Unsteady Free Convective Flow of a Temperature Varying Electrically Conducting Fluid

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Beverley Tyler
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1 Procdings of th World ongrss on Enginring 9 Vol II WE 9 July - 9 London U.K. Unstady Fr onvctiv Flow of a Tpratur Varying Elctrically onducting Fluid Krishna Gopal Singha and P. N. Dka bstract n unstady viscous incoprssibl fr convctiv flow of an lctrically conducting fluid btwn two hatd vrtical paralll plats is considrd in prsnc of an inducd agntic fild applid transvrsly to th flow. ssuing that th agntic fild inducs a fild along th lins of otion which varis transvrsly to th flow and th fluid tpratur changing with ti analytical solutions for vlocity inducd agntic fild and tpratur distributions ar obtaind for sall and larg agntic Rynolds nubrs. Th skin friction on th two plats ar obtaind and plottd graphically. Th probl has also bn solvd for throtric cas i.. whn th lowr plat is adiabatic. Indx Trs Hartann nubr inducd agntic fild agntic Rynolds nubr skin friction unstady fr convctiv flow. I. INTRODUTION Borkakati and Srivastava[] invstigatd fr and forcd convction and MHD flow. In a fluid th variation of tpratur causs variation of dnsity. This in turn raiss forc of buoyancy which govrns th fluid otion. This typ of unstady fluid otion undr th action of unifor agntic fild applid xtrnally rducs th hat transfr and th skin friction considrably. This procss of rduction of hat transfr and skin friction of th fluid otion has various nginring applications such as nuclar ractor powr transforation tc. Borkakati and hakraborty[] invstigatd th natur and bhaviour of a viscous incoprssibl lctrically conducting fluid ovr a flat plat which is oving with a unifor spd in a quiscnt fluid in prsnc of a unifor agntic fild. In thir conclusion thy hav found that for an incoprssibl fluid both th fluid vlocity and tpratur gradually dcrass with th incras of viscosity paratr. Elbashbshy[] studid hat and ass transfr in th sa probl in prsnc of variabl transvrs agntic fild. Th unstady probl in a channl was studid nurically by ttia[4] with tpratur dpndnc viscosity. H also considrd stady stat solution for vlocity and tpratur. In his study h analyzd th ffct of viscosity paratr dfind as ratio of viscosity of th fluid at two diffrnt tpraturs. In th rcnt yars ttia[5] studid an unstady agntohydrodynaic flow and hat transfr Manuscript rcivd January 7 9. Krishna Gopal Singha is with th Dpartnt of Mathatics Karanga Girls Highr Scondary School Jorhat -75 India (-ail : kgsingha@yahoo.co. P. N. Dka is with Dpartnt of Mathatics Dibrugarh UnivrsityDibrugarh764 India (-ail: pndka@yahoo.co.in. probl of dusty fluid btwn two paralll plats with variabl physical proprtis. Takhar[6] considrd th ffct of radiation on fr convction flow along siinfinit vrtical plat in prsnc of transvrs agntic fild. Vry rcntly Singha[7] invstigatd th ffct of hat transfr on unstady hydroagntic flow in a paralll-plat channl of an lctrically conducting viscous incoprssibl fluid. H found that vlocity distribution incrass nar th plats and thn dcrass vry slowly at th cntral portion btwn th two plats. Th principal nurical rsults prsntd in his work showd that th flow fild is apprciably influncd by th applid agntic fild. Gourla and Katoch[] discussd an unstady fr convction flow through th vrtical paralll plats in th prsnc of unifor agntic fild. In this papr w ar invstigating th fully dvlopd fr convction lainar flow of an incoprssibl viscous lctrically conducting fluid btwn two vrtical paralll plats in th prsnc of a unifor inducd agntic fild applid transvrsly to th flow. This inducs a fild along th lins of otion which varis transvrsly to th flow. Th tpratur of th fluid otion is assud to b changing with ti. Th analytical solutions for vlocity inducd agntic fild and th tpratur distributions ar obtaind for sall and larg agntic Rynolds nubrs R. Th skin frictions at th two plats ar obtaind for diffrnt agntic fild paratrs and ar plottd graphically. Th rat of hat transfr ar also obtaind and ar plottd graphically. Th probl has also bn solvd for throtric cas i.. whn th lowr plat is adiabatic. II. FORMULTION OF THE PROBLEM W ar considring an unstady lainar convctiv flow of a viscous incoprssibl lctrically conducting fluid btwn two vrtical paralll plats. Lt X -axis b takn along vrtically upward dirction through th cntral lin of th channl and Y -axis is takn prpndicular to th X -axis. Th plats of th channl ar at y = ± h.th unifor agntic fild B is applid paralll to Y -axis and th inducd fild so producd is along X -axis that varis along Y -axis. Th vlocity and V = u( y and agntic fild distributions ar [ ] B = B( y B applid and inducd agntic fild rspctivly. B and (y B ar ISBN: WE 9

2 Procdings of th World ongrss on Enginring 9 Vol II WE 9 July - 9 London U.K. Ra ( β gh T / ( να R αµ σ α = ρ = is th Rayligh nubr = is th Magntic Rynolds nubr k / ( p α is th thral diffusivity k is th thral conductivity ν = / ( σµ is th agntic diffusivity ν = µ / ρ is th kintic viscosity σ is th lctrical conductivity ρ is th fluid dnsity µ is th prability of th diu and µ is th co-fficint of viscosity. In ordr to driv th govrning quations of th probl th following assuptions ar ad Th non-dinsional boundary condition bcos t = : u = b = T = at y = ±h t > : u = b = B' / B T = nt at y = ±h (5 (i th fluid is finitly conducting and th viscous dissipation and th Joul hat ar nglctd (ii Hall ffct and polarization ffct ar ngligibl (iii initially (i.. at ti t = th plats and th fluid ar at zro tpratur (i.. T = and thr is no flow within th channl (iv at ti t > th tpratur of th plat ( y = ± h nt chang according to T = T ( whr T is a constant tpratur and n is a ral nubr dnoting th dcay factor and (v th plats ar considrd to b infinit and all th physical quantitis ar functions of y and t only. III. GOVERNING EQUTIONS Undr th abov assuptions th non-dinsional govrning quations ar as follows: u u Ra M b = + T t + y Pr R R R P r y ( b u b R = t y R P r y ( T T = t P r y ( Th following non-dinsional trs ar usd: t = νt / h b = B / B y = y / h u = u / u whr u T = ( T T / T (4 = gt h ν ( β / Th astrisks hav bn droppd with th undrstanding that all th quantitis ar now dinsionlss. whr P ν / α r M = ( B h σ / ( ρν is th Hartann nubr = is th Prandtl nubr R ( u h / ν = is th Rynolds nubr IV. SOLUTIONS To solv ( to ( subjct to th boundary condition (5 w apply th transforation of variabls u = f ( y -nt b = g( y -nt and T = φ( y -nt Substituting (6 in (- w hav fro ( f Ra M g + nf + φ + =. (7 y Pr R R R P r y Fro ( g f + ( nr P g ( R R P r + r = y y Fro ( φ + ( np r φ = y Th corrsponding boundary conditions ar : for t = : f = g = φ = at y = ± nt nt ( for t > : f = g = ( B' / B φ = at y = ± Th solutions of (7-9 subjct to th boundary conditions ( ar. (6 ( (9 cos( ay φ ( y = ( cos a ( α α ( α α ( cosh( β sinh( β ( cosh( β sinh( β f ( y = cosh( y sinh( y + cosh( y + sinh( y + y y + y + y 4 cos( ay ( g( y = sin y + cos y + ( ( a sin( ay ( ( ( α + ( β + + ( a ( α(cosh( α y α β + α + α sinh( y( ( (cosh( y sinh( y (cosh( y sinh( y( (cosh( y β y α 4 + β β β α + β β + sinh( ( + ( ISBN: WE 9

3 Procdings of th World ongrss on Enginring 9 Vol II WE 9 July - 9 London U.K. whr a = np r R / ( P R = a r = / ( r M R R P = nr P r n + 4 = R R P sc a 5 = r a = = 5 4 / ( = 7 a + n a 6 α 4 6 n = + 6 / 4 β = 6 + n + 6 / 9 = ( a ( α + ( β ( β + + β 6 cos a ( a ( a cosh α( α β ( α + β ( β + + β 6 cos a ( a ( a cosh α( α β ( α + β + ( + ( 4 + α α 7 6 cos a a ( a cosh β ( α β ( α + β + ( + ( 4 + α α 7 6 cos a a ( a 4 cosh β ( α β ( α + β (cos c ( 4 ( a sinh a ( α + 9 ( β + + ( a ( α sinh α( β + ( + β sinh β ( α + ( + 4 sc( {( a ( α cosh α ( β ( + β cosh β ( α + ( } whr 5 = sc[a] 9 α 47 + α4 = α α 4 47 = = ( a ( + α ( + β β a + ( a + β 6 5 = ( α β ( α + β α a + ( a + α 6 = ( + = ( 4 ( + coth[ a]sin[ a] + cos[ a] tanh[ α ] 7 (cosh[ α ] + sinh[ α ] 5 (cos[ a] sch[ α] cosch[ α]sin[ a] 4 (cos[ a] sc h[ β ] + cosch[ β ]sin[ a] cosch[β ] 5 ( cosh[ β ]sin[ a] cos[ a] sinh[ β ] 6 5 = (cosc[ ]( asin[a]( α + ( β + 4 ( a (sinh[ α ]( β sinh[ β ]( α + = (sc[ ]( asin[a]( α + ( β + 4 ( a (cosh[ α]( β + + cosh[ β ]( α + VI. SKIN FRITION Th skin friction at th plats y = ± is dfind as du τ µ dy ± Th non-dinsional skin friction aftr roving th astrisks taks th for: ( V. THERMOMETRI SE Lt us assu that th lowr plat is adiabatic i.. th plat y is adiabatic (thrally insulatd wall. Th boundary conditions in th throtric cas ar f = g = φ = at y = + f = g = φ' = at y = - (4 Th solutions of (7-9 subjct to th boundary conditions (4 ar givn by cos{ a( + y} φt ( y = (5 cos( a 7 ( α α ( α α 9 ( cosh( β sinh( β ( cosh( β sinh( β cos( a ( + y (6 f ( y = cosh( y sinh( y + cosh( y + sinh( y t + y y + y + y asin[ a( + y] 4 gt ( y = sin y cos y + + a sinh[ α y] 9 + cosh[ α y] sinh[ β y] + cosh[ β y] α + β + (7 µβ gt h u τ ν y y =± using (6w gt µβ gt h df nt τ ν dy y=± VII. SKIN FRITION FOR THERMOMETRI SE Th skin friction at th plats y = ± is dfind as du τ µ dy ± Th skin friction in th non-dinsional for for throtric cas is givn by µβgt h dft nt τt ν dy y=± (9 ( ( ( ISBN: WE 9

4 Procdings of th World ongrss on Enginring 9 Vol II WE 9 July - 9 London U.K. VIII. RESULTS ND DISUSSION Th vlocity distribution f against th distanc fro th fixd plats y ar plottd at diffrnt valus of agntic Hartann nubr ( M for sall and larg agntic Rynolds nubr ( R in th Fig. (a to Fig. (d. On th basis of sa considration Fig. (a to Fig. (d and Fig. (a to Fig. (b ar plottd. Th following fluid paratrs ar usd: R a =. R =. P r =.7 n =.. ll ths plotting ar don by using MTHEMTI V.. ISBN: WE 9

5 Procdings of th World ongrss on Enginring 9 Vol II WE 9 July - 9 London U.K. Th vlocity and inducd agntic fild distributions in throdynaic cas ar shown in Fig.4(a-d to Fig.6(a-b. ISBN: WE 9

6 Procdings of th World ongrss on Enginring 9 Vol II WE 9 July - 9 London U.K. Furthr it is also obsrvd that vlocitis at th cntral plan of th channl in both th cass ar axiu but opposit in dirction. In Fig. (d it is obsrvd that in th nonthrotric cas th vlocity gradually dcrass with th incras of R. But in Fig. 4(d it is obsrvd that in th throtric cas th vlocity gradually incrass with th incras of R. In Fig. (a-d and Fig.5(a-d th inducd agntic fild strngth ar plottd against distanc fro th plats at point qual distanc fro th plats and at points on th plats. It is obsrvd that th inducd fild strngth in throtric cas ar alost opposit in natur to thos in nonthrotric cas. Th ffct of M and R on th frictional factor at th plats in throtric cas ar alost opposit in natur to thos in non-throtric cas. Fro Fig.(a-d and Fig.6(a-b it is obsrvd that th skin-friction first incrass thn gradually dcrass with th incras of M whil in throtric cas skin-friction gradually dcrass with th incras of M at y = ±. REFERENES []. K. Borkakati and.. Srivastava Studis in fr and forcd convction and MHD flow Ph.D. Thsis Dib. Uni..976 pp []. K. Borkakati and S. hakrabarty Effct of variabl viscosity on lainar convction flow of an lctrically conducting fluid on a unifor agntic fild Thortical and pplid Mchanics vol 7 pp [] E.M.. Elbashhbshy Hat and ass transfr along a vrtical plat in th prsnc of agntic fild Indian J. Pur ppl. Math vol. 7 issu pp.6-6. [4] H.. ttia Transint MHD flow and hat transfr btwn two paralll plats with tpratur dpndnt viscosity Mchanics Rsarch ounications vol. 6 issu 999 pp.5-. [5] H.. ttia Unstady MHD flow and hat transfr of dusty fluid btwn paralll plats with variabl physical proprtis pplid Mathatical Modling vol. 6 issu 9 pp [6] H.S. Takhar R.S.R. Garla and V.M. Soundagkar. Short counication radiation ffcts on MHD fr convction flow of a gas past a si-infinit vrtical plat Int.J. Nu. Mthods for Hat and Fluid vol. 6 issu 996 pp [7] K. G. Singha Th ffct of hat transfr on unstady hydroagntic flow in a paralll plats channl of an lctrically conducting viscous incoprssibl fluid Int. J. Fluid Mch. Rsarch vol 5 issu pp. 7-6[onlin]. vailabl: [] M.G. Gourla and L. Suaha Katoch Unstady fr convction MHD flow btwn hatd vrtical plats Ganita vol. 4 issu 99 pp Th skin frictions at th plats y = ± in th nonthrotric and throtric cass ar shown in Fig.(ab and Fig.6(a-b for diffrnt valus M and R.Th rsults obtaind fro Fig. (a-d to Fig. 6(a-b ar as follows: Fro Fig (a-d and Fig.(a-d 4 it is obsrvd that th vlocity distributions in throtric cas ar alost opposit in natur to thos in non-throtric cas. ISBN: WE 9

Hall Effects on the Unsteady Incompressible MHD Fluid Flow with Slip Conditions and Porous Walls

Hall Effects on the Unsteady Incompressible MHD Fluid Flow with Slip Conditions and Porous Walls Applid Mathmatics and Physics, 3, Vol, No, 3-38 Availabl onlin at http://pubsscipubcom/amp///3 Scinc and Education Publishing DOI:69/amp---3 Hall Effcts on th Unstady Incomprssibl MHD Fluid Flow with Slip