Analysis on Variable Fluid Viscosity of Non-Darcian Flow over a Moving Vertical Plate in a Porous Medium with Suction and Viscous Dissipation

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1 IOR Jounl o Engineeing IORJEN IN e: 5- IN p: Vol. Iue 8 Augut. V 8- Anli on Vile Flui Vioit o Non-Din Flow ove Moving Vetil lte in oou Meium with ution n Viou Diiption Animun I. L. n Aluo O. B. Mthemtil iene Deptment Feel Univeit o ehnolog Aue Nigei. Mthemti/ttiti Deptment Oun tte oltehni Iee Nigei. Att: - An nli i ie out to tu ee onvetive low n het tne o viou inompeile lui ove linel moving vetil poou plte with ution n viou iiption. he lui vioit i ume to v line untion o tempetue. he govening oun le eqution e eue to oun vlue polem uing the onept o imilit tnomtion. he oeponing ouple oin ieentil eqution i olve numeill uing the Runge Kutt outh oe metho long with hooting tehnique. Gphil eult o the imenionle veloit n tempetue itiution e hown o viou vlue o the themophil pmete ontolling the low egime. Numeil vlue o phil quntitie uh the lol in-ition oeiient n the Lol Nuelt nume e peente in tul om. Kewo: - Vile lui vioit oou mei Viou Diiption Non Din low Ritive Het lu I. INRODUION he phi o lui low in ieent mei n onuit i well-eehe e in engineeing with gouneing wo pionee woe in the iel o engineeing. Liewie tnpottion o het though poou mei h gine oniele ttention ue to it vt pplition in the inut n lo ue to the ineing nee o ette unetning o the oite tnpot poee. hee e numeou ptil pplition in the inut whih n e moele o n e ppoimte tnpot though poou mei uh gin toge het ehnge evie petoleum eevoi hemil tlti eto n poee geotheml n geophil engineeing moitue migtion in iou inultion n nule wte ipol n othe. Bejn n Khi [] invetigte the ee onvetion oun le low in poou meium owing to omine het n m tne. Li n Kuli [8] ue the eie epnion metho to invetigte ouple het n m tne in ntul onvetion om phee in poou meium. he ution n lowing eet on ee onvetion ouple het n m tne ove vetil plte in tute poou meium wee tuie Rpti et l. [8]. he nli o onvetive tnpot in poou meium with the inluion o non-din eet h lo een mtte o tu in eent e. he ineti eet i epete to e impotnt t highe low te n it n e ounte o though the ition o veloit que tem in the momentum eqution whih i nown the Fohheime etenion o the D lw. A etile eview o onvetive het tne in D n non-d poou meium n e oun in the oo Niel n Bejn []. he polem o low n het tne in lmin oun le ove tething heet in tute poou meium h n impotnt pplition in the metllug n hemil engineeing iel. Le et l. [7] oniee the low n het tne oun le tgntion point low o n inompeile viou lui tow hete poou tething heet emee in poou meium ujet to ution/lowing with intenl het genetion o option. ii in 96 w the it peon to tu the lmin oun le low ue igi ue moving in it own plne. ne in 97 etene the wo o ii onieing the me moel ujet to tething plte. Gupt n Gupt [6] tuie the polem in the light o ution o lowing. Reentl nn et l. [7] tuie MHD Boun Le Flow o Het n M ne ove Moving Vetil lte in oou Meium with ution n Viou Diiption; the govening oun le eqution e eue to two-point oun vlue polem uing imilit tnomtion. he eultnt polem i olve numeill uing the Runge-Kutt outh oe metho long with hooting tehnique. he momentum oun le thine eee while oth theml n onenttion oun le thinee inee with n inee in the mgneti iel intenit. All the ove mentione invetigtion wee ie out o lui hving ontnt vioit thoughout the oun-le. Howeve it i nown tht thee phil popetie m hnge igniintl with tempetue. Fo intne the vioit o wte eee out % when the tempetue inee om to 5 5. he vioit o i i n.65t tempetue K K K n 8K epetivel eei n Bhw []. o Intentionl ogniztion o ientii Reeh 8 g e

2 Anli on Vile Flui Vioit o Non-Din Flow ove Moving Vetil lte in oou peit utel the low ehvio in the oun le it i nee to te the vition o vioit into ount. he tu o viou inompeile lui low with tempetue epenent popetie i o get impotne in inutie uh oo poeing oting n polme poeing whee tempetue i genete. In inutil tem lui low n e ujete to eteme onition uh high tempetue peue n he te. In ition etenl heting uh the mient tempetue n high he te n le to high tempetue eing genete within the lui. All thee onition nomll hve igniint eet on the lui popetie. In lui nmi it i well nown tht the popet whih i mot enitive to tempetue ie i vioit. Flui ue in inutie uh polme lui engine oil plm oil hve vioit tht vie pil with tempetue n thi eventull give ie to tong ee eet whih nomll le to igniint hnge in the tutue o the lui low le n Hung []. Muhoph et l. [] ie out ompehenive eeh on ee onvetive oun le low n het tne o lui with vile vioit ove poou tething vetil ue in peene o theml ition Lie goup tnomtion w opte n outh oe lil Runge Kutt metho w opte to olve the oeponing ouple oin ieentil eqution. he epote tht; with n inee o tempetue epenent lui vioit pmete the lui veloit inee ut the tempetue eee t ptiul point o the heet. In the peent eeh we tu ee onvetive low n itive het tne o viou inompeile lui hving vile vioit ove tething poou vetil plte whih i n etenion o Muhoph et l. []. e uthe ume tht the lui low i une the inluene o viou iiption n D-Fohheime. he i eqution govening the low iel e ptil ieentil eqution n thee hve een eue to et o oin ieentil eqution ppling uitle imilit tnomtion. he eultnt eqution e ouple n non-line n hene e olve numeill uing the outh oe Runge- Kutt metho long with hooting tehnique. he eet o viou govening pmete on the veloit n tempetue e peente gphill n iue quntittivel. II. MAHEMAIAL FORMULAION A te two-imenionl lmin ee onvetion low o viou inompeile eletill onuting lui long poou vetil tething heet in the peene o ution n viou iiption i oniee. he i i ten long the plte in the upw ietion n the i i noml to the heet. he veloit o the lui w om the plte ue i ume to e zeo. Auming tht the Bouineq n oun le ppoimtion hol n uing the D-Fohheime moel the i eqution whih goven the polem e given Muhoph et l. [] n nn et l. [6] n eomulte : u u u v u v u u g u u K K q q u u v ujet to oun onition m u B v V t u he het tne o viou inompeile lui low ove vetil tething heet emeging out o lit t oigin. he viou iiption tem in the eneg eqution i igniint we ume tht the lui veloit i high n epeent oointe e long the ontinuou ue in the ietion o motion n pepeniul to it epetivel u n v e the Din veloit omponent long n ietion Denit ume ontnt g i the gvit i the empiil ontnt i the vioit i the inemti vioit B i the oeiient o theml epnion K i the pemeilit o the poou meium i the theml onutivit i the peii het t ontnt peue q Dimenionl het option oeiient. Uing Bewte [] itive het lu q i eie the Roeln ppoimtion Intentionl ogniztion o ientii Reeh 9 g e

3 Anli on Vile Flui Vioit o Non-Din Flow ove Moving Vetil lte in oou Intentionl ogniztion o ientii Reeh g e q 5 whee i the ten-boltzmnn ontnt n i nown the option oeiient. Auming tht the tempetue ieene within the low i uh tht m e epne in lo eie n epning out n negleting highe oe. It i ume tht the tempetue ieene within the low e uiientl mll uh tht n e epee line untion o tempetue. hi i omplihe epning in lo eie out the ee tem tempetue n negleting highe-oe tem. he lo eie epnion o untion out!... ''! ' n n n...! 6 7 Ming ue o 5 n 7 eqution eome 6 u q v u 8 e now intoue the ollowing eltion o u v n u v n 9 K K g 6 q whee u i the tem untion. he temwie veloit n the ution veloit e ten m B U n V v. In thi eeh m i the powe lw eponent / m B i the wll tempetue n i the mient tempetue the lui vioit i ume to v line untion o tempetue. Aoing to Bthelo [] ] [ whee i the ontnt vlue o the oeiient o vioit om the heet eeene vioit n e ontnt. In thi eeh we onie the e when Fo viou lui Ling n D [9] ugget vioit epenene on tempetue o the om whee i theml popet o the lui n i the vioit w om the hot heet. Othe tempetue epenent vioit moel e Renol moel ep M n Vogel moel ep ee e.g. Moui n hite [] Neem n Ali [] Oo [5] n emili n Yil [6]. Ou opte moel oe not ie t ll with ll the moel mentione ove. he nge o tempetue i.e. tuie hee i. Ming ue o n thi eqution in 9

4 Anli on Vile Flui Vioit o Non-Din Flow ove Moving Vetil lte in oou Intentionl ogniztion o ientii Reeh g e Uing ove eltion in the oun le eqution n in the moiie eneg eqution 8 we get ] [ g ] [ K K 6 ] [ q whee ujet to m B V t o tnom eqution - into et o oin ieentil eqution the ollowing imilit tnomtion n imenionle vile e intoue g J G F K D K N p q E V whee E N D F n G J e imenionle tem untion imenionle tempetue imilit vile Lol Fohhemie pmete Lol D mete emeilit elte pmete o the poou meium Rition pmete ntl nume Het genetion/aoption pmete Eet elte mete ution pmete n Gho elte pmete epetivel. It i nown tht G G J Lol Gho nume in thi eeh. ] [ ] [ D F J G 5 ] [ E N 6 ujet to B t 7 8 III. NUMERIAL OLUION he et o ouple non-line govening oun le eqution 5 n 6 togethe with the oun onition 7 n 8 e olve numeill uing Runge-Kutt outh oe tehnique long with hooting metho. Fit o ll highe oe non-line ieentil Eqution 5 n 6 e onvete into

5 Anli on Vile Flui Vioit o Non-Din Flow ove Moving Vetil lte in oou imultneou ieentil eqution o it oe n the e uthe tnome into initil vlue polem ppling the hooting tehnique. he eultnt initil vlue polem i olve emploing Runge-Kutt outh oe tehnique. he tep ize. 5 i ue to otin the numeil olution with iteen eiml ple u the iteion o onvegene. Fom the poe o numeil omputtion the in-ition oeiient n the Nuelt nume whih e epetivel popotionl to '' n ' e lo ote out n thei numeil vlue e peente in tul om. IV. FIGURE AND DIUION he eet o Lol Fohheime pmete pemeilit pmete Rition pmete ntl nume Het genetion/aoption pmete Eet Nume n ution pmete h een omulte n olve numeill. In oe to unetn the low o the lui omputtion e peome o ieent pmete uh F D N E.In thi eeh B n J G i hoen to e equl to one. Figue ehiit the veloit poile o evel vlue o with. 7 in the peene o ution. when N.. In the e o uniom ution the veloit o the lui i oun to inee with the inee o the tempetue epenent lui vioit pmete. hi n e epline phill the pmete inee the on etween the lui eome wee n the vioit eee n the lui low t te te. In Figue vition o tempetue iel with o evel vlue o uing F.5 D. N..7. E. in the peene o ution. e hown. It i oeve tht the tempetue eee with the ineing vlue o. he inee o tempetue epenent lui vioit pmete le to eee o theml oun le thine whih eult in eee o tempetue poile. Deee in tempetue poile o the theml oun le men eee in the veloit o the lui popetie. A mtte o t in thi e the lui ptile unego two oppoite oe whih e: i One oe inee the lui veloit ue to eee in the lui vioit with inee in the vlue o ii he eon oe eee the lui veloit ue to eee in tempetue. {ine eee with ineing }.Ve ne the vetil ue. the tempetue i high the it oe ominte n w om the ue. 5 the tempetue i low; thi implie tht the eon oe ominte in tht egion. Figue illutte the eet o pemeilit pmete on the veloit. It i notie tht the pemeilit pmete inee the veloit eee. ine the pooit o the plte inee the lui low ove the poou plte the g ten to inee whih w the veloit. Alo Figue how the vition o the theml oun-le with the eet o pemeilit pmete. e notie tht the theml oun le thine inee with n inee in the pemeilit pmete. hill thi n e epline ollow the pooit pmete inee thi give oom to moe entne o het into the low. A the het inee the tempetue poile lo eome ete n ten to inee. Figue 5 n Figue 6 evel tht the eet o Fohhemie pmete i epeiene ne the plte onl inee in the vlue o F le to eee in veloit within 6.. Revee i the e on tempetue poile Fohhemie pmete F inee the tempetue inee thoughout the low egion. e hoe D n high vlue o Fohhemie nume F. 5 to nlze the eet o the lol D nume D on the veloit poile n tempetue iel hown in Figue 7 n 8. Figue 7 epit tht the veloit inee lightl with n inee in the vlue o D. he eet o lol D nume i epeiene within etin nge inee in the vlue o D le to inee in veloit within Revee i the e on tempetue poile D inee the tempetue eee lightl. he eee in o the low egion i o lowe te. hi eet i negligile on the tempetue ue to the itintion o the vlue o D. On the veloit poile eet o D i negligile w om the wll i.e. within In oe to tu the ehvio o meu nole ge with hogen n i i ujete to low une ou umption; Figue 9 n how tht lui hving mlle Intentionl ogniztion o ientii Reeh g e

6 Anli on Vile Flui Vioit o Non-Din Flow ove Moving Vetil lte in oou ntl nume Meu with. 5 e muh moe eponive to theml ition thn lui Ai with. 7 hving lge ntl nume. It i lo note om igue 9 n tht tempetue n veloit eee with the ineing vlue o ntl nume. Fom igue 9 eet o i negligile ve loe to the wll. 9 n om the wll. 5 e hve illutte non-imenionl veloit tempetue gint o ome epeenttive vlue o the het oue/option pmete..8. poitive vlue o epeent oue i.e. het genetion in the lui while negtive vlue o epeent option o the het tht eit within the lui. Figue n how eet o het oue pmete ove veloit n tempetue poile. It i oeve tht ue to the genetion o het into the lui low; the uon oe inee whih in tun give highe veloit in the veloit n theml oun le. Alo Figue n how eet o het option pmete ove veloit n tempetue poile ue to the option o het om the lui low; the uon oe eee whih in tun give lowe veloit in the veloit n theml oun le. Figue 5 how the vition o Eet nume E ove the momentum oun-le we oeve tht E oe not hve n utntil eet on veloit o the lui low. Alo Fig 6 how the vition o the theml oun-le with the Eet nume E. It i oeve tht the theml oun le thine inee lightl with n inee in the Eet nume E. he vition o veloit n tempetue itiution with ition pmete N e hown in Figue 7 n 8 epetivel. Fom Figue 7 n 8 we oeve tht the veloit n the tempetue poile eee with the inee o ition pmete N. Rition n e ue to ontol the veloit n the theml oun le quite eetivel. he eet o ition pmete N on the veloit oun le i hown in Figue 7 o. F.5. D.7. E...he veloit poile how eee om the plte with the inee o ition pmete N. It i note om Figue 8 tht the tempetue eee with the ineing vlue o the ition pmete N. he eet o ition pmete N i to eue the tempetue igniintl in the low egion. he inee in ition pmete men the elee o het eneg om the low egion n o the lui tempetue eee the theml oun le thine eome thinne. hi eult n e uthe epline the t tht eee in the vlue o N / o given n men eee in the Roeln ition optivit. Aoing to momentum n eneg eqution the ivegene o the itive het lu q / inee eee whih in tun inee the te o itive het tnee to the lui n hene the lui tempetue inee. In view o thi eplntion the eet o ition eome moe igniint N N n n e neglete when N. Figue 9 n epeent the eet o ution on lui veloit n tempetue when the lui vioit i uniom. ith n inee in the vlue o ution n F.5. D N..7. E. the hoizontl veloit i oun to eee. hi impl men ution ue to eee the veloit o the lui in the oun le egion. hi n e epline to lmn ollow; in e o ution the hete lui i puhe tow the wll whee the uon oe n t to et the lui ue to high inluene o the vioit. hi eet t to eee the wll he te. In ition inee in the vlue o ution pmete lo le to eee o tempetue o the lui in the oun le egion. he eplntion o uh ehvio i tht the lui i ought loe to the ue n eue the theml oun le thine ee igue. Intentionl ogniztion o ientii Reeh g e

7 ' ' Anli on Vile Flui Vioit o Non-Din Flow ove Moving Vetil lte in oou.8.6 =. =. =.6 = Figue : Veloit oile o ieent vlue o tempetue epenent vile lui vioit F.5 D. N..7. E =. =. =.6 = Figue : empetue oile o ieent vlue o tempetue epenent vile lui vioit F.5 D. N..7. E = =.5 =. = Figue : Veloit oile o ieent vlue o emeilit mete. F.5 D N..7. E.. Intentionl ogniztion o ientii Reeh g e

8 ' Anli on Vile Flui Vioit o Non-Din Flow ove Moving Vetil lte in oou.8.6. = =.5 =. = Figue : empetue oile o ieent vlue o emeilit mete. F.5 D N..7. E F = F = F = F = Figue 5 : Veloit oile o ieent vlue o Lol Fohheime mete F.. D N..7. E F = F = F = F = Figue 6 : empetue oile o ieent vlue o Lol Fohheime mete F.. D N..7. E.. Intentionl ogniztion o ientii Reeh 5 g e

9 ' ' Anli on Vile Flui Vioit o Non-Din Flow ove Moving Vetil lte in oou.8.6 D =.6 D =.5 D =.76 D = Figue 7 : Veloit oile o ieent vlue o Lol D mete D. F.5. N..7. E D =.6 D =.5 D =.76 D = Figue 8 : empetue oile o ieent vlue o Lol D mete D. F.5. N..7. E =.5 =. =.7 = Figue 9 : Veloit oile o ieent vlue o ntl mete. F.5. D N.. E.. Intentionl ogniztion o ientii Reeh 6 g e

10 ' Anli on Vile Flui Vioit o Non-Din Flow ove Moving Vetil lte in oou =.5 =. =.7 = Figue : empetue oile o ieent vlue o ntl mete. F.5. D N.. E = =. =.8 = Figue : Veloit oile o ieent vlue o Het Genetion mete. F.5. D N..7 E.... = =. =.8 = Figue : empetue oile o ieent vlue o Het Genetion mete. F.5. D N..7 E.. Intentionl ogniztion o ientii Reeh 7 g e

11 ' ' Anli on Vile Flui Vioit o Non-Din Flow ove Moving Vetil lte in oou = = -. = -.8 = Figue : Veloit oile o ieent vlue o Het Aoption mete. F.5. D N..7 E = = -. = -.8 = Figue : empetue oile o ieent vlue o Het Aoption mete. F.5. D N..7 E E = E = E = 6 E = Figue 5 : Veloit oile o ieent vlue o Eet mete E.F.5. D N..7.. Intentionl ogniztion o ientii Reeh 8 g e

12 ' Anli on Vile Flui Vioit o Non-Din Flow ove Moving Vetil lte in oou.8 E = E = E = 6 E = Figue 6 : empetue oile o ieent vlue o Eet mete E.F.5. D N N =. N =.5 N = 5. N = Figue 7 : Veloit oile o ieent vlue o Rition mete N. F.5. D.7. E N =. N =.5 N = 5. N = Figue 8 : empetue oile o ieent vlue o Rition mete N. F.5. D.7. E.. Intentionl ogniztion o ientii Reeh 9 g e

13 ' Anli on Vile Flui Vioit o Non-Din Flow ove Moving Vetil lte in oou.9.8 = = = = = = = = Figue 9 : Veloit oile o ieent vlue o ution mete. F.5. D N..7. E..9.8 = = = = = = = = Figue : empetue oile o ieent vlue o ution mete. F.5. D N..7. E. Intentionl ogniztion o ientii Reeh g e

14 Anli on Vile Flui Vioit o Non-Din Flow ove Moving Vetil lte in oou le. in Fition n Nuelt nume Fo Ving Vlue O ontolling mete V. ONLUION In thi ppe oun le nli o ntul onvetion het tne o vile vioit inompeile lui ove linel moving poou vetil ue i oniee. Fom the numeil eult petining to the peent tu inite tht the eet o ineing tempetue epenent lui vioit pmete on viou inompeile lui i to inee the low veloit whih in tun ue the tempetue to eee. he highe vlue o the Fohheime nume F inite lowe veloit ve loe to the wll ut negligile om the wll n lo inite igniint inee o tempetue o the low egion. Veloit poile inee with the ineing vlue o D nume D. Mgneti iel et the motion o the lui n ution tilize the honmi n theml oun le gowth. Hoizontl veloit ' n tempetue eee with the inee in ntl pmete n ition pmete. VI. AKNOLEDGEMEN One o the utho Animun I. L. gteull nowlege the uppot o o. O. K. Koio D. E. A. Aeile n D. A. J. Omowe o thei invetment oth ietl n inietl on thi wo. REFERENE [] Bthelo G.K An Intoution to Flui Dnmi mige Univeit e Lonon. [] Bejn A. n Khi K.R Het n m tne ntul onvetion in poou meium Int. J. Het M ne Vol. 8 pp [] BewteM. Q. 97. heml Ritive ne opetie John ile n on hihete. [] eei. n Bhw. 98. hil n omputtionl Apet o onvetive Het ne New Yo pinge. [5] ne L.J. 97. Flow pt tething plte Z. Angew. Mth. h Intentionl ogniztion o ientii Reeh g e

15 Anli on Vile Flui Vioit o Non-Din Flow ove Moving Vetil lte in oou [6] Gupt.. n Gupt A Het n m tne on tething heet with ution n lowing n. J. hem. Eng [7] Le G.. 7. Muhoph. n m. K. Int. omm. Het M n. 7. [8] Li F.. n Kuli F.A. 99. ouple het n m tne om phee uie in n ininite poou meium Int. J. Het M ne Vol. pp.9-5. [9] Ling J.X. n D A Foe onvetion ove lt plte umee in poou meium: vile vioit e Amein oiet o Mehnil Enginee NY pe 87-A/H-. [] Lognthn. n Au... Lie Goup Anli o the Eet o Vile Flui Vioit n heml Rition on Fee onvetive Het n M ne with Vile tem onition ientii Reeh Jounl Vol oi:.6/eng..88 [] Moui M. n hite I Eet o vile vioit n viou iiption on the low o thi ge lui in pipe Int. J. Non-Line Meh [] Muhoph. n Le G.. 8. Eet o theml ition n vile lui vioit on ee onvetive low n het tne pt poou tething ue Intentionl Jounl o Het n M ne [] Neem. n Ali M. 9. Anltil olution o pipe low o outh ge lui with Renol n Vogel moel o vioitie ommunition in Nonline iene n Numeil imultion [] Niel D.A. n Bejn A.6. onvetion in oou Mei pinge-velg New Yo. [5] Oo... Dippene o itilit o etive thi - ge lui with Renol moel vioit in lt hnnel Int. J. Non-Line Mehni [6] emili M. n Yil B.. 6. Entop genetion o pipe low o thi ge lui with Vogel moel vioit Int. J. Non-Line Mehni -7. [7] nn M. L. Bh N. R. n oonim.. MHD Boun Le Flow o Het n M ne ove moving Vetil lte in oou Meium with ution n Diiption Intentionl Jounl o Engineeing Reeh n Applition IJERA Vol. Iue 5 pp IN: [8] Rpti A. zivnii G. n Koui N. 98. Fee onvetion n m tne low though poou meium oune n ininite vetil limiting ue with ontnt ution Lette Het M ne Vol.8 pp.7-. [9] ii B Boun-le ehviou on ontinuou oli ue: I. he oun-le eqution o two imenionl n mmeti low AIhE J. 76. [] ii B Boun-le ehviou on ontinuou oli ue: II. he oun-le on ontinuou lt ue AIhE J. 7. [] le J.J. n Hung H. 7. Etenionl low with viou heting Jounl o lui mehni vol. 57pp Intentionl ogniztion o ientii Reeh g e

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