AMSE JOURNALS 2016-Series: Modelling B; Vol. 85; N 1; pp Submitted Oct. 2015; Revised Jan. 17, 2016; Accepted April 15, 2016

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1 AMSE JORNALS 6-Sis: Mdlling B; Vl. 85; N ; pp 8-4 Submittd Ot. 5; Rvisd Jan. 7, 6; Aptd Apil 5, 6 MHD F Cnvtin and Mass Tansf Fl thugh a Vtial Osillaty Pus Plat in a Rtating Pus Mdium ith Hall, In-Slip Cunts and Hat Su Md. Dla Hssain, Md. Abdus Samad and Md. Mahmud Alam * Dpatmnt f Applid Mathmatis, nivsity f Dhaka, Dhaka-, Bangladsh * Mathmatis Disiplin, Khulna nivsity, Khulna-98, Bangladsh * Cspnding auth: (alam_mahmud@yah.m) Abstat In this pap is an invstigatin f unstady sillaty MHD f nvtin, hat and mass tansf fl f an ltially nduting visus inmpssibl fluid thugh a pus mdium alng an infinit vtial pus plat ith th ffts f Hall and in-slip unts, hat su in a tating systm und th influn f St fft. Th plat is assumd t sillat in tim ith nstant fquny s that th slutins f th bunday lay quatins a th sam sillaty typ. Th gvning systm f patial diffntial quatins a tansfmd by usual tansfmatin. Th dimnsinlss quatins a thn slvd numially by finit diffn mthd. With th hlp f gaphs, th ffts f th vaius imptant paamts nting int th pblm n th pimay and snday vlitis, tmpatu and nntatin distibutins ithin th bunday lay a disussd. Als th ffts f th apppiat paamts n th skin fitin ffiint, ats f hat and mass tansf in tms f th Nusslt and Shd numbs a psntd gaphially. Ky ds: MHD, Hall ffts, sillatin, tatin, hat su. Intdutin Th MHD mass tansf fl und th atin f stng magnti fild plays a l in astphysial and gphysial pblms. Hall and in-slip unts a likly t b imptant in fls f labaty plasma. In th study f magnt hyddynami fluid fl in a tating systm has bn mtivatd by sval imptant pblms, suh as maintnan and sula vaiatins f ath s magnti fild, th intnal tatin at f sun, th stutu f tating stas, th plantay and sla dynam pblm, ntifugal mahins t. Cnvtin in pus mdium has appliatins in gthmal ngy vy, il xtatin, thmal ngy stag and fl thugh filting dvis. Th ffts f magnti fild n f nvtin fl a imptant in liquid-mtals, ltlyts and inizd gass. Th thmal physis f hyd magnti pblms ith mass tansf is f intst in p ngining and mtallugy. Th appliatins f th fft f Hall unt n th fluid fl ith vaiabl nntatin hav bn sn in MHD p gnats, 8

2 astphysial and mtlgial studis as ll as in plasma physis. Th Hall Efft is du mly t th sidays magnti f n th difting f hags. In nt yas, th analysis f hydmagnti fl invlving hat and mass tansf in pus mdium has attatd th attntin f many shlas baus f its pssibl appliatins in divs filds f sin and thnlgy suh as silsins, astphysis, gphysis, nula p ats t. In gphysis, it finds its appliatins in th dsign f MHD gnats and alats, undgund at ngy stag systm t. Effts f Hall unt and hat tansf n fl du t a pull f nti tating disk invstigatd by Asgha t al. (5). Singh (983) studid th ffts f Hall unts n an sillaty MHD fl in th Stks pblm past an infinit vtial pus plat. Th fist xat slutin f Navi-Stks quatin ith fl f visus inmpssibl fluid past a hizntal plat sillating in its n plan invstigatd by Stks (85). Natual nvtin ffts n Stks pblm as fist study by Sndalgka (979). Th sam pblm as nsidd by Rvanka () f impulsivly statd sillating plat. Tubatu t al. (998) invstigatd th fl f an inmpssibl visus fluid past an infinit plat sillating ith inasing dasing vlity amplitud f sillatin. Gupta t al. (3) hav analyzd fl in th Ekman lay n an sillating plat. Sundalgka t al. (997) fund an xat slutin f magnti f nvtin fl past an sillating plat. Mass tansf ffts n fl past an sillating plat nsidd by Lahuika t al. (995). Th study f hat gnatin absptin ffts in mving fluids is imptant in vi f sval physial pblms, suh as fluids undging xthmi ndthmi hmial atins. Vajavlu and Hadjinilau (993) studid th hat tansf haatistis in th lamina bunday lay f a visus fluid v a stthing sht ith visus dissipatin fitinal hating and intnal hat gnatin. Ziaul Haqu t al.() studid mipla fluid bhavis n stady MHD f nvtin and mass tansf fl ith nstant hat and mass fluxs, jul hating and visus dissipatin. Das t al (4) hav bn analyzd finit diffn analysis f hydmagnti fl and hat tansf f an lasti-visus fluid btn t hizntal paalll pus plat. Haqu and Alam() studid mipla fluid bhavius n unstady MHD hat and Mass tansf fl ith nstant hat and mass fluxs, jul hating and visus dissipatin. Haqu and Alam(9) hav bn invstigatd tansint hat and mass tansf by mixd nvtin fl fm a vtial pus plat ith indud magnti fild, nstant hat and mass fluxs. In this pap th ffts f Hall and in-slip unts n MHD fl in hat and mass tansf f an ltially nduting inmpssibl fluid alng an infinit sillaty vtial pus plat ith hat su in a tating systm hav bn nsidd. Als, th ffts f diffnt fl paamts nuntd in th quatins a studid. Th pblm is gvnd by systm f upld nn-lina patial diffntial quatins hs xat slutin is diffiult t btain. Hn, th pblm is slvd by finit diffn mthd and is psntd gaphially.. Mathmatial mdl f th fl Cnsid th unstady fl f an ltially nduting inmpssibl visus fluid past an infinit vtial pus plat y. Whn th plat vlity t () sillats in tim fquny n and is givn as t) snt 9 t ith a (. Th fl is assumd t b in th x ditin and hih is takn alng th plat in th upad ditin and y axis is nmal t it. Initially th

3 fluids as ll as th plat a at st but f tim nstant angula vlity abut th th fluid is at th sam tmpatu. Als it is assumd that th tmpatu f th plat and spis nntatin a aisd t T ( T ) and C ( C ) t th hl systm is alld t tat ith a y axis. Initially, it is nsidd that th plat as ll as sptivly, hih a th aft maintaind nstant, h T, Ca tmpatu and spis nntatin at th all and T, C a th tmpatu and th nntatin f th spis utsid th bunday lay sptivly, th physial nfiguatin f th pblm is shn in Fig.. A unifm magnti fild B is ating tansvs t th plat. sing th latin By B B B B B, B f th magnti fild, (,, ) x y z has bn nsidd vyh in th plat ( a nstant). Hv, f suh a fluid, th hall and inslip unts ill signifiantly afftd th fl in psn f lag magnti filds. Th indud magnti fild is ngltd sin th magnti Rynlds numb f a patially-inizd fluid is vy small. If J ( J x, J y, J z ) is th unt dnsity, fm th latin. J, J y nstant has bn btaind. Sin th plat is ltially nn-nduting, J y at th plat and hn z vyh. Sin th plat is infinit in xtnt, all physial quantitis, xpt pssu, a funtins f y v Cntinuity quatin: y Mmntum quatin: and t nly. Th quatins f th pblms a; u u u B ( u ) v g T T g C C u t y y k ( ) ( ) ( ) B ( u ) v u t y y k Engy quatin: Mass quatin: ( ) T T T v Q( T T) t y y p C C C Dk T v D m t y y T y m T m Th initial and bunday nditins f th mdl a; u( y, t), ( y, t), T( y, t) T, C( y, t) C f t (6) B is C T B O X C T Fig. Physial nfiguatin and dinat systm u v () () (3) (4) (5) 3

4 int int u( y, t), ( y, t), T( y, t) T, C( y, t) C at y, t (7) u( y, t), ( y, t), T( y, t) T, C( y, t) C at y, t h unt, i is vy small nstant quantity and is th in-slip unt, y is unifm vlity,, i is Hall is Catsian -dinat, u and a th mpnnts f fl vlity, g is th lal alatin du t gavity, is th thmal xpansin ffiint, nntatin xpansin ffiint, is th kinmati vissity, k * is th is th magnti pmability, is th dnsity f th fluid, is th ltial ndutivity, is th thmal ndutivity, th spifi hat at th nstant pssu, absptin quantity, kt is th thmal diffusin ati, D m is th ffiint f mass diffusivity, Q T m is th man fluid tmpatu and magnti mpnnt in y ditin. N a nvnint slutin f quatin () is; vv p is is th hat B is th (nstant) (8) h th nstant v psnts th nmal vlity at th plat hih is psitiv ngativ f sutin bling. sing quatin (8), th quatins ()-(5) bm; Mmntum quatin: u u u B ( u ) v g T T g C C u t y y k ( ) ( ) ( ) B ( u ) v u t y y k ( ) Engy quatin: T T T v Q( T T) t y y Mass quatin: p C C C Dk T v t y y T y m T Dm m Th bunday nditins f th mdl a; int int u( y, t), ( y, t), T( y, t) T, C( y, t) C at y, t (3) u( y, t), ( y, t), T( y, t) T, C( y, t) C at y, t (9) () () () 3. Mathmatial Fmulatins F th pups f slving th systm f quatins numially, th tansfm f gvning quatins int nn-dimnsin fm a ndd, th usual nn-dimnsinal vaiabls a intdud as; y u t T T C C,, W,, T, C T T C C 3 (4)

5 Thus intduing th latin (4) in quatin (9)-(), th flling dimnsinlss diffntial quatins hav bn btaind as; M W G T G m C RW M W W W W R W T T T T T T Q T, T P P C C C T S S h v p (Sutin Paamt), G g T T 3 (mdifid Gashf numb), M B (magnti paamt), S D m (Gashf numb), G p P m * g C 3 C (Pandtl numb), DmkT T T (Shmidt numb), S (St numb), R (tatinal paamt), C C Q p (Hat su paamt), (Pmability f th pus mdium) k (5) (6) (7) (8) Th spnding bunday nditins (3) bm as; i i (, ), W (, ), T (, ), C(, ) at (, ), W (, ), T (, ), C(, ) at, (9) 4. Sha stss, Nusslt and Shd numb u Th Sha stss alng x -axis is givn by x hih is pptinal t y y. Sha stss alng Th Nusslt numb z -axis is givn by z hih is pptinal t y N u hih is pptinal t Th Shd numb, S h hih is pptin t y T C. W. 3

6 5. Slutin Thniqu Expliit finit diffn mthd t slv th quatins (8)-(). F ths pups, th gin ithin th bunday lay is dividd by sm patiula lins f -axis, h -axis is nmal t th mdium as shn in Fig.. It is assum that th maximum lngth f th bunday lay is ( 5) as spnds t i.. max numb f gid spaing in m( 4) alng vais fm t 5 and th ditin is, hn th nstant msh siz -axis bms.3( 5) ith small tim stp, Lt t.. n n n n, W, T, C n n n n dntd th valus f, W, T, C at th nd f a tim-stp. Thn an apppiat st f finit diffn quatins spnding t th quatins (5)-(8) a as; n n n n n n n i i i i n n i i i n n GTi Ci RW i i M W n n i i W W W W W W W M n n n n n n n i i i i i i i n n n n R i Wi i Wi T T T T T T T Ti P P n n n n n n n i i i i i i i C C C C C C C T T T n n n n n n n n n n i i i i i i i i i i S S Th bunday nditins a btaind as n n n n n, W, T, C (4) n n n n, W, T, C L L L L h L Th numial valus f sha stss, Nusslt numb and Shd numb a valuatd by fivpint appximat fmula f th divativs. Th stability nditins f th pblm a as flls; P P,, S Hn th nvgn itia f th mthd a P. and S. (dtails a nt shn f bvity) O i Fig. Finit diffn gid spa i i i i i i 3 i m () () () (3) 33

7 6. Rsults and Disussin Justifiatin f Gid Spa m T vify th ffts f gid spa f, th d is tu ith th diffnt gid spa m 35,4,45. It is sn that th is a littl hang btn thm hih a shn in Fig.3. Ading t this situatin, th sults f vlity, tmpatu and nntatin hav bn aid ut f m 4 Stady stat slutin Th numial slutins f th nn-lina diffntial quatin (5)-(8) und th bunday nditins (9) hav bn pfmd by applying impliit finit diffn mthd. In d t vify th ffts f tim stp siz, th pgamming d is un u mdl ith svn diffnt tim stp sizs as, 4,8,9,,,. T gt stady stat slutins, th mputatins hav bn aid ut up t, W, T andc, hv shs littl hangs aft. It is bsvd that, th sult f mputatins f 8. Thus th slutins f all vaiabls f 9 a ssntially stady-stat. Hn th vlity, tmpatu and nntatin pfils a dan f hih is shn in Fig R..5 R. m m 4 m Fig.3 Pimay vlity f diffnt gid spa f tatinal paamt R 3 4 Fig.4 Pimay vlity f diffnt tim stp f tatinal paamt R T invstigat th physial situatin f th pblm, th numial alulatin has bn aid ut f dimnsinlss pimay vlity( ), snday vlity(w ), tmpatu(t ), nntatin(c ), sha stss in x -ditin( x ), sha stss in numb( 34 z -ditin( z ), Nusslt N u ) and Shd numb( S h ) f vaius valus f th paamts suh as Hall paamt ( ), in-slip paamt( i ), magnti paamt( M ), tatinal paamt ( R ), Pandtl numb( P ), sutin paamt ( ), Shmidt numb ( S ), Gashf numb( G ), mdifid Gashf numb( G m ), St numb( S ), pmability paamt( )and hat su( ). Th valus f th paamts a hsn abitaily in mst ass. Sm standad valus f f th

8 Pandtl numb( P spnds t ai, spnds t at at ) is nsidd baus f th physial imptan. Ths a P.7 P. C Fm Fig.5 (a,b) it is sn that pimay vlity ( spnds t ltlyt slutin suh as salt at and P. F mputatin,. hav bn hsn abitaily. 7. ) and sha stss in x-ditin ( x ) das ith inas f hat su paamt. Th fft f inasing th valu f is t das th bunday lay hih is as xptd du t th fat that hn hat is absbd th buyany f dass hih tads th fl at. It is sn that fm Fig.6 (a,b) snday vlity W and sha stss in z-ditin ( z ) hav ppsit bhavi ith an inas f. It an b laly sn that fm Fig.7 (a) th tmpatu pfils dass ith an inas f. Baus hn hat is absbd, th buyany f dass th tmpatu pfils. Nusslt numb T N u has vs fft hih is shn in Fig.5(b). Th fft f nntatin pfils has a ngligibl inasing fft but Shd numb hih is shn in Fig.8(a,b) ith an inas f. S h C dass It is bsvd that in Fig. 9(a,b) and Figs. (a,b), pimay vlity ( ) and sha stss in x- ditin ( x ) inas ith an inas f Hall paamt fftiv ndutivity das ith inas f and f n pimay vlity. Simila tnd aiss in snday vlity W ditin ( z ) pfils ith inasing hav dasing fft n W and z i and in-slip paamt i. Th hih dus th magnti damping and sha stss in z- hih is fund fm in Fig.. (a,b). It is fund that hih a shn in Fig. (a,b). Fm Fig.3 (a,b), it has bn sn that th pimay vlity and sha stss in x-ditin ( dass ith an inas in magnti paamt M. This is du t th fat that, th tansvs magnti fild nmal t th fl ditin, has a tndny t at th dag knn as th Lntz f hih tnds t sist th fl. Th snday vlity W and th sha stss z inas ith inas in M hih has bn illustatd in Fig.4 (a,b). Th sult indiats that th sulting Lntzian bdy f ill nt at as a dag f as in nvntinal MHD fls, but as an aiding bdy f. This ill sv t alat th snday fluid vlity. In Fig.5 (a,b) illustat that th pimay vlity and sha stss in x-ditin ( ) pfils das ith th inas f Pandtl numb P. This is baus in th f nvtin th plat vlity is high than th adjant fluid vlity and th mmntum bunday lay thiknss dass. Fig.6 (a,b), th snday vlity W and th sha stss a inasd ith th inas f P. In Fig.7 (a), th tmpatu pfils T das ith an inas f P. If P inass, th thmal diffusivity dass and ths phnmna lad t th dasing f ngy z x i x ) 35

9 ability that dus th thmal bunday lay. Th Nusslt numb ( appximatly any hang ith an inas f P hih is shn in Fig.7 (b). N u ) ds nt sh Fig.8 (a,b) a displayd th fft f tatinal paamt R n pimay vlity and sha stss in x-ditin ( x ) a dasd ith inas f R. In fat tatin paamt dfins th lativ magnitud f th Cilis f and th visus f, thus tatin tads pimay fl in th bunday lay. Simila bhavis a fund n snday vlity W and sha stss in z- ditin ( z ) hih a shn in Fig.9 (a,b) , i.3,.5 3., G 3.5,.7 M.5, P.7, R. S., S.5.,.5,. x , i.3,.5 3., G 3.5,.7 M.5, P.7, R. S., S.5.,.5,. 3 4 Fig. 5(a) Pimay vlity pfils f diffnt valus f hat su paamt Fig.5(b) Sha stss in x-ditin f diffnt valus f hat su paamt W -..,.5,. z -.7., i.3,.5 3., G 3.5,.7 M.5, P.7, R. S., S , i.3,.5 3., G 3.5,.7 M.5, P.7, R. S., S ,.5,. 3 4 Fig. 6(a) Snday vlity pfils f diffnt valus f hat su paamt Fig.6(b) Sha stss in z-ditin f diffnt valus f hat su paamt 36

10 T.8.6., i.3,.5 3., G 3.5,.7 M.5, P.7, R. S., S.5 N u , i.3,.5 3., G 3.5,.7 M.5, P.7, R. S., S.5.4.,.5,..3.,.5, Fig. 7(a) Tmpatu pfils f diffnt valus f hat su paamt Fig. 7(b) Nusslt numb f diffnt valus f hat su paamt C.8.6., i.3,.5 3., G 3.5,.7 M.5, P.7, R. S., S.5 S h.4., i.3,.5 3., G 3.5,.7 M.5, P.7, R. S., S.5.4.,.5,..,.5, Fig.8(a) Cnntatin pfils f diffnt valus f hat su paamt Fig. 8(b) Shd numb f diffnt valus f hat su paamt.5.5., i.3,.5 3., G 3.5,.7 M.5, P.7, R. S., S.5 x ,.,.4.5.,., , i.3,.5 3., G 3.5,.7 M.5, P.7, R. S., S Fig.9(a) Pimay vlity pfils f diffnt valus f Hall paamt Fig. 9(b) Sha stss in x-ditin f diffnt valus f Hall paamt 37

11 W , i.3,.5 3., G 3.5,.7 M.5, P.7, R. S., S.5.,.,.4 z -.8., i.3,.5 3., G 3.5,.7 M.5, P.7, R. S., S.5.,., Fig. (a) Snday vlity pfils f diffnt valus f Hall paamt Fig.(b) Sha stss in z-ditin f diffnt valus f Hall paamt.,.,.5 3., G 3.5,.7 M.5, P.7, R. S., S.5 x ,.6,.9 i.3,.6,.9 i ,.,.5 3., G 3.5,.7 M.5, P.7, R. S., S Fig. (a) Pimay vlity pfils f diffnt valus f in-slip paamt i Fig. (b) Sha stss in x-ditin f diffnt valus f in-slip paamt i W ,.,.5 3., G 3.5,.7 M.5, P.7, R. S., S.5.3,.6,.9 i z ,.,.5 3., G 3.5,.7 M.5, P.7, R. S., S.5.3,.6,.9 i Fig. (a)snday vlity pfils f diffnt valus f in-slip paamt i Fig. (b) Sha stss in z-ditin f diffnt valus f in-slip paamt i 38

12 ,., i.3.5, 3., G 3.5.7, P.7, R. S., S.5 M.5,.7,.9 x M.5,.7,.9.,., i.3.5, 3., G 3.5.7, P.7, R. S., S Fig. 3(a) Pimay vlity pfils f diffnt valus f magnti paamt M Fig. 3(b) Sha stss in x-ditin f diffnt valus f magnti paamt M M.5,.7, M.5,.7,.9 -. W ,., i.3.5, 3., G 3.5.7, P.7, R. S., S.5 z ,., i.3.5, 3., G 3.5.7, P.7, R. S., S Fig. 4(a) Snday vlity pfils f diffnt valus f magnti paamt M Fig. 4(b) Sha stss in z-ditin f diffnt valus f magnti paamt M ,., i.3.5, 3., G 3.5.7, M.5, R. S., S.5 P.7,.,7. x ,., i.3.5, 3., G 3.5.7, M.5, R. S., S.5 P.7,., Fig. 5(a) Pimay vlity pfils f diffnt valus f Pandtl numb P Fig. 5(b) Sha stss in x-ditin f diffnt valus f Pandtl numb P 39

13 P.7,., P.7,.,7. -. W ,., i.3.5, 3., G 3.5.7, M.5, R. S., S.5 z ,., i.3.5, 3., G 3.5.7, M.5, R. S., S Fig. 6(a) Snday vlity pfils f diffnt valus f Pandtl numb P Fig. 6(b) Sha stss in z-ditin f diffnt valus f Pandtl numb P T ,., i.3.5, 3., G 3.5.7, M.5, R. S., S.5 P.7,.,7. N u.9.8 P.7,.,7..,., i.3.5, 3., G 3.5.7, M.5, R. S., S.5 Fig. 7(a) Tmpatu pfils f diffnt valus f Pandtl numb P Fig. 7(b) Nusslt numb f diffnt valus f Pandtl numb P 4 R.,.4,.6 R.,.4, ,., i.3.5, 3., G 3.5.7, M.5, P.7 S., S.5 x 3..8.,., i.3.5, 3., G 3.5.7, M.5, P.7 S., S Fig. 8(a) Pimay vlity pfils f diffnt valus f tatinal paamt R Fig. 8(b) Sha stss in x-ditin f diffnt valus f tatinal paamt R 4

14 R.,.4, R.,.4,.6 W ,., i.3.5, 3., G 3.5.7, M.5, P.7 S., S.5 z ,., i.3.5, 3., G 3.5.7, M.5, P.7 S., S Fig.9(a) Snday vlity pfils f diffnt valus f tatinal paamt R Fig. 9(b) Sha stss in z-ditin f diffnt valus f tatinal paamt R 7. Cnlusins In this study, th finit diffn slutin f unstady MHD f nvtin and mass tansf fl thugh a vtial sillaty pus plat in a tating pus mdium ith hall, in-slip unts and hat su is invstigatd. Th flling nlusins a dan:. Pimay vlity, sha stss in x-ditin a inasd f inas th inas f R, P, M,., i hil das ith. Snday vlity, sha stss in z-ditin a inasd f inasing P M,, hil das ith th inas f i, R., 3. Tmpatu pfils a dasd f inasing P, hil Nusslt numb inass f But Pandtl numb yilds n fft n Nusslt numb. 4. Cnntatin pfils inass f inasing valus f hil vs fft n Shd numb is bsvd. H magnti and hat su paamt a usd signifiantly t ntl th fl and hat tansf haatistis. Rfns. Asgha S., Mhyuddin M.R., and Hayat T., Effts f Hall unt and hat tansf n fl du t a pull f nti tating disks, Intnatinal Junal f Hat and Mass tansf. 48, pp , 5.. Das, G. C.; Panda J. P; Das, S. S., Finit diffn analysis f hydmagnti fl and hat tansf f an lasti-visus fluid btn t hizntal paalll pus plats, Mdlling, Masumnt and Cntl B ; 73():3-44, Gupta A.S., Misa J.C., Rza M., Sundalgka V.M., Fl in th Ekman lay n an sillating pus plat, Ata Mhania, 65, pp. 6, 3. 4

15 4. Lahuika R. M., Phanka S. G., Sundalgka V. M., Biajda N. S., Mass tansf ffts n fl past a vtial sillating plat ith vaiabl tmpatu, Hat and Mass Tansf, 3(5), pp.39 3, Ziaul Haqu M, Alam M.M, Fds M. and Pstlniu A., Mipla fluid bhavis n stady MHD f nvtin and mass tansf fl ith nstant hat and mass fluxs, jul hating and visus dissipatin, Junal f King Saud nivsity-engining Sins, 4, 7-84,. 6. Rvanka S. T., F nvtin fft n fl past an impulsivly statd sillating infinit vtial plat, Mhanis Rsah Cmm., 7, 4 46,. 7. Singh, A. K., Hall ffts n an sillaty MHD fl in th Stks pblm past an infinit vtial pus plat. Astphysis and Spa Sin 93(), pp. 3, Stks G. G., On th fft f th intnal fitin f fluid n th mtin f pndulum, Tansatins Cambidg Philsphial Sity, IX, 8 6, Sundalgka V.M.,F nvtin ffts n th fl past a vtial sillating plat, Astphysis and Spa Sin, 64, pp.65 7, Sundalgka V. M., Das. N., Dka R. K., F nvtin ffts n MHD fl past an infinit vtial sillating plat ith nstant hat flux, Indian Junal f Mathmatis, 39(3), pp.95, Tubatu S., Bühl K., Zip J., N slutins f th II Stks pblm f an sillating flat plat, Ata Mhania, 9, pp.5 3, 998..Vajavlu K. and Hadjinilau A., Hat tansf in a visus fluid v a stthing sht ith visus dissipatin and intnal hat gnatin, Intnatinal Cmmuniatins in Hat and Mass Tansf, vl., n. 3, pp , Ziaul Haqu and Alam M. M., Mipla fluid bhavius n unstady MHD hat and Mass tansf fl ith nstant hat and mass fluxs, jul hating and visus dissipatin, AMSE Junal, Vl. 8(), 4. Haqu M. and Alam M. M., Tansint hat and mass tansf by mixd nvtin fl fm a vtial pus plat ith indud magnti fild, nstant hat and mass fluxs, AMSE Junal, Vl. 78(4), 9 4