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Numerical investigation on two-fluid model (micropolar-Newtonian) for pulsatile flow of blood in a tapered arterial stenosis with radially variable magnetic field and core fluid viscosity

Abstract

In the present study, an unsteady two-fluid model of blood through a tapered arterial stenosis with variable viscosity in the presence of variable magnetic field has been investigated. In this model, blood in the core region is assumed to be micropolar and plasma in the peripheral layer as Newtonian. Finite difference method is employed in solving the governing equations. The solutions for velocity, flow rate, wall shear stress and flow resistance are computed numerically. A comparison between the velocity profiles obtained by the present study and the experimental data is made and a good agreement between them is found. The model is used to study the effect of parameters such as taper angle, radially variable viscosity, hematocrit, the coupling number, the micropolar parameter, magnetic field and plasma layer thickness on physiologically important parameters such as wall shear stress distribution in the stenotic region and flow resistance. The results in the case of constant magnetic field and variable magnetic field are compared to study the effects of magnetic field on the flow of blood. It is found that the magnitudes of wall shear stress and flow resistance are higher in the case of variable magnetic field. It is important to note that the flow resistance is higher for magneto-micropolar fluid than the micropolar fluid. The wall shear stress decreases with increasing hematocrit parameter whereas flow resistance increases with hematocrit when the applied pressure gradient is held fixed. The model clearly shows the situation of a patient with a tapered arterial stenosis under feverish condition. Due care has been taken to compare the present theoretical results with the existing ones including experimental results and good agreement between them has been observed both qualitatively and quantitatively.

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Abbreviations

\(\bar{z}\) :

Axial distance

\(\bar{r}\) :

Radial distance

\(\bar{t}\) :

Time

\(\bar{R}_1(\bar{z})\) :

Radius of the central core region

\(\bar{R}(\bar{z})\) :

Radius of the artery in the stenotic region

\(\bar{k}(\bar{r})\) :

Rotational viscosity

\(\bar{u}_c\) :

Axial velocity of the core fluid

\(\bar{v}_c\) :

Radial velocity of the core fluid

\(\bar{u}_p\) :

Axial velocity of plasma

\(\bar{v}_p\) :

Radial velocity of plasma

\(\bar{v}_{\theta }\) :

Rotational velocity

\(\bar{j}\) :

Micro-gyration parameter

\(\bar{p}\) :

Pressure

\(\bar{H}_0^2(\bar{r})\) :

Variable magnetic field

\(h_m\) :

Hematocrit

\(m^2\) :

Micropolar spin parameter

N :

Coupling parameter

d :

Location of the stenosis

\(n_1\) :

Shape of stenosis

\(A_0\) :

Amplitude of steady pressure gradient

\(A_1\) :

Amplitude of pulsatile pressure

\({u_p}_0\) :

Initial velocity in the plasma region (without magnetic field)

\({u_c}_0\) :

Initial velocity in the core region (without magnetic field)

\({v_\theta }_0\) :

Initial rotational velocity (without magnetic field)

\(I_0\) :

Modified Bessel function of zero order

\(I_1\) :

Modified Bessel function of order 1

\(I_2\) :

Modified Bessel function of order 2

\(Q_c\) :

Flow rate in the core region

\(Q_p\) :

Flow rate in the peripheral plasma region

Q :

Total flow rate

\(\bar{\mu }(\bar{r})\) :

Consistency function

\(\bar{\mu }_c(\bar{r})\) :

Variable viscosity of the central core fluid

\(\bar{\mu }_p\) :

Viscosity of the plasma

\(\bar{\rho }_c\) :

Density of the central core fluid

\(\bar{\rho }_p\) :

Density of plasma

\(\bar{\sigma }\) :

Electrical conductivity of the fluid

\(\beta \) :

Material constant

\(\beta _1\) :

Constant in variable core viscosity

\(\beta _2\) :

Constant in variable rotational viscosity

\(\delta _s\) :

Maximum stenotic height

\(\psi \) :

Taper angle

\(\alpha _1\) :

Constant in variable magnetic field

\(\tau _{zr}^A\) :

Asymmetric part of shear stress

\(\tau _{rz}^S\) :

Symmetric part of shear stress

\(\tau _w\) :

Wall shear stress

\(\delta \) :

Plasma layer thickness

\(\lambda \) :

Flow resistance

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Acknowledgments

One of the authors S. Priyadharshini is thankful to the Ministry of Human Resource Development (MHRD), the Government of India for the grant of a fellowship through National Institute of Technology, Tiruchirappalli to carry out this work.

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Correspondence to S. Priyadharshini.

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Communicated by Florence Hubert.

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Ponalagusamy, R., Priyadharshini, S. Numerical investigation on two-fluid model (micropolar-Newtonian) for pulsatile flow of blood in a tapered arterial stenosis with radially variable magnetic field and core fluid viscosity. Comp. Appl. Math. 37, 719–743 (2018). https://doi.org/10.1007/s40314-016-0367-z

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Keywords

  • Micropolar fluid
  • Finite difference method
  • Hematocrit
  • Radially variable Magnetic field
  • Stenosed artery
  • Tapered artery

Mathematics Subject Classification

  • 92C10