotting fo such cass a caid out by using MTHEMTIC. In fig. th vaiations in vlocity fild du to vaiations in magmatic fild inclination with flow diction hav bn plottd. Whn th xtnal fild is mo inclind to flow diction th Hatmann ffct is considably ducd. Na is na zo th flow fild bhavs lik an odinay Coutt flow. With th incas of angl of inclination of magntic fild with flow diction th influnc of Hatmann ffct bcom mo and mo cla. In fig.3 it has bn obsvd that with th incas of Gashof numb th appa shap changs in vlocity fild. Whn th Gashof numb is incasd byond th citical valu G= vs flow phnomna na plat which is in motion with vlocity v=- is obsvd. W can conclud that fo small Gashof numb gims vlocity fild changs almost linaly. It indicats that th dominanc of buoyancy foc in th flow may hav nonlina influnc on flow fild. Plottings in fig.4 indicat that gat th angl of inclination of imposd fild with flow diction mo th cosponding changs in vlocity fild du to incas in Hatmann ffct. Th incasd changd in vlocity fild intnsifis th inducd fild. Fig. 5 shows ffct of Gashof numb on inducd magntic fild. With th incas in valu of Gashof numb it is obsvd that intnsity of inducd fild in flow in incasd and it is mo cla in th cntal gion of channl whas viscous ffct out fom th walls diminishs th diffnc in vlocity fild. Th lativ dominanc of bouncy foc and intia foc ov viscous foc in th flow may incas th intnsity of inducd magntic fild. Fom th plotting of Fig.6 it has bn obsvd that with th incas magntic Ryonld numb intnsity of inducd magntic fild is incasd. Fild intnsity in cntal flow gion of th channl is much high than na th bounday plats. This incas in intnsity in cntal flow gion of th channl is much 7 Pag

7 Study of Unstady Magntohy dodynamic Flow of n Incompssibl apid with incas in magntic Rynold numb. This indicats that stong magntic convction ffct ov magntic diffusion pocss may sult in incas in inducd fild intnsity in such a flow. In fig.7 it is obsvd that tmpatu fild changs much apidly with incas of Pandtl numb. In this vtical channl flow th tmpatu vaiation acoss th bounday plats is mo and mo apid with incas of momntum diffusion pocss in th flow along with dcas in hat diffusion pocss in th flow. Nomnclatu h half width btwn paalll plats channl (m) B xtnal unifom magntic fild (T) thmal conductivity (Wm - K - ) H magntic Hatmann numb a R Rynolds numb P Pandtl numb G Gashoff numb u h 3 B g h T T P Pclt numb P = P R R Rayligh numb a 3 g h T u R Magntic Rynolds numb m E lctic fild intnsity (NC - ) u f stam vlocity (ms - ) T tmpatu of th low plat (K) T tmpatu of th upp plat (K) t tim (s) c spcific hat at constant pssu ( J. k g p g acclation du to gavity ( ms ) Gk symbols thmal diffusitivity c lctical conductivity ( m ) co-fficint of viscosity ( k g m 3 dnsity of th fluid ( k g m ) kinmatic viscosity ( m s ) p s ) co-fficint of thmal xpansion of th fluid ( K ) pmability of th mdium magntic diffusitivity m Supscipt * non-dimnsional vaiabls dfind in Eq. () K ) 8 Pag

8 Study of Unstady Magntohy dodynamic Flow of n Incompssibl Rfncs []. G. S. Sth and S. K. Ghosh Unstady hydomagntic flow in a otating channl in th psnc of inclind magntic fild Intnational Jounal of Engining Scinc 4 (7) (986) []. S. K. Ghosh not on stady and unstady hydomagntic flow in a otating channl in th psnc of inclind magntic fild Intnational Jounal of Engining Scinc 9 (3.8) (99) 3-6. [3]. H. K. Yang and C. P. Yu Combind focd and f convction MHD channl flow in ntanc gion Intnational Jounal of Hat and Mass Tansf Vol: 7 No. 6 [974] pp [4]. Hazm li ttia and Mohamd Eissa Sayd-hmd Hall ffct on unstady MHD Coutt flow and hat tansf of a Bingham fluid with suction and injction pplid Mathmatical Modlling Vol: 8 No. [4] pp 7-45 [5].. Bhaali and. K. Bokakati Magntohydodynamical Flow with Hall Effcts Past an Infinit Poous Plat and Hat Tansf Jounal of th Physical Socity of Japan Vol: 49 No. 5 [98] pp [6].. Bhaali and. K. Bokakati Magntohydodynamic F Convction Flow with Hall Effcts past a Poous Vtical Plat Jounal of th Physical Socity of Japan Vol: 5 No. [983] pp-6-7. [7]. S. I Pai Magntogasdynamics and Plasma Physics Sping Vlag [96] p-54. [8]. J.C. Umavathi and M.S. Malashtty Magntohydodynamic mixd convction in a vtical channl Intnational Jounal of on- Lina Mchanics Vol: 4 No. [5] pp. 9-. [9]. K.G. Singha and P.N. Dka Magntohydodynamic hat tansf in two-phas flow in psnc of unifom inclind magntic fild Bull.Cal. Math. Soc. Vol. ()[9 pp []. Joaquín Zuco Jodán Numical study of an unstady f convctiv magntohydodynamic flow of a dissipativ fluid along a vtical plat subjct to a constant hat flux Intnational Jounal of Engining Scinc Vol: 44 No. 8-9[6] pp []. K. G. Singha Th ffct of hat tansf on unstady hydomagntic flow in a paalll plats channl of an lctically conducting viscous incompssibl fluid Int. J. Fluid Mch. Rsach vol 35 issu 8 pp. 7-86[onlin]. vailabl: []. K.G.Singha nalytical solution to th poblm of MHD f convctiv flow of an lctically conducting fluid btwn two hatd paalll plats in th psnc of an inducd magntic fild Intnational Jounal of pplid Mathmatics and Computation Volum (4)pp Pag

9 Study of Unstady Magntohy dodynamic Flow of n Incompssibl Intnational Jounal of Engining Scinc Invntion (IJESI) is UGC appovd Jounal with Sl. No. 38 Jounal no D.majyoti Goswami. Study of Unstady Magntohy dodynamic Flow of n Incompssibl Viscous Elctically Conducting Fluid Boundd By Two Non-Conducting Vtical Plats in Psnc of Inclind Magntic Fild. Intnational Jounal of Engining Scinc Invntion(IJESI) vol. 6 no. 9 7 pp.. Pag