A Casson Fluid Model for multiple Stenosed Artery in the Presence of Magnetic Field

Transcription

1 A Casson Fluid Model for multiple Stenosed Artery in the Presene of Magneti Field Rekha Bali* and Usha Awasthi ** Department of Mathematis Harourt Butler Tehnologial Institute, Nawabganj, Kanpur-8 (India) Abstrat The flow of blood through a multi-stenosed artery under the influene of external applied magneti field is studied. The artery is modeled as a irular tube. The effet of non-newtonian nature of blood in small blood vessels has been taken into aount by modeling blood as a Casson fluid. The effet of magneti field, height of stenosis, parameter determining the shape of the stenosis on veloities, volumetri flow rate in stenoti region and wall shear stress at surfae of stenosis are obtained and shown graphially. Some important observations regarding the flow of blood in multi stenosed artery are obtained leading to medial interest. Key Words: Blood flow, magneti effet, multiple stenosis artery, Casson fluid, shear stress, blood flow rate. AMS MSC No.: 76Z5, 9 CO5 Introdution Stenosis narrowing of body passage tube [] an ause series irulatory disorders by reduing the blood supply. Stenosis in the arteries supplying blood to the brain an ause erebral strokes, and in oronary arteries, myoardial infartion, leading to the heart failure. The atual auses of the stenosis are not known, but it has been suggested that holesterol deposition in arterial wall and profiliferation of onnetive tissues may be responsible [], vasular fluid dynamis is reported to play a signifiant role in the development and progression of the pathologial onditions [3]. Blood is suspension of ells in plasma. Due to the presene of hemoglobin (an iron ompound) in red ells, blood an be regarded as a suspension of magneti partile (red ells) in non-magneti plasma. The effet of a magneti field on blood flow has been analyzed theoretially by treating blood as an eletrially ondutive fluid [4].The ondutive flow in the presene of a magneti field indues voltage and urrents, resulting in a derease in the flow.the importane of heat transfer on artery diseases and blood flow was mentioned by several researher. Ugulu and Abby [5] laimed that, the heat transfer and a magneti field have a signifiant effet on blood flow through onstrited artery. An analytial solution for the steady flow of a visous fluid through an arbitrary shaped tube of variable ross-setion has been presented by Manton [6] using the ideas of steady lubriation theory Ramahandra Rao and Devanathan [7] and Hall [8] have extended the results of Manton [6] for unsteady pulsatile flows. The steady and unsteady flow through 53

2 hannels and tubes of variable ross-setion have been studied by Smith [9] and Duk [].Mathematial model for analyzing pulsatile flow in a single stenosed vessels have been proposed by Padmanabhan [], Mehrotha et al. [] and Mishra and Chakravorty [3]. The studies on the blood flow / unsteady blood flow through an artery with mild stenosis [4,5],effet of arterial dispensability on blood flow through model of mild axisymmetri arterial stenosis [6],flow of miro-polar fluid through a tube with a stenosis [7] non-newtonian aspets of blood flow through stenosed arteries [8],flow of ouple stress fluid through stenoti blood vessels [9], pulsatile flow of Casson s fluid through stenosed tube [],osillatory flow of blood in a stenosed artery [] and in a single onstrited blood vessels [],effet of erythroytes on blood flow harateristis in an indented tube [3],effet of an externally applied uniform magneti field on the blood flow in a single onstrited blood vessel [4] were also reported. In reent paper Manadal et al [5] developed a two dimension mathematial model to study the effet of externally imposed periodi body aeleration on non-newtonian blood flow through an elasti stenosed artery where the blood is haraterized by the generalized power-law model. In all the above studies none has applied magneti field.but the appliation of magneto hydrodynamis priniples in mediine and biology is of growing interest in the literature of bio-mathematis [6, 7, 8]. By Lenz s law, the Lorentz s fore will oppose the motion of onduting fluid. Sine blood is an eletrially onduting fluid, The MHD priniples may be used to deaelerate the flow of blood in a human arterial system and thereby it is useful in the treatment of ertain ardiovasular disorders and in the diseases whih aelerate blood irulation like hemorrhages and hypertension et. [9]. It is well known that blood being a suspension of ells, behaves as a non-newtonian fluid at low shear rates and during its flow through small blood vessels, espeially in diseased states, when lotting effets in small arteries are present.experimentaly behavior of blood at low shear rates an best be desribed by Casson model. Therefore, our aim to study the effet of an externally applied uniform magneti field on the multi-stenosed artery.blood is modeled as a Casson fluid by properly aounting for yield stress of blood in small blood vessel. The analytial expressions for the veloities (in normal and ore region), blood flow rate and wall shear stress are obtained. The effet of external magneti field and other parameter has been shown graphially in these results. Mathematial formulation Let us onsider the Casson fluid motion of blood through a multi-stenosed artery under the influene of an external applied uniform transverse magneti fluid. The geometry of the stenosis is as shown in fig.. We have taken some assumption for solving the model. () Let us take the flow of blood as axially symmetri and fully developed (i.e v r v flow in z-diretion only).this is entirely reasonable and reinfores the fat that in steady state inompressible flow in a irular tube of uniform ross-setion. The veloity does not hange in the diretion of the flow, exept near the entrane and exist regions. () Consider blood as a Casson fluid (non-newtonian) and magneti fluid.sine red ell is a major biomagneti substane and blood flow may be influened by the magneti field. (3) Consider the transverse magneti field.sine the biomagneti fluid (blood) is subjeted to a magneti field, the ation of magnetization will introdue a rotational motion to orient the magneti fluid partile with the magneti field). (4), 5 54

3 The above assumptions for Navier-Stokes equation is given by, P H r M z r r z () Where rand z denote the radial and axial oordinates respetively, magneti permeability, M magnetization, H magneti field intensity, P pressure and the shear stress. For Casson fluid the relation between shear stress and shear rate is given by Fung [3], u, if r u, if r () Where denotes yields stress and the visosity of blood. The boundary onditions appropriate to the problem under study are (i) u at r R( z) (3a) (ii) is finite at r (3b) (iii) In ore region u u at r R (3) Here u is ore veloity. Solution of the problem: Introduing the following non-dimensional sheme. r z R P r, z, R, P R R R U u H u,,, H U R U H (4) 55

4 Where H is external transverse uniform onstant magneti field. Using the non-dimensional sheme the governing equations from ()-(3) are written as: P H r F z r r z (5) u F, if r u, if r MH Where F, F U R U \ The boundary onditions (3a, 3b, 3) redue to (6) (i) u at r R( z) (7a) (ii) is finite at r (7b) (iii) In ore region u u at r R (7) The geometry of the stenosis in non-dimensional form is given as,, otherwise R( z) SL z b b z b b b b z b b S L (8) Where R S, (9) L is maximum height of stenosis S L z b b () Where S ( ) is the parameter for determining the shape of the stenosis.. R Solution: On using analytial method in equations (5), (7) and using boundary onditions (7a, 7b, 7) and (8) the expression for veloity u and ore veloity u are: 3 3 P dh P dh u F r R ( z) r R( z) F r R ( z) 3F z dz F 4F z dz () (4), 5 56

5 3 3 P dh P dh u F R R ( z) R R( z) F R R ( z) 3F z dz F 4F z dz The volumetri flow rate Q is given by: R Rz () Q r udr r u dr Q Q (3) R Where Q and Q are the flow rate in ore and annular region of stenoti tube. Using the equations () and () in equation (3) then, flow rate Q is: () P dh ( ) ( ) P dh Q F R R z R R z F R R ( z) F 7 z dz 6 8 z dz (4) The wall shear stress w is defined as: du w dr rr( z) (5) On differentiating eqn. () with respet to r and substituting in eqn. (4), then w is given by: P dh P dh ( ) ( ) F R z w F R z F z dz z dz (6) Results and Disussions Fig. (a) shows the axial veloity (u) with radial axis (r) for different values of indued magneti field gradient (H= dh dz ), when the magneti field gradient (H= dh dz ) inreases then the urve shifts towards the origin.this is due to the fat that as magneti field applied on the body, the Laurentz fore oppose the flow of blood and hene redues its veloity. This result ompare with Das [9] and Ponalagusamy [33]. 57

6 veloity (u) radial axis (r) Fig. (a) :variation of veloity (u) with radial axis (r) for different value of magneti field (H) veloity (u) radial axis (r) Fig. (b) :variation of veloity (u) with radial axis (r) for different value of the shape of the onstrition (S) Fig. (b) shows the axial veloity (u) with radial axis (r) for different values of shape of the onstrition (S). The higher values of shape of the onstrition (S) redue the axial veloity in a large extent. It is also lear that the streaming blood is muh higher than the non-newtonian values. Thus the non-newtonian harateristis of the flowing (4), 5 58

7 blood affet the axial veloity profile. This result agrees qualitatively with Nagarani and Sarojamma [34] veloity (u) radial axis (r) Fig. () :variation of veloity (u) with radial axis (r) for different values of stenosis height The variation of veloity (u) with radial axis (r) for different values of ratio of the maximum height of stenosis and radius of the normal tube ( R ) is shown in fig (,).Veloity dereases and it approahes zero when the ratio of the maximum height of stenosis and radius of the normal tube ( R ) inreased.where high shearing veloity produed in order to attain uniform flow rate at given parameter,so the severity of the multi-stenosis affets the axial flow distribution signifiantly. Figure 3,illustrate the variation of ore (plug) veloity (u) with ratio of the stenosis height and radius of the normal tube ( R ) for different values of indued magneti gradient (H= dh dz ).The urves are all featured to be analogous in the sense that they do drop to zero on the wall surfae from their maximum stenosis height( R =.5).The ore veloity dereases with inreasing the magneti gradient (H= dh dz ).This observation is in good agreement with those of Tzirtzilakis [35] although his studies were based on the Newtonian blood flow under the ation of an applied magneti field. The variation of the flow rate (Q) with radial axis (r) of different values of magneti gradient (H= dh dz ) and ratio of the stenosis height and radius of the normal tube ( R ) is shown in fig.4 (a) and fig. 4 (b). From fig. 4 (a), it is lear that the ratio of the stenosis height and radius of the normal tube ( R ) inreases the rate of flow diminishes appreiably for radial axis (r).the haraterization of blood irrespetive of the presene and absene of the magneti field ertainly ensures the importane of blood rheology in the flow phenomena. The flow rate diminishes as the artery gets narrowed gradually. It may be noted further that the 59

8 flow rate drops sharply with inreasing severity of the onstrition in the absene of the magneti field flow rate (Q) radial axis (r) Fig.4 (a) :variation of flow rate (Q) with radial axis (r) for differnt value of stenosis height In fig.4 (b),it is observed from the figure that in the presene of magneti field gradient (H= dh dz ) the rate of blood flow inreases at r = and then diminishes for beome the value of ( < r < ).The flow rate beomes higher in the absene of magneti field ant it gradually diminishes with inreasing magneti field gradient (H= dh dz ) whih is in good agreement with these of Haik et al [36 ]. (4), 5 6

9 .5 flow rate (Q) radial axis (r) Fig.4 (b) :variation of flow rate (Q) with radial axis (r) for different values of magneti field (H) Fig. 5(a),shows the result of the varaition of wall shear stress ( w ) with axial axis (z) for different values of yield stress ( ).It is noted that the wall shear stress inreases as the axial distane z inreases from ( to.5) and then it dereases as z inreases from (.5 to ). The maximum wall shear stress ours at the middle of the stenosis.the wall shear stress dereases when the yield stress ( ) inreases. The feature of these results is in good agreement with that of Srivastava and Saxena [37], whose studies were based on a one-dimensional Casson model of the blood flow in rigid arteries under steadystate onditions in the absene of magneti field. 6

10 Fig. 5(b), show the variation of wall shear stress ( w ) with axial axis (z) for different magneti field gradient (H= dh dz ).The wall shear stress inreases rapidly as the length of arterial inreases z from ( to.5) and gradually dereases for the length of arterial inreases z from (.5 to ).This is due to large veloity gradient and therefore the severity of the stenosis signifiantly affets the wall shear stress harateristis. It is also lear from the figure that when magneti gradient (H= dh dz ). Inreases then the wall shear stress dereases. However as the harateristi of the non-newtonian fluid hanges from shear-thinning to Newtonian. The wall shear stress is observed when the flowing blood is subjeted to externally applied transverse magneti field. The present stress distribution plays an important role in deteting the aggregation sites of platelets as mentioned by Fry [38]. Conlusions From the above disussion, it is lear that the magneti field, ratio of maximum height of stenosis and radius of normal tube and yield stress of the fluid are the strong parameters influening the flow. It is observed that, in the presene of magneti field the magnitude of veloity is dereased. The effet of yield stress and stenosis is to redue the wall shear stress and flow rate in the presene of magneti field. In view of these arguments, the present study may be more useful to ontrol the blood flow in diseased state. Referenes. D.F. Young, Fluid mehanis of arterial stenosis, J Biomeh. Eng,(979),, J.B. Shukla, R.S.Parihar & S.P.Gupta, Effet of peripheral layer visosity on blood flow through the artery, Bull Math Biol., (98),4, (4), 5 6

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