Effect of Applied Magnetic Field on Pulsatile Flow of Blood in a Porous Channel

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1 Dsmanta Kmar St et al, Int. J. Comp. Tec. Appl., ol 6, ISSN:9-69 Effect of Applied Magnetic Field on Plsatile Flow of Blood in a Poros Cannel Sarfraz Amed Dsmanta Kmar St Dept. of Matematics, Jorat Engineering College, Jorat, sarfrazamed.jrt@gmail.com Dept. of Matematics, N. N. Saikia College, Titabar, Jorat, stdk@aoo.com Abstract An approximate soltion is presented to te problem of plsatile flow of blood in poros cannel in presence of indced magnetic field. Te blood is assmed to be an incompressible Non Newtonian flid. To redce te eqation of motion to an ordinar differential eqation, a dimensionless ariable is sed. Nmerical reslts were obtained for different ales of te magnetic parameter, freqenc parameter and enolds nmber. It is obsered tat wen te magnetic parameter, te Hartmann nmber increases, te elocit as well as magnitde of mass flx decreases. For non magnetic case te paper redces to te problem inestigated b Wang.. Introdction Te application of Magneto drodnamics in psiological flow is of growing interest. Te flow of blood can be controlled b appling appropriate magnetic field. Man researcers ae sown tat blood is an electricall condcting flid Korceskii and Marocnlk [4], ardanan [8]. Te orentz s force will act on te constitent particles of blood and tis force will oppose te motion of blood and ts redce its elocit. Tis decelerated blood flow ma elp in te treatment of certain cardioasclar diseases and in te diseases wit accelerated blood circlation sc as pertension, emorrages etc. So, it is er essential to std te blood flow in presence of magnetic field. Man works ae been done in tis field b arios inestigators. Te plsatile flow of blood wit micro-organisms represented b two flid model trog essels of small exponential diergence nder te effect of magnetic filed as been stdied b atod and Gaatri [5]. A similar problem on blood flow trog a niform pipe wit sector of a circle as cross section in presence of transerse magnetic field as been stdied b atod and Pareen [6]. Exponential representation of blood flow goerning eqation nder external rnning plse magnitde field as been stdied b Jain, et. al. []. Flow in a poros cannel is important in transpiration cooling and gaseos diffsion process. Plsatile flow in a poros cannel, in particlar is also important in te dialsis of blood in artificial kidne Esmond and Clark []. Plsatile flow in a poros cannel as been inestigated b Wang [9] witot magnetic field. Plsatile flow in a poros cannel considering blood as Newtonian flid was inestigated b Ban and Hazarika []. Te same problem considering blood as Non Newtonian flid was inestigated b St and Hazarika [7]. Bt te aboe problem was inestigated nder transerse magnetic field. Here an attempt as been made in tis analsis to std te plsatile flow of blood in a poros cannel in presence of applied magnetic field. Here blood is assmed to be an incompressible non-newtonian flid.. Matematical Formlation of te Problem Here we consider a flid drien b an nstead pressre gradient p iwt A Be x between two poros plates at = and =. Here A and B are known constants and w is te freqenc. On one plate some flid is injected wit elocit and it is scked off at te opposite plate wit same elocit. De to continit, te elocit component in te - direction will be identicall eqal to eerwere. B and B are te components of magnetic field in x x and directions. So te goerning eqations of te flow nder te assmed conditions are te continit eqation and te Coll s eqation. nder te indced magnetic field applied in x and directions te goerning eqations becomes IJCTA NO-DEC Aailable online@ 779

2 ISSN:9-69 Dsmanta Kmar St et al, Int. J. Comp. Tec. Appl., ol 6, p x B Bx B p Bx4 Bx Bx 4 m Bx were is te elocit in te x direction and,, are te densit, electrical condctiit and kinematics iscosit. We se te following non-dimensional qantities. t ; ; t ; Y ; ; to P p o ;X Considering, t T, B,, t x ; ; ; ; B, B Te aboe eqation becomes, p x t B B B x B B wic implies, B ; Bx x ; e e ; B B e p, x ; t H sing te aboe non dimensional qantities te eqation becomes B B x B were t P x t p x B B x B B Mltipling bot side b te aboe eqation becomes B Bx BY 5 Now, m B x P t x, t t t wic implies, t B B x B B IJCTA NO-DEC Aailable online@ Bx Eqation 4 becomes m Bx wic implies, B B x B 78

3 ISSN:9-69 Dsmanta Kmar St et al, Int. J. Comp. Tec. Appl., ol 6, Bx Bx Bx 6 i.e., sing non-dimensional qantities te eqations and 4 respectiel becomes Bx i.e., we separate Eqations 5 and 6 into a stead part ~ nstead part. denoted b tidle A BY Bx BY 7 p x Be iwt B BY Bx BY Bx BY Bx For nstead part dependent of time BY B x BY Now we will separate eqation 6 into stead and nstead part For stead part independent of time Bx 9 Bx Bx 9 We introdce a new dimensionless ariable, were and f, Bx g Eqation 7 becomes f A f f A B f f f BY g f B f BY g f Mltipling b bot sides we get, A f f f For nstead part dependent of time IJCTA NO-DEC Aailable online@ Soltion of te Problem Now we separate Eqations 5 and 6 into a stead ~ nstead part. part denoted b tidle For stead part independent of time p A Be iwt x A BY Bx BY 7 Be iwt 5 B f BY g 78

4 ISSN:9-69 Dsmanta Kmar St et al, Int. J. Comp. Tec. Appl., ol 6, were, Magnetic parameter, M B Hartman nmber B or M Te aboe eqation becomes, A f f f Similarl eqation 8 becomes M f M g were l M f A f f f e M g mltipl bot sides b M f f f f e Eqation 9 becomes M g A Bx iwt e iwt f e f f iweiwt Be iwt t e B f e iweiwt f e iwt f BY g + iwt f f iwf f iw B t e B f BY g e iwt t M g A g g B f l B,, Te bondar condition are f=: at and f= at, g= at and g= at IJCTA NO-DEC Aailable online@ Diiding bot side b e we get f f f iw B T e were i.e., Ten te eqations 7 and 8 becomes M f f f f e were M BY Bx BY B f g g Be iwt f f g g g g B f Here dases represent differentiation wit respect to. We are not interested in discssion of te stead part so we sall not go for detail discssion ere. Te nstead eqation 8 & can be redced to ordinar differential eqation b introdcing non dimensional ariables and f e t, Bx g iwtf f B f g BY e it Y f g BY e it f g BY e it f f T =B were e e bot sides, e f e f e f Mltipling b 78

5 ISSN:9-69 Dsmanta Kmar St et al, Int. J. Comp. Tec. Appl., ol 6, e B e B g BY e wic implies e f it f e t i cos t i sin t e f + B and e f g BY e it B e f e f e f Eqation becomes, BY t e t - e e t i M e e i e t = B e t - e i M e e i t =B e et - e + - e Bx g g Mltipling b bot sides g B e t f g g g e t B f = 4 Wit te aboe sbstittions te eqation 8 and become respectiel, f e f e f M e e f B t M e + e =B e cos t t - e e - M e e + t 5 - e On ptting te ales of e t B f g g H f, f, f, f and g, g, g, g in, we get t e i i e Now eqating real and imaginar parts, M e e f B e t f e g B x ibx e i - BY g B x ibx i Again, H g Bx ibx On ptting te ales of B e B e e f. f + e g Bx =-B e sin t 6 f, f, f, f and g, g, g, g in 4, we get g - g e t B f = g - g M e t f B ib x = x B x ibx + M e t i e t = B x ibx - B x ibx + M i = Now eqating real and imaginar parts, B x B x M 7 were, e,, M B B x B x M 8 We pt, Te bondar conditions are ; at & B x ; B x at and g - g M e t f =, M B 4 f e t cos t i sin t i cos t i sin t Ten, f e t i cos t i sin t f e t i cos t i sin t IJCTA NO-DEC Aailable online@ B x and B x at 78

6 Dsmanta Kmar St et al, Int. J. Comp. Tec. Appl., ol 6, ISSN: eslt and Discssion Te problem nder inestigation is dominated mainl b te dimensionless parameters e te enolds nmber,,,, te cross flow enolds nmber, te Hartmann nmber M, Magnetic parameters,, t. Or interest is to inestigate te roll of magnetic parameters on te elocit field. From Fig.. it is obsered tat te flid elocit decreases as te magnetic parameter M increases. Te maxima of te elocit is sifted to te bondar laer in te region from. 5 to for all ales of M at M =, =, =., =., e =., s =.5, =., t =., wt=.79. Fig.. sows te effect of magnetic field for different ales of Hartmann Nmber M. It is obsered tat as Hartmann nmber increases, te effect of magnetic field is decreases. Fig.. sows te effect of magnetic field for different ales of M. It is obsered tat flid elocit decreases as M increases. Fig. 4 sows te elocit profiles wit effect of magnetic field for arios ales of e. It is seen tat te flid elocit, decreases as te enolds nmber e increases. Te maxima of te elocit is sifted to te bondar laer in te region from = for all ales of e wen M=.5, M =., =, =., =., s =.5, =., t =., wt=.79. Fig. 5 sows te elocit profiles wit effect of magnetic field for arios ales of s. It is seen tat te flid elocit increases as s increases. Te maxima of te elocit is sifted to te bondar laer in te region from =.5 to for all ales of s wen M=.5, M =., =, =., =., =., t =., wt=.79. In Fig. 6 te grap of ecloit profile for different ales of wt is gien As wt increases, te flid elocit is also increases and it is parabolic in natre. Here it is seen tat te flid elocit is greatl affected de to te presence of te magnetic field. Wen te Hartmann nmber increase, te flid elocit decreases. Also te magnitde of mass flx is dominated b te magnetic field. Te matematical expressions ma elp medical practitioners to control te blood flow of a patient wose blood pressre is er ig b appling certain magnetic field Fig.. Comparison of elocit profile for different Hartmann nmber at M'=, =, =., =., e=.,s=.5,=.,t=.,wt= Fig.. Comparison of te magnetic field for different Hartmann nmber at M'=, =, =., =., e=.,s=.5,=.,t=.,wt= Fig.. Comparison of elocit profile for different ales M' at M=.5, =, =., =., e=.,s=.5,=.,t=.,wt=.79 M= M= M= M= M=4 M=5 M= M= M= M= M=4 M=5 M= M= M= M= M=4 M=5 IJCTA NO-DEC Aailable online@ 784

7 ISSN:9-69 Dsmanta Kmar St et al, Int. J. Comp. Tec. Appl., ol 6, Fig. 4. Comparison of elocit profile for different ales e at M=.5, M'=., =, =., =., s=.5,=.,t=.,wt= e= e= e=4 e=6 e=8 5. eferences. Ban, B.C. and Hazarika, G.C., Proc. National Conference on Applied Matematics, pp. -7. Esmond, W.G. & Clark, H., Matematical Analsis and Mass Transfer Optimization of a compact, low cost, pmpless sstem for Hemodialsis Dialng. Proceedings, Biomedical flid Mecanics Smposim, ASME Jain,., onge, A.K., Ali, S.A. and Ali, S.N., Appl. Sc. Periodical, ol. 4, pp Korceski, E.M. and Marocnik,.S., Biofizika, ol., pp. pp atod,.p., Gaatri, Applied Science Periodical, ol., pp atod,.p., Pareen, S.., Te Matematics Edcation, ol., pp St D.K. and Hazarika G.C., Matematical Stdies of Certain Second Order Blood Flow Problems, DM erlag, German 8. ardanan,.a., Biopsics, ol. 8, pp Wang, C.Y., Plsatile flow in a poros cannel, Jornal of Applied Mecanics, ol. 8, pp Fig. 5. Comparison of elocit profile for different ales s at M=.5, M'=., =, =., =., =.,t=.,wt=.79 s=.5 s=. s=.5 s=. s=.5 s= wt= wt= wt=6 wt=9 wt= wt= Fig. 6. Comparison of elocit profile for different ales of wt at M=.5, M'=, =, =., =., e=.,s=.5,=.,t=. IJCTA NO-DEC Aailable online@ 785