This paper discuss the biomagnetic fluid aspect of blood flow through narrow tubes with mild stenosis. The blood flow in narrow arteries is treated as two-fluid model with non-Newtonian Jeffrey fluid in the core region and in peripheral region as Newtonian fluid. An analytical solution is obtained for the velocity, volume flow rate, and wall shear stress. The effect of Hartmann number, Jeffrey fluid parameter, shape and size of the stenosis on the velocity, volume flow rate and wall shear stress is discussed.
Keywords
Biomagnetic fluid flow Non-Newtonian fluid Stenosed artery Hartmann number
The application of magnetic field in blood flow is useful in the treatment of disease like cancer etc. Blood is consider as a bio-magnetic fluid, which exhibits the magnetic behaviour due to cell membrane, protein, and the haemoglobin, the form of iron ion which is present in the RBC and it behaves like diamagnetic when oxygenated and paramagnetic when de-oxygenated. Haik et al. [1] developed the fundamental theory of BFD and constitutive equation of motion for the flow of blood. The applied transverse magnetic field reduces the strength of blockage at the peach of the bifurcation, shear stress and altered the flow velocity [2]. Increasing magnetic field increases the pressure gradient, decreases the velocity and temperature [3, 4]. Haik et al. [5] experimentally showed that flow rate decreased up to 30% subjected to high magnetic field of 10T and it was due to an increase in apparent viscosity of blood. Tzirtzilakis [6, 7] developed the mathematical model of the biomagnetic fluid flow in a channel with stenosis and in a driven cavity in presence of magnetic field. The magneto hydrodynamics effect on blood flow by considering it as micropolar fluid was discussed by Abdullah et al. [8]. The presence of magnetic field significantly reduced the blood flow rate in a porous channel flow [9].
Shukla et al. [10] discussed that the resistance to flow and wall shear stress decreases as peripheral layer viscosity decreases but increases with size of stenosis. Srivastava and Sexena [11] described a two layered Casson fluid flow model through stenotic blood vessels. Sankar and Lee [12, 13] discussed the two layered theoretical model of Casson and Hershel Bulkley fluid through stenosed artery. Tzritzilakis [14, 15] described the biomagnetic fluid flow through aneurysm and discussed the effect of magnetization and electric conductivity. The effect of magnetic field on blood flow in stenosed arteries with radially variable viscosity was discussed and influence of magnetic field and heat transfer on two-phase fluid model for oscillatory blood flow in an arterial stenosis was studied by Ponalagusamy and Selvi [16, 17]. The present paper attempts to study the blood flow through stenotic artery. In this paper, we discuss the two layered Jeffrey fluid model of blood flow through mild stenotic tubes under the effect of magnetic field. This study will discuss the effect of Hartmann number, Jeffrey fluid parameter, shape and size of the stenosis velocity, flow rate and shear stress.
Let us consider the axially symmetric, laminar, steady, incompressible and fully developed flow under the effect of transverse magnetic field perpendicular to flow direction. In a two layered blood flow through mild stenotic artery, core region is assumed as non Newtonian Jeffrey fluid and peripheral region is assumed as Newtonian fluid of plasma layer (Fig. 1). The flow in tubes is governed by cylindrical coordinate system (r, θ, z). The radius R(z) and R_{1}(z) of artery with stenosis in peripheral region and core region respectively are defined as [18]:
where R_{o} is the radius of the normal artery, L_{o}, d are the length and location of the stenosis in the artery, β is the ratio of core region radius to normal artery radius, δ_{p} and δ_{c} are the maximum height of the stenosis in the peripheral region and core region respectively such that \( \frac{{\delta_{c} }}{{R_{o} }} \ll 1 \) and \( \frac{{\delta_{p} }}{{R_{0} }} \ll 1. \)
The governing equations for blood flow in presence of magnetic field under above assumption are as follows:
where \( \bar{V} = (u,\,\,v,\,\,w) \) is the velocity field, ρ, f be the density and body force per unit volume. \( \mu_{0} ,\,\,\,\bar{M},\,\,\,\bar{H},\,\,\,\bar{B} \) be the magnetic permeability, magnetization, magnetic field intensity, magnetic field induction respectively. \( \sigma ,\,\,\bar{J} \) is the electrical conductivity and intensity of electric current.
The equation for steady, incompressible, Jeffrey fluid are given as [19]:
where \( \vec{\tau } \) and \( \vec{S} \) are Cauchy stress tensor and extra stress tensor, λ_{1} and λ_{2} be relaxation and retardation, μ and \( \dot{\gamma } \) be the fluid viscosity and shear rate.
Let us assume that the applied magnetic field is uniform. Therefore the magnetization force due to magnetic field will vanish and hence the blood flow is affected by its electric conductivity (Lorentz force) only. We consider only z direction flow, thus the governing Eqs. (3)–(5) reduces:
where \( \vec{S}_{rz} \) is a extra stress component along z direction and its value under above assumption for Newtonian and non Newtonian fluid are \( \vec{S}_{rz} = \bar{\mu }_{1} \frac{{\partial \bar{v}_{1} }}{{\partial \bar{r}}} \) and \( \vec{S}_{rz} = \frac{{\bar{\mu }_{2} }}{{1 + \lambda_{1} }}\frac{{\partial \bar{v}_{2} }}{{\partial \bar{r}}} \) respectively, where \( \bar{v}_{1} \), \( \bar{v}_{2} \) are the velocity of peripheral and core region and \( \bar{\mu }_{1} \), \( \bar{\mu }_{2} \) are the viscosity of Newtonian and non Newtonian fluid.
where \( P = \frac{\partial p}{\partial z} \) is the constant pressure gradient, m_{1}, m_{2} are the Hartmann number in peripheral and core region respectively and a_{1}, b_{1}, c_{1}, d_{1} are the arbitrary constants.
The suitable boundary condition which are physically and mathematically consistent are taken as follows:
$$ v_{1} = 0\quad {\text{at}}\quad r = R $$
(16)
$$ v_{1} = v_{2} \quad {\text{at}}\quad r = R_{1} $$
The effect of core region magnetic number M_{2} on the velocity in radial direction for Jeffrey fluid parameter λ_{1} = 0.3, viscosity ratio μ = 0.8, peripheral magnetic number M_{1} = 2, and β = 0.98 is discussed in Fig. 2, It is found that the velocity decreases towards the wall of artery from the centre of the artery. From this figure, it is also noticed that the velocity decreases with the increase of core region magnetic number.
The variation of volume flow rate Q is discussed in Fig. 3. From Fig. 3, it is noticed that the volume flow rate decreases with the increase of magnetic number for λ_{1} = 9, μ = 0.9 and M_{1} = 9, further it decreases with height of stenosis and minimum at the peak of stenosis along direction z. Table 1 are prepared to see the effect of viscosity ratio, Jeffrey fluid parameter and magnetic number on wall shear stress. It shows that wall shear stress decreases up to z = 10 which is the peak of stenosis and then increases but decreases with increase of peripheral magnetic number. Same occur in case of Jeffrey fluid parameter λ_{1}. For viscosity ratio μ, it increases with increase of μ.
Table 1
Wall shear stress
z
Wall shear stress (τ_{w})
Wall shear stress (τ_{w})
Wall shear stress (τ_{w})
λ_{1} = 5
λ_{1} = 7
λ_{1} = 9
M_{1} = 5
M_{1} = 7
M_{1} = 9
μ = 0.9
μ = 0.7
μ = 0.5
5
0.775
0.723
0.682
1.091
1.065
1.031
0.775
0.759
0.733
7
0.731
0.677
0.640
1.040
1.022
0.998
0.731
0.718
0.696
9
0.648
0.594
0.556
0.927
0.921
0.910
0.648
0.640
0.626
10
0.632
0.578
0.540
0.902
0.895
0.888
0.632
0.625
0.612
11
0.649
0.594
0.556
0.927
0.920
0.911
0.649
0.640
0.626
13
0.731
0.677
0.640
1.039
1.022
0.998
0.731
0.718
0.696
15
0.775
0.723
0.682
1.092
1.065
1.031
0.775
0.759
0.733
The purpose of present paper is to study the magnetic effect on blood flow in stenotic artery and veins using Jeffrey fluid model of blood flow. The findings of this paper are three fold in nature. The most important finding of this paper is that we increase the magnetic number in core and peripheral region, the axial velocity decreases. Second, the volume flow rate decreases with an increase of magnetic number as well as with the Jeffrey fluid parameter, but increases with the increase of viscosity ratio. In last, the wall shear stress decreases with the increase of magnetic number and Jeffrey fluid parameter. It also decreases with the increase of viscosity ratio. These finding has practical implications in the medical fields, such as therapeutic therapy and dialysis of blood.
Notes
Acknowledgements
The second author is thankful to the Science and Engineering Research Board for providing the financial assistance under its Project No. SR/FTP/MS-47/2012 during the work.
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