S. Chakrabarti 1 1 Department of Mechanical Engineering, S. Kumar * * Department of Mechanical Engineering,

Transcription

1 A Numrical Study on Variation of Stramlin Contours, Rattachmnt Lngth, Wall Prssur, Static Prssur and Stagnation Prssur with th Configuration of a Suddn Expansion: Viwd from Bio-Mdical Application S. Kumar * * Dpartmnt of Mchanical Enginring, Hritag Institut of Tchnology, Kolkata, Pin , W.B, India S. Chakrabarti 1 1 Dpartmnt of Mchanical Enginring, Indian Institut of Enginring Scinc and Tchnology, Shibpur, Pin , W.B, India Abstract : In this work, an attmpt has bn takn to carry out a numrical study on fluid flow through a suddn xpansion. Two-dimnsional stady diffrntial quations for consrvation of mass and momntum hav bn solvd for th Rynolds numbr ranging from 50 to 50 and aspct ratio ranging from 1.5 to 6.0 whn fully dvlopd vlocity profil has bn considrd at th inlt. Stram functions in th flow domain ar computd and svral stramlin contours ar gnratd by MATLAB softwar. Th ffct of aspct ratio and Rynolds numbr on th stramlin contours, rattachmnt lngth, wall prssur, avrag static prssur, distanc of maximum avrag static prssur ris from throat, L * P, variation of maximum avrag static prssur ris ( p * ) at location L * P, ffctivnss, avrag stagnation prssur and maximum avrag stagnation prssur drop ( p * s ) at Lp * of th suddn xpansion configuration hav bn xtnsivly studid. Kywords: Suddn xpansion, Stramlin contour, Athrosclrosis, Backward-facing stp 1. INTRODUCTION Suddnly xpandd gomtry has immns importanc to rsarchrs involvd in bio-nginring and bio-mdical ara du to its rich faturs such as rcirculation, sparation and rattachmnt tc. Ths typs of phnomna may occur in vin and artrial systm du to som disas. On of th most important disass of vins and artrial systm is athrosclrosis. Early dvlopmnt of athrosclrosis oftn occurs in rgions of artrial branching and sharp curvatur. Many arly vnts in th pathognsis of athrosclrosis ar linkd, at last initially, to complx homodynamic forcs uniqu to athrosclrosis pron rgions of th vasculatur [1]. Flow charactristics such as flow sparation, rcirculation and rattachmnt, as it occurs clos to artrial bifurcations, may dirctly contribut to th initiation of focal athrognsis [,3]. An stablishd in vitro modl usd to simulat flow bifurcating artrial rgions in th backward-facing stp flow chambr [4]. In this modl, fluid flows from a narrow channl ovr a stp xpansion into a widr channl. Th asymmtric xpansion of th flow path lads to a sparation of th flow. Clos to th xpansion stp, thr is a rcirculating ddy with a flow dirction against th main flow. Farthr downstram, th flow rattachs and vntually r-stablishs a unidirctional parabolic flow profil. At physiological Rynolds numbrs, th flow is laminar in th ntir chambr. In th backward-facing stp flow chambr, ndothlial clls xposd to th sparatd flow strams xprinc larg spatial shar strss gradints, spcially clos to th scond stagnation point. Howvr, if th onst of flow in suddn xpansion in th backwardfacing stp flow chambr, th stady stat of th flow rquirs svral millisconds to dvlop fully [5], subjcting th clls to a larg tmporal gradint of shar strss. Extnsiv rsarch work on suddn xpansion configurations viwd as a diffusr as wll as industrial applications has bn carrid out by svral rsarchrs [6-18]. As pr brif litratur rviw, it has bn obsrvd that a numbr of rsarchrs hav mainly don numrical and xprimntal works on th suddn xpansion flow chambr from industrial viwpoint. Vry fw rsarchrs hav studid th configuration with rspct to mdical applications. Thrfor, in this work, flow charactristics of a suddn xpansion gomtry hav bn studid in connction with diffrnt bio-mdical applications. In this 144

2 papr, an attmpt has bn mad to invstigat th ffct of important paramtrs such as aspct ratio and Rynolds numbr on th stramlin contours, rattachmnt lngth, avrag wall prssur, avrag static prssur, and avrag stagnation prssur. Our own FORTRAN cod is dvlopd for this study.. MATHEMATICAL FORMULATION A schmatic diagram of th computational domain is illustratd in Figur 1. Th flow undr considration is assumd to b stady, two-dimnsional and laminar. Th fluid is considrd to b Nwtonian and incomprssibl. Th following dimnsionlss variabls ar dfind to obtain th govrning consrvation quations in th nondimnsional form; Lngths: x = x W 1, y = y W 1, L P = L P W1, L R = Boundary Conditions In th prsnt work four diffrnt typs of boundary conditions ar applid. Thy ar as follows, At th walls: No slip condition is usd, i.., u * = 0, v * = 0. At th inlt: Fully dvlopd vlocity profil is prscribd and th transvrs vlocity is st to b zro, i.., u * = y, v * = 0. (c) At th xit: Fully dvlopd condition is assumd and hnc gradints ar st to zro, i.. u x = 0, v x = 0. (d) At th lin of symmtry: Th normal gradint of th axial vlocity and th transvrs vlocity ar st to zro, i.., u y = 0, v* = NUMERICAL PROCEDURE L R W1, L i = L i W 1, L x = L x W1, Vlocitis; u = u V 1, v = v V 1 Prssur, p = p ρv 1 With th hlp of ths variabls, th non-dimnsional mass and momntum consrvation quations ar writtn as follows, u x + v y = 0 (1) u u u x + v y = p x + 1 R x u x + y u y () u v v x + v y = p y + 1 v R x x + v y y (3) Whr, th flow Rynolds numbr, R = ρv 1W 1 W1 / i,1 Primary duct y, y * Li A B L x LR C μ. Scondary duct D W / Th partial diffrntial quations (1), () and (3) hav bn discrtisd by a control volum basd finit diffrnc mthod. Powr law schm has bn usd to discrtis th convctiv trms (Patankar [19]). Th discrtisd quations hav bn solvd itrativly by SIMPLE algorithm, using lin-by-lin ADI mthod. Th convrgnc of th itrativ schm has bn achivd whn th normalizd rsiduals for mass and momntum quations summd ovr th ntir calculation domain will fall blow Th distribution of grid nods has bn considrd non-uniform and staggrd in both co-ordinat dirctions allowing highr grid nod concntrations in th rgion clos to th stp and th walls of th duct. In th prsnt computation work, th flow is assumd to b fully dvlopd at th xit and hnc th xit is chosn far away from th throat. For aspct ratio of 1.5 to 3.0, th xit lngth, L * * x is considrd to b 50 and th inlt lngth, L i has bn considrd to b 1 du to assumption of fully dvlopd vlocity profil at th inlt of th computational domain. During computations, th numrical msh comprisd of grid nods in th inlt sction and in th xit sction in x and y dirctions rspctivly hav bn considrd (Chakrabarti t al. [8]). Th siz of numrical msh in x and y dirctions for aspct ratio ranging from 4.0 to 6.0 has bn fixd aftr grid indpndnc tst. In th prsnt computation, th xit lngth, L * * x is considrd to b 10 and th inlt lngth, L i has bn considrd to b 1 du to assumption of fully dvlopd vlocity profil at th inlt of th computational domain. Th numrical msh comprisd of grid nods in th inlt sction and in th xit sction in x and y dirctions rspctivly provids grid indpndnt rsults. Inlt i,1 x, x * Lin of Symmtry Figur 1. Schmatic Diagram of th computational domain Exit 4. VALIDATION OF COMPUTATIONAL RESULTS In ordr to validat th accuracy of th numrical modl, comparisons ar mad btwn th numrical rsults and th xprimntal data rportd by Durst t al [0]. Th xprimntal gomtry usd in Durst t al. [0] is 145

3 considrd in th simulation, which is shown in Fig. 1. Th computations hav bn carrid out for Rynolds numbr of 56 for aspct ratio of 3. Axial locations of prsntd vlocity profil in th papr (Durst t al. [0]) hav bn convrtd to th locations in accordanc with our considration, and accordingly th rsults hav bn placd in Fig.. It shows th variations of axial vlocity profils at diffrnt locations of downstram of th suddn xpansion, along with th xprimntal masurmnts of Durst t al. [0]. As obsrvd, th numrical rsults ar in good agrmnt with th xprimntal data. 5. RESULTS AND DISCUSSIONS Th important rsults of th prsnt study ar rportd in this sction. Th paramtrs thos affct th flow charactristics ar indntifid as: Rynolds numbr, 50 R 50. Aspct ratio, 1.5 A * 6.0. (c) Inlt vlocity distribution- Fully dvlopd R=56, AR= u* x*=7 (Our) x*=7 (Durst, 1974) x*=11 (Our) x*=11(durst, 1974) Fig. Axial vlocity profils at diffrnt axial locations 5.1. Variation of stramlin contours and rattachmnt lngth Th ffct of Rynolds numbr and aspct ratio on th variation of stramlin contours has bn invstigatd. From th prsntd stramlin contours (Fig. 3 and Fig. 4), it is obsrvd that with incras in Rynolds numbr, rattachmnt point movs downstram of th throat. Th siz of rcirculating bubbl incrass with incras in Rynolds numbr for a fixd aspct ratio. For a fixd Rynolds numbr, th rcirculating bubbl siz also incrass with incras in aspct ratio. This is also supportd by th Fig. 5 and Fig. 5 rspctivly. Th siz and strngth of rcirculating bubbl formd du to ngativ vlocity zon in this configuration plays an important rol for diffrnt biomdical applications. In th artry of human body, th configuration of suddn xpansion ncountrs frquntly. In coronary artry disass, athrosclrosis is on of th most common manifstations of artrial disas and is charactrizd by dposits of yllowish plaqus containing cholstrol, lipid matrial, and lipophags formd within th intima and innr mdia of artris. Th rcirculation zon in artris is considrd to b an important phnomnon for th formation and propagation of athrosclrosis. Th physiological significanc of th rcirculation zon is that blood stram stagnats locally in this rgion and allows platlts and fibrin to form a msh at th innr wall in which lipid particls bcom trappd and vntually coalsc to form athromatous plaqu, this may tnd to caus a mor svr stnosis and hardning of th artris. Apart from that, th rattachmnt point is having also significanc on th formation and propagation of athrosclrosis. Th high cll turnovr rat taks plac nar th rattachmnt point du to high cll division and low cll dnsity nar that rgion. For this, a laky junction may dvlop which is considrd to b th possibl pathway for transport of low-dnsity lipoprotin through th artrial wall. Flow charactristics such as flow sparation, rcirculation and rattachmnt, as it occurs clos to artrial bifurcations, may dirctly contribut to th initiation of focal athrognsis. 5. Variation of wall prssur Th wall prssur plays an important rol in th artrial systms of th human organs. Sinc low prssur at th stnosis zon corrlats th taring action of ndothlium layr with subsqunt thickning of plaqu. Thrfor, th chancs of taring action and plaqu dposition ar vry high in th downstram away from th throat and vry nar to th throat and it incrass ithr with incras of Rynolds numbr or aspct ratio. Th ffct of flow Rynolds numbr on th variation of non-dimnsional wall prssur at th upstram and downstram of th suddn xpansion configuration for fiv Rynolds numbr, namly 80, 100, 150, 00 and 50 for a typical aspct ratio of 5 (Figur 6 ), has bn prsntd. Figur6 shows th ffct of aspct ratio on th variation of non-dimnsional wall prssur at th upstram and downstram of th configuration for 4.0, 5.0 and 6.0 for a typical Rynolds numbr of 00. From both th graph it has bn sn that natur ar vry similar. From th ovrall study of th ffct of aspct ratio and Rynolds numbr, it is notd that th axial location of th maximum prssur rcovry points movs towards th downstram with R and A *. An intrsting fatur is that th maximum prssur ris occurs arlir for lowr valus of R. This can b xplaind radily by noting that for vry low low Rynolds numbr flow, th fluid will almost adhr to th wall without showing any tndncy to sparation and th point of maximum prssur ris will b sufficintly clos to th throat. As th flow Rynolds numbr is incrasd, it is natural to xpct a largr rattachmnt lngth; consquntly prssur ris will b continud ovr a larg lngth, bfor a maxima is rachd. From th graph, it is also obsrvd that th maximum prssur ris occurs arlir for lowr valus of aspct ratio. This can b xplaind by th fact that for vry low aspct ratio, th siz of rcirculating bubbl is 146

4 vry small rsulting in lowr diffusion du to which th fluid will almost adhr to th wall without showing any tndncy to sparation and th point of maximum prssur ris will b sufficintly clos to th throat Variation of avrag static prssur along th axial lngth Th initiation and progrssion of athrosclrosis is dpndnt on th accumulation of low dnsity protin in th artry wall. On of th biomchanical forcs of th chancs of th dposition is dpnding on transmural fluid flux. Sinc th fluid flux dpnds on avrag prssur of th blood, thrfor, th avrag static of blood at any sction of th coronary artry may b considrd to b an important paramtr in assssing th xtnt of growth of stnosis. Th ris in static prssur is on of th important paramtrs in assssing th prformanc of suddn xpansion usd as a diffusr. A proprly dsignd suddn xpansion usd as a diffusr should nsur a high static prssur ris whil having minimum stagnation prssur loss. Thrfor, in this sction, an attmpt has bn mad to study th ffct of Rynolds numbr and aspct ratio sparatly on th avrag static prssur in th post throat zon. In th prsnt work, th avrag static prssur at any cross-sction is dtrmind by th following xprssion: pda p avg (4) da Th ffct of diffrnt Rynolds numbr and diffrnt aspct ratio on th avrag static prssur at downstram of th configuration has bn studid Figur 7 shows th variation in avrag static prssur for fiv Rynolds numbr ranging from 80 to 50 for a typical aspct ratio of 5. Th gnral trnd of ths curvs of Fig. 7 is notd to b mor or lss sam as that of th cass of wall prssur. From th graph, it is obsrvd that th avrag static prssur stply drops at th throat rgion, thraftr th prssur again riss. Th drop is du to fact that across th throat rgion, thr is an abrupt chang in cross-sctional ara of th suddn xpansion configuration. This incrass th dnominator of th quation (4) to an xtnt that vn a static prssur rcovry in this zon cannot adquatly compnsat and hnc th stp prssur drops. At th post-throat rgion, th numrator of quation (4) is influncd by both positiv prssur zon (cratd by th fluid in th main stram that dos not undrgo rcirculation) and ngativ prssur zon (cratd by th rcirculation fluid). Initially in th post-throat rgion, th ovrall cross-sctional ara of th rcirculating bubbl is high and so th static prssur ris is small. Furthr in downstram thr is mor positiv prssur and also incrasd kintic nrgy diffusion and so th avrag static prssur riss rsulting in significant prssur rcovry. Aftr a crtain maximum prssur, thr is a drop in th avrag static prssur as th viscous dissipativ ffcts suprsd at this stag. Also, it is sn that for lowr Rynolds numbr flow, thr is a sharp incras of avrag static prssur, in trms of distanc along th axis in th post-throat rgion i.., th pak avrag static prssur point is progrssivly shiftd to th right numbr flow is mor bcaus of highr diffusion. Figur 7 shows th variation of avrag static prssur along th lngth of suddn xpansion configuration for thr 4.0, 5.0 and 6.0 for a typical Rynolds numbr of 00. Th gnral trnd of th curv of Fig. 7 is mor or lss sam as that of th prvious cass. Initially in th post-throat rgion, th ovrall cross-sctional ara of th rcirculating bubbl is high and so th static prssur ris is small. Furthr in downstram thr is mor positiv prssur and also incrasd kintic nrgy diffusion and so th avrag static prssur riss rsulting in significant prssur rcovry. Aftr a crtain maximum prssur, thr is a drop in th avrag static prssur. Th dcras in avrag static prssur in suddn xpansion configuration may b xplaind by th fact that siz of rcirculating bubbl incrass with incras in aspct ratio, sinc this larg ddy is associatd with largr wall lngth, a substantial amount of prssur had, rcovrd du to diffusion, is usd up for providing th frictional drop along th suddn xpansion configuration wall. Also, it is sn that for lowr aspct ratio, thr is a sharp incras of avrag static prssur, in trms of distanc along th axis in th post-throat rgion i.., th pak avrag static prssur point is progrssivly shiftd to th right as th aspct ratio incrass; howvr th prssur rcovry is mor for highr aspct ratio Variation of avrag stagnation prssur along th axial lngth In this sction, an attmpt has bn mad to comput th avrag stagnation prssur at any sction by th following xprssion: p * s, avg A p 1 V A u da u da Whr th subscripts rfrs to th plan of masurmnts. Th dirction of th vlocity vctor, particularly for a rcirculating typ flow situation, has bn takn into account during th calculation of avrag stagnation prssur at any givn sction. Th dtails hav bn dscribd in Chakrabarti t al. [8]. Figur 8 shows th variation in avrag stagnation prssur for typically fiv Rynolds numbr, 80, 100, 150, 00 and 50 for th aspct ratio of 5.0. It is obvious that along a stramlin only, th stagnation prssur should rmain constant or thr may b som dcras (in th absnc of nrgy transfr). By using quation (5) for avrag stagnation prssur along th lngth, th xpctd bhaviour of variation of avrag stagnation prssur has bn obtaind. It has bn obsrvd that th avrag stagnation prssur incrass with incras in Rynolds numbr. This can b xplaind as, for highr Rynolds numbr flow, th kintic nrgy contribution towards th working fluid at a sction will b highr at that sction. Figur 8 shows th variation in avrag stagnation prssur for thr aspct ratio of 4.0, 5.0 and 6.0 for a typical Rynolds numbr of (5) 147

5 00. From th figur, it is sn that th avrag stagnation prssur incrass with incras in aspct ratio. This can b xplaind by using th quation (5). Th incras in avrag stagnation prssur may b attributd to th incrasing natur of rcirculating bubbl into th dvlopmnt of static prssur (p ) which incrass th numrator of quation (5), and as a rsult P s avg incrass. Th stagnation prssur along th axial distanc is gradually dcrasing in nar-throat zon mainly and thn it shows th asymptotic bhavior throughout th lngth of th configuration. Th gnration of high prssur nar th post-throat zon may comprss th stnosis, formd insid th artry which will lad to opning of th blockag in th artry Variation of maximum avrag stagnation prssur drop ( P * * S ) at location L P Figur 9 shows th variation of stagnation prssur drop with Rynolds numbr of 80, 100, 150, 00 and 50 for typical aspct ratio of 1.5,.0, 3.0, 4.0, 5.0 and 6.0 rspctivly. From th graph, it is obsrvd that stagnation prssur drop rmains narly asymptotic with th incras of Rynolds numbr for a fixd aspct ratio. Initially, static prssur riss du to diffusion of kintic nrgy. This gain is considrably gratr than ddy losss, thrfor, in vry low Rynolds numbr rgim th stagnation prssur drop dcrass. Evn at highr Rynolds numbr th diffusion of kintic nrgy occurs, but th gains ar offst by th ddy losss. Thus, in Rynolds numbr rgim th stagnation prssur drop curv shows asymptotic bhaviour. Figur 9 shows th variation of stagnation prssur drop with aspct ratio of 1.5,.0, 3.0, 4.0, 5.0 and 6.0 for typical Rynolds numbr of 80, 100, 150, 00 and 50 rspctivly. From th prsntd graph, it is rvald that thr is a sharp ris in th stagnation drop with incras in aspct ratio for a fixd valu of Rynolds numbr. This can b xplaind by quation (5). With th incras in aspct ratio, th shap and siz of th rcirculating bubbl incrass and sinc this larg ddy is associatd with largr wall lngth, a substantial amount of static prssur, rcovrd du to diffusion is usd up for providing th frictional drop along th suddn xpansion configuration and hnc numrator of quation (5) dcrass which incrass th avrag stagnation prssur drop. 6. CONCLUSION Th prsnt work, th flow charactristics of plain suddn xpansion in low Rynolds numbr rgim with fully dvlopd vlocity profil at th inlt has bn studid. Th ffct of Rynolds numbr and aspct ratio on stramlin contours, rattachmnt lngth, wall prssur, avrag static prssur, avrag stagnation prssur and stagnation prssur drop hav bn invstigatd in dtail and this lads to th following important conclusions: Th siz of rcirculating bubbl incrass with incras in Rynolds numbr for a fixd aspct ratio and it also incrass with incras in aspct ratio for a fixd Rynolds numbr. It is xpctd that th rat of hat transfr as wll as tmpratur of th flowing fluid will incras as th rcirculation zon incrass. It occurs mainly du to convrsion of kintic nrgy to th flowing fluid into hat nrgy. Du to th incrasing rcirculation zon, rtntion of blood at a particular ara will incras as a rsult of which, toxid mdicin, if injctd at that location, will gt tim to b sdimntd on th rquird ara, without affcting th important part of th body lik hart, kidny, livr, lung tc. Th prssur at th wall of th configuration is found to chang quit sharply in th rgion of throat. (c) Th avrag static prssur stply drops at throat rgion, thraftr prssur again riss. Initially in th post-throat rgion, th static prssur ris is small but furthr in downstram th avrag static prssur riss rsulting in significant prssur rcovry. (d) For all th cass, it is obsrvd that avrag stagnation prssur gradually dcrass. () Rynolds numbr has no strong ffct on th variation of maximum avrag stagnation prssur drop, but aspct ratio has significant ffct on it. Th maximum avrag stagnation prssur drop incrass with incras in aspct ratio. Th gnration of high prssur at th xpandd zon may comprss th stnosis, formd insid th artry which will lad to opning of th blockag in th artry. 148

6 R = 50 A * =.0 R = 100 A * = 4.0 R = 00 A * = 6.0 Figur 3. Stramlin Contour plotting at A * = 5.0 for diffrnt Rynolds numbr Figur 4. Stramlin Contour plotting at R = 00 for diffrnt aspct ratio 149

7 L * R A * _1.5 A * _.0 A * _ R Pw * Fig. 6 Variation of avrag static wall prssur S * L * R R_80 R_100 R_150 R_ A * P * avg X * R_80 R_100 R_150 R_50 Fig. 5 Variation of rattachmnt lngth Pw * A * = 5.0 R_80 R_100 R_150 R_50 P * avg X * S * Fig. 7 Variation of avrag static prssur 150

8 P * s avg R_80 R_100 R_150 R_50 X * Maximum avrag stagnation prssur drop FULLY DEVELOPED A * R_80 R_100 R_150 R_50 Fig. 9 Variation of maximum avrag stagnation prssur drop 0. P * s avg Fig. 8.Variation of avrag stagnation prssur along th axial lngth maximum avrag stagnation prssur drop X * R A * _1.5 A * _.0 A * _3.0 REFERENCES 1. Glagov, S., Zarins, C., Giddns, D. P. and Ku, D. N., 1988, Hmodynamics and athrosclrosis: Insight and prspctivs gaind from studis of human artris, Arch. Pathol. Lab. Md., 11, Davis, P. F., Barb, K. A., Volin, M. V., Robotwskyj, A., Chn, J., Josph, L., Grim, M. L., Wrnick, M. N., Jacobs, E., Polack, D. C., Dpaola, N. and Barakat, A. I., 1999, Spatial rlationships in arly signaling vnts of flow mdiatd ndothlial mchanotransduction, Annu. Rv. Physiol, 59, Davis, P. F., Polack, D. C., Handn, J. S., Hlmk, B. P. and Dpaola, N., 1999, A spatial approach to transcriptional profiling: Mchanotransduction and th focal origin of athrosclrosis, Trnds Biotchnol., 17, Trusky, G. A., Barbr, K. M., Roby, T. C., Olivir, L. A. and Combs, M. P., 1995, Charactrization of a suddn xpansion flow chambr to study th rspons of ndothlium to flow rcirculation,asme J. Biomch. Eng., 117, Durst, F. and Prira, J. C. F., 1988, Tim-dpndnt laminar backward-facing stp flow in a two-dimnsional duct, ASME J. Fluids Eng., 110, Baudoin, J., Cadot, O., Aidr, J. and Wsfrid, J., 004, Thrdimnsional stationary flow ovr a backward-facing stp, Europan Journal of Mchanical B/Fluids, 3, Blackburn, H. M., Barkly, D. and Shrwin, S. J., 008, Convctiv instability and transint growth in flow ovr a backward-facing stp, Journal of Fluids Enginring, 603, Chakrabarti S., Ray S. and Sarkar A., 003, Low Rynolds numbr flow through suddn xpansion - from a diffusr viwpoint, Journal of Enrgy, Hat and Mass Transfr, 5, Chakrabarti S., Ray S. and Sarkar A.,008, Numrical analysis for suddn xpansion with fnc in low Rynolds numbr rgim, Journal of Enrgy, Hat and Mass Transfr, 30, Chrdron, W., Durst, F. and Whitlaw, J. H., 1978, Asymmtric flows and instabilitis in symmtric ducts with a suddn xpansion, Journal of Fluids Enginring, 84, Djoan, A. and Lschzinr, M. A., 004, Larg ddy simulation of priodically prturbd sparatd flow ovr a backward-facing stp, Intrnational Journal for Hat and Fluid flow, 5, Ghosh, S., Pratihar, D. K., Maiti, B. and Das, P. K., 010, An volutionary optimization of diffusr shaps basd on CFD simulations, Intrnational Journal for Numrical Mthods in Fluids, 63(10),

9 13. Ma, H. K., Hou, B.R., Lin, C.Y. and Gao, J. J., 008, Th improvd prformanc of on-sid actuating diaphragm micropump for a liquid cooling systm, Intrnational Communications in Hat and Mass Transfr, 35, Nabavi, Majid, 010, Thr-dimnsional asymmtric flow through a planar diffusr: Effcts of divrgnc angl, Rynolds numbr and aspct ratio, Intrnational Communications in Hat and Mass Transfr, 37, Rvultaa, A, 005, On th two-dimnsional flow in a suddn xpansion with larg xpansion ratios, Physics of Fluids, 17, Schafr, F., Brur, M. and Durst, F.,009, Th dynamics of th transitional flow ovr a backward-facing stp, Journal of Fluids Enginring, 63, Shw, T. W. H. and Rani, H. P., 006, Exploration of vortx dynamics for transitional flows in a thr-dimnsional backwardfacing stp channl, Journal of Fluids Enginring, Viru, D. and Siddiqu, I., 010, Axial flow of svral nonnwtonian fluids through a circular cylindr, Intrnational Journal of Applid Mchanics, (3), Patankar, S. V Numrical Hat Transfr and Fluid Flow, Hmisphr Publication. 0. Durst, F., Mlling, A. and Whitlaw, J. H., 1974, Low Rynolds numbr flow ovr a plan Symmtrical suddn xpansion, Journal of Fluids Enginring, 64, p s * = Non-dimnsional avrag stagnation prssur drop R = Rynolds numbr S * = Non-dimnsional distanc along th wall u, v = Vlocity componnts in x and y dirctions of a Cartsian co-ordinat u *, v * = Non-dimnsional vlocity componnts in x and y dirctions V 1 = Avrag vlocity W 1, W = Width of inlt and xit sction x, y = Cartsian co-ordinats 1, i = Inlt Nomnclatur A * = Aspct ratio or ara ratio, givn by A /A 1 da= Elmntal ara A 1 = Cross-sctional ara at inlt sction A = Cross-sctional ara at xit sction L i = Inlt lngth (i.., lngth btwn inlt sction and throat) L i * = Non-dimnsional inlt lngth L x = Exit lngth (i.., lngth btwn throat and xit sction) L x * = Non-dimnsional xit lngth L P = Distanc of maximum avrag static prssur ris L P * = Non-dimnsional distanc of maximum avrag static prssur ris L R = Rattachmnt lngth L R * = Non-dimnsional rattachmnt lngth µ = Dynamic viscocity ρ = Dnsity p = static prssur p * = Non-dimnsional static prssur P avg * = Non-dimnsional avrag static prssur P w * = Non-dimnsional wall prssur P s * = Non-dimnsional stagnation prssur 15