Two-Layered Model of Blood Flow through Composite Stenosed Artery

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Avalable at http://pvamu.edu/aam Appl. Appl. Math. ISSN: 93-9466 Vol. 4, Iue (December 9), pp. 343 354 (Prevouly, Vol. 4, No.

1 Avalable at Appl. Appl. Math. ISSN: Vol. 4, Iue (December 9), pp (Prevouly, Vol. 4, No.) Applcaton Appled Mathematc: An Internatonal Journal (AAM) Two-ayered Model of Blood Flow through Compote Stenoed Artery Padma Joh *, Ahutoh Pathak ** B.K. Joh *** * Department of Mathematc Mahakal Inttute of Technology Ujjan-456, Inda E-mal: joh.padma@gmal.com ** School of Stude n Mathematc Vkram Unverty Ujjan-456, Inda autoh.pathak8@gmal.com *** Department of Mathematc Government Engneerng College Ujjan-456, Inda bk.joh684@gmal.com eceved: May 5, 9; Accepted: December 5, 9 Abtract In th paper a teady, axymmetrc flow, wth a contrcted tube ha been tuded. The artery ha been repreented by a two-layered model contng of a core layer a perpheral layer. It ha been hown that the retance to flow wall hear tre ncreae a the perpheral layer vcoty ncreae. The reult are compared graphcally wth thoe of prevou nvetgator. It ha been oberved that the extence of perpheral layer ueful n repreentaton of deaed arteral ytem. Keyword: Arteral wall, Blood rheology, Perpheral layer vcoty, Steno MSC #: 76Z5, 9C35. Introducton Steno mean contrcton or narrowng. An artery whch affected by th abnormal growth can lead to erou conequence uch a blockage of the artery, troke many other arteral deae. The tudy of blood flow through tenotc artere play an mportant 343

2 344 Joh et al. role n the dagnotc fundamental undertng of cardovacular deae. To undert the effect of mld teno, everal reearcher uch a Haldar (987), Mra Chakravarty (986), Young (968) nvetgated the flow of blood through contrcted tube treatng blood a a Newtonan flud. The model ha been extended by aumng that blood behave lke a non-newtonan flud Nakamura Sawada (988) Shukla et al. (98). In all above model, the blood flow repreented by ngle-layered model. Buglarello Sevlla (97) have hown expermentally that the blood flowng through narrow tube can be well repreented by a two-layered model ntead of one. In th type of model there a perpheral layer of plama a core regon of upenon of red blood cell. Shukla et al. (98) have taken two-layered model to analyze the perpheral layer vcoty. Ponalaguamy (7) focued on lp velocty, thckne of perpheral layer core layer vcoty at the veel wall. Srvatava (3) tuded analytcally numercally effect of mld teno on blood flow charactertc n a two-flud model. In th tudy we analyed the flow of blood through axymmetrc, mld compote haped teno. The reult obtaned are compared wth prevou nvetgator Shukla, et al. (98).. Formulaton of the Problem We condered the flow of blood n a tube havng axymmetrc mld teno. It aumed that blood an ncompreble flud whch repreented by a two-layered model. The external layer how perpheral layer of plama the nternal core layer decrbe the upenon of red blood cell. The chematc dagram howng the flow gven by the followng fgure: r d p (z) (z) p o z O the ymbol t for : adu of the non-tenotc regon z: ( ) adu of the tenotc regon ( ) z : adu of the central layer n tenotc regon : The length of the artery : The length of the teno Fgure : Geometry of tenoed artery,

3 AAM: Intern. J., Vol. 4, Iue (December 9) [Prevouly, Vol. 4, No. ] 345 d : p : p : : : : : : ocaton of teno Inlet flud preure Intantaneou outlet flud preure Intantaneou maxmum heght of the teno Maxmum bulgng of nterface Vcoty of flud n central core layer Vcoty of flud n perpheral layer ato of central core radu to the tube radu. The geometry of the tenotc tube wthout perpheral layer decrbed a follow: zd, d zd, z () co z d, d z d, (), Otherwe. The governng equaton of blood flow gven by Kapur (985), dp w rr r r r, () w axal velocty, p flud preure r vcoty of flud. The boundary condton are w at r ( z) (3) w r at r. (4) Solvng equaton () under boundary condton (3) (4), we get dp r w dr (5) r r The volumetrc flow rate gven by

4 346 Joh et al. Q rwdr, (6) whch on ung equaton (5) gve 3 r dr dp Q. (7) r Thu, the preure gradent can be obtaned a dp Q I( z), (8), 3 rdr I( z). (9) r Integratng equaton (8) ung condton p p at z p p at z,, we have Q p p. I( z) () The retance to flow defned by, p p. () Q From equaton (), () (), we can fnd dl I I( z), () d I 3 r dr. (3) r Now, the hear tre at wall gven by, r w r r( z). (4)

5 AAM: Intern. J., Vol. 4, Iue (December 9) [Prevouly, Vol. 4, No. ] 347 By ung equaton (5) (8) n (4), we can fnd hear tre at maxmum heght of teno.e. at z d, whch a follow zq ( ) I( z) zd. (5) For fndng the effect of perpheral layer vcoty, the vcoty functon () r can be defned a, r ( z) () r, ( z) r ( z), (6) are the vcote of the central the perpheral layer repectvely. The functon ( ) z repreent the hape of the central layer wth teno. The mathematcal repreentaton of th model can be decrbed a zd, d zd () z co zd, d zd, Otherwe. (7) By ung equaton (6) n (5), velocte w c correpondng volumetrc flow rate Qc wp Qp are obtaned a follow can be obtaned then the dp c 8 (8) Qc rw dr dp Qp rwp dr, 8 (9). Thu, the total volumetrc flow rate Q defned a dp Q Qc Qp ( ) ()

6 348 Joh et al. Equaton () can alo be obtaned by equaton (7) ung (6), whch how that Q a contant. Integratng equaton (8), (9) () acro the length of artery, aumng that preure drop ame n each cae, we obtan Q c 4 p p M 4 MG, () M, () G g g, (3) g d, (4) d g d (5) d Q p 4 p p M, (6) 8 MG M (7) G g g, (8) 3 4

7 AAM: Intern. J., Vol. 4, Iue (December 9) [Prevouly, Vol. 4, No. ] 349, g d (9) 3 d d 4 d g. (3) Now Q 4 p p M 8 MG, (3) 4 M (3) G g5 g6, (33) g g d (34) d d. (35) d From equaton () through (3) ung Q Qc Qp, we can fnd

8 35 Joh et al. M M M MG MG MG. (36) Now ung n equaton (7), we get zd, d zd z () co zd, d zd, Otherwe. (37) On comparng equaton () (37), we can oberve that. (38) Now, keepng n mnd the equaton (6), the dmenonle retance to flow the dmenonle hear tre can be obtaned by ung equaton (3) n equaton () (5) a follow MG M, (39) , (4) 4Q, 3 are the retance to flow wall hear tre for the cae of no teno repectvely, wth. Evaluatng the ntegral (34) (35) after ung equaton (38) rewrtng the expreon for a follow

9 AAM: Intern. J., Vol. 4, Iue (December 9) [Prevouly, Vol. 4, No. ] 35 3 M 5 8 ( 7/) (4) 3 M. (4) The obtaned n (4) the ame a n Shukla, et al. (98). If n equaton (4) (4), we get 3 ( 7/) 5 8, (43) whch the ame a the rato obtaned by Joh et al. (9), 3, whch the ame a the one obtaned by Young (968). 3. Concluon A two-layered model of blood flow through a tenoed artery ha been condered. The model cont of a central core layer of erythrocyte urrounded by a perpheral plama layer both wth dfferent vcote. The expreon for have been plotted for dfferent value of parameter. Fgure 3 repreent the varaton of wth for dfferent value of repectvely. It ha been oberved that ncreae wth

10 35 Joh et al. the ncreae n the heght of teno, th ncreae have alo been noted when ncreaed. It further noted that retance to flow (denoted by broken lne n the fg ) lower n the preent model a compared to prevou nvetgaton of Shukla et al. (98). Alo, ung the data.3,.,.95. n equaton (4) (4), t can be noted that are decreaed by 7% 4%, repectvely, when compared wth the cae of no teno wth. Agan, n the abence of perpheral layer thee charactertc are ncreaed by 5% 37% repectvely for the ame teno ze. Thu, t eem that the reult of preent analy of two-layered model can better explan the flow behavour of tenotc artere. Fgure.

11 AAM: Intern. J., Vol. 4, Iue (December 9) [Prevouly, Vol. 4, No. ] 353 Fgure 3. Acknowledgement: We are thankful to the anonymou revewer for ueful uggeton. EFEENCES Buglarello, G. Sevlla, J. (97). Velocty Dtrbuton other Charactertc of Steady Pulatle Blood Flow n Fne Gla Tube, Borheology, 7, pp Haldar, K. (987). Ocllatory Flow of Blood n a Stenoed Artery, Bulletn of Mathematcal Bology, 49, pp Joh, P., Pathak, A. Joh, B. K. (9), Effect of Blood Flow n a Compote Stenoed, Artery Varahmhr Journal of Mathematcal Scence, to appear. Kapur, J. N. (985). Mathematcal Model n Bology Medcne, Afflated Eat-Wet Pre, New Delh. Mra, J. C. Chakravarty, S. (986). Flow n Artere n the Preence of Steno, Journal of Bomechanc, 9, pp Nakamura, M. Sawada, T. (988). Numercal Study on the Flow of a Non-Newtonan Flud through an Axymmetrc Steno, Journal of Bomechancal Engneerng,, pp

12 354 Joh et al. Ponalaguamy,. (7). Blood Flow through an Artery wth Mld Steno: A Two ayered Model, Dfferent Shape of Stenoe Slp Velocty at the Wall, Journal of Appled Scence, 7, pp Shukla, J. B., Parhar. S. ao, B.. P. (98). Effect of Steno on Non-Newtonan Flow of the Blood n an Artery, Bulletn of Mathematcal Bology, 4, pp Shukla, J. B., Parhar,. S. Gupta, S. P. (98). Effect of Perpheral ayer Vcoty on Blood Flow through the Artery wth Mld Steno, Bulletn of Mathematcal Bology, 4, pp Srvatava, V. P. (3). Flow of a Couple Stre Flud epreentng Blood through Stenotc Veel wth a Perpheral ayer, Indan Journal of Pure Appled Mathematc, 34, pp Young D. F. (968). Effect of a Tme- Dependent Steno on Flow through a Tube, Journal of Engneerng Indutral Tranacton ASME, 9, pp

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