Simulation of Variable Viscosity and Jeffrey Fluid Model for Blood Flow Through a Tapered Artery with a Stenosis

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1 Commun. Theor. Phys. 57 (2012) Vol. 57 No. 1 January Simulation of Variable Viscosity and Jeffrey Fluid Model for Blood Flow Through a Tapered Artery with a Stenosis Noreen Sher Akbar 1 and S. Nadeem 12 1 Department of Mathematics Quaid-i-Azam University Islamabad Pakistan 2 Department of Mathematics & Natural Sciences Prince Mohammad Bin Fahd University P.O. Box 1664 Al Khobar Kingdom of Saudi Arabia (Received April ; revised manuscript received May ) Abstract Non-Newtonian fluid model for blood flow through a tapered artery with a stenosis and variable viscosity by modeling blood as Jeffrey fluid has been studied in this paper. The Jeffrey fluid has two parameters the relaxation time λ 1 and retardation time λ 2. The governing equations are simplified using the case of mild stenosis. Perturbation method is used to solve the resulting equations. The effects of non-newtonian nature of blood on velocity profile temperature profile wall shear stress shearing stress at the stenotsis throat and impedance of the artery are discussed. The results for Newtonian fluid are obtained as special case from this model. PACS numbers: d Key words: Jeffrey two constant fluid model blood flow tapered artery stenosis variable viscosity analytical solution 1 Introduction Blood is a multi-component material in which a plethora of chemical reactions take place. Jesty and Namerson [1] made an excellent review of the extrinsic and intrinsic coagulation pathways and found the various positive and negative feedback mechanisms and the inhibitory reactions that control coagulation. The effects of overlapping stenosis on arterial flow problem have successfully been carried out analytically by Chakravarty and Mandal. [2] Casson [3] examined the validity of Casson fluid model in studies pertaining to the flow characteristics of blood and reported that at low shear rates the yield stress for blood is non-zero. Three-dimensional MRI-based computational models of stenotic arteries were studied by Tang et al. [4] Steady and pulsatile simulations of turbulent flow were carried out by Stroud et al. [5] They examined that in large arteries with severe stenosis turbulent flow patterns appear in the post-stenotic area for upstream Reynolds numbers Re > 500. Varghese and Frankel [6] investigated pulsatile turbulent flow in stenotic vessels using the Reynolds-averaged Navier Strokes approach. Mekheimer and El Kot [7] studied the micropolar fluid model for blood flow through a tapered artery with a stenosis. In the above mentioned studies fluid viscosity is assumed to be constant. There are few article which have been made to study effects of variable viscosity on the blood flow through the tapered arteries. Bali and Awasthi [8] have analyzed the effect of external magnetic field on blood flow in stenotic artery and considered the viscosity of blood as radial co-ordinate dependent. Unsteady viscous flow with variable viscosity in a vascular tube with an overlapping constriction was investigated by Layek et al. [9] In the above studies the variable viscosity is considered to be a function of space variable. In a typical situation most of the fluids have temperature dependent viscosity and this property varies significantly when large temperature difference exists. Important studies to the topic include the works in [10 14]. In the view of above analysis we have considered the Jeffrey fluid model of blood flow through a tapered artery with a stenosis and variable viscosity. The constitutive equations are model and simplified using the assumptions of mild stenosis. Analytical solutions of the governing equations along with the boundary conditions of stenosed symmetric artery have been calculated. The expressions for velocity temperature resistance impedance wall shear stress and shearing stress at the stenosis throat have been evaluated. The graphical behavior of different type of tapered arteries have been examined for different parameters of interest. 2 Formulation of the Problem Consider the flow of an incompressible Jeffrey fluid lying in a tube having length L. We are considering the cylindrical coordinate system (r θ z) in such a way that ū v and w are the velocity component in r θ and z directions respectively. Heat transfer phenomena is taken into account by giving temperature T 0 to the wall of the tube while at the centre of the tube we are considering symme- Corresponding author noreensher@yahoo.com c 2011 Chinese Physical Society and IOP Publishing Ltd

2 134 Communications in Theoretical Physics Vol. 57 try condition on temperature. The equations governing the steady incompressible Jeffrey fluid are given as ū r + ū r + w = 0 (1) z ( ρ ū r + w ) ū z = p r + 1 r r ( r S r r ) + z ( S r z ) S θ θ r (2) ( ρ ū r + w ) w z = p z + 1 r r ( r S r z ) + z ( S z z ) (3) [ ρc p ū T T ] + w r z [ 2 T =k r r T r + 2 T ] z 2 + Q 0. (4) The extra stress tensor S for Jeffrey fluid [10] is defined as S = µ 1 + λ 1 ( γ + λ 2 γ). (5) In the above equation µ is the viscosity λ 1 is the ratio of relaxation times γ (vector quantity) the shear rate λ 2 the retardation time. Fig. 1 Geometry of an axially nonsymmetrical stenosis in the artery. with The geometry of the stenosis is defined as [7] h(z) =d(z)[1 η(b n 1 (z a) (z a) n )] a z a + b =d(z) otherwise (6) d(z) = d 0 + ξz (7) in which d(z) is the radius of the tapered arterial segment in the stenoic region d 0 is the radius of the non-tapered artery in the non-stenoic region ξ is the tapering parameter b is the length of stenosis (n 2) is a parameter determining the shape of the consriction profile and referred to as the shape parameter (the symmetric stenosis occurs for n = 2) and a indicates its location as shown in Fig. 1. The parameter η is defined as η = δnn/(n 1) d 0 b n (n 1) (8) where δ denotes the maximum height of the stenosis located at z = a + b n n/(n 1). We introduce the non-dimensional variables r = r d 0 z = z b w = w u 0 u = bū u 0 δ p = d2 0 p u 0 bµ 0 Re = ρbu 0 µ 0 S rr = b S rr u 0 µ 0 S zz = b S zz u 0 µ 0 h = h d 0 Srz = d 0 S rz u 0 µ 0 S θθ = b S θθ λ 2 = λ 2 u 0 u 0 µ 0 b θ = ( T T 0 ) T 0 Pr = µ 0c p k µ( T) µ(θ) = µ 0 β 1 = Q 0d 2 0 k T 0 G r = αgd3 0 T 0 ν 2 (9) here u 0 is the velocity averaged over the section of the tube of the width d 0. Making use of Eqs. (5) and (9) and after adopting the additional conditions [7] (i) (ii) Reδ n (1/(n 1)) 1 b d 0 n (1/(n 1)) O(1) b (10a) (10b) Eqs. (1) to (4) for the case of mild stenosis (δ /d 0 1) take the form u r + u r + w = 0 (11) z p = 0 (12) r p z = 1 [ µ(θ)r ) ( 2 w )}] + λ 2 w (13) r r 1 + λ 1 r r z 0 = 2 θ r θ r r + β 1. (14) The corresponding boundary conditions are in which w r = 0 θ = 0 at r = 0 (15a) r w = 0 θ = 0 at r = h(z) (15b) h(z) = (1 + ξz)[1 η 1 ((z σ) (z σ) n )] σ z σ + 1 (16) η 1 = δnn/(n 1) (n 1) δ = δ d 0 σ = a b ξ = ξb d 0 (17) where (ξ = tanφ) φ is called tapered angle and for converging tapering (φ < 0) non-tapered artery (φ = 0) and the diverging tapering (φ > 0) as discussed in [7].

3 No. 1 Communications in Theoretical Physics 135 Fig. 2 Geometry of the axially stenosied tapered artery for different tapered angle. In the present problem we have considered the Reynold s model of viscosity which is defined as follows [10 14] µ(θ) = e βθ. (18) 3 Solution of the Problem 3.1 Perturbation Solution Exact solution of Eq. (14) yields ( r 2 h 2 ) θ(r z) = β 1. (19) 4 To get the perturbation solution we expand w p and F by taking λ 2 as a perturbation parameter as follow w = w 0 + λ 2 w 1 + O(λ 2 ) 2 p = p 0 + λ 2 p 1 + O(λ 2 ) 2 F = F 0 + λ 2 F 1 + O(λ 2 ) 2 (20a) (20b) (20c) With the help of Eqs. (19) to (20c) the expression for velocity field and pressure gradient for small λ 2 can be written as w(r z) = dp dz (a 1(r 4 h 4 ) + a 2 (r 2 h 2 )) + λ 2 (a 21 (r 12 h 12 ) + a 22 (r 10 h 10 ) + a 23 (r 8 h 8 ) + a 24 (r 7 h 7 ) + a 25 (r 6 h 6 ) + a 26 (r 5 h 5 ) + a 27 (r 4 h 4 ) + a 28 (r h)) (21) dp dz =F a 30 + λ 2 (a 31 ). (22) a 29 The pressure drop ( p = p at z = 0 and p = p at z = L) across the stenosis between the section z = 0 and z = L is obtain from (22) as done by [7] p = L 0 ( dp ) dz. (23) dz 3.2 Resistance Impedance The expression for resistance impedance is obtain from Eq. (23) as where λ = p { a F = R(z) h=1 dz + + L a+b 0 R(z) h=1 dz a+b a R(z)dz } (24) R(z) = 1 + a 30/F + λ 2 (a 31 )/F a 29. (25) The resistance impedance in simplified form is written as { ( 1 + a30 /F + λ 2 (a 31 )/F ) h=1 λ = (L b) a 29 a+b } + R(z)dz. (26) a 3.3 Expression for Wall Shear Stress The nonzero dimensionless shear stress is given by [ µ(θ) ) ( S 2 w )}] rz = + λ 2 w. (27) 1 + λ 1 r r z From Eq. (27) we can find the expression for wall shear stress by [ µ(θ) ) ( S 2 w )}] rz = + λ 2 w 1 + λ 1 r r z r=h. (28) The shearing stress at the stenosis throat i.e. the wall shear at the maximum height of the stenosis located at is defined as z = a b + 1 n n/(n 1) τ s = S rz h=1 δ. (29) The final expressions for the dimensionless resistance to λ wall shear stress S rz and the shearing stress at the throat τ s are defined as λ = 1 3 {( 1 b L a+b + 1 L a S rz = 1 [ µ(θ) 4F 1 + λ 1 r τ s = 1 [ µ(θ) 4F 1 + λ 1 r where λ = λ λ 0 )( 1 + a30 /F + λ 2 (a 31 )/F ) h=1 a 29 } R(z)dz (30) S rz = S rz τ 0 ) ( 2 w + λ 2 w r z ) + λ 2 w )}] r=h (31) ( 2 w )}] r z r=h=1 δ (32) τ s = τ s τ 0 λ 0 = 3L τ 0 = 4F and λ 0 τ 0 are the resistance to flow and the wall shear stress for a flow in a normal artery (no stenosis).

4 136 Communications in Theoretical Physics Vol Numerical Results and Discussion The quantitative effects of the relaxation time λ 1 retardation time λ 2 the stenosis shape n viscosity parameter β and maximum height of the stenosis δ for converging tapering diverging tapering and non-tapered arteries are observed physically through Figs The variation of axial velocity for β λ 1 λ 2 δ and n for the case of a converging tapering diverging tapering and non-tapered arteries are displayed in Figs. 3 to 7. In Figs. 3 7 we observe that with an increase in β and λ 1 velocity profile increases while decreases with an increase in λ 2 δ and n. It is also seen that for the case of converging tapering velocity gives larger values as compared to the case of diverging tapering and non-tapered arteries. In Figs we notice the impedence resistence increases for converging tapering diverging tapering and non-tapered arteries when we increase λ 2 and n while decreases when we increase λ 1 and β. We also observe that resistive impedence in a diverging tapering apear to be smaller than those in converging tapering because the flow rate is higher in the former than that in the latter as anticipated and impedence resistence attains its maximum values in the symmetric stenosis case (n = 2). Figures are prepared to see the variation of the shearing stress at the stenotsis throat τ s with δ. It is analyzed through figures that shearing stress at the stenotsis throat increses with an increase in λ 1 and decreases with an increase in λ 2 and β. It can also be depicted that shearing stress at the throat τ s possess an inverse variation to the flow resistence λ with respect to relaxation time λ 1 and retardation time λ 2 and β. Finally Figs show how the converging tapering diverging tapering and non-tapered arteries influence on the wall shear stress S rz. It is observed that with an increase in λ 1 and β shear stress increases while decreases with an increase in λ 2 n and δ the stress yield diverging tapering with tapered angle φ > 0 converging tapering with tapered angle φ < 0 and non-tapered artery with tapered angle φ = 0. Figure 20 is prepared for the variation of temperature profile. It is analyzed that with an increase in β 1 temperature profile increases. Trapping phenomena have been discussed through Figs It is observed that with an increase in relaxation time λ 1 the stenosis shape n and viscosity parameter β number of trapping bolus increases and size of the trapping bolus decreases while with an increase in retardation time λ 2 number and the size of the trapping bolus decreases. Fig. 3 Variation of velocity profile for F = 0.3 β 1 = 0.5 G r = 0.02 δ = 0.6 σ = 0.0 n = 2 z = 0.5 λ 2 = 0.5 Fig. 4 Variation of velocity profile for F = 0.3 β 1 = 0.5 G r = 0.02 β = 0.6 σ = 0.0 n = 2 z = 0.5 λ 2 = 0.6 Fig. 5 Variation of velocity profile for F = 0.3 β 1 = 0.5 G r = 0.02 β = 0.6 σ = 0.0 n = 2 z = 0.95 λ 2 = 0.4 δ = 0.5.

5 No. 1 Communications in Theoretical Physics 137 Fig. 6 Variation of velocity profile for F = 0.3 β 1 = 0.5 G r = 0.02 β = 0.6 σ = 0.0 n = 2 z = 0.95 λ 1 = 0.2 δ = 0.3. Fig. 9 Variation of resistence for F = 0.03 β 1 = 0.5 b = 1 G r = 0.02 n = 2 σ = 0.0 λ 2 = 0.4 z = 0.5 β = 0.3 L = 1. Fig. 7 Variation of velocity profile for F = 0.3 β 1 = 0.5 G r = 0.02 β = 0.6 σ = 0.0 λ 2 = 0.4 z = 0.95 λ 1 = 0.3 δ = 0.5. Fig. 10 Variation of resistence for F = 0.03 β 1 = 0.5 b = 1 G r = 0.02 n = 2 σ = 0.0 λ 1 = 0.4 z = 0.5 β = 0.3 L = 1. Fig. 8 Variation of resistence for F = 0.3 β 1 = 0.5 b = 1 G r = 0.02 n = 2 σ = 0.0 λ 2 = 0.4 z = 0.5 λ 1 = 0.3 L = 1. Fig. 11 Variation of resistence for F = 0.03 β 1 = 0.5 b = 1 G r = 0.02 λ 2 = 0.5 σ = 0.0 λ 1 = 0.3 z = 0.5 β = 0.2 L = 1.

6 138 Communications in Theoretical Physics Vol. 57 Fig. 12 Variation of shear stress at the stenosis throat for F = 0.01 β 1 = 0.5 G r = 0.02 λ 2 = 0.4 Fig. 15 Variation of wall shear stress for F = 0.3 β 1 = 0.5 G r = 0.02 δ = 0.5 σ = 0.0 n = 2 λ 2 = 0.4 Fig. 13 Variation of shear stress at the stenosis throat for F = 0.01 β 1 = 4 G r = 0.02 λ 2 = 0.4 β = 0.8. Fig. 16 Variation of wall shear stress for F = 0.3 β 1 = 5 G r = 0.02 β = 0.2 σ = 0.0 n = 2 λ 2 = 0.4 Fig. 14 Variation of shear stress at the stenosis throat for F = 0.01 β 1 = 0.5 G r = 0.02 λ 1 = 0.4 β = 0.3. Fig. 17 Variation of wall shear stress for F = 0.3 β 1 = 5 G r = 0.02 β = 0.2 σ = 0.0 n = 2 δ = 0.4

7 No. 1 Communications in Theoretical Physics 139 Fig. 18 Variation of wall shear stress for F = 0.3 β 1 = 5 G r = 0.02 β = 0.2 σ = 0.0 λ 2 = 0.4 δ = 0.5 Fig. 21 Stream lines for different values of β (a) β = 0.1 (b) β = 0.2 other parameters are φ = π λ 2 = 0.1 δ = 0.01 σ = 0.4 λ 1 = 0.3 n = 2 F = 0.3. Fig. 19 Variation of wall shear stress for F = 0.3 β 1 = 5 G r = 0.02 β = 0.2 σ = 0.0 λ 2 = 0.4 δ = 0.5 n = 2. Fig. 20 Variation of temperature profile for δ = 0.6 σ = 0.0 n = 2 z = 0.5. Fig. 22 Stream lines for different values of n (a) n = 2 (b) n = 4 other parameters are φ = π λ 2 = 0.1 δ = 0.01 σ = 0.4 λ 1 = 0.3 F = 0.2 β = 0.2.

8 140 Communications in Theoretical Physics Vol. 57 Fig. 23 Stream lines for different values of λ 1 (a) λ 1 = 0.1 (b) λ 1 = 0.2 other parameters are φ = π λ 2 = 0.1 δ = 0.01 σ = 0.4 n = 2 F = 0.2 β = 0.2. Fig. 24 Stream lines for different values of λ 2 (a) λ 2 = 0.15 (b) λ 2 = 0.20 other parameters are φ = π λ 1 = 0.1 δ = 0.01 σ = 0.4 n = 2 F = 0.2 β = Conclusion The main points of the performed analysis are as follows: (i) We observe that with an increase in viscosity parameter and relaxation time velocity profile increases. (ii) The effects of retardation time stenosis shape and height of stenosis on the velocity are opposite as compared to viscosity parameter and relaxation time. (iii) It is also seen that for the case of converging tapering velocity gives larger values as compared to the case of diverging tapering and non-tapered arteries. (iv) We notice that the impedence resistence increases for converging tapering diverging tapering and nontapered arteries when we increase retardation time and stenosis shape while decreases when we increase relaxation time and viscosity parameter: We also observe that resistive impedence in a diverging tapering appear to be smaller than those in converging tapering. (v) It is analyzed that shearing stress at the stenosis throat increases with an increase in relaxation time and decreases with an increase in retardation time. (vi) It is observed that with an increase in relaxation time stenosis shape and height of stenosis wall shear stress increases and decreases with an increase in retardation time. (vii) Temperature profile increases with an increase in Brinkmann number and retardation time while decreases with an increase in relaxation time. (viii) Concentration profile has an opposite behavior as compared to the temperature profile. (ix) It is observed that with an increase in relaxation time stenosis shape and flow rate number of trapping bolus increases and size of the trapping bolus decreases. (x) Number and the size of the trapping bolus decreases with an increase in retardation time. References [1] J. Jesty and Y. Nemerson The Pathways of Blood Coagulation in E. Beutler M.A. Lichtman B.S. Coller and T.J. Kipps (Eds.) Williams Hematology McGraw Hill 5 (1995) [2] S. Chakravarty and P.K. Mandal Math. Comput. Model. 19 (1994) 59. [3] N. Casson Rheology of Disperse Systems C.C. Mill (Ed.) Pergamon Press London (1959). [4] D. Tang C. Yang J. Zheng P. Woodard G. Sicard J. Saffitz and C. Yuang Annals of Biomedical Engineering 32 (2004) 947. [5] J. Stroud S. Berger and D. Saloner J. Biomechanical Engineering 124 (2002) 9. [6] S. Varghese and S. Frankel Journal of Biomechanical Engineering 125 (2003) 445. [7] Kh. S. Mekheimer and M.A. El Kot Acta Mech. Sin. 24 (2008) 637. [8] R. Bali and U. Awasthi Appl. Math. and Comput. 188 (2007) [9] G.C. Layek S. Mukhopadhyay and R.S.R. Gorla Int. J. Eng. Sci. 47 (2009) 649. [10] S. Nadeem and Noreen Sher Akbar Z. Naturforsch (2009) in press. [11] S. Nadeem T. Hayat Noreen Sher Akbar and M.Y. Malik Int. J. Heat Mass Transfer 52 (2009) [12] S. Nadeem and M. Ali Commun. Nonlinear Sci. Numer. Simulat. 14 (2009) [13] S. Nadeem and Noreen Sher Akbar Commun. Nonlinear Sci. Numer. Simulat. 14 (2009) [14] S. Nadeem and Noreen Sher Akbar and M. Hameed Int. J. Numer. Meth. Fluids 63 (2010) 1375.