International Journal of Innovative Research in Advanced Engineering (IJIRAE) ISSN: Issue 10, Volume 4 (October 2017)

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1 Iteratioal Joural of Iovative esearch i Advaced Egieerig (IJIAE) ISSN: Issue 1, Volume 4 (October 17) OSCILLATOY FLOW OF BLOOD IN POOUS VESSEL OF A STENOSED ATEY WITH VAIABLE VISCOSITY: EFFECTS OF MAGNETIC FIELD M. Chitra 1, D. Karthikeya 1Associate Professor, esearch Scholar, Departmet of Mathematics, Thiruvalluvar Uiversity, Vellore , Tamiladu, Idia chitratvu@gmail.com, karthikeyad9@gmail.com Mauscript History Number: IJIAE/S/Vol.4/Issue19/OCAE19 eceived:, September 17 Fial Correctio: 3, September 17 Fial Accepted: 5, October 17 Published: October 17 Editor: Dr.A.Arul L.S, Chief Editor, IJIAE, AM Publicatios, Idia Copyright: 17 This is a ope access article distributed uder the terms of the Creative Commos Attributio Licese, Which Permits urestricted use, distributio, ad reproductio i ay medium, provided the origial author ad source are credited. Abstract: This paper presets a theoretical study of oscillatory flow of blood i steosed artery i the presece of Magetic field with variable viscosity. The fluid (blood) medium is assumed to be porous i ature. The goverig equatio for lamiar icompressible fully developed ad Newtoia fluid by assumig time depedet expoetial pressure gradiat subject to the boudary coditios is solved by usig the Frobeius method. The variable viscosity of blood depedig o hematocrit is take ito accout i order to improve resemblace to the real situatio. It is assumed that the surface roughess is cosie shaped ad the maximum height of roughess is very small compared with radius of the ucostructed tube. The aalytical expressio for velocity compoet (V), Volumetric flow rate (Q) ad wall shear stress ( ) are obtaied. The effect of magetic field (B ), Hartma umber (M) ad maximum Hematocrit at the ceter of the arterial segmet (H) o velocity, flow rate ad stress are computed graphically. Key Words: Magetic field, Oscillatory flow, Porous, Hematocrit, Frobeius method. I. INTODUCTION The fluid dyamical behavior of blood through a arterial segmet havig steosis plays a vital role i cardiovascular disease. The arrowig i the artery, commoly referred to as steosis, is a dagerous disease ad is caused due to the abormal growth i the lume of the arterial wall. Steosis may be formed at oe or more locatios of the result cardiovascular system. As a result of such udesirable formatio at the edothelium of the vessel wall reductio of regular blood flow is likely to take place ear the steosis. If this disease takes a severe form, it may lead to stroke, heart attack ad various cardiovascular disease. May cardiovascular diseases such as due to the leadig cause of death worldwide. The partial occlusio of the arteries due to steotic obstructio ot oly restrict the regular blood flow but also characterizes the hardeig ad thickeig of the arterial wall. However, the mai cause of the formatio of steosis is still ukow but it is well established that the fluid dyamical factors play a importat role as to further developmet of steosis. The blood flow through a artery depeds upo the pumpig actio of the heart gives rise to a pressure gradiat which produces a oscillatory flow i the blood vessel (Haldar, 1987). I fact blood is a suspesio of cells i plasma. IJIAE: Impact Factor Value SJIF: Iospace, Morocco (16): PIF:.469 Jour Ifo: 4.85 ISAJIF (16): Idexcopericus: (ICV 15): IJIAE 14-17, All ights eserved Page -5

2 Iteratioal Joural of Iovative esearch i Advaced Egieerig (IJIAE) ISSN: Issue 1, Volume 4 (October 17) The plasma which is a solutio of proteis, electrolyte ad other substaces, is a icompressible virtually Newtoia fluid. From biomechaical poit of view, blood is cosidered as a itelliget fluid, probably the most oe i the ature, capable of adaptig itself i a great extet i order to provide utriets to the orgas. Huma body experieces magetic fields of moderate to high itesity i may situatios of day to day life. I recet times, may medical diagostic devices especially those used i diagosig cardiovascular disease make use of magetic fields. It is kow from the mageto-hydrodyamics that whe a statioary, trasverse magetic field is applied exterally to a movig electrically coductig fluid, electrical currets are iduced i the fluid. Sice blood is electrically coductig fluid, its flow characteristics is iflueced by the applicatio of magetic field. If a magetic field is applied to a movig ad electrically coductig fluid. it will iduce electric as well as magetic fields. The iteractio of these fields produces a body force per uit volume kow as Loretz force, which has sigificat impact o the flow characteristics of blood. Such a aalysis may be useful for the reductio of blood flow durig surgery ad Magetic esoace Imagig (MI). Hematocrit is the most importat determiat of whole blood viscosity. Therefore, blood viscosity ad vascular resistace affect total peripheral resistace to blood flow, which is abormally high i the primary stage of hypertesio. Agai hematocrit is a blood test that measures the percetage of red blood cells preset i the whole blood of the body. The percetage of red blood cells blood cells may as i adult huma body is approximately 4-45% [18]. ed blood cells may affect the viscosity of whole blood ad thus the velocity distributio depeds o the hematocrit. So blood caot be cosidered as homogeeous fluid []. Due to the high shear rate ear the arterial wall, the viscosity of blood is low ad the cocetratio of red blood cells is high i the cetral core regio. Therefore, blood may be treated as Newtoia fluid with variable viscosity particularly i the case of large blood vessels. The effect of magetic field o blood flow has bee aalyzed theoretically ad experimetally by may ivestigators [6-9)] uder differet situatios. Shit ad his coivestigators [1-14] explored variety of flow behavior of blood i arteries by treatig Newtoia, o-newtoia model i the presece of a uiform magetic field. It is well kow that blood behaves differetly whe flowig i large vessels, i which Newtoia behaviour is expected ad i medium ad small vessels where o-newtoia effects appear [1]. May researchers studied the pulsatile flow of blood i steosed artery [4]. Misra ad Shit studied the effect of magetic field o blood flow through a artery i usteady situatio ad observed the effect of magetic parameter[15], usteady parameter ad the radius phase agle o the flow charecteristics. Sayal et.al studied the effect of magetic field o pulsatile blood flow through a iclied circular tube with periodic body acceleratios ad discussed the effect of magetic field[16], gravitatioal parameter, iclied agle, body acceleratio, time etc. o axial blood flow, flow rate ad acceleratio. Bhuya ad Hazarika studied the magetic effect o flow through circular tube of o-uiform cross-sectio with permeable walls[17]. I the preset ivestigatio, we cosider the usteady oscillatory flow of blood through a steosed artery uder the effect trasverse magetic field. The study pertais to a situatio i which the variable viscosity of blood depedig upo hematacrit is take ito cosideratio. It is assumed that the arterial segmet is cylidrical tube with time depedet expoetial pressure gradiat ad goverig equatios are solved by usig Frobeius method. The effect of Hartma umber, magetic field ad maximum Hematocrit at the ceter of the arterial segmet o velocity profile, volumetric flow rate ad wall shear stress are computed graphically. II. FOMULATION OF THE POBLEM.1. Flow Geometry The Idealized geometry of steosis is give by fig. 1 ( z) 1 [1 cos l l z d whe d z d l 1 ; otherwise (1) ( )]; IJIAE: Impact Factor Value SJIF: Iospace, Morocco (16): PIF:.469 Jour Ifo: 4.85 ISAJIF (16): Idexcopericus: (ICV 15): IJIAE 14-17, All ights eserved Page -53

3 Iteratioal Joural of Iovative esearch i Advaced Egieerig (IJIAE) ISSN: Issue 1, Volume 4 (October 17) Where (z) is the radius of the artery i the steotic regio, is the radius of the ormal artery, l the legth of steosis, d legth of o-steosis ad δ the maximum height of steosis such that 1(fig 1).. Flow aalysis ad coordiate system : Let us cosider the oscillatory flow of blood through a artery with mild costrictio. The flow is assumed to be lamiar, Newtoia, viscous, icompressible, usteady ad axially symmetric by assumig ( r) variable viscosity ad desity is costat. We assumed that blood s compressible, suspesio of erythrocytes i plasma ad has uiform dese throughout but the viscosity µ(r) varies i the radial directio. Accordig to Eistei s formula for the variable viscosity of blood take to be µ r µ 1 h r () where µ is the coefficiet of viscosity of plasma, β is a costat ad h(r) stads for the hematocrit. The aalysis will be carried out by usig the flowig empirical formula for hematocrit. r h( r ) H 1 m i which represets the radius of a ormal arterial segmet, H is the maximum hematocrit at the ceter of the artery ad (m ) a parameter that determies the exact shape of the velocity profile for blood. The shape of the hematocrit profile give by equatio (3) is valid oly for very dilute suspesios of erythrocytes, which are cosidered to be of spherical shape. Accordig to our cosideratios, the equatio that govers the flow of blood uder the actio of a exteral magetic field may be put as the goverig equatio of motio i axial directio is v 1 p ( r) v 1 v ( r) B v v t z r r r k The boudary coditios are No slip coditio: v= at r= (z) v r Symmetry coditio: at r= (5) III. METHOD OF SOLUTION It is coveiet to write these equatios i dimesioless form by meas of the followig trasformatio variables. r z v t p ( r) k r ', z ', ', v ', t ', p ', '( r), k ' v t v (6) Where v,, p,, ad B are the average velocity, radius i the uobstructed tube, pressure, desity, viscosity of blood ad the applied magetic field i respectively. The the equatio (5) reduces to the form, by droppig primes, we get 1 v p ( r) v 1 v 1 ( r) M v t t z e r r r e k Where M B v The boudary correspodig coditios becomes No slip coditio : v= at r=(z) v r e, tv t ad v=v(r,t) is the velocity i the axial directio. Symmetry coditio : at r= (8) Let the solutios for v ad p be set i the forms IJIAE: Impact Factor Value SJIF: Iospace, Morocco (16): PIF:.469 Jour Ifo: 4.85 ISAJIF (16): Idexcopericus: (ICV 15): IJIAE 14-17, All ights eserved Page -54 (7) (3) (4)

4 Iteratioal Joural of Iovative esearch i Advaced Egieerig (IJIAE) ISSN: Issue 1, Volume 4 (October 17) it v r e v r, t p e i t z Sub. equatio (9) i equatio (7), we get (9) v ( ) 1 1 r r r 1 ( r ) 1 v P z M v e r r e k t t ( ) with the use of the trasform defied i the equatios () ad (3), the goverig equatio (7) reduces to 1 m v 1 1 m e e r a1 ar M a1 ar i v r r r k (11) t With a =1+a, a =βh. The equatio (11) ca be solved subjected to the boudary coditios (8) usig Frobeius method. For this, of course, v has to be bouded at r=, the oly admissible series solutio of the equatio (11) will exists ad ca put i the form e v( r) C A r Br 4a (1) 1 (1) where C, A ad B are arbitrary costats. To fid the arbitrary costat C, we use the o-slip boudary coditio (8) ad obtaied as e B C (13) 4a1 A Substitutig the value of v from equatio (1) ito equatio (11) o simplificatio, we get m m a1 a m C ( 1)( a1 ar ) A r ( a1 ( m1) ar ) A r M A r Br k k e a1 a 1 ( a1 ar ) Br ( )( a1 ( m 1) ar ) Br M Ar Br 4a1 k k e 4a1 (14) e Equatig the coefficiet of C ad i equatio (14) we have 4a Ad m m m 1 a a ( 1)( ) ( ( 1) ) m m 1 m a1 ar Ar a1 m ar Ar M Ar Br k k (15) m m a 1 ( 1 a m a1 ar ) Br ( )( a1 ( m1) ar ) B r M Ar Br 1 k k (16) Hece the costats A ad B are obtaied by equatig the coefficiets of r ad r from both side of equatios (15) ad (16) respectively, we get a1 a a ( 1) A 1 m m a A 1 m M A 1 A 1 m k k A 1 (17) a (1 ) 1 IJIAE: Impact Factor Value SJIF: Iospace, Morocco (16): PIF:.469 Jour Ifo: 4.85 ISAJIF (16): Idexcopericus: (ICV 15): IJIAE 14-17, All ights eserved Page -55

5 Iteratioal Joural of Iovative esearch i Advaced Egieerig (IJIAE) ISSN: Issue 1, Volume 4 (October 17) B a a a ( 3 ) a m B M B B k k 1 1 m 1 1 m 1 a 1 (3 ) With A B 1 (19) Substitutig the expressio of C i the equatio (1), we have the velocity profile the arterial segmet i the radial directio is B A r Br A e () v( r) 4a1 A r Where V ( z). The average velocity V has the form e (1) 8 Where P(z) is the pressure gradiet of the flow field i the ormal artery i the absece of magetic field. The o-dimesioal expressio for v is give by B Ar Br A V V () V a1 A The volumetric flow rate across the arterial segmet is give by Q rv ( r) dr (3) (18) Substitutig v from Eq. () ito Eq. (3) ad the itegratig with respect to r, we obtai 3 e P ( z ) Q a 1 4 A B A B A 4 (4) If Q be the volumetric flow rate i the ormal portio of the artery, i the absece of magetic field ad porosity effect the 3 e Q (5) 8 Therefore, the o-dimesioal volumetric flow rate has the followig form Q Q 4 P ( z ) 4 A B B A 4 Q a1 P ( z ) A The wall shear stress o the edothelial surface is give by dv ( r ) (7) dr r ( z ) IJIAE: Impact Factor Value SJIF: Iospace, Morocco (16): PIF:.469 Jour Ifo: 4.85 ISAJIF (16): Idexcopericus: (ICV 15): IJIAE 14-17, All ights eserved Page -56 (6)

6 Iteratioal Joural of Iovative esearch i Advaced Egieerig (IJIAE) ISSN: Issue 1, Volume 4 (October 17) Substitutig v from Eq. () ito Eq. (7), we obtai m H 1 B Ar ( ) Br A P ( z ) e o 4a1 A r If τ = P(z) be the shear stress at the ormal portio of the arterial wall, i the absece of magetic field, the o-dimesioal form of the wall shear stress is give by m 1 1 (9) 1 H 1 B Ar ( ) B r A 1 o P ( z) a N 1 Ar IV. ESULTS AND DISCUSSION The theorectical study of oscillatory flow of blood i steosed artery i the presece of Magetic field with variable viscosity have bee discussed. The study pertais to a situatio i which the variable viscosity of blood depedig upo hematocrit is take ito cosideratio. It is assumed that the arterial segmet is i cylidrical tube with time depedet expoetial pressure gradiat. The variable viscosity of blood depedig o hematocrit is take ito accout i order to improve resemblace to the real situatio. It is assumed that the surface roughess is cosie shaped ad the maximum height of roughess is very small compared with radius of the ucostructed tube. The aalytical expressio for velocity compoet, Volumetric flow rate ad wall shear stress are obtaied (8) Fig. : Velocity distributio at z=.5 with r for differet values Hartma umber M, whe hemotocirt H=., frequecy parameter β=.5, eyolds umber e=1. Fig. 3 : Velocity distributio at z=.5 with r for differet values hematocrit H, whe Hartma umber M=.5, frequecy parameter β=.5, eyolds umber e=1. IJIAE: Impact Factor Value SJIF: Iospace, Morocco (16): PIF:.469 Jour Ifo: 4.85 ISAJIF (16): Idexcopericus: (ICV 15): IJIAE 14-17, All ights eserved Page -57

7 Iteratioal Joural of Iovative esearch i Advaced Egieerig (IJIAE) ISSN: Issue 1, Volume 4 (October 17). The effect of magetic field, Hartma umber ad maximum Hematocrit at the ceter of the arterial segmet o velocity, flow rate ad stress are computed graphically. I the previous sectio we have obtaied aalytical expressios for differet flow characteristics of blood through the actio of a exteral magetic field. I this sectio we are to discuss the effect of various parameter o the flow characteristics graphically with the use of followig umerical data which is applicable to blood. d=.5, H=., m=, M=.5, β=.5, l=.5, e =1. Fig. 4 : Velocity distributio at z=.5 with r for differet values eyolds umber e, hematocrit H=., whe Hartma umber M=.5, frequecy parameter β=.5,. Fig. 5 : Variatio of the rate of flow with for differet Hematocriy H, Hartma umber M=.5, frequecy parameter β=.5. Fig. 6: Variatio of the rate of flow with for differet Hartma umber M, Hematocrit H=., frequecy parameter β=.5. IJIAE: Impact Factor Value SJIF: Iospace, Morocco (16): PIF:.469 Jour Ifo: 4.85 ISAJIF (16): Idexcopericus: (ICV 15): IJIAE 14-17, All ights eserved Page -58

8 Iteratioal Joural of Iovative esearch i Advaced Egieerig (IJIAE) ISSN: Issue 1, Volume 4 (October 17) Fig. 7 : Variatio of the wall shear stress with time for differet Hartma umber M, Hematocrit H=., frequecy parameter β=.5. Fig. 8 : Variatio of the wall shear stress of flow with time for differet Hematocrit H, Hartma umber M=.5, frequecy parameter β=.5. Figure shows that the velocity profile gradually decreases at the ceterli e of the artery icrease of Hartma umber. Figure 3 gives the distributio of axial velocity for differet values of the hematocrit H, that the velocity decreases at the steosed artery with the icrease of hematocrit level H. This fact lies withi the hematocrit as the blood viscosity is high i the steosed artery due to the aggregatio of blood cells rather tha low viscosity i the plasma ear the arterial wall. Figure 4 shows that the velocity profile gradually decreases at the icrease of eyolds Number. Figure 5 & 6 shows that the volume flow rate icreases with the icrease of Magetic field stregth B ad hematocrit H. Figure 7 ad 8 gives the distributio of the wall shear stress for differet values of the hematocrit H ad Hartma umber M. We observe from figure 7 & 8 shows that wall shear stress icreases as the hematocrit H ad Hartma umber icreases. Oe ca ote from this figures that the wall shear stress is low at the throat of the secodary steosis as well as at dowstream of the artery. V. CONCLUSION A theoretical study of blood flow through a steosed artery i the presece magetic field has bee carried out. I this study the variable viscosity of blood depedig o hematocrit. The problem is solved aalytically by usig the Frobeius method. The mai fidigs of the preset study may be listed as follows: The flow velocity at the cetral regio decreases gradually with the icrease of magetic field stregth. The hematocrit ad the pressure has a liear relatioship as reported. The lower rage of hematocrit may leads to the further depositio of cholesterol at the edothelium of the vascular wall. Hematocrit cotributes to the regulatio of blood pressure. IJIAE: Impact Factor Value SJIF: Iospace, Morocco (16): PIF:.469 Jour Ifo: 4.85 ISAJIF (16): Idexcopericus: (ICV 15): IJIAE 14-17, All ights eserved Page -59

9 Iteratioal Joural of Iovative esearch i Advaced Egieerig (IJIAE) ISSN: Issue 1, Volume 4 (October 17) Fially we ca coclude that further potetial improvemet of the model are aticipated. Sice the hematocrit positively affects blood pressure, further study should examie the other factors such as diet, tobacco, smokig, overweight etc. from a cardiovascular poit of view. Moreover o the basis of the preset results, it ca be cocluded that the flow of blood ad pressure ca be cotrolled by the applicatio of a exteral magetic field. All the flow characteristics are foud to be affected by the ifluece of applied magetic field with profile velocity steosis. VI. EFEENCES 1. D. Biswas, B.. Laskar, Ifluece of magetic field ad slip velocity o pulsatile blood flow through a costricted artery, (4) (11), G.C. Layek, S. Mukhopadhaya ad.s.. Gorla, Usteady viscous flow with variable viscosity i a vascular tube with a double costrictio Iteratioal Joural of Egieerig Sciece 47 (9), G.C. Shit, M. oy ad A. Siha, Mathematical modelig of blood flow through a tapered overlappig steosed artery with variable viscosity, Applied Bioics ad Biomechaics 11 (14) DOI 1.333/ABB D. Biswas, Chakraborty, Steady flow of blood through a catheterized tapered artery with steosis, A Theoritical model, Assam Uiversity Joural of Sciece ad Techology, 4(), (9) pp G.C. Hazarika ad Barali Sharma, Magetic field effect o oscillatory flow of blood i a steosed artery, Iteratial coferece o frotiers i Mathematics (15), ISBN: pp G. Varshey, V.K. Katiyar ad S. Kumar, Effect of magetic field o the blood flow i artery haig multiple steosis: A umerical study, Iteratioal Joural of Egieerig Sciece ad Techology () (1), Y. Jaik, V. Pai ad C.J. Che Apperet Viscosity of huma blood i a static magetic field, Joural of Magetism ad Magetic Materials 5 (1), D.W.Chakeras, A. Kargarlu, H. Boudoulas ad D.C. Youd, Effect of static magetic field exposure of upto 8 Tesla o sequetial huma vital sig measuremets, Joural of Magetic esoace Imagig 18 (3), Y. Kiouchi, H. Yamaguchi ad T.S. Teforde, Theretical aalysis of magetic field iteractios with aortic blood flow, Bioelecomagetics 17 (1996), G.C. Shit ad M. oy, Pulsatile flow ad heat trasfer of a mageto micropolar fluid through a steosed artery uder the ifluece of body acceleratio, Joural of Mechaics i Medicie ad Biology 11(11), J.C. Misra, A. Siha ad G.C. Shit, Mathematical modellig of blood flow i porous vessel havig dourble steoses i the presece of a exteral magetic field, Iteratioal Joural of Biomathematics 4 () (11), J.C. Misra, G.C. Shit, S. Chadra ad P.K. Kudu, Hydromagetic flow ad heat trasfer of a secod grade viscoelastic fluid i a chael with oscillatory strethig walls: Applicatio to the dyamics of blood flow, Joural of Egieerig Mathematics 69 (11), J.C. Misra, G.C. Shit, Flow of a biomegetic fluid i a chael with stretchig walls, ASMEJ Appl Mecch, 76(9) 616 (9 pages). 14.J.C. Misra, G.C. shit add H.J. ath, Flow add heat trasfer of a MHD viscoelastic fluid i a chael with stretchig walls: Some applicatios to hemodyamics, Computers & fluids 37(8), J.C. Misra, ad G.C. Shit, Effect of magetic field o blood flow through a artery: A umercal model, 1(4) (7), D.C. Sayal, K. Das ad S. Debath Effect of magetic field o pusatile blood flow through a iclied circular tbe with periodic body acceleratio, Joural of Physical Scieces, 11, (7) pp B.C. Bhuya ad G.C. Hazarika Magetic effect o flow through circular tube of o-uiform crossectio with permeable walls, Applied Sciece Periodicall, 5(1) (3), S. Oka. Cardiovascular hemorhology, Cambridge Uiversity Press, Lodo, 1981, 8. IJIAE: Impact Factor Value SJIF: Iospace, Morocco (16): PIF:.469 Jour Ifo: 4.85 ISAJIF (16): Idexcopericus: (ICV 15): IJIAE 14-17, All ights eserved Page -6