Unsteady Casson Fluid Flow through Parallel Plates with Hall Current, Joule Heating and Viscous Dissipation

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1 AMSE JOURNALS 015-Sis: Modlling B; Vol. 84; N 1; pp 1- Submittd Aug. 014; Rvisd Fb. 015; Aptd Fb Unstdy Csson Fluid Flow though Plll Plts with Hll Cunt Joul Hting nd Visous issiption *Md. Fisl Kbi **Md. Mhmud Alm Abstt *Mthmtis Sin Engining nd Thnology Shool Khuln Univsity Khuln Bngldsh (fislkbi08ku@gmil.om) ** Mthmtis Sin Engining nd Thnology Shool Khuln Univsity Khuln Bngldsh (lm_mhmud000@yhoo.om) Th unstdy mgntohydodynmi flow of n ltilly onduting visous inompssibl non-nwtonin Csson fluid boundd by two plll non-onduting poous plts hs bn studid with Hll unt Joul hting nd Visous dissiption Th dvlopd modl hs bn dimnssionlizd by usul tnsfomtion thniqu. Th obtind non-simil oupld non-lin ptil diffntil qutions hv bn solvd by using xpliit finit diffn thniqu. Th pimy nd sondy vloity pofils nd tmptu distibutions disussd fo th diffnt vlus of dimnsionlss pmt vss dimnsionlss oodint. Th sh stss nd Nusslt numb hv lso bn invstigtd. Th obtind sults hv bn disussd with th hlp of gphs to obsv ffts of vious pmts on th bov mntiond quntitis. Th stbility onditions nd onvgn iti of th xpliit finit diffn shm stblishd fo finding th stition of th vlus of vious pmts to gt mo uy. Ky wods Csson fluid Hll unt Joul hting nd Visous dissiption 1. Intodution In fluid dynmis Coutt flow is th lmin flow of visous fluid in th sp btwn two plll plts on of whih is moving ltiv to th oth. Th flow is divn by vitu of visous dg fo ting on th fluid nd th pplid pssu gdint plll to th plts. Th mgntohydodynmi (MH) flow btwn two plll plts on in unifom motion nd th 1

2 oth hld t st known s MH Coutt flow is lssil poblm tht hs mny pplitions in MH pow gntos nd pumps ltos odynmi hting ltostti pipittion polym thnology ptolum industy puifition of ud oil nd fluid doplts nd spys ltostti pipittion polym thnology flow mts nd nul tos using liquid mtl oolnts. Th most impotnt non-nwtonin fluid possssing yild vlu is th Csson fluid whih hs signifint pplitions in polym possing industis nd biomhnis. Csson fluid is sh thinning liquid whih hs n infinit visosity t zo t of sh yild stss blow whih no flow ous nd zo visosity t n infinit t of sh suh s Nil polish whippd m kthup molsss syups pp pulp in wt ltx pint i blood som silion oils som silion otings. Csson's onstitutiv qution psnts nonlin ltionship btwn stss nd t of stin nd hs bn found to b utly pplibl to silion suspnsions suspnsions of bntonit in wt nd lithogphi vnishs usd fo pinting inks Csson [1] nd Wlwnd t l. []. Bt nd Jn [3] s nd Bt [4] Syd-Ahmd nd Atti [5] nd Atti [6] hv nlyzd th flow o/nd ht tnsf of Csson fluid in diffnt gomtis. Shoo t l. [7] hv studid th MH mixd onvtion stgntion point flow nd ht tnsf in poous mdium. Pnd t l. [8] hv nlyzd ht nd mss tnsf on MH flow though poous mdi ov n lting suf in th psn of sution nd blowing. Pnd t l. [9] hv studid hydomgnti flow nd ht tnsf though poous mdium of lsto-visous fluid ov poous plt in th slip flow gim. Atti [10] hs studid th influn of th Hll unt on th vloity nd tmptu filds of n unstdy Htmnn flow of onduting Nwtonin fluid btwn two infinit nononduting hoizontl plll nd poous plts. Atti nd Syd-Ahmd [11] hs studid tnsint MH Coutt flow of Csson fluid btwn plll plts with ht tnsf. Syd- Ahmd t l. [1] hs nlyzd tim dpndnt pssu gdint fft on unstdy MH outt flow nd ht tnsf of sson fluid. Ou im of this sh is to xtnd th wok of Atti nd Syd-Ahmd [11] in s of on dimnsionl flow. In this study th unstdy mgntohydodynmi flow of n ltilly onduting visous inompssibl non-nwtonin Csson fluid boundd by plll non-onduting poous plts hs bn studid with Hll unt Joul hting nd Visous dissiption. Th govning qutions of th poblm ontin systm of non-lin oupld ptil diffntil qutions whih tnsfomd by usul tnsfomtion into non-dimnsionl systm of ptil oupld non-lin diffntil qutions. Th obtind non-simil ptil

3 diffntil qutions hv bn solvd numilly by xpliit finit diffn thniqu. Th pimy nd sondy vloity pofils nd tmptu distibutions disussd fo th diffnt vlus of dimnsionlss pmt vss dimnsionlss oodint. Th sh stss nd Nusslt numb hv lso bn invstigtd. Th sults of this study disussd fo th diffnt vlus of th wll known pmts nd shown gphilly.. Mthmtil Modl of th Flow Th fluid is ssumd to b lmin inompssibl nd obying Csson modl nd flows btwn two infinit hoizontl plts lotd t th nd fom z unifom vloity U 0 y h plns nd xtnd fom x to to. Th upp plt is suddnly st into motion nd movs with whil th low plt is sttiony. Th upp plt is simultnously subtd to stp hng in tmptu fom T 1 to T. Thn th upp nd low plts kpt t two onstnt tmptus T 1 nd T sptivly with T > T 1. Th fluid is td upon by n xponntilly dying pssu gdint in th x-dition nd unifom sution fom bov nd intion fom blow whih pplid t t = 0. A unifom mgnti fild B 0 is pplid in th positiv y-dition. Th physil modl of this study is funishd in th following fig.1. Fig. 1 Gomtil onfigution of boundy ly A unifom mgnti fild is ssumd undistubd s th indud mgnti fild is ngltd by ssuming vy smll mgnti Rynolds numb. Th Hll fft is tkn into onsidtion nd onsquntly z-omponnt fo th vloity is xptd to is. Th unifom intion 3

4 implis tht th y-omponnt of th vloity v 0 is onstnt. Th fluid motion hs bn sttd fom st t t 0 nd th no-slip ondition t th plts in z-dition implis tht th fluid vloity hs no z-omponnt t y h. Th initil tmptu of th fluid is ssumd to b qul to T 1. Sin th plts infinit in th x nd z- ditions th physil quntitis do not hng in ths ditions. Th qution of onsvtion of lti hg J y. J 0 givs onstnt wh th unt dnsity J J J J ) bus th dition of popgtion is onsidd only long th y-xis nd J ( x y z plt is ltilly non-onduting th onstnt is zo i.. dos not hv ny vition long th y-xis. Sin th J y 0 t th plt nd vywh. Sin th plts infinitly xtndd nd th fluid motion is unstdy so ll th flow vibls funtion of y nd t. Thus odn with th bov ssumptions lvnt to th poblm nd Boussinsq s ppoximtion th bsi boundy ly qutions givn blow; Continuity qution v 0 y Momntum qution in x-dition u u 1 P 1 u 1 B 0 v0 ( u mw) t y x y y 1 m Momntum qution in z-dition w w 1 P 1 w 1 B 0 v0 ( w mu) t y z y y 1 m Engy qution T T T u w 1 B0 v0 u w t y p y p y y p 1 m Appnt visosity (1) () (3) (4) K 0 u w y y with th osponding initil nd boundy onditions ; 1 1 t 0 u 0 w 0 T T vywh (6) 4 (5)

5 t 0 u 0 w 0 T T fo 1 y h Wh u U w 0 T T fo x 0 y h y tsin oodint systm; u v x (7) y omponnt of flow vloity sptivly; is th kinmti visosity; is th dnsity of th fluid; m is th Hll pmt; is th ltil ondutivity; is th thml ondutivity; p is th spifi ht t th onstnt pssu; is th ppnt visosity. K is th Csson s offiint of visosity; 0 is th yild stss; 3. Mthmtil Fomultion Sin th govning qutions (1)-(5) und th initil (6) nd boundy (7) onditions hv bn bsd on th finit diffn thniqu it is quid to mk ths qutions dimnsionlss. Fo this pupos th following dimnsionlss quntitis intodud; x X h y Y h u U U 0 w W U Th bov dimnsionlss vibls bom; x hx y hy u U0U w U0W 0 tu 0 h h t U p T T1 T T P U nd. K 0 p U P T T T T 1 1 nd. K 0 Now th vlus of th bov divtivs substitutd into th qutions (1)-(5) nd ft simplifition th following nonlin oupld ptil diffntil qutions intms of dimnsionlss vibls obtind; V 0 (8) Y U S U dp 1 U H ( U mw ) R Y dx R Y Y 1 m W S W 1 W H ( W mu ) R Y R Y Y 1 m S 1 U W H E E U W R Y P y Y Y 1 m (9) (10) (11) 5

6 Wh 1 U W Y Y vh S K ( Sution Pmt ) U h R ( Rynold s Numb ) K (1) H B h ( Htmnn Numb Squd ) 0 K U h p 0 ( Pndtl Numb ) P E U0 K h T T p 1 ( Ekt Numb ) m B 0 ( Hll Pmt ) h 0 ( Csson Numb ) U0 K Also th ssoitd initil (6) nd boundy (7) onditions bom; 0 U 0 W 0 0 vywh (13) 0 U 0 W 0 0 fo Y 1 U 1 W 0 1 fo Y 1 (14) 4. Sh Stss nd Nusslt Numb Th quntitis of hif physil intst sh stss nd Nusslt numb. Fom th vloity fild th ffts of vious pmts on th plt sh stss hv bn invstigtd in s of moving plt. Th pimy sh stss u x y yh nd th sondy sh stss w z y yh whih popotionl to U Y Y 1 nd W Y Y 1 sptivly. Fom th 6

7 tmptu fild th ffts of vious pmts on Nusslt numb in s of moving plt hv bn lultd. Nusslt numb N u T y yh whih is popotionl to Y Y Numil Thniqu In this stion th govning sond od nonlin oupld dimnsionlss ptil diffntil qutions with th ssoitd initil nd boundy onditions ttmptd to solv. Th xpliit finit diffn mthod hs bn usd to solv qutions (8)-(1) with th hlp of th onditions givn by (13) nd (14). Th psnt poblm quis sst of th finit diffn qutions. In this s th gion within th boundy ly is dividd by som ppndiul lins of Y-xis wh Y-xis is noml to th mdium s shown in fig.. Fig.. Finit diffn sp gid It is ssumd tht th mximum lngth of boundy ly is Ymx i.. Y vis fom -1 to 1 nd th numb of gid sping in Y -xis boms Y y 1 Lt n 1 U dition is p( 98) hn th onstnt msh siz long Y with smll tim sp n 1 W nd n 1 dnot th vlus of n U n W nd n t th nd of tim-stp sptivly. Using th xpliit finit diffn ppoximtion th st of finit diffn qutions obtind s; 7

8 n1 n n n n n n U U S U 1 U dp 1 U 1 n U U 1 R Y dx R Y n n n n 1 1 U 1U H n n ( U ) mw R Y Y 1 m n1 n n n n n n U U S U 1 U 1 U 1 n U U 1 o P R Y R Y n n n n 1 1 U 1U H n n ( U ) mw R Y Y 1 m dp dx P ( sy ) (15) n1 n n n n n n W W S W 1 W 1 W 1 n W W 1 R Y R Y n n n n 1 1 W 1W H n n ( W ) mu R Y Y 1 m (16) n1 n n n n n n n n n n S U n 1 U W 1 W E R Y P Y Y Y EH U W 1 m n n (17) n 1 n n n n U 1 U W 1 W Y Y nd th initil nd boundy onditions with th finit diffn shm ; (19) U 0 W U 0 W 0 0 n n n n n n U 1 W 0 1 (0) H th subsipts dsignts th gid points with Y oodint nd th supsipt n psnts vlu of tim n wh n Fom th initil ondition (19) th (18) 8

9 vlus of n U n W nd n known t 0. At th nd of ny tim-stp th nw pimy vloity n 1 U th nw sondy vloity n 1 W nd th nw tmptu n 1 T t ll intio nodl points my b obtind by sussiv pplitions of qutions (15) (16) nd (17) sptivly. This poss is ptd in tim nd povidd th tim-stp is suffiintly smll n U n W nd n should vntully onvg to vlus whih ppoximt th stdy-stt solution of qutions (9)-(11). Also th numil vlus of th Sh Stss nd Nusslt numb vlutd by fiv-point ppoximt fomul fo th divtiv. 6. Stbility nd onvgn nlysis H n xpliit finit diffn mthod is bing usd; th nlysis will min inomplt unlss th stbility nd onvgn of th finit diffn shm disussd. Fo th onstnt msh sizs th stbility iti of th shm my b stblishd s follows. Th gnl tms of th Foui xpnsion fou W nd ll i Y t tim bitily lld t 0 pt fom onstnt wh i 1. At timt ths tms bom; U W : : i Y i Y i Y : (1) : i Y nd ft th tim-stp ths tms will bom; U W : : : : i Y i Y i Y () i Y Substituting (1) nd () into qutions (15)-(17) th stbility onditions of th poblm s funishd blow; S H Y R R 1m 1 nd (3) 9

10 S 1 Y R ( Y ) P 1 (4) Fom qutions (3) nd (4) th onvgn iti of th poblm R H 199 S 500 nd P Rsults nd isussion In this stion it hs bn psntd tht th sults obtind using th sussiv xpliit finit diffn numil thniqu. To invstigt th physil onditions of th dvlopd mthmtil modl it hs bn obtind th numil vlus of th 10 x nd z omponnts of vloity ommonly known s pimy nd sondy vloity nd tmptu within th boundy ly fo th lmin boundy ly flow. In od to nlyz th physil sitution of th modl it hs bn omputd th stdy stt numil vlus of th non-dimnsionl pimy vloity tmptu sh stss nd Nusslt numb vlus of Sution pmt numb numb H Pndtl numb. S N u U sondy vloity W within th boundy ly fo diffnt Rynold s numb Hll pmt P R nd Ekt numb E m Htmnn with th fixd vlu of Csson Th tnsint pimy vloity sondy vloity tmptu pofils stdy stt sh stss nd Nusslt numb hv bn shown in Figs. 4.3 to 4.38 fo diffnt vlus of S R m H P nd E. Th vlus of S In s of Sution pmt S ; Th ffts of th Sution pmt R m S H P nd E hosn bitily. on th pimy vloity sondy vloity tmptu fild pimy sh stss sondy sh stss nd Nusslt numb psntd in Figs. 4.3 to 4.8 sptivly. It hs bn obsvd tht pimy vloity U inss with th ins of Sution pmt S in Fig At fist th sondy vloity W inss thn dss with th ins of Sution pmt S tht mns sondy vloity pofil is oss flow in Fig Figs psnt th tmptu pofil pimy sh stss x sondy sh stss z nd Nusslt numb In s of Rynold s numb N u ins fo th insing vlus of S. R ;

11 Th ffts of th Rynold s numb R on th pimy vloity sondy vloity tmptu fild pimy sh stss sondy sh stss nd Nusslt numb psntd in Figs. 4.9 to 4.14 sptivly. stss x It hs bn obsvd tht th pimy vloityu sondy vloityw pimy sh nd sondy sh stss z hv insd gdully s th is of in Figs nd 4.13 sptivly. Th tmptu pofil nd Nusslt numb N u R s illusttd hv dsd with th insing vlus of R shown in Figs nd 4.14 sptivly. In s of Pndtl numb P ; Th pimy vloity U sondy vloity W pimy sh stss x nd sondy sh stss z unhngd with th is of Pndtl numb P s shown in Figs nd 4.19 sptivly. T h tmptu pofil nd Nusslt numb N u hv insd with th insing vlus of Pndtl numb P shown in Figs nd 4.0 sptivly. In s of Hll pmt m ; Th ffts of th Hll pmt m on th pimy vloity sondy vloity tmptu fild pimy sh stss sondy sh stss nd Nusslt numb psntd in Figs. 4.1 to 4.6 sptivly. stss It hs bn obsvd tht th pimy vloityu sondy vloityw pimy sh x nd sondy sh stss nd 4.5 sptivly. z hv insd s th is of m s illusttd in Figs. Th ngligibl fft of m on tmptu pofil hs bn found with th insing vlus of m shown in Fig Th Nusslt numb shown in Fig In s of Htmnn numb H ; N u inss with th ins of m s 11

12 Th ffts of th Htmnn numb H on th pimy vloity sondy vloity tmptu fild pimy sh stss sondy sh stss nd Nusslt numb psntd in Figs. 4.7 to 4.3 sptivly. Th pimy vloity U nd pimy sh stss x hv dsd with th ins of Htmnn numb H s shown in Figs. 4.7 nd 4.30 sptivly. At fist th sondy vloity W inss thft dss with th ins of Htmnn numb H tht mns sondy vloity pofil is oss flow in Fig It is obsvd fom Fig th sondy sh stss z dss with th ins of Htmnn numb H. Th ngligibl fft of H on tmptu pofil hs bn found with th insing vlus of H shown in Fig Th Nusslt numb N u dss with th ins of H s shown in Fig In s of Ekt numb E Th ffts of th Ekt numb ; E on th pimy vloity sondy vloity tmptu fild pimy sh stss sondy sh stss nd Nusslt numb psntd in Figs to 4.38 sptivly It hs bn obsvd tht th pimy vloityu sondy vloityw pimy sh stss x nd sondy sh stss z unhngd with th is of Ekt numb E s shown in Figs nd 4.37 sptivly. Th tmptu pofil nd Nusslt numb N u hv insd with th insing vlus of Ekt numb E s shown in Figs nd 4.38 sptivly. Hn it is onludd tht th mximum vloity ous in th viinity of th plt. 1

13 R 1.00 P 1.00 m 1.00 H 3.00 E R 1.00 P 1.00 m 1.00 H 3.00 E Fig. 4.3 Pimy Vloity Pofils fo diffnt vlus of imnsionlss Sution Pmt S Fig.4.4 Sondy Vloity Pofils fo diffnt vlus of imnsionlss Sution Pmt S R 1.00 P 1.00 m 1.00 H 3.00 E R 1.00 P 1.00 m 1.00 H 3.00 E Fig.4.5 Tmptu Pofils fo diffnt vlus of imnsionlss Sution Pmt S Fig.4.6 Pimy Sh Stss Pmt S in s of moving plt x Sution P 1.00 R 1.00 m 1.00 H 3.00 E P 1.00 R 1.00 m 1.00 H 3.00 E Fig.4.7 Sondy Sh Stss z fo Sution Pmt S in s of moving plt Fig.4.8 Nusslt Numb N u fo Sution Pmt S in s of moving plt 13

14 S 5.00 m 1.00 P 1.00 H 3.00 E S 5.00 m 1.00 P 1.00 H 3.00 E Fig.4.9 Pimy Vloity Pofils fo diffnt vlus of imnsionlss Rynolds Numb R Fig.4.10 Sondy Vloity Pofils fo diffnt vlus of imnsionlss Rynolds Numb R S 5.00 m 1.00 P 1.00 H 3.00 E S 5.00 m 1.00 P 1.00 H 3.00 E Fig.4.11 Tmptu Pofils fo diffnt vlus of imnsionlss Rynolds Numb R Fig.4.1 Pimy Sh Stss Numb R x in s of moving plt fo Rynolds 14

15 S 5.00 m 1.00 P 1.00 H 3.00 E S 5.00 m 1.00 P 1.00 H 3.00 E Fig.4.13 Sondy Sh Stss Numb R z in s of moving plt fo Rynolds Fig.4.14 Nusslt Numb Numb R N u in s of moving plt fo Rynolds R 1.00 S 5.00 m 1.00 H 3.00 E R 1.00 S 5.00 m 1.00 H 3.00 E Fig.4.15 Pimy Vloity Pofils fo diffnt vlus of imnsionlss Pndtl Numb P Fig.4.16 Sondy Vloity Pofils fo diffnt vlus of imnsionlss Pndtl Numb P R 1.00 S 5.00 m 1.00 H 3.00 E R 1.00 S 5.00 m 1.00 H 3.00 E Fig.4.17 Tmptu Pofils fo diffnt vlus of imnsionlss Pndtl Numb P Fig.4.18 Pimy Sh Stss x fo Pndtl Numb P in s of moving plt 15

16 R 1.00 S 5.00 m 1.00 H 3.00 E R 1.00 S 5.00 m 1.00 H 3.00 E Fig.4.19 Sondy Sh Stss Numb P in s of moving plt z fo Pndtl Fig.4.0 Numb P Nusslt Numb N u in s of moving plt fo Pndtl R 1.00 P 1.00 S 5.00 H 3.00 E R 1.00 P 1.00 S 5.00 H 3.00 E Fig.4.1 Pimy Vloity Pofils fo diffnt vlus of imnsionlss Hll Pmt m Fig.4. Sondy Vloity Pofils fo diffnt vlus of imnsionlss Hll Pmt m 16

17 R 1.00 P 1.00 S 5.00 H 3.00 E R 1.00 P 1.00 S 5.00 H 3.00 E Fig.4.3 Tmptu Pofils fo diffnt vlus of imnsionlss Hll Pmt m Fig.4.4 Pimy Sh Stss x Pmt m in s of moving plt fo Hll R 1.00 P 1.00 S 5.00 H 3.00 E R 1.00 P 1.00 S 5.00 H 3.00 E Fig.4.5 Sondy Sh Stss z Pmt m in s of moving plt fo Hll Fig.4.6 Nusslt Numb N u fo Hll Pmt m in s of moving plt S 5.00 R 1.00 P 1.00 m1.00 E S 5.00 R 1.00 P 1.00 m1.00 E Fig.4.7 Pimy Vloity Pofils fo diffnt vlus of imnsionlss Htmnn Numb H Fig.4.8 Sondy Vloity Pofils fo diffnt vlus 17 of imnsionlss Htmnn Numb H

18 S 5.00 R 1.00 P 1.00 m1.00 E S 5.00 R 1.00 P 1.00 m1.00 E Fig.4.9 Tmptu Pofils fo diffnt vlus of imnsionlss Htmnn Numb H Fig.4.30 Pimy Sh Stss Numb H x in s of moving plt. fo Htmnn S 5.00 R 1.00 P 1.00 m1.00 E S 5.00 R 1.00 P 1.00 m1.00 E Fig.4.31 Sondy Sh Stss z fo Htmnn Numb in s of moving plt H Fig.4.3 Nusslt Numb Numb H N u in s of moving plt fo Htmnn 18

19 S 5.00 R 1.00 P 1.00 m1.00 H S 5.00 R 1.00 P 1.00 m1.00 H Fig.4.33 Pimy Vloity Pofils fo diffnt vlus of imnsionlss Ekt Numb E Fig.4.34 Sondy Vloity Pofils fo diffnt vlus of imnsionlss Ekt Numb E S 5.00 R 1.00 P 1.00 m1.00 H S 5.00 R 1.00 P 1.00 m1.00 H Fig.4.35 Tmptu Pofils fo diffnt vlus of imnsionlss Ekt Numb E Fig.4.36 Pimy Sh Stss Numb E x in s of moving plt fo Ekt S 5.00 R 1.00 P 1.00 m1.00 H S 5.00 R 1.00 P 1.00 m1.00 H Fig.4.37 Sondy Sh Stss z fo Ekt Numb E in s of moving plt Fig.4.38 Nusslt Numb N u fo Ekt Numb E in s of moving plt 19

20 8. Conlusions In this study th quid physil poblm hs bn studid mthmtilly fo diffntil psptivs mostly onnd with thi solutions. Th oupld ptil diffntil qutions hv bn solvd numilly by xpliit finit diffn thniqu. Th unstdy mgntohydodynmi flow of n ltilly onduting visous inompssibl non-nwtonin Csson fluid boundd by plll non-onduting poous plts hs bn studid with Hll unt Joul hting nd Visous dissiption. Th sults disussd fo diffnt vlus of impotnt dimnsionlss pmts s Sution pmt ( S ) Rynold numb ( Htmnn numb ( H ) nd Ekt numb ( R ) Hll pmt ( m E ) Pndtl numb ( ) with th fixd vlu of Csson numb ( Som of th impotnt findings obtind fom th gphil psnttion of th sults listd hwith; 1. Th pimy vloity inss with th ins of S R P nd m. It dss with th ) ). ins of H nd no hng fo P nd E.. Th sondy vloity inss with th ins of R nd m. It shows th oss flow with th ins of S nd H nd no hng fo P nd E. 3. Th tmptu inss with th ins of S P nd E. It dss with th ins of R nd shows th mino fft with th ins of m nd H. 4. Th pimy sh stss in s of moving plt inss with th ins of S R nd m. It dss with th ins of H nd no hng fo P nd E. 5. Th sondy sh stss in s of moving plt inss with th ins of S nd m. It dss with th ins of H nd no hng fo P nd 6. Th Nusslt numb in s of moving plt inss with th ins of S E. R E P nd m whil it dss with th ins of R nd H. As th bsis fo mny sintifi nd ngining pplitions fo studying mo omplx poblms involving th flow of ltilly onduting fluids it is hopd tht th psnt invstigtion of th study of pplid physis of flow though th plll plts n b utilizd. In th puifition of ud oil nd fluid doplts nd spys s wll s in th polym possing 0

21 industis nd biomhnis th findings my b usful fo study of movmnt nd flow of sh thinning liquids. This wok hs bn don fo dmi point of viw. Rfns 1. N. Csson A Flow Eqution fo Pigmnt Oil-Suspnsions of th Pinting Ink Typ in Rholgy of isps Systms (C.C. Mill Ed.) Pgmon Pss London p W.P. Wlwnd T.Y. Chn nd.f. Cl An Appoximt Csson Fluid Modl fo Tub Flow of Blood Biohology vol. 1 no. pp R.L. Bt nd B. Jn Flow of Csson Fluid in Slightly Cuvd Tub Intntionl Jounl of Engining Sin. vol. 9 no. 10 pp B. s nd R.L. Bt Sondy Flow of Csson Fluid in Slightly Cuvd Tub Intntionl Jounl of Non-Lin Mhnis vol. 8 no. 5 pp M.E. Syd-Ahmd nd H.A. Atti Mgntohydodynmi Flow nd Ht Tnsf of Non-Nwtonin Fluid in n Enti Annulus Cndin Jounl of Physis vol. 76 no. 5 pp H.A. Atti nd M.E. Syd-Ahmd Hydodynmi Impulsivly Lid-ivn Flow nd Ht Tnsf of Csson Fluid T. Jounl of Sin nd Engining vol. 9 no. 3 pp S.N. Shoo J.P. Pnd nd G.C. sh Th MH mixd onvtion stgntion point flow nd ht tnsf in poous mdium Podings of Ntul Admi Sin vol. 83 no. 4 pp J.P. Pnd N. sh nd G.C. sh Ht nd mss tnsf on MH flow though poous mdi ov n lting suf in th psn of sution nd blowing Jounl of Engining Thmophysis vol. 1 no. pp J.P. Pnd N. sh G.C. sh Hydomgnti flownd ht tnsf though poous mdium of lsto-visous fluid ov poous plt in th slip flow gim AMSE Jounls sis Modlling B vol. 80 no. 1- pp H.A. Atti Hll Cunt Effts on th Vloity nd Tmptu Filds of n Unstdy Htmnn Flow Cndin Jounl of Physis vol. 76 no. 9 pp

22 11. H.A. Atti nd M.E. Syd-Ahmd Tnsint MH Coutt Flow of Csson Fluid btwn Plll Plts with Ht Tnsf Itlin Jounl of pu nd pplid Mthmtis no. 7 pp M.E. Syd-Ahmd H.A. Atti nd K.M. Ewis Tim pndnt Pssu Gdint Efft on Unstdy MH Coutt Flow nd Ht Tnsf of Csson Fluid Cndin Jounl of Physis vol. 3 no.1 pp