# Magnetic Nanoparticles for Drug Delivery through Tapered Stenosed Artery with Blood Based Non-Newtonian Fluid

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## Abstract

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## 1. Introduction

## 2. Materials

## 3. Methods and Results

#### 3.1. Zeroth Order System

#### 3.2. First Order System

#### 3.3. Second Order System

## 4. Discussion

#### 4.1. Velocity Mechanism−w

#### 4.2. Temperature Mechanism−θ

#### 4.3. Impedance Profile−I_{m}

#### 4.4. Wall Shear Stress−S_{rz}

#### 4.5. Trapping Mechanism

## 5. Conclusions

- (i)
- The Sutterby fluid parameter opposes the flow negligibly, whereas the Hartmann number and thermal Grashof number strengthen the flow field. In addition, the thermal Grashof number and the Hartmann number exhibit a decreasing tendency closer to the walls.
- (ii)
- It is also observed that the addition of gold nanoparticles (mono nanofluids) results in a greater magnitude of velocity than copper nanofluids.
- (iii)
- The thermal profile exhibits a diminishing trend owing to greater values of the Sutterby fluid parameter, whereas a rising trend is detected due to the significant impact of the magnetic field, Brinkman number, and thermal Grashof number.
- (iv)
- Copper nanoparticles (in the absence of gold nanoparticles) are observed to deplete the thermal profile substantially more than gold nanoparticles. Nevertheless, the thermal profile is enhanced by the presence of both nanoparticles (hybrid nanofluids).
- (v)
- It is observed that the impedance profile has a dual pattern for various values of the Sutterby fluid parameter, but the thermal Grashof number and magnetic field exhibit a uniformly declining tendency.
- (vi)
- When the impact of both nanoparticles rises, the impedance profile grows while the amplitude of the impedance profile for mono nanofluids decreases.
- (vii)
- For greater values of the Sutterby fluid parameter, the wall shear stress has been observed to rise considerably, whereas the inverse is true for the Hartmann number and the thermal Grashof number.
- (viii)
- The trapping mechanism demonstrates that the fluid parameters influence the size and frequency of the bolus. However, for other parameters, the trapped bolus manifested for certain values, although for converging and non-tapered arteries, it did not.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Geometrical structure of the blood flow through a stenosed tapered artery filled with Au-Cu Nanoparticles.

**Figure 2.**Velocity mechanism for multitudinous values of $\beta $. Black line is for $\beta =1$, Red line is for $\beta =3$, Green line is for $\beta =4$.

**Figure 3.**Velocity mechanism for multitudinous values of $\gamma $. Black line is for $\gamma =2$, Red line is for $\gamma =2.5$, Green line is for γ = 3.

**Figure 4.**Velocity mechanism for multitudinous values of $\xi $. Black line is for $\xi =1$, Red line is for $\xi =1.2$, Green line is for $\xi =1.4$.

**Figure 5.**Velocity mechanism for multitudinous values of ${\varphi}_{\mathrm{Gold}},{\varphi}_{\mathrm{Copper}}$. Black line is for ${\varphi}_{\mathrm{Gold}}=0,{\varphi}_{\mathrm{Copper}}=0.1$, Red line is for ${\varphi}_{\mathrm{Gold}}=0.1,{\varphi}_{\mathrm{Copper}}=0$, Green line is for ${\varphi}_{\mathrm{Gold}}=0.3,{\varphi}_{\mathrm{Copper}}=0.3$.

**Figure 6.**Temperature mechanism for multitudinous values of $\beta $. Black line is for $\beta =1$, Red line is for $\beta =1.5$, Green line is for $\beta =2$.

**Figure 7.**Temperature mechanism for multitudinous values of ${\beta}_{m}$. Black line is for ${\beta}_{m}=11$, Red line is for ${\beta}_{m}=13$, Green line is for ${\beta}_{m}=15$.

**Figure 8.**Temperature mechanism for multitudinous values of $\gamma $. Black line is for $\gamma =2$, Red line is for $\gamma =2.5$, Green line is for $\gamma =3$.

**Figure 9.**Temperature mechanism for multitudinous values of $\xi $. Black line is for $\xi =1$, Red line is for $\xi =1.2$, Green line is for $\gamma =1.4$.

**Figure 10.**Temperature mechanism for multitudinous values of ${\varphi}_{\mathrm{Gold}},{\varphi}_{\mathrm{Copper}}$. Black line is for ${\varphi}_{\mathrm{Gold}}=0.05,{\varphi}_{\mathrm{Copper}}=0$, Red line is for ${\varphi}_{\mathrm{Gold}}=0,{\varphi}_{\mathrm{Copper}}=0.05$, Green line is for ${\varphi}_{\mathrm{Gold}}=0.3,{\varphi}_{\mathrm{Copper}}=0.3$.

**Figure 11.**Impedance profile for multitudinous values of $\beta $. Black line is for $\beta =1$, Red line is for $\beta =2$, Green line is for $\beta =3$.

**Figure 12.**Impedance profile for multitudinous values of $\gamma $. Black line is for $\gamma =2$, Red line is for $\gamma =2.5$, Green line is for $\gamma =3$.

**Figure 13.**Impedance profile for multitudinous values of $\xi $. Black line is for $\xi =1$, Red line is for $\xi =1.2$, Green line is for $\gamma =1.4$.

**Figure 14.**Impedance profile for multitudinous values of ${\varphi}_{\mathrm{Gold}},{\varphi}_{\mathrm{Copper}}$. Black line is for ${\varphi}_{\mathrm{Gold}}=0.1,{\varphi}_{\mathrm{Copper}}=0$, Red line is for ${\varphi}_{\mathrm{Gold}}=0,{\varphi}_{\mathrm{Copper}}=0.1$, Green line is for ${\varphi}_{\mathrm{Gold}}=0.15,{\varphi}_{\mathrm{Copper}}=0.15$.

**Figure 15.**Wall shear stress for multitudinous values of $\beta $. Black line is for $\beta =1$, Red line is for $\beta =1.5$, Green line is for $\beta =2$.

**Figure 16.**Wall shear stress for multitudinous values of $\gamma $. Black line is for γ = 2, Red line is for γ = 2.5, Green line is for $\gamma =3$.

**Figure 17.**Wall shear stress for multitudinous values of $\xi $. Black line is for $\xi =1$, Red line is for $\xi =1.2$, Green line is for $\xi =1.4$.

**Figure 18.**Wall shear stress for multitudinous values of ${\varphi}_{\mathrm{Gold}},{\varphi}_{\mathrm{Copper}}$. Black line is for ${\varphi}_{\mathrm{Gold}}=0.1,{\varphi}_{\mathrm{Copper}}=0$, Red line is for ${\varphi}_{\mathrm{Gold}}=0,{\varphi}_{\mathrm{Copper}}=0.1$, Green line is for ${\varphi}_{\mathrm{Gold}}=0.1,{\varphi}_{\mathrm{Copper}}=0.1$.

**Figure 19.**Streamlines for multitudinous values of $\beta $. (

**a**) $\beta =1.7$, (

**b**) $\beta =2$, (

**c**) $\beta =2.5$.

**Figure 20.**Streamlines for multitudinous values of $\psi $. (

**a**) $\psi =-0.1$, (

**b**) $\psi =0$, (

**c**) $\psi =0.1$.

**Figure 21.**Streamlines for multitudinous values of $\gamma $. (

**a**) $\gamma =1.8$, (

**b**) $\gamma =2$, (

**c**) $\gamma =2.2$.

**Figure 22.**Streamlines for multitudinous values of $\xi $. (

**a**) $\xi =2.7$, (

**b**) $\xi =3$, (

**c**) $\xi =3.2$.

**Figure 23.**Streamlines for multitudinous values of ${\varphi}_{\mathrm{Gold}},{\varphi}_{\mathrm{Copper}}$. (

**a**) ${\varphi}_{\mathrm{Gold}}=0.1,{\varphi}_{\mathrm{Copper}}=0$, (

**b**) ${\varphi}_{\mathrm{Gold}}=0,{\varphi}_{\mathrm{Copper}}=0.2$, (

**c**) ${\varphi}_{\mathrm{Gold}}=0.1,{\varphi}_{\mathrm{Copper}}=0.1$.

Nanofluid | Hybrid Nanofluid | |
---|---|---|

Dynamic viscosity | ${\mu}_{nf}=\frac{{\mu}_{\mathrm{Blood}}}{{\left(1-{\varphi}_{\mathrm{Gold}}\right)}^{2.5}}$ | ${\mu}_{hnf}=\frac{{\mu}_{nf}}{{\left(1-{\varphi}_{\mathrm{Copper}}\right)}^{2.5}}$ |

Density | ${\rho}_{nf}=\left(1-{\varphi}_{\mathrm{Gold}}\right){\rho}_{\mathrm{Blood}}+{\rho}_{\mathrm{Gold}}{\varphi}_{\mathrm{Gold}}$ | ${\rho}_{hnf}=\left(1-{\varphi}_{\mathrm{Copper}}\right){\rho}_{nf}+{\rho}_{\mathrm{copper}}{\varphi}_{\mathrm{Copper}}$ |

Electrical conductivity | ${W}_{nf}={W}_{f}\left[\frac{{W}_{\mathrm{Gold}}\left(1+2{f}_{\mathrm{Gold}}\right)+2{W}_{\mathrm{Blood}}\left(1-2{f}_{\mathrm{Gold}}\right)}{{W}_{\mathrm{Gold}}\left(1-{f}_{\mathrm{Gold}}\right)+{W}_{\mathrm{Blood}}\left(2+{f}_{\mathrm{Gold}}\right)}\right]$ | ${W}_{hnf}={W}_{nf}\left[\frac{{W}_{\mathrm{Copper}}\left(1+2{f}_{\mathrm{Copper}}\right)+2{W}_{\mathrm{Blood}}\left(1-2{f}_{\mathrm{Copper}}\right)}{{W}_{\mathrm{Copper}}\left(1-{f}_{\mathrm{Copper}}\right)+{W}_{\mathrm{Blood}}\left(2+{f}_{\mathrm{Copper}}\right)}\right]$ |

Thermal conductivity | ${k}_{nf}={k}_{f}\left[\frac{2{k}_{\mathrm{Blood}}+{k}_{\mathrm{Gold}}-2\left({k}_{\mathrm{Blood}}-{k}_{\mathrm{Gold}}\right){f}_{\mathrm{Gold}}}{2{k}_{\mathrm{Blood}}+{k}_{\mathrm{Gold}}+\left({k}_{\mathrm{Blood}}-{k}_{\mathrm{Gold}}\right){f}_{\mathrm{Gold}}}\right]$ | ${k}_{hnf}={k}_{nf}\left[\frac{2{k}_{f}+{k}_{\mathrm{Copper}}-2\left({k}_{f}-{k}_{\mathrm{Copper}}\right){f}_{\mathrm{Copper}}}{2{k}_{f}+{k}_{\mathrm{Copper}}+\left({k}_{f}-{k}_{\mathrm{Copper}}\right){f}_{\mathrm{Copper}}}\right]$ |

Heat capacity | ${(\rho {C}_{p})}_{nf}=(1-{\varphi}_{\mathrm{Gold}}){(\rho {C}_{p})}_{\mathrm{Blood}}+{\varphi}_{\mathrm{Gold}}{(\rho {C}_{p})}_{\mathrm{Gold}}$ | ${(\rho {C}_{p})}_{hnf}=(1-{\varphi}_{\mathrm{Copper}}){(\rho {C}_{p})}_{nf}+{\varphi}_{\mathrm{Copper}}{(\rho {C}_{p})}_{\mathrm{copper}}$ |

Thermal expansion | ${\left(rb\right)}_{nf}=\left(1-{f}_{\mathrm{Gold}}\right){\left(rb\right)}_{\mathrm{Blood}}+{f}_{\mathrm{Gold}}{\left(rb\right)}_{\mathrm{Gold}}$ | ${\left(rb\right)}_{hnf}=\left(1-{f}_{\mathrm{Copper}}\right){\left(rb\right)}_{nf}+{f}_{\mathrm{Copper}}{\left(rb\right)}_{\mathrm{Copper}}$ |

**Table 2.**Computational values of Thermal & physical properties of hybrid nanofluid used for computational results [52].

Physical Characteristics | Base Fluid (Blood) | Copper Nanoparticles | Gold Nanoparticles |
---|---|---|---|

${C}_{p}\left[\mathrm{J}/\mathrm{Kg}\cdot \mathrm{K}\right]$ | 3617 | 385 | 129.1 |

$\rho \left[\mathrm{Kg}/{\mathrm{m}}^{3}\right]$ | 1050 | 8933 | 19300 |

$\sigma \left[\mathrm{S}/\mathrm{m}\right]$ | 1.33 | $5.96\times {10}^{7}$ | $4.5\times {10}^{7}$ |

$\beta \left[1/\mathrm{K}\right]$ | 0.18 | 16.65 $\times $ 10^{−6} | 0.0000142 |

$k\left[\mathrm{W}/\mathrm{m}\cdot \mathrm{K}\right]$ | 0.52 | 400 | 320 |

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**MDPI and ACS Style**

Bhatti, M.M.; Sait, S.M.; Ellahi, R.
Magnetic Nanoparticles for Drug Delivery through Tapered Stenosed Artery with Blood Based Non-Newtonian Fluid. *Pharmaceuticals* **2022**, *15*, 1352.
https://doi.org/10.3390/ph15111352

**AMA Style**

Bhatti MM, Sait SM, Ellahi R.
Magnetic Nanoparticles for Drug Delivery through Tapered Stenosed Artery with Blood Based Non-Newtonian Fluid. *Pharmaceuticals*. 2022; 15(11):1352.
https://doi.org/10.3390/ph15111352

**Chicago/Turabian Style**

Bhatti, Muhammad Mubashir, Sadiq M. Sait, and Rahmat Ellahi.
2022. "Magnetic Nanoparticles for Drug Delivery through Tapered Stenosed Artery with Blood Based Non-Newtonian Fluid" *Pharmaceuticals* 15, no. 11: 1352.
https://doi.org/10.3390/ph15111352