Ieraioal Joural of Mahemaics reds ad echology (IJM) Volume 53 Number 5 Jauary 18 Effecs of ime Depede acceleraio o he flow of Blood i rery wih periodic body acceleraio mi Gupa #1, Dr. GajedraSaraswa *, Dr. Ravedra Sigh $3 #1 Deparme of Mahemaics, Magalayaa Uiversiy, ligarh (UP), Idia *Deparme of Mahemaics, Magalayaa Uiversiy, ligarh (UP), Idia $3 K R Magalam Uiversiy, Soha,Palwal, (Haryaa), Idia bsrac- he aim of his paper is o develop a mahemaical model describig he effec of ime depede acceleraio wih periodic body acceleraio o he flow of blood i a arery. he flowig blood is reaed o be Newoia i characer ad he aalyical soluios are obaied for his blood flow problem. he soluio valid for he fas oscillaios ad a small exeral acceleraio, are obaied for he velociy, flux ad sress field. compuaioal aalysis for he fluid mechaics of blood flow is also performed for he assumed siuaio. he effec of periodic body acceleraio o he isaaeous flow rae, acceleraio ad shear sress are obaied ad observed ha i icreases if we icrease he magiude of periodic body acceleraio. Keywords:- Blood flow, areries, acceleraed moio, body acceleraio ad periodic exeral acceleraio. INRODUCION: he flow of blood hrough a arery i huma beig is a prese difficul o measure wihou major surgery. I is herefore ecessary o model blood problems hrough he arerial, eiher heoreically or experimeally. Whe developig a heoreical model, oe mus simplify he equaios of moio sufficiely o permi he calculaio of he required flow variables while a he same ime maiaiig he realism of he model. Various aalyical ad umerical approaches have bee made usig differe simplifyig assumpios. he effec of acceleraed blood flood flow i huma beig ca be very serious, which may cause a icrease i pulse rae loss of visio ad veous poolig of blood i exremiies.rzeiusee. al. [1] ad Verdouw e al. [] obaied a very good resul i his direcio ha idicaes ha blood pressure ad cardiac oupu are raised whe body acceleraio sychroous wih he hear bea is applied i a fooward direcio. Sud[3] made a aalysis of blood flow uder ime depede acceleraio ad obaied a resul which shows ha high blood velociies ad high shear rae capable of harmig he circulaio are produced uder he ifluece of such ime depede acceleraio. Sud e al. [4] agai worked o he flow hrough seosed arery subjec o periodic body acceleraio ad shows ha body acceleraio icreases he flow rae. he pulsaile flow of blood hrough rigid ube uder he ifluece of body acceleraio was sudied by Chaurai[6].Madal [7] observed he effec of body acceleraio o seady pulsaile flow of o Newoia fluid hrough a seosedarey. Sharma M. K. e. al.[11] sudied aboupulsaile blood flow hrough seosed arery wih axial raslaio. here is lo of ivesigaio, which was made for blood flow wih ime depede acceleraio, ad i is well kow ha he vibraio ampliudes of mechaical equipme e.g. a aeroplae, he effec of such vibraios o he huma sysem ca be quie closely approximaed by imposig a siusoidal velociy whose ampliude grows wih ime o he liear acceleraio of he body. hus a heoreical aalysis for predicig he ime depede acceleraio of blood flow is very impora subjec o ivesigaio for he desig of ai-gsuie ad cordie assis devices. herefore i his chaper a sudy which deals wih he problem of blood uder ime depede acceleraio uder periodic body acceleraio has bee made o fid a mahemaical model for compuaioal resul for he effec of hese facor o he blood flow velociy, flow rae ad shearig sress wih respec o radial disace. FORMION OF HE PROBLEM: o simplify he aalysis. We addiioally make he followig supposiios: 1. he flow is lamiar ad here is roaioal symmery of flow.. he frequece of body acceleraio is so small ha wave effec ca we egleced. 3. he variaio of velociy alog he ube legh is small compared wih he rae of chage of velociy wih respec o ime. 4. he arery is sufficiely alog ha he flows of blood alog ha he ed effecs ca be igored. 5. For simpliciy cosider f fb i.. e bwhere f & f b he frequecies i Hz be. ISSN: 31-5373 hp://www.ijmjoural.org Page 49
Ieraioal Joural of Mahemaics reds ad echology (IJM) Volume 53 Number 5 Jauary 18 GEOMERY OF BLOOD FLOW IN RERIES Cosider he flow of blood i he ube of radius R. he ube is iiially a ime. ime i suddely sars oscillaig alog is logiudial direcio wih velociy V cos a [3]Le a is he acceleraio i m / s, f is he agular frequecy i R/sec ad f is he frequecy i Hz. he imposed acceleraio herefore is a (cos si ). he V has ampliude a, which icreases liearly wih respec o ime. Now le us cosider he sysem.subjeced o periodic body acceleraiof()[4], is give by F( ) cos( ) Where b b b f is he circular frequecy i Hz. is he lead agle of F() wih respec o hear acio. he basic equaio goverig he flow of blood alog he logiudial direcio i he ube ca be wrie as (Bachlor, 1967) w w 1 w. ( ). (1). r r r While equaio(1)subjec o periodic body acceleraio may be wrie as: 1 w cos( ) ( w w ) r r r w 1 w w cos( ) i.e. r r r... () Where w is he axial velociy, is he desiy, is he viscosiy of he blood r is he radial disace. he presece of hepressuregradie i he Navier sokes equaio () was also used by womersley (1955) for aalyzig he oscillaory blood flow. he iiial ad boudary codiios of he problem are[3]: w( r, ) a For all r... (3) w( r, ) Fiie value as r for all... (4) V w( R, ) a cos w r=r, for >... (5) he imposed velociy[3] V is such ha: o 3 5, ad whev While, 3, whev MEHOD OF SOLUION: By applyig Laplace rasform ad followig Carslaw(1963) heory, ad omiig he calculaios, he soluio for he flow velociy ca be fially wrie as: ISSN: 31-5373 hp://www.ijmjoural.org Page 41
Ieraioal Joural of Mahemaics reds ad echology (IJM) Volume 53 Number 5 Jauary 18 k e j( y)si 3/ 1/ 3/ 3/ 1 1 a ( ) 1 j i y j i i j i y i w( y, ) wcos k si 4 e R y 3/ 3/ 3/ 1 j1 ( )( k ) j( i ) j( i y) j( i ) 3/ 1/ 1/ 1/ a iw j( i y) 1 i j1 i j1 i e R y 3/ 1/ 1/ j( i ) j( i y) j( i ) ka e k j (, y) k 4 4 1 j1 ( ) k Where 1 (6) j ad j are Bessel fucios of zero ad firs order respecively, he kiemaic viscosiy ad k R r Dimesioless umber R ad y R he expressio for he rae Q ca be wrie as[7]: (7) 1/ R Q rw( r, ) dr are he zeroes of j i R Now usig equiio (6)i equaio (7) we ge he expressio for he flow rae: 1 / 1 / 1 / 3 / 3 / 1 / 3 / j ( i y) j i j i ( ) 1 / 1 a j i j 1 i j 1 i 1 i i iw i 1 i ir 1 / 1 / 1 / 3 / 3 / 3 / j ( i ) j ( i ) j ( i ) j ( i ) j ( i ) j ( i ) Q Ra e R R e k 4 e k k r r 1 e R ka R w cos k si 4 1 4 1 j ( )( k ) r r k 1 r r. (8) he aalyical soluio for he velociy w(r, ) ad flow rae Q ()coais Bessel fucios wih complex argumes hece we shall obai explici soluios for small ad large values of various argumes of Bessel fucios. CSE: (a) IF. For small values of he dimesioless umber, he zero ad firs order Bessel fucios correspodig o he above argumes, up o wo erms ca be approximae as followig: x x j( x) 1, j1( x) 4 Where x is he approximaed argume. Subsiuig he velociy profile ca be wrie afer some simplificaios as: ISSN: 31-5373 hp://www.ijmjoural.org Page 411
Ieraioal Joural of Mahemaics reds ad echology (IJM) Volume 53 Number 5 Jauary 18 R a w y, a y 1 cos y 1si 4 4 k 4 e k 4 k j ka j y 1 1 k e j y si cos k si 1 j1 4 k (9) he expressio of he fluid acceleraio f ca be obaied from equaio (9) d i is as: a a f acos a si ( y 1) si ( y 1) cos k k e j y j k 3 4 ka 4 1 1 j y j 4 k 1 1 k ke si cos k si cos k e k cos si si cos... (1) Usig equaio (9) ad (1) we also calculae he values of shear sress.he shear sress ca be defied as: dw df (11) CSE: (b) IF obaied by employig he asympoic expressio of he Bessel fucio. he Bessel fucio j (x) ad argume x ca be wrie as: he soluio valid for he large values of he dimesioless variable, ca be 1/ 1 j x coa x x of order Followig Mchachla (1955) ad usig he asympoic argumes ad order as required i he equaio (9) subsiuig he approximaios he velociy profile ca be wrie afer some simplificaios as: b1 b 1 b1 b1 w y, b cos cos si k 4 e k 4 1 k j1 ka j y k e j y si cos k si 1 j1 4 k ISSN: 31-5373 hp://www.ijmjoural.org Page 41
Ieraioal Joural of Mahemaics reds ad echology (IJM) Volume 53 Number 5 Jauary 18 f (13) (1) Where for simpliciy we cosider: a y1 1 1 b e ad b y 1 y he expressio of he fluid acceleraio f ca be obaied from equaio (11) ad ha is: b1 b cos si 1 b1 b si 1 b b cos 1 k a k k e j y j k 3 4 4 1 j y j 4 k 1 1 k ke si cos k si cos k e k cos si si cos Usig equaio (11) ND (1) we also calculae he value of shear sress. he shear sress ca be defied as: dw df... (14) RESULS ND DISCUSSION: o evaluae he soluio we cosider ha he case of blood flow i small ad large areries ad for he case of blood flow we cosider ha =15kg m -3, =.4 kg m -1 s -1 ad he value of a is akig as 4.95 ms -1 while he frequecy f is 1. Hz. he equaio (9),(1) ad (11) represes soluio for he small arery whereas he equaio (1),(13) ad (14) represes soluio for he large arery. Here we cosider a =.g For he differe values of we plo he variaio of velociy w wih respec o radial disace a four pois i cycle by ake R=.1 m ad =14.1 ad for differe values of he chage of shear sress wih respec o radial disace. I is clear ha whe blood flood flow i a arery uder he ifluece of a ime depede acceleraio he here are some subsaial disurbaces. From he above calculaio we foud he i his chaper ha due o he periodic body acceleraio he flow velociy, flow acceleraio as well as shear sress icreases. From figure i is also clear ha here flucuaio become larger wih ime, as does he exeral acceleraio.... ISSN: 31-5373 hp://www.ijmjoural.org Page 413
Ieraioal Joural of Mahemaics reds ad echology (IJM) Volume 53 Number 5 Jauary 18 ISSN: 31-5373 hp://www.ijmjoural.org Page 414
Ieraioal Joural of Mahemaics reds ad echology (IJM) Volume 53 Number 5 Jauary 18 REFERENCES: 1. rzeius,.c., Laird, J.D., (197); Body acceleraio sychroous wih he hear bea, Bibl. Cardiol., 9, 1-5.. Verdouw, P. D., Noordergraaf,., rzeius,.c. (1973); Relaive moveme bewee subjec ad suppor i body acceleraio appliedsychroous wih he hear bea Bibl. Cardiol., 31, 57-6 3. Sud, V.K., Gierke, H.E., Kaleps, I. ad Oesreicher (1985); alysis of blood flow uder ime depede acceleraio, Med.&Biol.Eg.&Compu., vol. 3, pp.69-73. 4. Sud, V.K. ad Sekho, G.S. (1985); rerial flow uder periodic body acceleraio, Bullei of Mahemaical Biology, vol.47,pp.35-5 5. Kapur,J.N. (1985); Mahemaical models i Biology ad medicie, Eas- wes press Pv Ld (idia) 6. Chaurai, P. ad Upadhya, V.S. (1981); wo- fluid model for blood flow hrough small diameer ubes wih o zerocouole sress boudary codiio a he ierface, Biorheology, vol.18,pp.45.53 7. Madal, P.K., Chakravarhy, S. Madal,., ad mi, N. (7); Effec of body acceleraio o useady pulsaile flow of oewoia fluid hrough a seosed arery, pplied Mahemaics ad compuaio, Vol.189,pp.766-779 8. orrisi M., racia R. ad Valei. (1996); group aalysis approach for a oliear differeial sysem arisig i diffusio pheomea, Joural of Mahemaical Physics, Vol. 37, pp. 4758 4767. 9. Cheriha R. (1); New exac soluios of oe oliear equaio imahemaical biology ad heir properies, Ukraiia Mahemaical Joural, Spriger New York, Vol. 53, No. 1, pp. 171 177. 1. Kumar D. ad Kumar S. (6); compuaioal model for he ieracio bewee cell desiy ad immue respose, cacieciaidica, Vol. XXXII M, No., PP. 549 556. 11. Sharma M. K. e. al.(15); Pulsaile blood flow hrough seosed arery wih axial raslaio, I J. Biomah 8,1558(15)[1 pages] 1. Kumar S. ad Kumar S. (6); Numerical sudy of he axisymmeric blood flow i a cosriced rigid ube, Ieraioal review of pure ad applied mahemaics, vol. (),pp.99-19 ISSN: 31-5373 hp://www.ijmjoural.org Page 415