Effects of wall properties and heat transfer on the peristaltic transport of a jeffrey fluid through porous medium channel

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1 Mhemicl heor nd Modeling ISSN -58 Per ISSN 5-5 Online Vol. No.9 Effecs of wll roeries nd he rnsfer on he erislic rnsor of jeffre flid hrogh oros medim chnnel Dhei G. Slih Al-Khfj College of Comer Science nd Mhemics Uniersi of Al-Qdissi Diwni-Irq. E-mil: Absrc Ahmed M. Abdlhdi College of Science Uniersi of Bghdd Bghdd-Irq. E-mil: A mhemicl model is consrced o sd he effec of he rnsfer nd elsici of fleible wlls wih oros medim in swllowing of food bols hrogh he oesohgs. he food bols is sosed o be Jeffre flid nd he geomer of wll srfce of oesohgs is considered s erislic we hrogh oros medim. he eressions for emerre field il eloci rnserse eloci nd srem fncion re obined nder he ssmions of low Renolds nmber nd long welengh. he effecs of herml condcii Grshof nmber rc nmber mgne rigidi siffness of he wll nd iscos dming force rmeers on eloci emerre nd srem fncion he been sdied. I is noiced h increse in herml condcii rc nmber Grshof nmber nd he Jeffre rmeer resls in increse of eloci disribion. I is fond h he size of he red bols increses wih increse in he Jeffre rmeer rigidi nd siffness. Kewords: Mgneohdrodnmic Perislic rnsor Oesohgs Jeffre flid Poros medim Food bols.. Inrodcion Perislic rnsor is mechnism of ming flids in bes when rogressie we of re conrcion or ension roges long he lengh on he bondr of disensible be conining flid. Perislsis hs qie imorn licions in mn hsiologicl ssems nd indsr. I occrs in swllowing food hrogh he oesohgs chme moion in he gsroinesinl rc in he somoion of smll blood essels sch s enles cillries nd rerioles rine rnsor from kidne o bldder. In iew of hese biologicl nd indsril licions he erislic flow hs been sdied wih gre ineres. Mn of he hsiologicl flids re obsered o be non-newonin. Perislic flow of single flid hrogh n infinie be or chnnel in he form of sinsoidl we moion of he be wll is inesiged b Brns nd Prkes Shiro e l. ec. In he lierre some imorn nlicl sdies on erislic rnsor of non Newonin flids re ilble Dei nd Denhn Shkl nd G Sris nd Sris Ush nd Ro Vjrel e l. 5 5bH e l. 8b. Frher n ineresing fc is h in oesohgs he moemen of food is de o erislsis. he food moes from moh o somch een when side down. Oesohgs is long msclr be commences he neck oosie he long border of cricoids crilge nd eends from he lower end of he hrn o he crdic orifice of he somch. he swllowing of he food bols kes lce de o he eriodic conrcion of he esohgel wll. Pressre de o refleie conrcion is eered on he oserior r of he bols nd he nerior orion eeriences relion so h he bols moes 8

2 Mhemicl heor nd Modeling ISSN -58 Per ISSN 5-5 Online Vol. No.9 hed. he conrcion is rcicll no smmeric e i conrcs o zero lmen nd sqeezes i mrelosl wiho leing n r of he food bols sli bck in he oosie direcion. his shows he imornce of erislsis in hmn beings. Mir nd Prsd sdied he inflence of wll roeries on he Poiseille flow nder erislsis. Mhemicl model for he esohgel swllowing of food bols is nlzed b Mishr nd Pnde. Kih e l. nlzed he erislic flow of microolr flid in ericl chnnel wih longwe lengh roimion. Redd e l. sdied he effec of hickness of he oros meril on he erislic ming when he be wll is roided wih non-erodible oros lining. Lkshminrn e l. sdied he erislic ming of condcing flid in chnnel wih oros eriherl ler. Rdhkrishnmchr nd Srinisl sdied he inflence of wll roeries on erislic rnsor wih he rnsfer. Rhod e l. sdied he inflence of wll roeries on MHD erislic rnsor of ds flid. A new model for sd he effec of wll roeries on erislic rnsor of iscos flid hs been inesiged b Mokhr nd Hron Srinis e l. sdied he effec of sli wll roeries nd he rnsfer on MHD erislic rnsor. Sreendh e l. sdied he effecs of wll roeries nd he rnsfer on he erislic rnsor of food bols hrogh oesohgs. Afsr Khn e l. nlzed he erislic rnsor of Jeffre flid wih rible iscosi hrogh oros medim in n smmeric chnnel. Moied b his we consider mhemicl model o sd he effec of wll roeries nd he rnsfer on swllowing he food bols hrogh he oesohgs. he resls re nlzed for differen les of rmeers nmel Grshof nmber rc nmber herml condcii mgne rigidi siffness nd iscos dming forces of he chnnel wll hrogh oros medim.. Mhemicl Formlion Consider he erislic flow of n incomressible Jeffre flid in fleible chnnel wih fleible indced b sinsoidl we rins roging wih consn seed c long he chnnel wlls. he wll deformion is gien b H cos c where h nd c reresen rnserse ibrion of he wll il coordine ime hlf widh of he chnnel mlide of he we welengh nd we eloci reseciel.. Bsic eqions he bsic eqions goerning he non-newonin incomressible Jeffre flid re gien b: he conini eqion is gien b: 87

3 Mhemicl heor nd Modeling ISSN -58 Per ISSN 5-5 Online Vol. No.9 88 he momenm eqions re: k B g k he emerre eqion is gien b: c k 5 where is he il eloci rnserse eloci rnserse coordine flid densi ressre flid iscosi g ccelerion de o gri coefficien of liner herml ension of flid B mgneic field emerre c secific he consn ressre k is he herml condcii nd consn he ddiion/bsorion. he eloci nd emerres he cenrl line nd he wll of he erislic chnnel re gien s: h where is he emerre cenre is line nd is he emerre on he wll of erislic chnnel. he goerning eqion of moion of he fleible wll m be eressed s: * L where * L is n oeror which is sed o reresen he moion of sreched membrne wih iscosi dming forces sch h C m L * 7 where is he elsic ension in he membrne m is he mss er ni re C is he coefficien of iscos dming forces. Conini of sress h nd sing momenm eqion ield k B g h L * 8 In order o simlif he goerning eqions of he moion we m inrodce he following dimensionless rnsformions s follows:

4 Mhemicl heor nd Modeling ISSN -58 Per ISSN 5-5 Online Vol. No.9 89 k c k B M c g Gr c k s Q Q c c c c c Pr Re 9 where is he lengh of he chnnel Srem fncion Q Volme flow re rc nmber Re Renolds nmber Gr Grshof nmber dimensionless emerre M mgneic rmeer dimensionless he sorce/sink rmeer nd Pr is Prndl nmber. Sbsiing 9 ino eqions -8 we obin he following non-dimensionl eqions nd bondr condiions: cos h M c g Re Re Pr Re Re h E h E h E M Gr 5 = = he reglri condiion = = h he no sli condiion = = he bsence of rnserse eloci = = nd = = h. Solion of he roblem he generl solion of he goerning eqions -5 in he generl cse seems o be imossible; herefore we shll confine he nlsis nder he ssmion of smll dimensionless we nmber. I follows h. In oher words we considered he long-welengh roimion. Along o his ssmion eqions -5 become:

5 Mhemicl heor nd Modeling ISSN -58 Per ISSN 5-5 Online Vol. No.9 h cos 7 Gr M 8 9 M Gr E h E h E h Eqion shows h deendens on onl. he closed form solion for eqions 8- wih he bondr condiions Eq. is gien b h h B e M M B e hgr h M M E h M h Gr E h Cos E h Sin E M M {hgr h GrM [Gr h Sin ]} where B nd B re consns cn be deermines b sing he bondr condiions Eq.. he corresonding Srem fncion cn be obined b inegring Eq. nd sing he condiion. I is gien b GrM M M E E M h Gr h M E M Cos Be M M Gr M Sin Be M E B B M M 5 5. Resls nd Discssion In his secion he nmericl nd comionl resls re discssed for he roblem of n incomressible non-newonin Jeffre flid in he chnnel wih he nd mss rnsfer hrogh he 9

6 Mhemicl heor nd Modeling ISSN -58 Per ISSN 5-5 Online Vol. No.9 grhicl illsrions. he nmericl elions of he nlicl resls nd some imorn resls re disled grhicll in Figre -. MAHEMAICA rogrm is sed o find o nmericl resls nd illsrions. From Figre disls he effec of rigidi rmeer in he resence of siffness E nd iscos dming force E. I is noiced h he eloci increses wih increse in rigidi rmeer. A similr obserion is mde for differen les of E in he resence of oher rmeers i.e. rigidi nd iscos dming force which is shown in Figre. 8 E. E.5 E E Fig. Veloci disribion for differen les of E wih.5. E.5 E.5 Gr...7 M.9. E. 5 E.5 E E Fig. Veloci disribion for differen les of E wih.. E.7 E.5 Gr...7 M.9. 9

7 Mhemicl heor nd Modeling ISSN -58 Per ISSN 5-5 Online Vol. No E. E.5 E E Fig. Veloci disribion for differen les of E wih.. E.7 E.5 Gr...7 M.9. From figre we cn see he inflence of iscos dming force on eloci disribion in he resence of rigidi nd siffness. One cn obsere h he eloci decreses wih he increse in E. Figre5 illsres he effec of he rmeer Grshof nmber Gr on eloci disribion we see h increses wih he incresing of Gr when <. Figres nd 7 i is obsered h increse in Jeffre rmeer nd herml condcii resls in increse of eloci disribion 8 Gr Gr Gr Gr Fig 5. Veloci disribion for differen les of Gr wih.. E.7 E.5 E M.9. 9

8 Mhemicl heor nd Modeling ISSN -58 Per ISSN 5-5 Online Vol. No Fig. Veloci disribion for differen les of wih.. E.7 E.5 E.5 Gr..7 M Fig 7. Veloci disribion for differen les of wih.. E.7 E.5 E.5 Gr...7 M Fig 8. Veloci disribion for differen les of wih.. E.7 E.5 E.5 Gr M

9 Mhemicl heor nd Modeling ISSN -58 Per ISSN 5-5 Online Vol. No M M.5 M M Fig 9. Veloci disribion for differen les of M wih.. E.7 E.5 E.5 Gr Fig. Veloci disribion for differen les of wih.. E.7 E.5 E.5. Gr. M.9. Figre 8 show h eloci disribion increses wih he incresing of. Figre 9 show h eloci disribion decreses wih he incresing of mgneic rmeer M while Figre i is obsered h increse in rc nmber resls in increse of eloci disribion. he riion in emerre for rios les of herml condcii is shown in Figre. he emerre increses wih he increse in.he riion in emerre for rios les of herml condcii is shown in Figre. he emerre increses wih he increse in Fig. Veloci disribion for differen les of wih... 9.

10 Mhemicl heor nd Modeling ISSN -58 Per ISSN 5-5 Online Vol. No.9 ring henomenon he formion of n inernll circling bols of flid b closed sremlines is clled ring nd his red bols is shed hed long wih he erislic we. he effecs of E E E Gr M nd D on ring cn be seen hrogh Figres -. Fig. show h he size of he red bols increse wih he increse in E. Fig. is loed he effec of E on ring he size of he red bols increse wih he increse in E. Fig. show h he size of he red bols decrese wih he increse in E. he effec of herml condcii on ring is nlzed in Figre 5. I cn be conclded h he size of he red bols in he lef side of he chnnel decreses when increses where s i hs oosie behior in he righ hnd side of he chnnel. he inflence of Grshof nmber Gr on ring is nlzed in Figre. I shows h he size of he lef red bols decreses wih increse in Gr where s he size of he righ red bols increses wih increse in Gr. he effec of on ring cn be seen in Figre 7. We noice h he size of he bols increses wih increse. he effec of on ring is nlzed in Figre 8. We noice h he size of he bols increses wih increse. Fig. Grh of he sremlines for for differen les of E ; E =.5 b. E.5 E.5.9 M.9 Gr... E =.5 nd c E = g. Grh of he sremlines for for differen les of E ; E =. b E =.5 nd c. E.7 E.5.9 M.9 Gr... E =.5 Fi 95

11 Mhemicl heor nd Modeling ISSN -58 Per ISSN 5-5 Online Vol. No.9 Fig. Grh of he sremlines for for differen les of E ; E =.5 b E =.5 nd c. E.7 E.5.9 M.9 Gr... E = Fig. 5 Grh of he sremlines for for differen les of ; = b = nd c =. E.7 E.5 E.5.9 M.9 Gr... Fig. Grh of he sremlines for for differen les of Gr; Gr = b Gr = nd c Gr =. E.7 E.5 E.5.9 M

12 Mhemicl heor nd Modeling ISSN -58 Per ISSN 5-5 Online Vol. No.9 Fi g.7 Grh of he sremlines for for differen les of ; = b =. nd c. E.7 E.5 E.5.9 M.9 Gr.. =. Fig.8 Grh of he sremlines for for differen les of ; =.5 b =. nd c =.5. E.7 E.5 E.5.9 M.9 Gr.. he inflence of rc nmber on ring is nlzed in Figre 9. I shows h he size of he lef red bols decreses wih increse in where s he size of he righ red bols increses wih increse in. And Figre show h inflence of M on ring. I shows h he size of he lef red bols increses wih increse in M where s he size of he righ red bols decreses wih increse in M. 97

13 Mhemicl heor nd Modeling ISSN -58 Per ISSN 5-5 Online Vol. No.9 Fig.9 Grh of he sremlines for for differen les of ; =. b =.9 nd c =.5. E.7 E.5 E.5 M.9 Gr... Fig. Grh of he sremlines for for differen les of M; M =.5 b M = nd c M =.5. E.7 E.5 E.5.9 Gr Conclding remrks he resen sd dels wih he combined effec of wll roeries nd he rnsfer on he erislic rnsor of Jeffre flid hrogh oros medim chnnel. We obined he nlicl solion of he roblem nder long welengh nd low Renolds nmber ssmions. he resls re nlzed for differen les of erinen rmeers nmel Grshof nmber rc nmber herml condcii rigidi siffness mgne nd iscos dming forces of he chnnel wll. Some of he ineresing findings re;. he il eloci increses wih he increse in E E Gr nd. Frher he il eloci decreses wih increse in E nd M. nd ins is mimm heigh = s secified b he bondr condiions.. he olme of he red bols increses wih increse in E E nd. Moreoer more red bols ers wih increse in E E nd.. he olme of he red bols decreses wih increse in E.. he olme of he lef red bols decreses wih increse in Gr where s i hs oosie behior in he righ hnd side of he chnnel. And inersion wih resec o nd M. 5. he coefficien of emerre increses wih incresing les of herml condcii. REFERENCES Brns JC nd Prkes 97 Jornl of Flid Mechnics

14 Mhemicl heor nd Modeling ISSN -58 Per ISSN 5-5 Online Vol. No.9 Shiro AH Jffrin MY nd Weinberg SL 99 Jornl of Flid Mechnics Dei G nd Denhn R 975 Proceedings of Indin Acdem Science 8A 9-. Shkl JB nd G SP 98 Jornl of Biomechnicl Engineering 8-8. Sris LM nd Sris VP 98 Jornl of Biomechnics Ush S nd Ro AR 995 Jornl of Biomechnics Vjrel K Sreendh S nd Rmesh Bb V 5 Inernionl Jornl of Nonliner Mechnics 8-9. Vjrel K Sreendh S nd Rmesh Bb V 5b Alied Mhemics nd Comion H nd Ali N 8 Commnicion in Nonliner Science nd Nmericl Simlion - 5. H W Sllem N nd Ali N Commnicion in Nonliner Science nd Nmericl Simlion H Sjjd R nd Asghr S b Commnicion in Nonliner science nd Nmericl simlion 5 -. Mir K nd Prsd S N 97 Jornl of Biomechnics 8-9. Mishr JC nd Pnde SK Mhemicl nd comer modelling Kih A Hemdri Redd R Sreendh S Srn R nd Srinis ANS Adnces in lied Science Reserch Hemdri Redd R Kih A Sreendh S nd Hrirbhkrn P Adnces in lied Science Reserch Lkshminrn P Sreendh S nd Schrih G Adnces in Alied Science Reserch Rdhkrishnmchr G nd Srinisl Ch 7 Comer Rends Mecniqe Rhod VP nd Plli Klkrni Adnces in lied Science Reserch Mokhr A Abd Elnb nd Hron MH 8 Commnicion in Nonliner Science nd Nmericl simlion Srinis S Ghri R nd Kohndni M 9 Comer Phsics Commnicions 8 -. Sreendh S Um Shnkr C nd Rg Plli A In.J.of Al.Mh nd Mech Afsr khn A Ellhi R nd Vfi.K. Adnces in Mhemicl Phsics

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