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Computational model on pulsatile flow of blood through a tapered arterial stenosis with radially variable viscosity and magnetic field

Abstract

An unsteady two-fluid model of blood flow through a tapered arterial stenosis with variable viscosity in the presence of variable magnetic field has been analysed in the present paper. In this article, blood in the core region is assumed to obey the law of Jeffrey fluid and plasma in the peripheral layer is assumed to be Newtonian. The values for velocity, wall shear stress, flow rate and flow resistance are numerically computed by employing finite-difference method in solving the governing equations. A comparison study between the velocity profiles obtained by the present study and the experimental data represented graphically shows that that the rheology of blood obeys the law of Jeffrey fluid rather than that of Newtonian fluid. The effects of parameters such as taper angle, radially variable viscosity, hematocrit, Jeffrey parameter, magnetic field and plasma layer thickness on physiologically important parameters such as wall shear stress distribution and flow resistance have been investigated. The results in the case of radially variable magnetic field and constant magnetic field are compared to observe the effect of magnetic field in driving the blood flow. It is observed that increase in hematocrit increases the wall shear stress. The values of wall shear stress and flow resistance are obtained at various time instances and compared. It is pertinent to note that the magnitudes of flow resistance are higher in the case of converging tapered than non-tapered and diverging tapered artery.

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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

\(\bar{u}_c\) :

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

\(\bar{p}\) :

pressure

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

variable magnetic field

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

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

\(\bar{\lambda }_2\) :

retardation time

\(\delta _s\) :

maximum stenotic height

d :

location of the stenosis

\(n_1\) :

shape of stenosis

\(\psi \) :

taper angle

\(\lambda _1\) :

ratio of relaxation to retardation times

\(A_0\) :

amplitude of steady pressure gradient

\(A_1\) :

amplitude of pulsatile pressure gradient

\(\beta \) :

constant in variable core viscosity

\(h_m\) :

hematocrit

\(Q_c\) :

flow rate in the core region

\(Q_p\) :

flow rate in the peripheral plasma region

Q :

total flow rate

M :

magnitude of magnetic field strength

\(\alpha _1\) :

constant in variable magnetic field

\(S_{rz}\) :

shear stress

\(\tau _w\) :

wall shear stress

\(\delta \) :

plasma layer thickness

\(\lambda \) :

flow resistance

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Acknowledgements

One of the authors (S Priyadharshini) is thankful to the Ministry of Human Resource Development, the Government of India, for the grant of fellowship. The authors thank the editor and reviewers for their valuable suggestions.

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Correspondence to S PRIYADHARSHINI.

Appendix A

Appendix A

The continuity and momentum equations governing the pulsatile flow of Jeffrey fluid in the presence of magnetic field are given by (Eqs. (2)–(4)) in section 2)

$$\begin{aligned}&\frac{\partial \bar{v}_c}{\partial \bar{r}}+\frac{\bar{v}_c}{\bar{r}}+\frac{\partial \bar{u}_c}{\partial \bar{z}} = 0 \end{aligned}$$
(A-1)
$$\begin{aligned}&\quad \bar{\rho }_c \left( \frac{\partial \bar{u}_c}{\partial \bar{t}}+\bar{v}_c \frac{\partial \bar{u}_c}{\partial \bar{r}}+\bar{u}_c\frac{\partial \bar{u}_c}{\partial \bar{z}}\right) = -\frac{\partial p}{\partial z}+\frac{1}{\bar{r}}\frac{\partial }{\partial \bar{r}}(\bar{r}\bar{S}_{rz})+\frac{\partial }{\partial \bar{z}}(\bar{S}_{zz})-\bar{\sigma }\bar{B}_0^2(\bar{r})\bar{u}_c \end{aligned}$$
(A-2)
$$\begin{aligned}&\bar{\rho }_c \left( \frac{\partial \bar{v}_c}{\partial \bar{t}}+\bar{v}_c \frac{\partial \bar{v}_c}{\partial \bar{r}}+\bar{u}_c\frac{\partial \bar{v}_c}{\partial \bar{z}}\right) = -\frac{\partial p}{\partial r}+\frac{1}{\bar{r}}\frac{\partial }{\partial \bar{r}}(\bar{r}\bar{S}_{rr})+\frac{\partial }{\partial \bar{z}}(\bar{S}_{rz})-\frac{\bar{S}_{\theta \theta }}{\bar{r}} \end{aligned}$$
(A-3)

where the stress components are expressed as follows:

$$\begin{aligned} \bar{S}_{rr}= & {} \frac{2\bar{\mu }_c(\bar{r})}{1+\lambda _1}\left[ 1+\bar{\lambda }_2\left( \bar{v}_c\frac{\partial }{\partial \bar{r}}+\bar{u}_c\frac{\partial }{\partial \bar{z}}\right) \right] \frac{\partial \bar{v}_c}{\partial \bar{r}}, \end{aligned}$$
(A-4)
$$\begin{aligned} \bar{S}_{rz}= & {} \frac{\bar{\mu }_c(\bar{r})}{1+\lambda _1}\left[ 1+\bar{\lambda }_2\left( \bar{v}_c\frac{\partial }{\partial \bar{r}}+\bar{u}_c\frac{\partial }{\partial \bar{z}}\right) \right] \left( \frac{\partial \bar{v}_c}{\partial \bar{z}}+\frac{\partial \bar{u}_c}{\partial \bar{r}}\right) , \end{aligned}$$
(A-5)
$$\begin{aligned} \bar{S}_{zz}= & {} \frac{2\bar{\mu }_c(\bar{r})}{1+\lambda _1}\left[ 1+\bar{\lambda }_2\left( \bar{v}_c\frac{\partial }{\partial \bar{r}}+\bar{u}_c\frac{\partial }{\partial \bar{z}}\right) \right] \frac{\partial \bar{u}_c}{\partial \bar{z}}, \end{aligned}$$
(A-6)
$$\begin{aligned} \bar{S}_{\theta \theta }= & {} \frac{2\bar{\mu }_c(\bar{r})}{1+\lambda _1}\left[ 1+\bar{\lambda }_2\left( \bar{v}_c\frac{\partial }{\partial \bar{r}}+\bar{u}_c\frac{\partial }{\partial \bar{z}}\right) \right] \frac{\bar{u}_c}{\bar{r}}. \end{aligned}$$
(A-7)

Using non-dimensionalization as in Eq. (9), the governing equations in dimensionless form are given as follows:

$$\begin{aligned} \delta _s \left( \frac{\partial v_c}{\partial r}+\frac{v_c}{r}\right) +\frac{\partial u_c}{\partial z} = 0 \end{aligned}$$
(A-8)
$$\begin{aligned}&\frac{\alpha ^2}{\rho _0}\frac{\partial u_c}{\partial t}+\frac{Re \bar{\delta }_s}{\bar{L}_0}v_c\frac{\partial u_c}{\partial r}+Re \epsilon u_c\frac{\partial u_c}{\partial z} = -\frac{\partial p}{\partial z}+\frac{1}{r}\frac{\partial }{\partial r}(rS_{rz})+\epsilon ^2\frac{\partial }{\partial z}(S_{zz}) \\&-M^2u_c \end{aligned}$$
(A-9)
$$\begin{aligned}&\frac{\alpha ^2}{\rho _0}\delta _s \epsilon ^2 \frac{\partial v_c}{\partial t}+\frac{Re}{\rho _0}\delta _s^2 \epsilon ^3 v_c\frac{\partial v_c}{\partial r}+Re\delta _s\epsilon ^3u_c\frac{\partial v_c}{\partial z} = -\frac{\partial p}{\partial r}+\epsilon ^2\frac{1}{r}\frac{\partial }{\partial r}(rS_{rr})+\epsilon ^2\frac{\partial }{\partial z}(S_{rz}) \\&-\epsilon ^2\frac{S_{\theta \theta }}{r} \end{aligned}$$
(A-10)

where

$$\begin{aligned} S_{rr}= & {} \frac{2\delta _s}{1+\lambda _1}f_1(r)\left\{ 1+\lambda _2\delta _s\left( v_c\frac{\partial }{\partial r}+\frac{u_c}{\delta _s}\frac{\partial }{\partial z}\right) \right\} \left( \frac{\partial v_c}{\partial r}\right) \end{aligned}$$
(A-11)
$$\begin{aligned} S_{rz}= & {} \frac{f_1(r)}{1+\lambda _1}\left\{ 1+\lambda _2\delta _s\left( v_c\frac{\partial }{\partial r}+\frac{u_c}{\delta _s}\frac{\partial }{\partial z}\right) \right\} \left( \delta _s\epsilon ^2 \frac{\partial v_c}{\partial z}+\frac{\partial u_c}{\partial r}\right) \end{aligned}$$
(A-12)
$$\begin{aligned} S_{zz}= & {} \frac{2}{1+\lambda _1}f_1(r)\left\{ 1+\lambda _2\delta _s\left( v_c\frac{\partial }{\partial r}+\frac{u_c}{\delta _s}\frac{\partial }{\partial z}\right) \right\} \left( \frac{\partial u_c}{\partial \bar{z}}\right) \end{aligned}$$
(A-13)
$$\begin{aligned} S_{\theta \theta }= & {} \frac{2\epsilon }{1+\lambda _1} f_1(r)\left\{ 1+\lambda _2\delta _s\left( v_c\frac{\partial }{\partial r}+\frac{u_c}{\delta _s}\frac{\partial }{\partial z}\right) \right\} \left( \frac{u}{r}\right) \end{aligned}$$
(A-14)

and \(f_1(r) = 1+\beta _1h_m(R_1^{m_2}-r^{m_2})\). The non-dimensional retardation time may be defined as \(\lambda _2 = \frac{\bar{\lambda }_2\bar{u}_0}{\bar{L}_0}\).

In the initial stage of mild stenosis, \(\delta _s = \frac{\bar{\delta }_s}{\bar{R}_0}<< 1\); in this case, from Eq. (A-8), \(\frac{\partial u_c}{\partial z}<< 1\). Using the assumptions in [1, 4, 10] subject to the conditions

(i) \(\frac{Re \bar{\delta }_s n_1^{\frac{1}{n_1-1}}}{\bar{L}_0}<< 1\) and (ii) \(\epsilon n_1^{\frac{1}{n_1-1}}\sim {\mathrm{O(1)}} \, (\epsilon = \frac{\bar{R}_0}{\bar{L}_0})\),

the lumen radius is very small compared to the wavelength of pressure wave, equation of motion in radial direction (A-10) will reduce to \(\frac{\partial p}{\partial r} = 0\) [53] and low-Reynolds-number approximation [54, 55], and the momentum equations become

$$\begin{aligned} \frac{\alpha ^2}{\rho _0}\frac{\partial u_c}{\partial t}&= -\frac{\partial p}{\partial z}+\left( \frac{1}{1+\lambda _1}\right) \frac{1}{r}\frac{\partial }{\partial r}\left[ 1+\beta h_m\left( R_1^{m_2}-r^{m_2}\right) \right] \frac{\partial u_c}{\partial r}\\&-M^2(r)u_c, \end{aligned}$$
(A-15)
$$\begin{aligned} \frac{\partial p}{\partial r} = 0. \end{aligned}$$
(A-16)

Similarly, the governing equation for Newtonian fluid in peripheral plasma region can be obtained by performing an order of magnitude analysis as done here. For further details regarding the order of magnitude analysis, see references [56, 57].

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PRIYADHARSHINI, S., PONALAGUSAMY, R. Computational model on pulsatile flow of blood through a tapered arterial stenosis with radially variable viscosity and magnetic field. Sādhanā 42, 1901–1913 (2017). https://doi.org/10.1007/s12046-017-0734-5

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Keywords

  • Jeffrey fluid
  • radially variable viscosity
  • hematocrit
  • radially variable magnetic field
  • tapered arterial stenosis