Mathematical modelling for pulsatile flow of Casson fluid along with magnetic nanoparticles in a stenosed artery under external magnetic field and body acceleration

Abstract

In the present paper, the magnetohydrodynamics effects on flow parameters of blood carrying magnetic nanoparticles flowing through a stenosed artery under the influence of periodic body acceleration are investigated. Blood is assumed to behave as a Casson fluid. The governing equations are nonlinear and solved numerically using finite difference schemes. The effects of stenotic height, yield stress, magnetic field, particle concentration and mass parameters on wall shear stress, flow resistance and velocity distribution are analysed. It is found that wall shear stress and flow resistance values are considerably enhanced when an external magnetic field is applied. The velocity values of fluid and particles are appreciably reduced when a magnetic field is applied on the model. It is significant to note that the presence of nanoparticles, magnetic field and yield stress tend to increase the plug core radius. Increased wall shear stress and flow resistance affects the circulation of blood in the human cardiovascular system. The results obtained from the study can be used in normalizing the values of the model parameters and hence can be used for medical applications. The presence of magnetic field helps to slow down the flow of fluid and magnetic particles associated with it. The magnetic particles of nanosize developed in recent days are biodegradable and used in biomedical applications. Biomagnetic principles and biomagnetic particles as drug carriers are used in cancer treatments.

Appendix

The continuity and momentum equations governing the pulsatile flow of Casson fluid through stenosed artery with the influence of magnetic field and periodic body acceleration may be written as

$$\begin{aligned}&\frac{\partial \bar{v}_f}{\partial \bar{r}}+\frac{\bar{v}_f}{\bar{r}}+\frac{\partial \bar{u}_f}{\partial \bar{z}} = 0 \end{aligned}$$

(35)

$$\begin{aligned}&\frac{\partial \bar{u}_f}{\partial \bar{t}}+\bar{v}_f \frac{\partial \bar{u}_f}{\partial \bar{r}}+\bar{u}_f\frac{\partial \bar{u}_f}{\partial \bar{z}} = -\frac{1}{\bar{\rho }_f}\frac{\partial \bar{p}}{\partial \bar{z}}\nonumber \\&\quad -\frac{1}{\bar{\rho }_f}\left[ \frac{1}{\bar{r}}\frac{\partial }{\partial \bar{r}}(\bar{r}\bar{\tau }_{rz})+\frac{\partial }{\partial \bar{z}}(\bar{\tau }_{zz})\right] +\frac{1}{\bar{\rho }_f} \bar{K}\bar{N}\left( \bar{u}_p-\bar{u}_f\right) -\frac{\bar{\sigma }\bar{B}_0^2\bar{u}_f}{\bar{\rho }_f}+\bar{F}(\bar{t}) \end{aligned}$$

(36)

$$\begin{aligned}&\frac{\partial \bar{v}_f}{\partial \bar{t}}+\bar{v}_f\frac{\partial \bar{v}_f}{\partial \bar{r}}+\bar{u}_f\frac{\partial \bar{v}_f}{\partial \bar{z}} = -\frac{1}{\bar{\rho }_f}\frac{\partial \bar{p}}{\partial \bar{r}}\nonumber \\&\quad -\frac{1}{\bar{\rho }_f}\left[ \frac{1}{\bar{r}}\frac{\partial }{\partial \bar{r}}(\bar{r}\bar{\tau }_{rr})+\frac{\partial }{\partial \bar{z}}(\bar{\tau }_{rz})\right] +\frac{1}{\bar{\rho }_f} \bar{K}\bar{N}\left( \bar{v}_p-\bar{v}_f\right) \end{aligned}$$

(37)

where $\bar{u}_f$ and $\bar{v}_f$ are the axial and radial components of fluid velocity, respectively, $\bar{\rho }_f$ is the density of fluid, $\bar{p}$ is the pressure, $\bar{K}$ denotes the Stoke’s constant, $\bar{N}$ represents the number of nanoparticles per unit volume, $\bar{u}_p$ and $\bar{v}_p$ are the axial and radial components of particle velocity, respectively, $\bar{\sigma }$ is the electrical conductivity, $\bar{B_0}$ is the strength of external magnetic field and the body acceleration is expressed as $\bar{F}(\bar{t}) = \bar{A} \cos (\bar{\omega _b}\bar{t}+\phi )$, $\bar{A}$ is the amplitude of body acceleration, $\phi$ is the phase lag and $\bar{\omega }_b = 2\pi \bar{f}_b$, $\bar{f}_b$ is the frequency of body acceleration which is assumed to be small and hence the wave effects can be neglected.

The pulsatile pressure gradient acting on the fluid is expressed as $-\frac{\partial \bar{p}}{\partial \bar{z}} = \bar{A}_0+\bar{A}_1 \cos (\bar{\omega }_p\bar{t})$, where $\bar{A}_0$ and $\bar{A}_1$ are the constant amplitudes of steady-state and pulsatile pressure gradients, respectively, $\bar{\omega }_p$ is the angular frequency and is given by $\bar{\omega }_p = 2\pi \bar{f}_p$, $\bar{f}_p$ being the frequency of heart pulse.

The stress components for the rheology of Casson fluid are given by

$$\begin{aligned} \bar{\tau }_{zz}= 2\bar{\mu }(\bar{J}_2)\left( \frac{\partial \bar{u}_f}{\partial \bar{z}}\right) , \end{aligned}$$

(38)

$$\begin{aligned} \bar{\tau }_{rr}= 2\bar{\mu }(\bar{J}_2)\left( \frac{\partial \bar{v}_f}{\partial \bar{r}}\right) , \end{aligned}$$

(39)

$$\begin{aligned} \bar{\tau }_{rz}= \bar{\mu }(\bar{J}_2)\left( \frac{\partial \bar{u}_f}{\partial \bar{r}}+\frac{\partial \bar{v}_f}{\partial \bar{z}}\right) , \end{aligned}$$

(40)

$$\begin{aligned} \bar{\mu }(\bar{J}_2)= (\bar{\mu }_c^{\frac{1}{2}}\bar{J}_2^{\frac{1}{4}}+2^{-\frac{1}{2}}\bar{\tau _y}^{\frac{1}{2}})^2 \bar{J}_2^{-\frac{1}{2}}, \end{aligned}$$

(41)

$$\begin{aligned} \bar{J_2}= 2\left\{ \left( \frac{\partial \bar{v}_f}{\partial \bar{r}}\right) ^2 + \left( \frac{\bar{v}_f}{\bar{r}}\right) ^2 + \left( \frac{\partial \bar{u}_f}{\partial \bar{z}}\right) ^2\right\} +\left( \frac{\partial \bar{v}_f}{\partial \bar{z}}+\frac{\partial \bar{u}_f}{\partial \bar{r}}\right) ^2, \end{aligned}$$

(42)

where $\bar{\mu }(\bar{J}_2)$ is the apparent viscosity, $\bar{\tau _y}$ is the yield stress and $\bar{\mu _c}$ is Casson viscosity ($'-'$ over the letter denotes the corresponding dimensional quantity).

Using non-dimensionalization defined in (5), the governing equations in dimensionless form are

$$\begin{aligned}&\delta _s\left( \frac{\partial v_f}{\partial r}+\frac{v_f}{r}\right) +\frac{\partial u_f}{\partial z} = 0 \end{aligned}$$

(43)

$$\begin{aligned}&\alpha ^2\frac{\partial u_f}{\partial t}+Re\left[ \delta _sv_f\frac{\partial u_f}{\partial r}+u_f\frac{\partial u_f}{\partial z}\right] = -\frac{\partial p}{\partial z}-\left[ \frac{1}{r}\frac{\partial }{\partial r}(r\tau _{rz})+\frac{\partial \tau _{zz}}{\partial z}\right] \nonumber \\&\quad +R_1(u_p-u_f)-M^2u_f+F(t) \end{aligned}$$

(44)

$$\begin{aligned}\delta _s\alpha ^2 \frac{\partial v_f}{\partial t}+Re\left[ \delta _s^2 v_f\frac{\partial v_f}{\partial r}+ \delta _s u_f \frac{\partial v_f}{\partial z}\right] &= -\frac{\partial p}{\partial r}-\left[ \frac{1}{r}\frac{\partial }{\partial r}(r\tau _{rr})+\frac{\partial \tau _{rz}}{\partial z}\right] \nonumber \\&\quad +\,\delta _s R_1(v_p-v_f) \end{aligned}$$

(45)

and the dimensionless stress components are

$$\begin{aligned} \tau _{zz}= 2\mu (J_2)\frac{\partial u_f}{\partial z} \end{aligned}$$

(46)

$$\begin{aligned} \tau _{rr}= 2\mu (J_2)\delta _s \frac{\partial v_f}{\partial r} \end{aligned}$$

(47)

$$\begin{aligned} \tau _{rz}= \mu (J_2)\left( \frac{\partial u_f}{\partial r}+\delta _s \frac{\partial v_f}{\partial z}\right) \end{aligned}$$

(48)

$$\begin{aligned} \mu (J_2)= \left( J_2^{1/4}+2^{-1/2}\tau _y^{1/2}\right) ^2 J_2^{-1/2} \end{aligned}$$

(49)

$$\begin{aligned} J_2= 2\left\{ \delta _s \left[ \left( \frac{\partial v_f}{\partial r}\right) ^2+\left( \frac{v_f}{r}\right) ^2\right] +\left( \frac{\partial u_f}{\partial z}\right) ^2\right\} +\left[ \delta _s \frac{\partial v_f}{\partial z}+\frac{\partial u_f}{\partial r}\right] ^2 \end{aligned}$$

(50)

During the initial development of stenosis (mild stenosis), which means only minimal narrowing of the arterial lumen $\delta _s = (\frac{\bar{\delta }_s}{\bar{R}_0})<<1$, then from (43), $\frac{\partial u_f}{\partial z}<<1$. If $Re \delta _s<<1$,then from (44), it is significant to note that the effects of convective part (inertial terms) are smaller when compared with viscous terms and the radial velocity is negligibly small compared to the axial velocity. Equation (45) indicates that $\frac{\partial p}{\partial r}<<\frac{\partial p}{\partial z}$ and the variation of pressure along r-direction is negligibly small and can be neglected. Further, the radius of lumen is very small compared to the wavelength of pressure wave, the equation of motion in the radial direction is reduced to $\frac{\partial p}{\partial r} = 0$. For more details, refer [1, 3, 40, 49].

Using the assumptions and simplifications discussed in Ponalagusamy [3], Mekheimer and EI Kot [48], the appropriate momentum equations describing the pulsatile flow of Casson fluid through stenosed artery in the presence of periodic body acceleration under constant magnetic field in the case of a mild stenosis ($\delta _s<< 1$) along with the conditions (Young [1], $\delta _s Re<< 1$) are given by

$$\begin{aligned}&\alpha ^2 \frac{\partial u_f}{\partial t} = -\frac{\partial p}{\partial z}-\left[ \frac{1}{r}\frac{\partial }{\partial r}(r\tau )\right] +R_1(u_p-u_f)-M^2u_f+F(t) \end{aligned}$$

(51)

$$\begin{aligned}&-\frac{\partial p}{\partial r} = 0 \end{aligned}$$

(52)

$\tau$ is the shear stress and

$$\begin{aligned} \tau ^{\frac{1}{2}}= \theta _1^{\frac{1}{2}} + {\left( \frac{\partial u_f}{\partial r}\right) }^{\frac{1}{2}}, \tau \ge \theta _1 \end{aligned}$$

(53)

$$\begin{aligned} \frac{\partial u_f}{\partial r}= 0, \tau < \theta _1 \end{aligned}$$

(54)

where $\theta _1 = \frac{\bar{R}_0\bar{\tau }_y}{\bar{u}_0\bar{\mu }_c}$.