# MHD Pulsatile Flow of Blood-Based Silver and Gold Nanoparticles between Two Concentric Cylinders

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## Abstract

**:**

## 1. Introduction

_{2}nanoparticles and gold nanoparticles, per Yamaguchi et al. [36], can be used in the biomedical sciences, such as in photodynamic and ultrasound treatments for cancer.

## 2. Mathematical Formulation

#### 2.1. Physical Problem

#### 2.2. Governing Equations

#### 2.3. Boundary Conditions

## 3. Solution of the Problem

#### Pressure Calculation

## 4. Code Validation

## 5. Results and Discussion

## 6. Conclusions and Prime Findings

- ▪
- Because of the reduction in wave amplitude at high frequencies, the maximum velocity and temperature tend to stay constant.
- ▪
- It is likewise seen that the expansion of Ag-Au hugely expands the temperature of the base liquid.
- ▪
- HNF are more potent coolants than standard base liquids since they can dispose of more warmth than typical base liquids. Because the Womersley number is the ratio of throb to thick powers, an increase in the Womersley number gradually reduces the gooey powers.
- ▪
- It is likewise found that decreasing viscous powers improve the movement of the liquid particles to become quicker, and, therefore, result in a incremental change in the speed profile.
- ▪
- Taking into account viable warm conductivity, it is additionally resolved that blood-based Ag-Au has a higher heat transfer rate when contrasted with the pure fluid.
- ▪
- We may investigate physical properties such as the skin friction coefficient, as well as assess the problem using radiation and magnetohydrodynamic effects to identify the causes of stenosis, which may aid in the treatment of arterial stenosis.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

${\rho}_{hnf}$ | density of HNF $\left(Kg{\mathrm{m}}^{-3}\right)$ |

${\mu}_{hnf}$ | viscosity of HNF $\left(Kg{m}^{-1}{\mathrm{s}}^{-1}\right)$ |

${C}_{hnf}$ | specific heat of HNF $\left(J{K}^{-1}k{g}^{-1}\right)$ |

${k}_{hnf}$ | thermal conductivity of HNF $\left(W{m}^{-1}{K}^{-1}\right)$ |

${\sigma}_{hnf}$ | electric conductivity of HNF $\left(S{m}^{-1}\right)$ |

${\rho}_{f}$ | density of fluid $\left(Kg{m}^{-3}\right)$ |

${\rho}_{p}$ | density of NPs $\left(Kg{m}^{-3}\right)$ |

${C}_{f}$ | specific heat of the fluid $\left(J{K}^{-1}k{g}^{-1}\right)$ |

${C}_{p}$ | specific heat of NPs $\left(J{K}^{-1}k{g}^{-1}\right)$ |

${\sigma}_{f}$ | electric conductivity of fluid $\left(S{m}^{-1}\right)$ |

${\sigma}_{p}$ | electric conductivity of NPs $\left(S{m}^{-1}\right)$ |

${K}_{f}$ | thermal conductivity of fluid $\left(W{m}^{-1}{K}^{-1}\right)$ |

${K}_{p}$ | thermal conductivity of NPs $\left(W{m}^{-1}{K}^{-1}\right)$ |

${\mu}_{f}$ | dynamic viscosity of fluid $\left(J{K}^{-1}k{g}^{-1}\right)$ |

${\mu}_{p}$ | dynamic viscosity of NPs $\left(J{K}^{-1}k{g}^{-1}\right)$ |

$\alpha $ | Womersley number $\left(=\sqrt{R{e}_{\beta}}\right)$ |

$\gamma $ | temperature gradient $\left(K{m}^{-1}\right)$ |

$A$ | Amplitude (m) |

$\omega $ | pulsation |

$P$ | pressure $\left(Kg{m}^{-1}{s}^{-2}\right)$ |

M | Hartmann number |

$Pr$ | Prandtl number |

$R{e}_{\beta}$ | Reynolds number |

$t$ | time |

$D$ | diameter $\left(m\right)$ |

$r$ | radius $\left(m\right)$ |

$T$ | temperature $\left(K\right)$ |

$u,v,w$ | velocity components $\left(m\right)$ |

$\overrightarrow{J}$ | current density $\left(A{m}^{-1}\right)$ |

$\overrightarrow{B}$ | magnetic field |

$H$ | magnetic field intensity (T) |

$C$ | velocity of light $\left(m{s}^{-1}\right)$ |

$L$ | length of the cylinder $\left(m\right)$ |

$\varphi $ | volume fraction of the nanoparticles |

Used Indexes | |

i | internal |

e | external |

f | fluid |

p | particle |

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**Figure 2.**Comparison of the velocity profile of the present study with the velocity profile of Mohamed Deghmoum [37] for the different values of t ($M=10,R{e}_{\beta}=1$, ${\varphi}_{h}=0.01)$.

**Figure 3.**(

**a**) Variation of velocity profile when $M=0,R{e}_{\beta}=1$, $\left({\varphi}_{h}=0\right)$. (

**b**) Variation of velocity profile when $M=0,R{e}_{\beta}=1$, $\left({\varphi}_{h}=0.01\right)$.

**Figure 4.**(

**a**) Variation of velocity profile when $M=0,R{e}_{\beta}=10$, $\left({\varphi}_{h}=0\right)$. (

**b**) Variation of velocity profile when $M=0,R{e}_{\beta}=10$, $\left({\varphi}_{h}=0.01\right)$.

**Figure 5.**(

**a**) Variation of velocity profile when $M=0,R{e}_{\beta}=30$, $\left({\varphi}_{h}=0\right)$. (

**b**) Variation of velocity profile when $M=0,R{e}_{\beta}=30$, $\text{}\left({\varphi}_{h}=0.01\right)$.

**Figure 6.**(

**a**) Variation of velocity profile when $M=10,R{e}_{\beta}=1$, $\text{}\left({\varphi}_{h}=0\right)$. (

**b**) Variation of velocity profile when $M=10,R{e}_{\beta}=1$, $\text{}\left({\varphi}_{h}=0.01\right)$.

**Figure 7.**(

**a**) Variation of velocity profile when $M=30,R{e}_{\beta}=30$, $\left({\varphi}_{h}=0\right)$. (

**b**) Variation of velocity profile when $M=30,R{e}_{\beta}=30$, $\text{}\left({\varphi}_{h}=0.01\right)$.

**Figure 8.**(

**a**) Variation of vortex profile $t$ for Pure fluid $\left({\varphi}_{h}=0\right)$ when $R{e}_{\beta}=10,M=5$. (

**b**) Variation of vortex profile $t$ for hybrid fluid $\left({\varphi}_{h}=0.01\right)$ when $R{e}_{\beta}=10,M=5$.

**Figure 9.**(

**a**) Variation of a velocity profile for various values of ${R}^{*}$ when $\left({\varphi}_{h}=0\right)$. (

**b**) Variation of a velocity profile for various values of ${R}^{*}$ when $\left({\varphi}_{h}=0.01\right)$.

**Figure 10.**(

**a**) Variation of temperature profile for various values of pressure gradient $A$. (

**b**) Variation of temperature profile for various values of Hartmann number $M$. (

**c**) Variation of temperature profile for various values of nanoparticles ${\varphi}_{h}$.

**Figure 11.**(

**a**) Streamlines for pure fluid $\left({\varphi}_{h}=0\right)$. (

**b**) Streamlines for Ag-Au/blood hybrid fluid for $\left({\varphi}_{h}=0.01\right)$.

**Table 1.**Thermo-physical features of hybrid nanofluid [38].

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

$\mathrm{Viscosity}\left(\mu \right)$ | ${\mu}_{hnf}={\mu}_{f}{(1-{\varphi}_{A})}^{-2.5}{(1-{\varphi}_{B})}^{-2.5}$ |

$\mathrm{Density}\text{}\left(\rho \right)$ | ${\rho}_{hnf}$$=[\left(1-{\varphi}_{B}\right)\left\{\left(1-{\varphi}_{A}\right){\rho}_{f}+{\varphi}_{A}{\rho}_{{p}_{1}}\right\}]+{\varphi}_{B}{\rho}_{{p}_{2}}$ |

$\mathrm{Heat}\text{}\mathrm{capacity}\text{}\left(\rho {C}_{p}\right)$ | ${(\rho {C}_{p})}_{hnf}=[\left(1-{\varphi}_{B}\right)\{\left(1-{\varphi}_{A}\right){(\rho {C}_{p})}_{f}+{\varphi}_{A}{(\rho {C}_{p})}_{{p}_{1}}\}]+{\varphi}_{B}{(\rho {C}_{p})}_{{p}_{2}}$ |

$\mathrm{Thermal}\text{}\mathrm{conductivity}\text{}\left(\kappa \right)$ | $\frac{{\kappa}_{hnf}}{{\kappa}_{gf}}=\left[\frac{\left({\kappa}_{{p}_{2}}+2{\kappa}_{gf}\right)-2{\varphi}_{B}\left({\kappa}_{gf}-{\kappa}_{{p}_{2}}\right)}{\left({\kappa}_{{p}_{2}}+2{\kappa}_{gf}\right)+{\varphi}_{B}\left({\kappa}_{gf}-{\kappa}_{{p}_{2}}\right)}\right],$ $\frac{{\kappa}_{gf}}{{\kappa}_{f}}=\left[\frac{\left({\kappa}_{{p}_{1}}+2{\kappa}_{f}\right)-2{\varphi}_{A}\left({\kappa}_{f}-{\kappa}_{{p}_{1}}\right)}{\left({\kappa}_{{p}_{1}}+2{\kappa}_{f}\right)+{\varphi}_{A}\left({\kappa}_{f}-{\kappa}_{{p}_{1}}\right)}\right]$ |

$\mathrm{Electrical}\text{}\mathrm{conductivity}\text{}\left(\sigma \right)$ | $\frac{{\sigma}_{hnf}}{{\sigma}_{f}}$$=\left[1+\frac{3\left(\frac{{\varphi}_{A}{\sigma}_{{p}_{1}}+{\varphi}_{B}{\sigma}_{{p}_{2}}}{{\sigma}_{f}}-\left({\varphi}_{A}+{\varphi}_{B}\right)\right)}{\left(\frac{{\varphi}_{A}{\sigma}_{{p}_{1}+{\varphi}_{2}{\sigma}_{{p}_{2}}}}{\left({\varphi}_{A}+{\varphi}_{B}\right){\sigma}_{f}}+2\right)-\left(\frac{{\varphi}_{A}{\sigma}_{{p}_{1}}+{\varphi}_{B}{\sigma}_{{p}_{2}}}{{\sigma}_{f}}-\left({\varphi}_{A}+{\varphi}_{B}\right)\right)}\right]$ |

Thermophysical | $\mathit{\rho}\left(\mathit{k}\mathit{g}/{\mathit{m}}^{3}\right)$ | ${\mathit{c}}_{\mathit{p}}\left(\mathit{J}/\mathit{k}\mathit{g}\mathit{K}\right)$ | $\mathit{\sigma}\left(\mathit{\Omega}\text{}\mathit{m}\right)$ | $\mathit{k}\left(\mathit{W}/\mathit{m}\mathit{K}\right)$ | $\mathit{P}\mathit{r}$ |
---|---|---|---|---|---|

Blood | 1063 | 3594 | 0.667 | 0.492 | 21 |

Silver (Ag) | 10,500 | 235 | 6.3 $\times {10}^{7}$ | 429 | - |

Gold (Au) | 19,282 | 129 | 4.1 $\times {10}^{6}$ | 310 | - |

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## Share and Cite

**MDPI and ACS Style**

Shahzad, F.; Jamshed, W.; Aslam, F.; Bashir, R.; Tag El Din, E.S.M.; Khalifa, H.A.E.-W.; Alanzi, A.M.
MHD Pulsatile Flow of Blood-Based Silver and Gold Nanoparticles between Two Concentric Cylinders. *Symmetry* **2022**, *14*, 2254.
https://doi.org/10.3390/sym14112254

**AMA Style**

Shahzad F, Jamshed W, Aslam F, Bashir R, Tag El Din ESM, Khalifa HAE-W, Alanzi AM.
MHD Pulsatile Flow of Blood-Based Silver and Gold Nanoparticles between Two Concentric Cylinders. *Symmetry*. 2022; 14(11):2254.
https://doi.org/10.3390/sym14112254

**Chicago/Turabian Style**

Shahzad, Faisal, Wasim Jamshed, Farheen Aslam, Rasheeda Bashir, El Sayed M. Tag El Din, Hamiden Abd El-Wahed Khalifa, and Agaeb Mahal Alanzi.
2022. "MHD Pulsatile Flow of Blood-Based Silver and Gold Nanoparticles between Two Concentric Cylinders" *Symmetry* 14, no. 11: 2254.
https://doi.org/10.3390/sym14112254