Life Science Journal 2014;11(7)

Transcription

1 Lif in Jonl 04;7 h://.lifsinsi.om Homooy Pbion Mhod Fo H nsf Of Pisli Flo Of Visolsi Flid In Eni Cylind Wih Vibl Effs gd Albi Assm F. Elsyd nd O An Bég ^ Mh. D. Fly of lid sin ibh Univsiy Almdinh Al Mon di Abi Mh D. Fly of Edion Ain shms Univsiy Hlioolis Cio Egy. ssm_lsyd@yhoo.om Dmn of Engining nd Mhmis hffild Hllm Univsiy UK goob@gmil.om Abs: In his h Homooy bion mhod HPM hs bn sd o solv h nomlid sysm of non-lin il diffnil qions dsibing h momnm ngy nd mss nsf nd oi bondy ondiions of isli visolsi Jffy flid flo ih omlin ll. h infln of Pndl nmb Ek nmb dis im on vibl visosiy vloiy m nd xil ss gdin disibions sdid ghilly. [gd Albi nd Assm F. Elsyd nd O An Bég. Homooy Pbion Mhod Fo H nsf Of Pisli Flo Of Visolsi Flid In Eni Cylind Wih Vibl Effs. Lif i J 04;7:97-06]. IN: h://.lifsinsi.om. 4 Ky ods: Homooy bion mhod HPM ni ylind Pisli flo Jffys flid vibl visosiy.. Inodion Biohysil flid dynmis onsis ih of modn sh in biomhnil ngining. In il non-nonin nso hnomn in h hmn body [] onin o siml h nion of ngining siniss nmil nlyss lid hysiiss nd mhmil biologiss. h song nonliniy of flos nond in non-nonin dynmis gnlly qis ofl nmil mhods fo obs nd gmi solions []. Biohologil flos [] xmlifid by bil blood ms digsiv flids synovil lbins [4]. hs flids dvi fom h lssil Nonin lin lionshi bn h sh sss nd sh. Rhologil hisis xhibid by sh flids xmly divs nd o ddss lisi flos id sm of non-nonin modls hv bn dvlod ov h s fiv dds. hs inld mio-sl flids [5] hih mimik oy moions of il ssnsions nd yhoys nd bsd on Eingn s mio-mohi hoy [6]. Visolsi flids [7] off diffn fomlion fo hologil ffs nd imd simling mmoy ffs nsil sss lxion lsiiy nd oh fs of l biologil liqids. Exmls of sh modls s imlmnd in mdil ngining inld h Klvin-Voig modl [8] in synovil flos Mxll s modl [9] in blood flos miil modls [0] fo y flid dynmis nd Oldoyd-B modls fo mbyologil nso []. In n ys isli nso hs lso gnd signifin nion. Pislsis is nl mhnism in biology sd o m liqids by mns of moving onion on h bs o hnnl lls. I iss in food migion vi h sohgs hym moion in h gsoinsinl vsomoion of smll blood vssls sh s vnls illis nd iols nd fhmo is fndmnl o l nso fom h kidnys o h bldd. Non- Nonin sdis of islsis hv hivd in n ys. A divs y of hologil modls hv bn imlmnd in isli nso inlding Mxll flids [] Jffys flids [] Willimson flids [4] Oldoyd-B flids [5] Johnson-glmn flids [6] nnoflids [7] liqid ysl nmogns [8] ol sss flids [9] nd visolsi Binghm yild sss flids [0]. All hs modls involv mndos ng of onsiiv qions nd mimik xnsiv fs ising in l hologil flids. In h vs mjoiy of simlions of non- Nonin isli flos h flid ois sh s visosiy ssmd o b onsn. In gnl h offiins of visosiy fo l flids fnions of s oodin m nd ss. An xml of biologil oblm involving vibl visosiy nd flid-s inion is h bhvio of d blood lls in illis. Rd blood lls hih hv n inio visosiy oximly 5 ims g hn h sonding blood lsm knon o ndgo lg dfomions s hy sq hogh vssls h n b s lil s hlf hi dim. h viion of visosiy d o s oodin nd m hng is mo dominn hn oh ffs. hfo i is highly dsibl o inld h ff of vibl visosiy insd of onsiding h visosiy of h flid o b onsn. vl shs hv onsidd vibl visosiy in isli flid 97

2 Lif in Jonl 04;7 h://.lifsinsi.om dynmis inlding ihi l. [] ho sd n xonnil modl. Fh sdis inld []-[0]. Mos invsigions of isli nso lso onsid sigh ondis nd ngl v hih is ky f in l mdil liions. h infln of v n b vy ominn on flo s nd v ogion. Insing sdis in his gd hv bn ommnid by visky l. [] sing fini lmn od nd Rh [] ho sd bion mhods o solv h ini-f Nvi-oks qions. Mo nly hologil sdis of vd b isli nso hv ivd som nion. Rlvn oks in his gd inld Hin l. [] ho sd h Osld-dWl sh hinning modl Hin l. [4] ho mloyd Johnson-glmn visolsi modl Ali l. [5] ho ilid Rin-Rivlin hid od modl nd Ali l [6] ho imlmnd h Eingn miool modl in vd b hyhmi ll flos. In n ys mny divs nmil mhods hv bn imlmnd o nly omlx isli flos. hs inld fini lmn ods sh s ADINA-FI [7] COMO [8] sl fini lmns [9] viionl fini lmns [40] inding non-iiv fini lmn nlysis [4] Adomin s domosiion mhod ADM [4] Li Bolmnn mhods LBM [4] nd fini volm omionl flid dynmis [44]. In n ys h so-lld homooy smi-nmil mhods hv lso mgd s signifin ool in bioflid mhnis nd oh s. On il mhod H s homooy bion mhod HPM [45] hih is sil s of Lio s homooy nlysis mhod HAM [46] hs ovd immnsly ol nd sssfl in solving highly nonlin biologil nso oblms. I hs nly bn imlmnd in ls bioh nsf by El yd [47] nd in isli visolsi flo in mbl mils by ihi nd Bég [48]. h onvgn of homooy bion mhod HPM hs bn invsigd by Bi nd Ghvini [49]. In h sn sdy xmin h isli flo of Jffys visolsi flid hogh ni ylindil ondi ih omlin lls in h sn of h nsf ffs. hml diffsion nd hml ondiviy onsidd o b fnions of onnion nd flid m nd fhmo m-dndn visosiy is ilid in h modl. Visos hing ffs lso inood in h sn simlions. h non-lin nondimnsionlid il diffnil qions sbj o hysilly lisi bondy ondiions solvd by mns of H s Homooy Pbion Mhod HPM. h ffs of h hysil ms on h hml ondiviy nd flid m hv bn skhd ghilly. h sn sdy is lvn o novl isli m dsigns.. Mhmil nso Modl L s onsid h isli flo of visolsi Jffys flid ih vibl hysil ois flid visosiy hml ondiviy nd mss diffsiviy hogh nifom oxil bs. h inn b is kn s igid nd h o dfombl b hs sinsoidl v vlling don is ll. h inn b is lod h osiion 0 s shon in Fig.. Fig. Gomy of h flo nd o-odin sysm W sl ylindil oodin sysm ih dnoing dil imhl nd xil oodins sivly. h gomil shs of h o lls dfind mhmilly by h lions: b Cos[ ] Wh nd h dii of h inn nd o bs b is mlid of h v is h v lngh is h ogion vloiy nd is h im. Fh ssm h h bondy of h inn b is h m nd h o b is minind m. If V v is h vloiy. h flid ois ssmd o isooi nd onsn x fo h flid visosiy hml ondiviy. h oniniy momnm ngy nd onnion qions fo n inomssibl Jffy flid dsibd s follo; V 0 V V V g V k V

3 Lif in Jonl 04;7 h://.lifsinsi.om 99 Wh is h dnsiy is h offiin of hml xnsion. is h m nd k is h hml diffsiviy K is h hml diffsion io nd fo Jffy flid is givn by h lion 6 h io of lxion o dion ims h dion im d d is sh d d nd h mil im diffniion. h isli flo of Jffy flid is nsod by ogssiv v of onion o xnsion long omlin ll h qion osonding o h omlin ll is [6] 7 L i Wh d M L is n oo.h is h lsi nsion in h mmbn M is h mss ni d is h offiin of visos dming nd is h ss on h osid sf of h ll. I is ssmd h 0 nd h lls of h hnnl inxnsibl so h only hi ll moion ks l nd h hoionl dislmn of h ll is o hs h no-sli bondy ondiion fo h vloiis is i g V d M i l s ssm h h vloiy fild fo h flo is 0 V. h dimnsionlss ms sd in h oblm dfind s follos 9 * 0 * * * R k k K b Wh is h mlid io is h dimnsionlss v nmb. Using h bov dimnsionlss ms h govning qions dd o h folloing fom f doing s mk 0 0 R G R E P k K P R h dimnsionlss bondy ondiions i G R i h R is h Rynolds nmb g G is h Gshof nmb k P is h Pndl nmb D is h hmid nmb E is h Ek nmb d E m E E h non-dimnsionlss ll omlin ms. h omonns of non-dimnsionl ssss fo Jffy flid vld s 5 Fo h sn nlysis ssm h dynmi visosiy h hml ondiviy K [4-0] 6...!! f b b K. mi-anyil Hm olions In od o solv h nonlin bondy vl oblm dsibd by qns. 0 nd bondy ondiions 4 nd h long vlngh oximion 0 0. Uiliing qns. 6 ons h folloing homooy mloying h HPM oh [4748]:

4 Lif in Jonl 04;7 h://.lifsinsi.om = 0 7 = h foms of h solions of h qions 7-8 s follos = 9 = Wh q [0] is n mbdding m = =. l s sggs h iniil gss Wh = = + = + = + = [ ] + 0 Log Log Log Log Log + Log 4 + Log Log Log Log Log + Log 4 + Log Log + + Cos[ + ] Log Log Log + Log Log Log Cos[ + ] γ = β β = Log Log 5 η η η F[] 6 bsiing h solions 0- ino qions. 7-8 nd qing h ms of h sm o of q by qing ll ms of g h folloing sysm of odiny diffnil qion Zo Od ysm = 0 = 0 7 Wih h folloing bondy ondiions = 0 = 0 = = = + + = + + = = = 8 00

5 Lif in Jonl 04;7 h://.lifsinsi.om h solions of h bov o od sysms n b obind by sing qions 0- s folloing = + [ ] + Log Log 9 = Log Log Fis Od ysm = + = + + Wih h folloing bondy ondiions = 0 = = 0 = = 0 = = = h solion of h bov fis od sysm n b obind by sing qions s folloing = Log+ = + Log+ 0 h = + = = = = = 5 = 6 = = + 7 0

6 Lif in Jonl 04;7 h://.lifsinsi.om = = + = + 9 = = = 4 Log + Log α = α = 4 Log Log Log Log hn h solions of qions 7-8 oding o h bondy qions 4 by sing HPM n b in s follos; = lim = lim 4 4. Rsls And Inion ld omions snd in Figs -. In ll ss smooh nd sbl solions hv bn obind ih HPM. Figs. - ills h dynmi visosiy nd hml ondiviy Wih n ls in im h is signifin dssion in dynmi visosiy fig.. Fhmo sdy sn in visosiy omnis is in vl of h dishov iil vl of dis is ind osonding o h oimm visosiy ind. Visosiy k is lso obsvd o b disld slighly o high dis ih ds in im. Fig. shos h hml ondiviy is dd ih boh n ins in Pndl nmb nd Ek nmb. hml ondiviy is lso mximid dis vls of oximly 60 s ih visosiy. Hov ofils bom insingly skd o h igh ih lo Pndl nmbs. h insd im fo lxion of h flid lds o dssion in visosiy. his nd hs lso bn idnifid in nmb of oh sdis inlding Ndm nd Akb []. In hology [7] h dd visosiy hs lso bn ssoid ih do in nsil ssss in h visolsi flid ih ogssion in im hih ids lxion of h liqid nd onibs o ds in blk visosiy of h flid. I old h h Jffys visolsi modl lso obys his hysil hisi. Evn hogh livly smll im lss hv bn onsidd in fig. h sls sm omising fo nso of hologil liqids in il fo biohmil s isli ms h insd ffiiny is hivd ih lo visosiy s lbod by fo xml ihi nd Bég [] fo sigh ondis nd lso li by Rh [] fo vd b nso. h dion in visosiy ill sv lso o ds h fiionl sisn on h ll inios mnifsing in nhnd ming ffiinis. h omions in fig. osond o non-o v nd his shos h isli ffiiny n b nhnd in vd sysms. h hysil inion fo h ds in hml ondiviy ih insing Pndl nmb P ls o h invs lionshi of hs o vibls. P is h io of h od of dynmi visosiy nd sifi h iy o h hml ondiviy. As Pndl nmb inss fo onsn visosiy nd sifi h iy h hml ondiviy ms d. I lso mbodis h liv of visos momnm diffsion o ngy diffsion. h momnm diffsion ill hfo ins s h hml ngy diffsion flls. Fo P < s invsigd in fig. h momnm diffsion is lys lo hn hml diffsion. Hov h ins in Pndl nmb lvs h mgnids of h fom ih gd o h l. h imliion fo l isli osss.g. gsi nso o indsil ho s nso is h lss h is diffsd fo high Pndl nmb ssnsions. his ill invibly lso infln h liv s ih hih mils n b md in h onfigion. Ho flids ill sli ih g s. h Ek nmb E is lso ky m hih is ssoid ih visos hing i.. kini ngy dissiion nd onvsion o h. As E is insd h kini ngy onvd o h is nhnd. G h hs hfo o b nsod in h gim nd his ill dss hml ondiviy s obsvd in fig.. 0

7 Lif in Jonl 04;7 h://.lifsinsi.om Fig. h visosiy disibion dis fo diffn ims. vss Fig 6. Pss gdin vss fo diffn vls of ll omlin m. Fig. hml ondiviy vss dis fo diffn vls of Pndl nmb nd Ek nmb. Fig 7. Pss gdin vss fo diffn vls of ll omlin m. Fig 4. m disibions vss dis fo diffn vls of ll omlin m. Fig 8. Pss gdin vss fo diffn vls of ll omlin m. Fig 5. Vloiy disibion vss dis fo diffn vls of ll omlin m. Fig 9. Pss gdin vss fo diffn vls of hml Gshof nmb. 0

8 Lif in Jonl 04;7 h://.lifsinsi.om Figs 4 5 di h ff of h ll omlin m E on h m nd xil vloiy disibions in h gim sivly vss dil E oodin. his m dfind by iss in h lsi ll bondy ondiions 6 nd is offiin of hid od m E - s sh i ill x onsidbl infln on flo hisis. Wih insing E m is nd in h gim signifinly imlying h ih g ll nsion h gim is hd mo sbsnilly. his is no dob ssoid inimly ih h lion in h xil flo hih lso omnis n ins in h omlin m E s obsvd in fig. 6. G nsion in h ondi ll lly ids in momnm dvlomn nd ill lso ld o n nhnd isli ming ffiiny. imil obsvions hv bn md by Alsdi l. [] fo Osld-dWl sh-hinning flids Hin l. [4] fo Nonin flids nd Ali l. [5] fo hid gd visolsi flids. Fig. 6 sns h sons in xil ss gdin disibion o viion in h fis omlin m E. Wih insing xil oodin i.. longidinl oodin h ss gdin iniilly snds smoohly ks nd hn dys lg vls. As E inss h ss gdin is lly sbsnilly lvd. his is hysilly ih h xil lion omd li in fig. 5. G nsion in h ondi lls ill s lbod li o l h flo nd nhn boh xil vloiy nd ss gdin. his ill ffivly boos h isli ming ffiiny fo h sysm. Alhogh h l hs no bn lod fo bviy h olion bn xil vloiy xil ss gdin nd hydomhnil ffiiny is lly domnd in mny sdis inlding visky l. [] ihi l. Bég l. [4] Ndm nd Akb [] hih hv mloyd divs ng of hologil modls. Fig. 7 illss h ff of h sond ll omlin m E on xil ss gdin. h n omd is vy simil o fig. 8 i.. is in h ins E is obsvd o nhn h ss gdin. E iss lso in h ll bondy ondiions 6 b in mixd diviv vi E -i lly inflns h flid dynmis s songly s h fis omlin m E nd in f gns onsidbly high mgnids of ss gdin. E is ssoid ih h mss ni of h omlin ll no h lsi nsion. A dns ll onsiion hfo lso lvs ss gdin. Fig. 8 lso shos h n ins in h hid ll omlin m E sls in onsidbl lvion in xil ss gdin. E is onnd o h visos dming of h ll. Mgnids of d/d hov fo of 0 lo in fig. 0 hn in fig. 9. Fig.9 shos h hml Gshof nmbs G h xil ss disibions. A mkd dssion is sd in d/d boh ih insing G vl. 5. Conlsions A HPM homooy bion mhod smiomionl solion hs bn dvlod fo nonlin visolsi isli flo h nd mss nsf ih oss-diffsion nd visos hing ffs in n ni vd o-xil ylindil ondi ih omlin lls. h non-dimnsionlid onsvion qions hv bn solvd. Vibl flid ois hv lso bn inood fo visosiy hml ondiviy nd mss diffsiviy of h sis oxygn. Fll dils of HPM hv bn ovidd inlding slion of mbdding ms. h ghil solions hv shon h: i Insing hml nd soll Gshof nmb songly ds xil ss gdin ny vl of dis. ii An ins in h lsi inil mss nd visos dming ll omlin ms onsisnly nhn h xil ss gdin. iii Insing lsi ll omlin m nd soll Gshof nmb sivly l nd dl h xil flo. iv Insing lsi ll omlin m nhns ms. v Insing im dss dynmi visosiy of h Jffys visolsi flid. vi Insing Ek nmb nd Pndl nmb boh d h hml ondiviy of h visolsi flid. vii Insing o nmb dss sis diffsiviy hs h onvs sons is omd ih n ins in Ek nmb. h sn sdy is lvn o gsi flid mhnis nd omlx isli ms fo biohmil s onvyn. Alhogh ing nd bols fomion hv no bn ddssd in h sn sdy hs ss nd invsigion nd ill b dsibd in n nsing il. Aknoldgmns h fis o hos old lik o xnd hi hnks nd iion o h Dnshi of inifi Rsh ibh Univsiy KA fo soing his oj by gn No.4/4 h ls ho OAB is 04

9 Lif in Jonl 04;7 h://.lifsinsi.om gfl o h l Pof. D. Hns J. Rh Cn of Alid hnology nd Miogviy ZARM Univsiy of Bmn Gmny fo som sfl disssions in h sing of 0 gding liions of nonlin isli hology in biohmil ngining. Rfns. R.K. Hobbi nd B. Roh Inmdi Physis fo Mdiin nd Biology ing N Yok 4 h dn M.M. Rshidi M. Kimnsh O. An Bég nd.k. Hng Mgnohydodynmi biohologil nso hnomn in oos mdim: A simlion of mgni blood flo onol nd filion In. J. Nm. Mh. Biomd. Eng. 7: G.W. o Bli nd D.C. nn An Inodion o Biohology Elsvi in Pblishing Comny N Yok H.L. Goldsmih Biohology in h s ny Biohology A.C. Eingn Mio-oninm Fild hois- Vol II: Fln Mdi ing N Yok A.C. Eingn iml mioflids In. J. Engining in Jn d Vin Edio Visolsiiy - Fom hoy o Biologil Aliions Inh UA D.V. Dvis nd A J Plfy Visolsi ois of som hologil hmn synovil flids Ann. Rhm. Dis. 8: G.B. hson nd N.M. Hndson Effs of flo gomy on blood visolsiiy Biohology P. hif-kshni J-P. Hbshmn D. ssoon nd H. Pio Kvho Rhology of h vios gl: Effs of momoll ogniion on h visolsi ois J. Biomhnis H. C. F C. W. Wolgmh nd. R. Pos imming sds of filmns in nonlinly visolsi flids Phys. Flids D. ihi nd O. An Bég A nmil sdy of osilling isli flo of gnlid Mxll visolsi flids hogh oos mdim nso in Poos Mdi D. ihi.k. Pndy nd O. An Bég Mhmil modlling of h nsf ffs on slloing dynmis of visolsi food bols hogh h hmn osohgs In. J. hml ins O. An Bég M. Kimnsh M. M. Rshidi nd M. Dvoodi Mli-s DM simlion of mgno-isli flo of onding Willimson visolsi flid In. J. Al. Mh. Mh D. ihi O. An Bég nd J.L. Cil-os Pisli flo of gnlid Oldoyd-B flids ih sli ffs Com Mhods in Biomhnis Biomdil Engining 0. DOI:0.080/ Hin. Hy nd A. Alsdi H nd mss nsf ffs on h isli flo of Johnson- glmn flid in vd hnnl ih omlin lls In. J. H Mss n O. An Bég nd D. ihi Mhmi simlion of isli ming ih dobldiffsiv onvion in nnoflids: bionnongining modl Po. IMhE. P N: J. Nnongining nd Nnosysms J. G. Cnn A. E. Vsdkis L. D io nd D. Pslis Ooflidi modlo bsd on isli nmogn mioflos N Phoonis D. ihi nd O. An Bég Mgnohydodynmi isli flo of ol sss flid hogh oxil hnnls onining oos mdim J. Mhnis Mdiin Biology D. ihi nd O. An Bég Mhmil modling of isli ming of visolsi flids Po. IMhE. P H: J. Engining in Mdiin 0. In ss. D. ihi.k. Pndy A. iddiqi nd O. An Bég Non-sdy isli olsion ih xonnil vibl visosiy: A sdy of nso hogh h digsiv sysm Com Mhods Biomhnis nd Biomdil Engining DOI.ORG/ / Ndm nd N. Akb Pisli flo of Jffy flid ih vibl visosiy in n symmi hnnl Zishif f Nfoshng Ndm nd N. Akb Effs of h nsf on Pisli nso of MHD Nonin flid ih vibl visosiy: Aliion of Adomin domosiion mhod Comm. Nonlin. i. Nm. im Ndm nd N. Akb Effs of m dndn visosiy on Pisli flo of Jffy-six onsn flid in non-nifom vil b Comm. Nonlin. i. Nm. im Ndm Non h Akb nd M. Hmd Pisli nso nd h nsf of MHD 05

10 Lif in Jonl 04;7 h://.lifsinsi.om Nonin flid ih vibl visosiy Innionl Jonl fo Nmil Mhods in Flids Ndm Non h Akb Infln of m dndn visosiy on isli nso of Nonin flid: Aliion of n ndoso Alid Mhmis nd Comion Ndm. Akb N.. nd Hy. Effs of vibl visosiy on h isli moion in hid od flid Zishif f Nfoshng A Ndm Non h Akb Infln of h nsf nd vibl visosiy in vil oos nnls ih islsis Jonl of oos mdi Non h Akb. Ndm imlion of vibl visosiy nd Jffy flid modl fo blood flo hogh d y ih snosis Commniions in hoil Physis visky E.O. Mgno nd J. Chisnsn Fini lmn nlysis of flo indd by onions lik hos of h insin J. Biomhnis H.J. Rh Pisli flo hogh lob-shd b In. J. Mh. i A. Alsdi. Hy M. Msf nd. Hin Pisli flo of sdolsi flid in vd hnnl ih ll ois AME J Al. Mhnis Hin. Hy. Asgh nd A. A. Hndi Infln of omlin lls on isli moion ih h/mss nsf nd hmil ion In. J. H Mss nsf N. Ali M. jid Z. Abbs. Jvd Non- Nonin flid flo indd by isli vs in vd hnnl E. J. Mh.-B: Flids N. Ali M. jid. Jvd Z. Abbs An nlysis of isli flo of miool flid in vd hnnl Chin. Phys. Ls A. K. Ydnnh-Adkni nd H. Nioomnd- Osii N ADINA-FI oh in modling isli nso of non-nonin flid J. Mh. Md. Biol. 0. DOI: 0.4/ C. R d Lim l. A biomimi ioli m: omionl nd ximnl hiion nsos nd Aos A: Physil lvjn E.G. lkb nd V. Vsn Rm A nmil sdy of flo hogh vylld hnnls In. J. Nm. Mh. Flids R. Bhgv. hm H.. kh.a. Bég O. An Bég nd.k. Hng Pisli ming of miool flid in oos hnnl- modl fo snosd is J. Biomh B.V.Rhish Km nd K.B. Nid A nmil sdy of isli flos Com. Flids A. Ebid A n nmil solion fo h MHD isli flo of bio-flid ih vibl visosiy in il ylindil b vi Adomin domosiion mhod Phys. Ls-A K. Conningon Q. Kng H. Visnhn A. Abdl-Fh nd. Chn Pisli il nso sing h li Bolmnn mhod Phys. Flids H. Ko I. Kobyshi M. Nkjim K. Um Anlysis of flo hnomn in gsi onns indd by hmn gsi islsis sing CFD Food Biohysis J.H. H Homooy bion hniq Com Mhods in Alid Mhnis nd Engining A. B. Ps M. M. Rshidi O. An Bég nd. di mi-omionl simlion of mgno-hmodynmi flo in smi-oos hnnl sing oiml homooy nd diffnil nsfom mhod Com. Biol. Md. 0. h://dx.doi.og/ 0.06/j.ombiomd i 46. A. F. Elsyd Comison bn viionl iion mhod nd homooy bion mhod fo hml diffsion nd diffsion hmo ffs of hixooi flid hogh biologil isss ih ls diion xisn Al. Mh. Modl D. ihi nd O. An Bég Homooy bion simlion of nonlin isli nso hnomn in visolsi-sd oos mdi ondis Mh. Biosi. 0. In Pss. 48. J. Bi nd H. Ghvini. Convgn of h homooy bion mhod fo il diffnil qions Nonlin Anlysis: Rl Wold Aliions /5/04 06