MHD Effect on the Couple Stress Fluid Flow Through a ... - Core

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ScienceDirect Procedia Engineering 127 (2015) 877 – 884

International Conference on Computational Heat and Mass Transfer-2015

MHD Effect on the Couple Stress Fluid Flow Through a Bifurcated Artery D. Srinivasacharyaa,∗, G. Madhava Raoa a Department

of Mathematics, National Institute of Technology, Warangal, Telangana State,506004,India.

Abstract This paper aims at presenting numerical solutions for steady MHD blood flow through a bifurcated artery with mild stenosis in parent lumen with heat transfer assuming blood as couple stress fluid. The arteries design of bifurcation is referred to be symmetric about the axis of the artery and straight cylinders of limited length. The governing equations are non-dimensionalized and coordinate transformation is used to convert the irregular boundary to a well defined boundary. The resulting system of equations is solved numerically using the finite difference method. The variation of shear stress, flow rate and impedance near the flow divider with relevant physical parameters are presented graphically. © by by Elsevier Ltd.Ltd. This is an open access article under the CC BY-NC-ND license c 2015 2015The TheAuthors. Authors.Published Published Elsevier (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of ICCHMT – 2015. Peer-review under responsibility of the organizing committee of ICCHMT – 2015

Keywords: Couple stress; Bifurcated artery; Mild stenosis; Back and Secondary flow; Numerical Solution.

1. Introduction It is well known that the study of bio-fluid dynamical aspects of the human cardiovascular system have gained much attention in recent decades with respect to the diagnosis and the genesis of atherosclerosis. Usually formation of fatty material like calcium on their inner walls, is known as arterial stenosis. The deposition of atherosclerotic plaque depends on the geometry of the arteries. The most common locations of formation of stenosis are the curvatures, junctions and bifurcations of large and medium arteries. In recent years, considerable attention has been given to the study of atherosclerosis and blood flow dynamics in the stenosed or bifurcated artery. This is because of the blood flow characteristics in the arterial system strongly depends on the nature of the blood, the concentration of the cell, the geometry of the artery, curvature, size and shape. Several researchers have studied the flow of blood in the stenosed or bifurcated artery by modelling the blood as a Newtonian or non-Newtonian fluid. The couple stress fluid model introduced by Stokes [1] has distinct features, such as the presence of couple stresses, body couples and non-symmetric stress tensor. The main feature of couple stresses is to introduce a size dependent effect. Classical continuum mechanics neglect the size effect of material particles within the continua. These fluids are capable of describing various types of lubricants, blood, suspension fluids, etc. Srinivasacharya and Srikanth [2] ∗

Corresponding author. Tel.: +91-9849187249. E-mail address: [email protected]; [email protected]

1877-7058 © 2015 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license

(http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of ICCHMT – 2015

doi:10.1016/j.proeng.2015.11.425

878

D. Srinivasacharya and G. Madhava Rao / Procedia Engineering 127 (2015) 877 – 884

analyzed the effect of couple stress fluid parameters, the size of the catheter on physical quantities like impedance and shear stress in the couple stress fluid flow in constricted annulus. Mekheimer and Abd elmaboud [3] discussed the peristaltic flow of a couple stress fluid in an annulus with the application of an endoscope. Nadeem and Akram [4] considered the peristaltic flow of couple stress fluid in an asymmetric channel under the influence of the induced magnetic field. Reddy et al. [5] investigated the slip velocity effects on the couple stress fluid flow in the constricted tapered artery. The application of magnetic fields plays important role in the flow of blood in the human arterial system. MHD application reduces the flow rate, which is useful in the treatment of hypertension and certain cardiovascular disorders. Several magnetic devices have been developed for drug carriers, cell separation, and treatment of cancer tumour. The MHD fluid flow in curved and bifurcated channel is an interesting and important area in the study of blood flow mechanics due to its applications in medical sciences. The theory of electromagnetic field in medical research has been first initiated by Kolin [6] and later the possibility of regulating the movement of blood in arterial system by applying magnetic field has been discussed by Korchevskii et al. [7]. The peristaltic transport of a couple stress fluid in a porous channel theoretically investigated by Maiti and Mista [8]. Singh and Singh [9] suggested that MHD principle may be used to reduce the acceleration of the flow of blood in a human arterial system. The pumping characteristics, velocity field, axial pressure gradient and trapping phenomena have been discussed by Akram et al. [10] to explain the physical features of emerging parameters of couple stress fluid. Beg et al. [11] studied the oscillatory hydro magnetic flow of couple stress fluid in an inclined, rotating channel with non-conducting walls. Sajid et al. [12] considered the mathematical model for the two-dimensional MHD peristaltic flow of a couple stress fluid in a channel with the induced magnetic field. Ramesh and Devakar [13] shown that an increase of couple stresses and heat generation parameter increase the size of the trapped bolus, peristaltic pumping and temperature. The present article deals with MHD couple stress fluid through a bifurcated artery with mild stenosis in the parent lumen. The variation of flow rate, impedance, shearing stress and temperature profiles are analyzed numerically for different values of physical parameters involved in the present study. 2. Mathematical formulation Mathematical model of blood flow through a bifurcated artery with mild stenosis in parent lumen is consider to study the numerical solution of steady MHD flow of an incompressible electrically conducting couple stress fluid on the physiological properties of blood flow near the apex. The stenosis over a length of the artery is assumed to have developed in an axi-symmetric manner and the parent aorta have a single mild stenosis in its lumen as shown in fig.(1). The fluid properties, including the electrical conductivity are considered to be constant, except for the density, so that the Boussinesq approximation is used. Neglecting the radiation of the heat transfer and Joule heating. Let (r, θ, z) is being the co-ordinates of any point in the cylindrical polar co-ordinate system, in which z is assumed to be the central axis of the parent artery. The arteries, forming bifurcations are symmetrical about the central axis of the parent artery and are straight circular cylinders of restricted length. Curvature is introduced at the lateral junction and the flow divider so that the flow separation zones (if any occurs)can be eliminated. The governing equations of the flow are ∇·q=0 ρ

dq = −∇P + μ (∇ × ∇ × q) − η(∇ × ∇ × ∇ × ∇ × q) + ρgβ2 (T − T 0 ) − σ1 B20 w dt

(1) (2)

dT = k0 ∇2 T + Q1 (T − T 0 ) (3) dt where k0 is the thermal conductivity, g is the gravitational acceleration, c p specific heat at constant pressure, Q1 is the heat generation, β2 is the volumetric expansion parameter, B0 is the applied magnetic field, ρ is the density of the couple stress fluid, η is the couple stress viscosity parameter, P is the fluid pressure, q is the velocity vector and μ is the blood viscosity. The geometry of the bifurcated artery with mild stenosis in the parent lumen mathematically given by the outer wall R1 (z) and the inner wall R2 (z) ρc p

879


Fig. 1: Schematic diagram of stenosed bifurcated artery.

⎧ a 0 ≤ z ≤ d and d + l0 ≤ z ≤ z1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ (a − 42 (l0 (z − d ) − (z − d )2 ) d ≤ z ≤ d + l0 l0 R1 (z) = ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ (a + r0 − r02 − (z − z1 )2 ) z1 ≤ z ≤ z2 ⎪ ⎪ ⎪ ⎪ ⎩ (2r secβ + (z − z )tanβ) z2 ≤ z ≤ zmax 1 2 ⎧ ⎪ 0 0 ≤ z ≤ z3 ⎪ ⎪ ⎪ ⎪ ⎨ 2 − (z − z − r )2 ) z ≤ z ≤ z + r (1 − sinβ) R2 (z) = ⎪ ( (r ) 3 3 3 ⎪ 0 0 0 ⎪ ⎪ ⎪ ⎩ (r cosβ + z4 ) z + r (1 − sinβ) ≤ z ≤ zmax 3 0 0

(4)

(5)

where r1 is the radius of the daughter artery, a is the radius of the parent artery at non-stenosed portion, β is the half of the bifurcation angle, l0 is the length of the stenosis at a distance d from the origin, is the maximum height of the stenosis at z = d + l0 /2 and zmax represents the maximum length of the bifurcated artery. z1 , z2 , z3 are the location of the onset, offset of the lateral junction, flow divider respectively, these are defined as z2 = z1 + r0 sin β, z3 = z2 + q1 , z4 = (z − z3 − r0 (1 − sinβ))tanβ, where q1 is a small number lying in between 0.1 and 0.5, this is defined for compatibility of the geometry. r0 , r0 are the radii of curvatures for the lateral junction and flow divider respectively as (z3 − z2 )sinβ a − 2r1 secβ and r0 = (6) r0 = cosβ − 1 1 − sinβ The boundary conditions associated with the physical problem are ⎫ ∂w ∂2 w σ ∂w ⎪ T = T 0 on r = 0 for 0 ≤ z ≤ z3 ⎪ ⎪ ∂r = 0, ∂r2 − r ∂r = 0, ⎪ ⎬ ∂2 w σ ∂w (7) w = 0, ∂r2 − r ∂r = 0, T = T w on r = R1 (z) for all z ⎪ ⎪ ⎪ 2 ⎪ ∂ w σ ∂w ⎭ w = 0, ∂r2 − r ∂r = 0, T = T 0 on r = R2 (z) for z3 ≤ z ≤ zmax Where σ = ηη is the couple stress fluid parameter which is responsible for the effect of local viscosity of particles apart from the bulk viscosity of the fluid μ. If η = η, effects of couple stresses will be disappear in fluid, which implies that couple stress tensor is symmetric. In this case the equation (7) shows that, the couple stresses are disappearing on the inner and outer walls of the bifurcated artery. Since the flow is assumed to be symmetric about the z-axis, all the variables are independent of θ. Hence, for this flow the velocity is given by q = (u(r, z), 0, w(r, z)). It is evident to note that the radial velocity is very small and can be neglected for flow in an artery with mild stenosis [2]. Now (1)-(3) in non-dimensional form reduced to

2 2

∂ 1 ∂2 1 ∂ 1 ∂ Gr dp w− 2 (8) + + w+ Θ − H2w = Re dz ∂r2 r ∂r α ∂r2 r ∂r

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where Gr =

ρ2 gβ2 a3 (T w −T 0 ) μ2

1 ∂ ∂2 Θ + sΘ = 0 + ∂r2 r ∂r

is the Grashof number, Re =

ρaw0 μ

(9)

is the Reynolds number, H = B0 a

σ1 μ

is the Hartmann

number, s = Qk10a is the Heat source parameter, α2 = μaη is the couple stress fluid parameter, w = w0 w is the 0 non-dimensional velocity, Θ = TTw−T −T 0 is the non-dimensional temperature and w0 is the characteristic velocity. The associated boundary conditions in non-dimensional form are 2

2

∂w ∂r

= 0, w = 0, w = 0,

∂2 w − σr ∂w ∂r = 0, ∂r2 ∂2 w σ ∂w − r ∂r = 0, ∂r2 ∂2 w σ ∂w − r ∂r = 0, ∂r2

⎫ ⎪ Θ = 0 on r = 0 for 0 ≤ z ≤ z3 ⎪ ⎪ ⎪ ⎬ Θ = 1 on r = R1 (z) for all z ⎪ ⎪ ⎪ ⎪ Θ = 0 on r = R2 (z) for z3 ≤ z ≤ zmax ⎭

(10)

The effect of R1 (z) and R2 (z) from the boundary can be transferred into the governing equations by the radial 2 coordinate transformation ξ = r−R R , given by [14] where R(z) = R1 (z) - R2 (z). Using this transformation in (8)-(9) take the form

∂3 w 1 ∂4 w 2R 1 2 ∂2 w 2 ∂ξ 4 + α2 (ξR+R ) ∂ξ 3 − 1 + α2 (ξR+R )2 R ∂ξ 2 + α 2 2

(11) 4 Gr R 1 1 2 4 4 dp − (ξR+R R3 ∂w ∂ξ − Re Θ + H R w = −R dz α2 (ξR+R2 )3 2) ∂2 Θ ∂Θ R + R2 sΘ = 0 + ∂ξ2 (ξR + R2 ) ∂ξ

(12)

The corresponding boundary conditions in transformed coordinates are ∂w ∂ξ

= 0, w = 0, w = 0,

∂2 w σR ∂w − (ξR+R = 0, ∂ξ2 2 ) ∂ξ 2 ∂ w σR ∂w − = 0, (ξR+R2 ) ∂ξ ∂ξ2 ∂2 w σR ∂w − (ξR+R2 ) ∂ξ = 0, ∂ξ2

⎫ Θ = 0 on ξ = 0 for 0 ≤ z ≤ z3 ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ Θ = 1 on ξ = 1 for all z ⎪ ⎪ ⎪ ⎪ ⎭ Θ = 0 on ξ = 0 for z3 ≤ z ≤ zmax ⎪

(13)

The physical quantities to be analyzed are the temperature, flow rate, impedance and shear stress for both parent and daughter arteries. The flow rate for both parent and daughter arteries are determined by using

1

Q p = 2πRi Ri

ξi wi, j dξi + R2i

wi, j dξi

1

Qd = πRi [Ri

(14)

0

0

and

1

ξi wi, j dξi + R2i

0

1

wi, j dξi

(15)

0

The resistance to the flow (resistive impedance) in parent and daughter artery is calculated using d p z3 dz f or z < z3 (λ p )i = Q p (zmax − z3 ) ddzp f or z ≥ z3 (λd )i = Qd

(16)

(17)

The mean value shear stress for shear stress is calculated by using τi j =

1 1 1 ∂ ∂2 w ∂w 1 ∂w ∂2 w − 2 2 + − . 2 2 2 3 2 R ∂ξ 4Rα (ξR + R2 ) ∂ξ 4α R ∂ξ ∂ξ 4R α (ξR + R2 ) ∂ξ2

(18)

881


0

0

-100

-200

-200 -400

-300

impedance

impedance

-600 -800

-1000

α=1 α=2 α=3 α=4

-1200 -1400

-400 -500 -600

σ=0 σ=0.2 σ=0.4 σ=0.6

-700 -800

-1600

-900 -1000

-1800 20

22

24

z

26

28

20

30

22

24

(a)

z

26

28

30

(b)

Fig. 2: Effect of (a) α and (b) σ on impedance near to the apex.

-100

0

-200 -200

impedance

impedance

-300 -400

-400

-500

-600

-600

H=1 H=1.5 H=2 H=2.5

-800

s=0.5 s=1 s=1.5 s=2

-700 -800 -900

-1000 20

22

24

z

26

28

20

30

22

24

(a)

z

26

28

30

28

30

(b)

Fig. 3: Influence of (a) H and (b) s on impedance near to the apex.

5

200

β=π/12 β=π/6 β=π/4 β=π/3

4

0

-200

impedance

flow rate

3

-400

2

-600

1

β=π/12 β=π/6 β=π/4 β=π/3

-800

-1000

0

-1

-1200 20

22

24

(a)

z

26

28

30

20

22

z

24

26

(b)

Fig. 4: variations of(a) impedance and (b) flow rate with β near to the apex.

3. Results and Discussion The equations (11)-(12) along with boundary conditions (13) is solved numerically using finite difference method. First, the equations (11) and (12) is converted into a system of six first order differential equations. Then these equations are replaced with equivalent central finite difference approximations. The deriving system of algebraic equations forms a block tridiagonal matrix and is solved using block elimination method. We used the following data: a = 0.005 π , r1 =0.51a, τm =2a, α = 1.5, σ = 0.2, Re = 10, Gr = 2, H = 2.0, s = 1.5. m, d = 0.01 m, l0 = 0.005 m, β = 10

882


0.25

0.35

α=1 α=2 α=3 α=4

0.30

σ=0 σ=0.2 σ=0.4 σ=0.6

0.20

flow rate

flow rate

0.25

0.20

0.15

0.15

0.10

0.10

0.05 0.05

0.00

0.00 20

22

z

24

26

28

20

30

22

24

(a)

z

26

28

30

(b)

Fig. 5: Effect of (a) α and (b) σ on flow rate near to the apex.

0.25

0.30

H=1 H=1.5 H=2 H=2.5

0.25

s=0.5 s=1 s=1.5 s=2

0.20

0.15

flow rate

flow rate

0.20

0.15

0.10

0.10 0.05

0.05

0.00

0.00 20

22

z

24

26

28

30

20

22

24

(a)

z

26

28

30

(b)

Fig. 6: Influence of (a) H and (b) s on flow rate near to the apex.

1

-0.2 -0.4

0

shear stress

shear stress

-0.6

-1

-2

α=1 α=2 α=3 α=4

-3

-0.8 -1.0 -1.2

α=1 α=2 α=3 α=4

-1.4 -1.6 -1.8

-4 22

24

26

(a)

z

28

30

22

24

26

z

28

30

(b)

Fig. 7: Variations of shear stress along the (a)inner and (b)outer walls of the daughter artery with α.

Figures 2a and 2b explores the influence of α and σ respectively on impedance on both sides of the apex. From fig. 2a impendence is enhanced with advancement in the value of α and from fig. 2b impedance is enhanced in the parent artery and diminishing in the daughter artery with increase in the value of σ. The effect of H and s on impedance on both sides of the apex is shown in figures 3a and 3b. From fig. 3a impedance is diminishing with advancement in the value of H this implies that impedance is decreasing with strong magnetic field. From fig. 3b the effect of s on impedance is similar to that of H, but the effect is almost negligible. From figures 2 and 3, it is to be identified that impedance is increasing with an increase in the value of z, until inset of lateral junction, then a slight decrease occurred suddenly, and after that gradually increasing till the apex, and then a sudden decrease is identified. This is because

883

D. Srinivasacharya and G. Madhava Rao / Procedia Engineering 127 (2015) 877 – 884 1

-0.25

-0.50

shear stress

shear stress

0

-1

-2

β=π/12 β=π/6 β=π/4 β=π/3

-3

-0.75

-1.00

β=π/12 β=π/6 β=π/4 β=π/3

-1.25

-1.50

-4 22

24

26

z

28

23

30

24

(a)

25

26

z

27

28

29

30

(b)

Fig. 8: Influence of β on shear stress along the (a)inner and (b)outer walls of the daughter artery.

-0.6

0.0

-0.5

shear stress

shear stress

-0.8 -1.0

-1.5

-2.0

H=1 H=1.5 H=2 H=2.5

-2.5

-1.0

H=1 H=1.5 H=2 H=2.5

-1.2

-3.0

-1.4 22

24

z

26

28

30

22

24

(a)

26

z

28

30

(b)

Fig. 9: Variations of shear stress along the (a) inner and (b) outer walls of the daughter artery with H.

-0.8

0

-0.9

shear stress

shear stress

1

-1

-2

σ=0.0 σ=0.2 σ=0.4 σ=0.6

-3

-1.0

-1.1

σ=0.0 σ=0.2 σ=0.4 σ=0.6

-1.2

-1.3

-4 23

24

(a)

z

25

26

23

24

z

25

26

(b)

Fig. 10: Variations of shear stress along the (a) inner and (b) outer walls of the daughter artery with σ.

of diverging of blood flow at the bifurcation of the artery. Thereafter, it is found that the impedance is uniform till zmax . Figures 4a and 4b illustrated the effect of β on impedance and flow rate. It is noticed from fig. 4a and fig. 4b that impedance and flow rate are enhanced with advancement in the value of β. The effect of α and σ on flow rate on both sides of the apex is illustrated in fig 5a and fig 5b. From these figures, it is noticed that the flow rate is increasing with the increase in the value of α and σ on both sides of the apex, but effect of σ is almost negligible. The effect of H and S on flow rate on both sides of the apex is shown in fig. 6a and fig. 6b. The flow rate is diminishing with enhancement in the value of H and s on both sides of the apex, but the effect with s is almost insignificant. From figures 5 and 6 flow rate profiles are perturbed largely near to the apex in the parent artery due to the presence of backflow at the start of

884


the flow divider. All the profiles are locally increasing till inset of lateral junction, then a small decrease is identified and then increase until the apex. Thereafter, these profiles found to be steady till zmax . The influence of α on shear stress along the inner and outer walls of the daughter artery is depicted in fig. 7a and fig. 7b. From these figures shear stress is enhancing along the inner and outer walls of the daughter artery with advancement in the value of α. Figures 8a and 8b represented the effect of β on shear stress along the inner and outer walls of the daughter artery respectively. It is seen that shear stress is advancing with advancement in the value of β along both the inner and outer walls of the daughter artery. Figures 9a and 9b illustrated the influence of H on shear stress along the inner and outer walls of the daughter artery. From these figures shear stress is increases with an increase in the value of H along the inner and outer walls of the daughter artery. The influence of σ on shear stress along the inner and outer walls of the daughter artery is depicted in fig. 10a and fig. 10b. From fig. 10a shear stress increases along the inner wall and from fig. 10b shear stress decreases along the outer wall with an increase in the value of σ.

4. Conclusion The present results leads us to understand, numerically as well as physically, the effect of β, α, σ, H and s on couple stress blood flow through bifurcated artery, which is of great importance in the medical sciences and biomedical engineering. • The flow rate and impedance both have been increasing with the increase in the value of α, β, σ and decrease in the value of H, s both in parent and daughter arteries. • The shear stress has been increasing with increase in the values of α, β, H along the inner and outer walls of the daughter artery. But the shear stress has been increasing along the inner wall and decrease along the outer wall with an increase in the value of σ. References [1] V.K. Stokes, Couple stesses in fluids, Physics of Fluids, Vol. 9, (1966), pp. 1709–1715. [2] D. Srinivasacharya and D. Srikanth, Effect of couple stresses on the flow in a constricted annulus, Arch Appl Mech, vol. 78, (2008), pp. 251-257. [3] Kh.S. Mekheimer and Y. Abd elmaboud, Peristaltic flow of a couple stress fluid in an annulus: Application of an endoscope, Phys. A., Vol. 387, (2008), pp. 2403–2415. [4] S. Nadeem and S. Akram, Peristaltic flow of a couple stress fluid under the effect of induced magnetic field in an asymmetric channel, Arch. Appl. Mech., Vol. 81, (2011), pp. 97-109. [5] J. V. R. Reddy, D. Srikanth and S.K. Murthy, Mathematical modelling of couple stresses on fluid flow in constricted tapered artery in presence of slip velocity-effects of catheter, Appl. Math. Mech. -Engl. Ed., Vol. 35 (8), (2014), pp. 947–958. [6] A. Kolin, An electromagnetic flow meter: Principle of the method and its application to blood flow measurements, Experimental Biology and Medicine, Vol. 35, (1936), pp. 53-56. [7] E. M. Korchevskii and L.S. Marochnik, Magnetohydrodynamic version of movement of blood, Biophysics, Vol. 10, (1965), pp. 411413. [8] S. Maiti and J.C. Misra, Peristaltic transport of a couple stress fluid: some applications to hemodynamics, J. Mech. Med. Biol., Vol. 12, (2012), id 1250048 [21 pages] [9] A.K. Singh and D.P. Singh, MHD Flow of Blood through Radially Non-symmetric Stenosed Artery: a Hershcel-Bulkley Model, International Journal of Engineering, Vol. 26, (2013), pp. 859-864. [10] Akram Safia, Kh.S. Mekheimer and S. Nadeem, Influence of Lateral Walls on Peristaltic Flow of a Couple Stress Fluid in a Non-Uniform Rectangular Duct, Applied Mathematics and Information Sciences An International Journal, vol. 8 (3), (2014), pp. 1127-1133. [11] Ahmed Sahin, O. Anwar Beg, S.K. Ghosh, A couple stress fluid modeling on free convection oscillatory hydromagnetic flow in an inclined rotating channel, Ain Shams Engineering Journal, vol 5, (2014), pp. 12491265. [12] M. Sajid, N. Ali, Z. Abbas and T. Javed, On modelling of two-dimensional MHD flow with induced magnetic field: solution of peristaltic flow of a couple stress fluid in a channel, Iranian Journal of Science and Technology, Vol. 39 (A1), (2015), pp. 35-43. [13] K. Ramesh and M. Devakar, Effects of Heat and Mass Transfer on the Peristaltic Transport of MHD Couple Stress Fluid through Porous Medium in a Vertical Asymmetric Channel, Journal of Fluids, ID 163832, (2015), 19 pages http://dx.doi.org/10.1155/2015/163832. [14] G.C. Shit and M. Roy, Pulsatile flow and heat transfer of a magneto-micropolar fluid through a stenosed artery under the influence of body acceleration, J. of Mech. in Medicine and Biology, Vol. 11, (2011), pp. 643-661.