Pulsatile flow of Herschel-Bulkley fluid through an inclined multiple stenoses artery with periodic body acceleration

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1 Available online at Advances in Applied Science Reseach 6 7():- ISSN: CODEN (USA): AASRFC Plsatile flow of eschel-blkley flid thogh an inclined mltiple stenoses atey with peiodic body acceleation Raja Agawal and N. K. Vashney Depatment of Mathematics S. V. College Aligah ABSTRACT The plsatile flow of eschel Blkley flid thogh an inclined mltiple stenoses atey with peiodic body acceleation has been investigated in this pape. Assming the stenoses to be mild the nonlinea eqations govening the flow ae solved sing petbation techniqe. Analytical expessions ae obtained fo axial velocity plg velocity wall shea stess and flow ate. Thei vaiations with diffeent flow paametes ae plotted in figes. It is noticed that the velocity inceases as body acceleation inceases bt it deceases as yield stess inceases and wall shea stess inceases as body acceleation inceases. Keywods: Plsatile flow Body acceleation eschel Blkley flid stenosed atey. INTRODUCTION The cadiovascla system pimaily fnctions in ntient and waste tanspot thoghot the body. The blood vessels distibte blood to diffeent ogans and spply themselves with ntition. The ateies fa fom inet tbes adapt to vaying flow and pesse conditions by enlaging o shinking to meet changing hemodynamic demands. The systemic flow is chaacteised pedominantly by its plsatile nate and the many levels of banching of the vascla netwok. The heat ejects and fills with blood in altenating cycles called systole and diastole. Blood is pmped ot of the heat ding systole. The heat ests ding diastole and no blood is ejected. In sitations like taveling in vehicles o aicaft jackleg dills opeating jackhamme o the sdden movements of the body ding spots activities the hman body expeiences extenal body acceleation. Polonged expose of a healthy hman body to extenal acceleation may case seios health poblem like headache loss of vision abdominal pain and incease plse ate. De to physiological impotance of body acceleation many mathematical models have been poposed fo blood flow with body acceleation. Chatani and Palanisami[] discssed Casson flid model of plsatile flow of blood flow nde peiodic body acceleation. Chatani and Palanisami[] investigatedplsatile flow of powe law flid model fo blood flow nde peiodic body acceleation. Chatani and Palanisamy [] consideed plsatile flow of blood with peiodic body acceleation. Chatani and Samy[4] stdied Plsatile flow of Casson's flid thogh stenosed ateies with applications to blood flow. Chien[5] discssed hemoheology in clinical medicine Recent Advances in Cadiovascla Diseases. El-Shehed [6] consideed plsatile flow of blood thogh a stenosed poos medim nde peiodic body acceleation. Elshehawey et.al [7] discssed plsatile flow of blood thogh a poos medim nde peiodic body acceleation. Mandalet.al [9] consideed effect of body acceleation on nsteady plsatile flow of non-newtonian flid thogh a stenosed atey. Mathi Pasad and Radhakishnamachaya [] stdiedeffect of mltiple stenoses on eschel-blkley flid thogh a tbe with nonnifom coss-section. Mathi Pasad and Radhakishnamachaya [] stdied flow of eschel-blkley flid thogh an inclined tbe of non-nifom coss-section with mltiple stenoses. Meill et.al [] consideed pesse

2 Raja Agawal and N. K. Vashney Adv. Appl. Sci. Res. 6 7():- flow elations of hman blood in hollow fibe at low shea ates. Nagaani and Saojamma[] stdied effect of body acceleation on plsatile flow of Casson flid thogh a mild stenosed atey. Sanka and emalatha[4] investigated plsatile flow of eschel-blkey flid thogh stenosed ateies a mathematical model. Saojamma and Nagaani[5] consideed plsatile flow of Casson flid in a homogeneos poos medim sbject to extenal acceleation.shkla et.al [6]investigated Effects of stenosis on non-newtonian flow thogh an atey with mild stenosis. Siddiqi and Misha [7] stdied a stdy of modified Casson s flid in modeled nomal and stenotic capillay tisse diffsion phenomena. Siddiqiet.al [8] consideed Mathematical modelling of plsatile flow of Casson s flid in ateial stenosis. T and Deville [9] discssedplsatile flow of non-newtonian flids thogh ateial stenoses. Vajavel et.at [] investigated peistaltic tanspot of a eschel-blkley flid in an inclined tbe. In the pesent investigation an effot has been made to stdy the plsatile flow of eschel Blkley flid thogh an inclined mltiple stenoses atey with non-nifom coss-section sbject to peiodic body acceleation assming that the stenoses ae mild. Analytical expessions fo axial velocity and flow ate have been deived and the effects of vaios paametes on these flow vaiables have been stdied. MATEMATICAL FORMULATION R : z d δ π L R cos z d : d z d L L R d L z B R( z ) = δ π L R cos ( z B ) : B z B L L * δ π L R ( z ) cos ( z B ) : B z B L * L R ( z ) : B z B. : The following estictions fo mild stenoses [] ae spposed to be satisfied: ( R R ) δ << min i ot δ << L whee R = R( z) at z = B. i i ot L () δ ( i = ) ae the lengths and maximm heights of two stenoses (the sffixes and efe to the ee L i and i fist and second stenosis espectively). The pesse gadient and body acceleation ae given by: p = A A cos( ω pt ) z () G( t ) = a cos( ω t φ) () b

3 Raja Agawal and N. K. Vashney Adv. Appl. Sci. Res. 6 7():- Whee A and A ae pesse gadient of steady flow and amplitde of oscillatoy pat espectively a is the amplitde of body acceleation ω p = π f p ωb = π fb with f p is the plse feqency and f b is body acceleation feqency φ is the phase angle of body acceleation with espect to the pesse gadient and t is time. β Fige: Geomety of an inclined tbe with mltiple stenoses The govening eqation of motion fo flow in cylindical pola coodinates can be witten in the fom: p ρ = ( ) G( t ) ρ g sin β t z ρ = A A cos( ω pt ) ( ) a cos( ωbt φ) ρ g sin β t p = (4) (5) Whee z denote the adial and axial coodinates espectively and ρ denote density axial velocity of blood t time p pesse and the shea stess and β be the small angle of inclination g is acceleation de to gavity. Fo eschell-blkley flid the elation between shea stess and shea ate is given by n = µ if > (6) 4

4 Raja Agawal and N. K. Vashney Adv. Appl. Sci. Res. 6 7():- = if < Whee is the total velocity is the yield stess n is the powe law index and µ is the coefficient of viscosity fo eschell-blkley flid. When (7) coesponds to vanishing velocity gadient in that egion. oweve the flid behavio is indicated wheneve >. < i.e. the shea stess is less than the yield stess thee is a coe egion which flows as a plg and Eq. The bonday conditions ae: is finite at = (8) = at = R( z ) (9) Intodcing the non-dimensional vaiables: z z t t δ = = = ω δ = = A R / 4 R R A R / p µ R( z ) d L = R( z) = = d = L = A R / R R B B L B a A ωb L = B = a = e = ω = ω B B A A A n = ( ) F = () R A 4ρg µ µ p (7) The non-dimensional momentm eqation (4) becomes sin β α = 4( ecost ) ( ) 4acos( ωt φ) t F ω p Whee α = R α is Womesley feqency paamete. µ / ρ () Eqations (6) and (7) can be witten as if = if < ( ) n = The bonday conditions (eqations 8 and 9) edce to > () () 5

5 Raja Agawal and N. K. Vashney Adv. Appl. Sci. Res. 6 7():- is finite at = (4) = at = R( z) (5) The geomety of the stenosis in non-dimensional fom is given by : z d δ π L cos z d : d z d L L R( z) = δ π L cos ( z B ) : B z B L L * δ π L R ( z) cos ( z B ) : B z B L * L R ( z) : B z B. : d L z B L (6) METOD OF SOLUTION On sing petbation method the velocity and shea stess ae expanded as follows in tems of α (whee α <<) ( z t) = ( z t) α ( z t)... (7) ( z t) = ( z t) α ( z t)... (8) p( z t) = p( z t) α p( z t)... (9) R ( z t) = R ( z t) α R ( z t)... () p p p Sbstitting (7) and (8) in eqation () and eqating the constant tem and α tem we get sin β ( ) = ( ecost ) acos( ωt φ) 4F = ( ) t Integate eqation () and sing bonday condition (4) = () Whee ( ) sin β f t = ecost acos( ωt φ) 4F Sbstitting (7) and (8) in () () () 6

6 Raja Agawal and N. K. Vashney Adv. Appl. Sci. Res. 6 7():- k = ( k ) k = k ( k ) Whee k = /n ( ) Integating eqation (4) sing the elation () and the bonday condition (5) we obtain k k k k ( ) ( ) = A R A R (6) Whee k k = = ( ) A A f t n The plg coe velocity p can be obtained fom eqation (6) as k k k k ( ) ( ) = A R R A R R (7) p p p Neglecting the tems of O ( α ) and highe powes of α in eqation () R (4) (5) R p can be obtained fom () as p = (8) Using eqation () we get the soltion fo as = a a a (9) k k k k Whee a a a nr ( k ) R ( k ) = = = ( k ) k k a a a a f t f t ( k )( k ) k 4 = 5 = ( ) ( ) ( ) a = a f ( t) a = a f ( t) k k Similaly sing eqations (5) and (9) we can obtain the soltion fo as k k k k = {( b b ) R b } {( b b4 ) ( b5 b6 )} R ( b7 b8 ) b R b b R b R b R b R k k k k k k k k k k 9 4 k k k k p = {( ) p } {( 4) ( 5 6)} ( 7 8) p b b R b R b b b b R b b R b R b R b R R b R R b R R b R R k k k k k k k k k k 9 p p p p 4 p () () 7

7 Raja Agawal and N. K. Vashney Adv. Appl. Sci. Res. 6 7():- k k k n na ka4 ka na Whee b = b = b = b4 = k k k n k k k( k ) a k( k ) a4 b5 = b6 = k k k k ka k( k ) a4 b7 = b8 = k k a a k k b9 = k( k ) b = ( k ) a k k k k ka ka k b = b = b = ( k ) a k k k b = ( k ) a 4 Using eqations (7) and (8) the total velocity distibtion and shea stess can be witten as ( ) ( ) k k k k k k = A R A R α {( b b ) R b } {( b b ) ( b b )} R ( b b ) k k k k k k k k k k k k b9 R b b R b R b R b4 R ( ) ( ) k k k k k k p = A R R p A R R p α {( b b ) R br p } {( b b ) ( b b )} R ( b b ) R k k p k k k k k k k k k k b9 R b R p b R R p b R R p b R R p b4 R R p ( ) = R k k R α { a5r a6r a7r } = α = The second appoximation plg coe adis of α in eqation () as R ( R ) () () (4) 4 R p can be obtained by neglecting tems of O ( α ) and highe powes p p = (5) With the help of eqations () (8) and (5) α p p p p Rp can be given by k k ( ) R = a R a R a R (6) The volmetic flow ate Q is given by R( z) Q = 4 ( z t) d 8

8 Raja Agawal and N. K. Vashney Adv. Appl. Sci. Res. 6 7():- k ( k ) k k k 4 ( b b ) Q = 4 A R A R α R ( k ) ( k ) b b b b b b R R k 4 k k 4 k b b b b b b b R k k k k k 7 8 k 9 4 (7) e = e = e = e = Fige : Vaiation of axial velocity with adial distance fo =. a = ω = φ =. β = a = a = a = a = Fige : Vaiation of axial velocity with adial distance fo =. e = ω = φ =. β =. 9

9 Raja Agawal and N. K. Vashney Adv. Appl. Sci. Res. 6 7(): t =.4 t =.6 t =.8 t = Fige 4: Vaiation of axial velocity with adial distance fo =. a = ω = φ =. β = φ =. φ =.4 φ =.6 φ = Fige 5: Vaiation of axial velocity with adial distance fo =. a = ω = e = β =.

10 Raja Agawal and N. K. Vashney Adv. Appl. Sci. Res. 6 7(): β =. β =. β =.5 β = Fige 6: Vaiation of axial velocity with adial distance fo =. a = ω = φ =. e = = =. =. = Fige 7: Vaiation of axial velocity with adial distance fo e = a = ω = φ =. β =.

11 Raja Agawal and N. K. Vashney Adv. Appl. Sci. Res. 6 7():-.5 a = a =.5.5 Fige 8: Vaiation of shea stess with time t fo =. e = ω = φ =. β =. RESULTS AND DISCUSSION The velocity pofile fo the plsatile flow of eschel Blkley flid thogh an inclined mltiple stenoses atey with peiodic body acceleation is compted by sing () fo diffeent vales of paamete e body acceleation paamete a time t phase angle φ inclination angle β yield stess have been shown thogh figes - 7. Fige - shows that the vaiation of velocity pofile fo diffeent vales of paamete e. It can be noted hee that as the paamete e inceases the velocity pofile inceases. In the pesence of body acceleation velocity inceases apidly. As the body acceleation inceases the plg egion shinks and hence moe flow takes place (fige.). It can easily be seen fom figes 4 & 5 that an incease in the time t and phase angle φ leads to decease in the velocity pofile. Fom figes 6 and 7 it can be obseved that an incease in the inclination angle β and yield stess t lead to an incease in velocity pofile. Vaiation of wall shea stess with time t is pesented in fige - 8. Fom this fige it can be clealy obseved that fo any vale of body acceleation paamete a wall shea stess gadally deceases as time t inceases ntil it attains its minimm at t = 8 wheefom it gadally inceases with time and eaches its appoached magnitde at t = 6. CONCLUSION The pesent stdy deals with a theoetical investigation of the chaacteistics of the plsatile flow of blood thogh an inclined mltiple stenoses atey with peiodic body acceleation. Blood is epesented by eschel Blkley flid model. Using appopiate bonday conditions analytical expessions fo the velocity and flow ate have been obtained. It is clea fom the above eslt and discssions that the body acceleation effects lagely on the axial velocity of blood flow. A pope ndestanding of inteactions of body acceleation with blood flow in pesence of inclination cold be sefl in the diagnosis and theapetic teatment of some health poblems (joint pain vision loss and vascla disode) to bette design of potective pads and machines.

12 Raja Agawal and N. K. Vashney Adv. Appl. Sci. Res. 6 7():- ence fom all the above discssions we can conclde that a caefl choice of the vales of the paametes of body acceleation yield stess and inclination angle will affect the flow chaacteistics and hence can be tilised fo medical and engineeing applications. REFERENCES [] Chatani P Palanisami VBioheol [] Chatani P PalanisamiVBioheol [] Chatani P Palanisamy V Int. J. Eng. Sci [4] Chatani P Samy RP Bioheol [5] Chien S Recent Advances in Cadiovascla Diseases [6] El-Shahed M Applied Mathematics and Comptation [7] Elshehawey EF Elbabay EMEElsayed M EAfifiNAS El-ShahedM Int. Jonal of theoetical Physics (). [8] Blai GWSSpanne DCAn Intodction to Bioheology Elsevie Amstedam 974. [9] Mandal PK ChakavathySMandal A AminN Applied Mathematics and Comptation [] Mathi Pasad KRadhakishnamachayaG Intenational e-jonal of engineeing mathematics: Theoy and Application [] MathiPasad K RadhakishnamachayaG Ach. Mech [] Meill EW Benis AM Gilliland ER ShewoodTKSalzman EW Appl. Physiol [] Nagaani PSaojamma GKoea- Astalia Rheology Jonal (4). [4] SankaDS emalatha K Intenational Jonal of Non-Linea Mechanics (8). [5] Saojamma GNagaani P Int. J. Non-Linea Diffe. Eqns. Theo. Models Appl [6] ShklaJBPaiha RS Rao BRP Bll. Math. Biol [7] Siddiqi SU Misha S Appl. Math. Compt [8] Siddiqi SU Gpta RS VemaNK Misha S Appl. Math. Compt.9 - (). [9] T C Deville MJ. Biomech [] VajavelKSeenadh S Ramesh BabV Int. J. of Non-Linea Mech