Effect of Peclet Number on the Dispersion of a Solute in a Blood flowing in Non-Uniform Tube

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1 Intenatinal Junal f Cmputatinal and Applied Mathematics. ISSN Vlume 1, Numbe (017), pp Reseach India Publicatins Effect f clet Numbe n the Dispesin f a Slute in a Bld flwing in Nn-nifm Tube Syed Waseem Raja (1) M.V.Ramana Muthy ().Mhammed Abdul Rahim (3) (1) Reseach Schla, Depatment f Mathematics, Rayalaseema nivesity, Kunl, A.P, India syed.aja@ediffmail.cm () Pf. f Mathematics,Osmania nivesity,hydeabad-7. (3) Asst.Pf., Depatment f Geneal Sciences, YIC, Yanbu,K.S.A. Abstact: The effect f peclet numbe n bld flw in a tube f vaying css sectin is analyzed mathematically, teating bld as a hmgeneus Newtnian fluid. The nnlinea diffeential equatin ae slved by using finite diffeence scheme and the effect f peclet numbe n cncentatin in diffeent axial psitin ae calculated. Keywds: Dispesin, clet Numbe, cncentatin, Reynlds numbe. 1. INTRODCTION: The pincipal task f bld is t pvide xygen and nuishment t the tissues and the gans and t cllect waste substances. The study f mass tansfe and diffusin phenmena inside the ateial lumen and thugh the vascula wall in Bld vessels ccupies an imptant psitin. The cncentatin in the fluid is btained by tw phenmena- diffusin and cnvectin. Cnvectin is the pcess f the slute being caied dwnsteam because f the flw, while diffusin is a mlecula mechanism Batchel [14].A numbe f mdels n Slute dynamics have been develped. Tayl [17] studied the dispesin due t cnvectin alne, and utlined an elegant mathematical appximatin f the case whee mlecula diffusin plays an imptant le. Ais [16] studied the pblem

2 450 Syed Waseem Raja, M.V.Ramana Muthy & Mhammed Abdul Rahim in tubes f abitay css-sectin. Lighthill [15] btained an elegant slutin t the dispesin f a slute in a lamina tube. Gill and Sankaasubamanian [13] btained the exact slutin t the unsteady cnvective diffusin equatin f immiscible displacement in fully develped lamina flw in tubes. Hunt [11] btained an analytical slutin. Recently sme mathematical mdels cupling 3D flw and slute dynamics have been develped. They ae defined in a finite ateial segment f abitay shape, whee inflw slute distibutin is pvided. Rappitsch et al.[10] studied numeically mdelling f shea-dependent mass tansfe in lage ateies. They descibe bld flw by the unsteady 3D incmpessible Navie-Stkes equatins f Newtnian fluids; slute tanspt is mdelled by the advectindiffusin equatin. They, btained slutin by using finite element methd. A numeical simulatin is caied ut f pulsatile mass tanspt in a 3D ateial bend t demnstate the influence f the ateial flw pattens in wall pemeability chaacteistics and tansmual mass tansfe. In the cntent f athegenesis, mass tanspt efes t the mvement f athegenic mlecules fm flwing bld int the atey wall, vice vesa. Psi et al. [6] cnside absptin and exchange thugh the vascula tissues. All these mdels pvide the lcal cncentatin patten and useful t undestand the elatinship between the lcal flw pattens, the nuishing f ateial tissues and pssible pathlgies deived when such a pcess is alteed. Quateni et al.[7] studied mathematically and numeically, mdeling f slute Dynamics in bld flw and Ateial walls. They cnsideed the Navie- Stkes equatin f an incmpessible fluid, descibing the bld velcity and pessue fields, ae cupled with an advectin, diffusin equatin f the slute cncentatin. It is knwn that gemetical effects, such as cuvatue, will stngly affect the flw patten and cnsequently the cncentatin f gases and substances disslved in the bld, Me et al. [9]. It is wth t investigate, and t what extent, the gemety and haemdynamic facts ae espnsible f anmalus accumulatin and alteed absptin f substances n the ateial wall, leading t athescletic lesins and degeneative pcess, Stangeby et al. [8]. Pntelli et al. [5] studied mass tanspt and diffusin pcesses f a substance disslved in the bld and shwn the chaacteistics f the lng wave ppagatin f slute in a cuved vessel. A cmpaative study n Mathematical analysis f unsteady dispesin f slutes in bld steam studied by D.S Sanka et.al [11] teating bld as Heschel-Bulkley fluid. Nuul Aini et.al [] investigated dispesin f slute in bld thugh cicula channel plates with the effect f chemical eactin, teating bld as cassn fluid mdel. Jagadeesha et al. [3] studied slute tansfe in a pwe law fluid flw thugh pemeable tube.

3 Effect f clet Numbe n the Dispesin f a Slute in a Bld flwing 451 Pntelli et al. [4] studied a mathematical mdel f mass tanspt and diffusin phenmena in the ateial lumen and btained analytical and numeical slutins f the chaacteistics f the lng wave ppagatin and the le played by the gemety n the slute distibutin and demnstate the stng influence f cuvatue in atey. Hweve, all these analyses have been develped f igid tubes f unifm csssectin. The study f flw thugh tubes f nn-unifm css-sectin has attacted many eseach wkes. Thus, we have examined the pblem numeically f lw Reynld s numbe flw f a viscus incmpessible fluid in a cicula tube f nnunifm css-sectin t analyze the effect f clet numbe n cncentatin.. MATHEMATICAL FORMLATION: The bld flw in the atey is assumed t be a hmgeneus Newtnian fluid, which flws in a tube f slwly vaying css-sectin. The cylindical cdinate system R,,X is chsen, in which X- axis cincides with the axis f the tube. The flw f the bld is assumed t be steady and axi-symmetic. Let and W be the velcity cmpnents in the R and X diectin espectively. The gvening equatin f such flws in nn-dimensinal fm ae 1 P 1 W (1) R X R R R R R X W R W W X 1 P X W R 1 W R R W X () R R W X 0 (3) The specific tanspt equatin f such flw is given by C C W R X C 1 C D R R R (4) Diffeent substances ae disslved in bld, tanspted thugh the steam and pssibly exchanged thugh the ateial wall. F simplicity, the pesences f ne slute nly is cnsideed and its cncentatin dented by C. let assume that, the axial diffusin is small cmpaed t adial diffusin and the tube length t diamete is s lage that thee ae n end effects. Whee, and D ae espectively the density, cefficient f kinematic viscsity and cefficient f diffusivity ae taken cnstant.

4 45 Syed Waseem Raja, M.V.Ramana Muthy & Mhammed Abdul Rahim The cespnding bunday cnditins ae F fluids: = 0, W=0 n R = S(X) (5) F slute: C=C1= input cncentatin at X = 0 (6) C R C R 0 0 n R= 0 (7) n R = S(X) (8) Whee bunday cnditin (6) assumes a unifm and cnstant cncentatin f slute at the entance f the tube. The bunday cnditins (7) and (8) assues an axisymmety cnditin f the cncentatin pfile. The apppiate nn-dimensinal vaiables ae u, W w, C c, C DX, a x R a S x Whee, C ae the epesentative velcity and cncentatin espectively, and a is the adius f the tube at X = 0. Whee a = S( x ) a S D x a,= clet numbe = D, Re= Reynlds numbe The gvening equatin f such flws in nn-dimensinal fm ae F fluid: x u w u p S 1 u 1 u u u u (9) pe x Re S x x

5 Effect f clet Numbe n the Dispesin f a Slute in a Bld flwing 453 x w w w 1 p S 1 w 1 w 1 w u pe x pe x Re S x pe x (10) u u Sx w 0 pe x (11) F Slute: S 1 u c c 1 c 1 c w x pe x S x The bunday cnditins ae (1) u = 0, w=0 n =1 (13) 3. SOLTIONS: T btain the cncentatin, the axial and adial velcities f the flw thugh a tube ae btained fm equatins (9--11) with apppiate bunday cnditins by Mantn s wk [1]. Hence, withut giving a desciptive methd f slutin, we diectly wite the simplified slutin f velcity adial axial velcities as fllws: 3 4 (14) pes u x 1 Sc w (15) S x S x 9 9 Whee Sc=/Re is the Schmidt numbe, is small paamete, which incpates the slw vaiatin in css-sectin. Bunday cnditin t be impsed n the tanspt equatin in dimensinal ae c = 1 at x =0 (16) c c 0 0 at =0 (17) at =1. (18)

6 454 Syed Waseem Raja, M.V.Ramana Muthy & Mhammed Abdul Rahim The expessins (14) and (15), when substituted in equatin (1) and slved alng with the bunday cnditins (16--18) will yield the cncentatin f the slute. Hweve, analytical slutin f the esulting diffeential equatin is nt pssible; we have pted f numeical slutins. A finite diffeence scheme that can be used t numeically appximate a given diffeential equatin. T btain a discetizatin f a diffeential equatin, it is pssible t btain a finite diffeence fmula f evey tem in the diffeential equatin and then cmbine these fmulas in the bvius manne. Just eplace each tem in the diffeential equatin with its finite diffeence appximatin. We used Cank-Niclsn appximatin. 4. RESLT AND DISCSSION: Ou gal is t study the effect f clet numbe n cncentatin c in diffeent axial psitin in a tube. The values f clet numbe ae taken in the ange Hee we take gemety as S(x) 1 (0.1) x a The clet numbe, is the ati f the species tanspt by fluid cnvective D mtin t the species tanspt by mlecula diffusin that is, is measue f the mass tansfe by cnvectin cmpaed t that due t diffusin. It is lage, then bth cnvective and diffusin play key les in diffeent pats f the flw filed. If is small, then species tanspt is dminated by diffusin. If the cnvective velcity is ze, then =0, mass tansfe ccus nly by diffusin and the cncentatin pfile f the tanspt species can be btained fm a slutin f the diffusin equatin. The cncentatin pfiles in the adial diectin f diffeent values f clet numbe () at diffeent axial psitins ae shwn in figues 1(a)- 1(f). it is inteesting t nte that f all values f, the influence f n cncentatin is negligible in the entance egin f the tube. This tend is evesed twads the dischage end. Hweve f lage values f, the vaiatin f cncentatin with is significant. Als, it may be bseved that as is inceases, cncentatin deceases at the dischage end f the tube. In each case cncentatin deceases with adial distance fm tube axis t the wall. The cncentatin pfiles becmes flat nea the exist f the tube. This decease in cncentatin is expected due t the lss f slute and it is nt unifm all alng the adial cdinate. Simila studies have been epted by Hunt [11] in tems f the values f. Lage which have been cnsideed imply that cnvectin is me pnunced than diffusin. Thus, the slute initially nea wall and centeline mve dwnsteam as a

7 Effect f clet Numbe n the Dispesin f a Slute in a Bld flwing 455 esult f the flw athe than getting diffused. The slute in the end f the tube is cnvicted and diffused..0 s:=1+0.1*x e:=1 1.8 R:= Effect f when x= Effect f when x=0. 1. pe=100,00,300,400, pe=100,00,300,400, Fig.1(a) Effect f when x= x Fig.1(b) Effect f when x= x10 4 pe= x pe=100,00,300,400, x10 5 pe= x x Fig.1(c) x x10 5 Fig.1(d).0x10 1 Effect f when x=0.8.0x10 18 Effect f when x= x x x x x10 11 pe=00 5.0x10 17 pe= x x10 17 pe=300 pe= x x x x10 18 Fig.1(e) Fig.1(f) REFERENCES [1] Sanka., D.S., Jaffa., N.A., and Yatim., Y., 016., Mathematical analysis f unsteady dispesin f slutes in bld steam- A Cmpaataive Study, Glbal Junal f pue and Applied Mathematics, 1(), pp

8 456 Syed Waseem Raja, M.V.Ramana Muthy & Mhammed Abdul Rahim [] Jaffa., N.A., Yatim., Y., and Sanka., D.S., 016., Influence f Chemical Reactin n the steady dispesin f slute in bld flw- A mathematical mdel, Fa East Jun. Of Mathematical Sciences, 100(4) pp [3] Jagadeesha., S., and Rama Ra., I., 01., Slute Tansfe in a Pwe law fluid flw thugh pemeable tube, Adv. The. Appl. Mech., 5(7), pp [4] Pntelli.,G., 007, Cncentatn wave f a slute in an atey: the influence f cuvatue, Cmpute methds in Bimechanics and Bimedical Engg., 10(), pp [5] Pntelli., G., and Tatune., A., 005, Ppagatin f a slute wave in a cuved vessel. Advanced cuse and wkshp n Bld Flw, wasaw, pp [6] Psi, M., ktld., K., Ding., Z., and Fiedman., M.H., 004, Influence f cuvatue dynamics n pulsatile cnay atey flw in a ealistic bifucatin mdel, J.Bimech., 37, pp [7] Quakteni., A., Veneziani,, A., and Zunin., P., 00, Mathematical and numeical mdeling f slute dynamics in bld flw and ateial walls. SIAM, J. Num. Anal., 39(5), pp [8] Stangeby., D.K., and Ethie., C.R.,00, Cmputatinal analysis f cupled bld wall ateial LDL tanspt, ASME J.Bimech., Eng.14, pp.1-8. [9] Me., J.A., and Ethie., C.R.,1997, Oxygen mass tansfe calculatins in lage ateies, J. Bimech. Engg.,119, pp [10] Rappitch.,G.,ktld., K., and hkpe., F., 1997, Numeical mdelling f shea-dependent mass tansfe in lage ateies, Int. J.Num. Meth., Fluids, 5, pp [11] Hunt, B.,1977, Diffusin in lamina pipe flw, Int.J. Heat and Mass Tans., 0, pp [1] Mntn., M.J.,1971, Lw Reynld's numbe axisymmetic flws in slwly vaying tubes, J. Fluid Mech., 49, pp [13] Gill., W.N., and Sankaasubamaniam.,1970, Exact analysis f unsteady cnvective diffusin, Pc, Ry., Sc., Sec.A, 316, pp [14] Batchel., A., 1967, An Intductin t fluid Dynamics, Cambidge nivesity, Pp,Lands. [15] Lighthill, M.J., 1966, Initial develpment f diffusin in Piseuille flw, J.Inst. Math. Applics.,, pp [16] Ais,R., 1956, On the dispesin f a slute in a fluid flwing thugh a tube, Pc, Ry., Sc., Sec.A, 35, pp [17] Tayl, A. I., 1953, Dispesin f sluble matte in slvent flwing slwly thugh a tube, Pc, Ry., Sc., Sec.A, 19, pp