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Heat Transport Exploration for Hybrid Nanoparticle (Cu, Fe_{3}O_{4})—Based Blood Flow via Tapered Complex Wavy Curved Channel with Slip Features

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## Abstract

**:**

_{3}O

_{4}) and copper (Cu) are used in the analysis. The fundamental laws of mass, momentum and energy give rise the system of nonlinear coupled partial differential equations which are normalized using the variables, and the resulting set of governing relations are simplified in view of a smaller Reynolds model approach. The numerical simulations are performed using the computational software Mathematica’s built-in ND scheme. It is noted that the velocity of the blood is abated by the nanoparticles’ concentration and assisted in the non-uniform channel core. Furthermore, the nanoparticles’ volume fraction and the dimensionless curvature of the channel reduce the temperature profile.

## 1. Introduction

- ➢
- The presentation of a mathematical model for the peristaltic transport of Casson fluid with the interaction of hybrid nanofluid containing the ferro nanoparticles and copper nanomaterials in a curved channel.
- ➢
- The role of slip effects and Hall current is also observed.
- ➢
- The highly nonlinear system of the obtained model is numerically solved with the ND-Solver.
- ➢
- The physical thermal impact of hybrid nanoparticles is focused to control the blood flow properties. The current investigation presents novel applications for human blood flow, thermal systems, various engineering processes, extrusion systems, human endoscopy, the control of heating phenomena, chemical processes and biomedical applications [37,38,39,40,41,42].

## 2. Mathematical Modeling of Hybrid Nanofluid

## 3. Solution of the Problem

## 4. Discussion

#### 4.1. Axial Velocity Profile

_{3}O

_{4}nanoparticles for both $q=0.1$ and $q=-5.0$. This indicates that due to the interaction of the nanoparticles, which control the thermal transport near the micro channel, there is a decrease in velocity in the center of the channel, and due to the slip features at the walls, the velocity rises. Figure 2c is plotted to analyze the change in the axial velocity component due to increasing fluctuation of the Hartmann number. The reduced results are observed in the upper regime when enhancing values are being assigned to the Hartmann number. However, a rising velocity in the lower regime is observed for the same Hartmann constant variation. The change in the Hartmann number against the axial velocity is described via Figure 2d. Reduced velocity in the core of the channel is observed. As a result, a depressive velocity trend occurs and the velocity fluctuates toward the upper regime of the channel. The reduction in velocity flow is due to the application of magnetic force which results in a resistive Lorentz force. The control of velocity due to ${m}_{1}$ has been noted in Figure 2e. The enhanced change in hybrid nanofluid movement against ${m}_{1}$ in the core regime is noticed. Moreover, the channel diameter begins to decrease when larger measurements are assigned to ${m}_{1}$. Figure 2f presents important observations of the change in velocity caused by the impact of curvature $k.$ The results are further observed for an infinite range of curvature. The response of velocity is similar for non-similar values of flow rate $q$.

#### 4.2. Temperature Profile

#### 4.3. Trapping Phenomena

#### 4.4. Pressure Rise

#### 4.5. Heat Transfer Coefficient and Wall Shear Force

## 5. Conclusions

- ❖
- The declining change in velocity associated with the larger Lorentz force is exhibited.
- ❖
- Upon enhancing the Hall parameter, the role of magnetic force is controlled.
- ❖
- The change in the non-uniformity of a curved surface and the increment in the rate of velocity are observed.
- ❖
- A reduction in temperature results from a larger nanoparticle volume fraction.
- ❖
- The heat transfer is increased for the curved configuration while lower results are noted for the planner channel.
- ❖
- In both regimes of the symmetric channel, the disappearance of bolus trapping due to magnetic force is noted.
- ❖
- A progressive skin friction for hybrid nanofluid at low scales is given for Lorentz force.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**(

**a**–

**f**): (

**a**) Change in u

_{2}(η) for ${\alpha}_{1}$, (

**b**) change in u

_{2}(η) for ${\alpha}_{2}$, (

**c**) change in u

_{2}(η) for $m$, (

**d**) change in u

_{2}(η) for $Ha$, (

**e**) change in u

_{2}(η) for m

_{1}, (

**f**) change in u

_{2}(η) for m

_{1}/.

**Figure 3.**(

**a**–

**f**): (

**a**) Change in $\theta $ for ${\gamma}_{1}$ (

**b**) change in $\theta $ for ${\beta}_{1}$, (

**c**) change in $\theta $ for $m$, (

**d**) change in $\theta $ for $Ha$, (

**e**) change in $\theta $ for ${m}_{1}$ and (

**f**) change in $\theta $ for $k$.

**Figure 4.**Stream lines for (

**a**) ${\alpha}_{1}=0.0$ and (

**b**) ${\alpha}_{1}=0.1$ the other parameters are $Ha=3.0,\text{}k=3.0,\text{}m=2.0,\text{}q=-0.1,{\gamma}_{1}=1.0,{\beta}_{1}=0.1$.

**Figure 5.**Stream lines for (

**a**) $Ha=0.0$ and (

**b**) $Ha=3.0$ the other parameters are ${\alpha}_{1}=0.05,\text{}k=3.0,\text{}m=2.0,\text{}q=-0.1$ and ${\alpha}_{2}=0.1$.

**Figure 6.**Stream lines for (

**a**) $m=0.0$ and (

**b**) $m=2.0$ the other parameters are ${\alpha}_{1}=0.05,\text{}k=3.0,\text{}Ha=3.0,\text{}q=-0.1$ and $\text{}{\alpha}_{2}=0.1$.

**Figure 7.**Stream lines for (

**a**) ${m}_{1}=0.0$ and (

**b**) ${m}_{1}=0.1\text{}$ where the other parameters are ${\alpha}_{2}=0.1,\text{}k=3.0,\text{}Ha=3.0,\text{}q=-0.1,m=1.0$ and ${\alpha}_{1}=0.05$.

**Figure 8.**(

**a**–

**d**): (

**a**) Change in pressure due to ${\alpha}_{1},$ (

**b**) change in pressure due to ${\alpha}_{2},$ (

**c**) change in pressure due to $Ha$ (

**d**) change in pressure due to $m$.

**Table 1.**Hybrid nanofluid different consequences with mathematical forms [5].

Density | ${\rho}_{hnf}=\left(1-{\alpha}_{1}-{\alpha}_{2}\right){\rho}_{f}+{\alpha}_{1}{\rho}_{{s}_{1}}+{\alpha}_{2}{\rho}_{{s}_{2}}$. |

Viscosity | ${\mu}_{hnf}=\frac{{\mu}_{f}}{{\left(1-{\alpha}_{1}-{\alpha}_{2}\right)}^{2.5}}$ |

Effective heat capacity | ${\left(\rho {C}_{p}\right)}_{hnf}=\left(1-{\alpha}_{1}-{\alpha}_{2}\right){\rho}_{f}{\left({C}_{p}\right)}_{f}+{\alpha}_{1}{\rho}_{{s}_{1}}{\left({C}_{p}\right)}_{{s}_{1}}+{\alpha}_{2}{\rho}_{{s}_{2}}{\left({C}_{p}\right)}_{{s}_{2}}$ |

Thermal conductivity | $\frac{{K}_{hnf}}{{K}_{f}}=\frac{\frac{{\alpha}_{1}{K}_{{s}_{1}}+{\alpha}_{2}{K}_{{s}_{2}}}{{\alpha}_{1}+{\alpha}_{2}}+2{K}_{f}-2{K}_{f}\left({\alpha}_{1}+{\alpha}_{2}\right)+2\left({\alpha}_{1}{K}_{{s}_{1}}+{\alpha}_{2}{K}_{{s}_{2}}\right)}{\frac{{\alpha}_{1}{K}_{{s}_{1}}+{\alpha}_{2}{K}_{{s}_{2}}}{{\alpha}_{1}+{\alpha}_{2}}+2{K}_{f}+K\left({\alpha}_{1}+{\alpha}_{2}\right)-\left({\alpha}_{1}{K}_{{s}_{1}}+{\alpha}_{2}{K}_{{s}_{2}}\right)}$ |

Electric conductivity | $\frac{{\sigma}_{hnf}}{{\sigma}_{bf}}=\frac{{\sigma}_{{s}_{1}}+2{\sigma}_{bf}-2{\alpha}_{2}\left({\sigma}_{bf}-{\sigma}_{{s}_{1}}\right)}{{\sigma}_{{s}_{1}}+2{\sigma}_{bf}+{\alpha}_{2}\left({\sigma}_{bf}-{\sigma}_{{s}_{1}}\right)}$ $\mathrm{where}\text{}{\sigma}_{bf}=\frac{{\sigma}_{{s}_{2}}+2{\sigma}_{f}-2{\alpha}_{1}\left({\sigma}_{f}-{\sigma}_{{s}_{2}}\right)}{{\sigma}_{{s}_{2}}+2{\sigma}_{f}+{\alpha}_{1}\left({\sigma}_{f}-{\sigma}_{{s}_{2}}\right)}{\sigma}_{f}$ |

Material | Cu | Blood | Fe_{3}O_{4} |
---|---|---|---|

$\rho \left(\mathrm{kg}/{\mathrm{m}}^{3}\right)$ | 8933 | 1063 | 5200 |

$K\left(\mathrm{W}/\mathrm{mk}\right)$ | 401 | $0.492$ | 6 |

$C\left(\mathrm{J}/\mathrm{kgK}\right)$ | 385 | 3594 | 670 |

$\sigma \left(\mathrm{S}/\mathrm{m}\right)$ | $5.96\times {10}^{7}$ | $0.8$ | 25,000 |

$\mathit{H}\mathit{a}$ | $\mathit{m}$ | $\mathit{B}\mathit{r}$ | $\mathit{k}$ | ${\mathit{\beta}}_{1}=0.0$ | ${\mathit{\beta}}_{1}=0.1$ | ||
---|---|---|---|---|---|---|---|

Cu Nanofluid | Hybrid Nanofluid | Cu Nanofluid | Hybrid Nanofluid | ||||

0.0 | 1.0 | 1.0 | 3.0 | $1.946431$ | $1.803762$ | $1.313924$ | $1.305188$ |

1.0 | $1.947308$ | $1.804217$ | $1.314652$ | $1.305564$ | |||

2.0 | $1.949939$ | $1.805583$ | $1.316832$ | $1.306690$ | |||

0.0 | $1.952836$ | $1.807087$ | $1.319297$ | $1.307965$ | |||

1.0 | $1.949939$ | $1.805583$ | $1.316832$ | $1.306690$ | |||

2.0 | $1.947920$ | $1.804535$ | $1.315149$ | $1.305820$ | |||

0.0 | $1.283249$ | $1.283249$ | $1.283249$ | $1.283249$ | |||

2.0 | $2.56784366$ | $2.28255142$ | $1.30959600$ | $1.290532$ | |||

4.0 | $3.85243826$ | $3.28185379$ | $1.33594274$ | $1.29781534$ | |||

2.5 | $1.95869391$ | $1.81494916$ | $1.32416001$ | $1.31492136$ | |||

5.0 | $1.92554637$ | $1.78290025$ | $1.29642262$ | $1.28689076$ | |||

$\infty $ | $1.88993547$ | $1.74844033$ | $1.26657588$ | $1.25666942$ |

$\mathit{H}\mathit{a}$ | $\mathit{m}$ | $\mathit{k}$ | ${\mathit{\beta}}_{1}=0.0$ | ${\mathit{\beta}}_{1}=0.1$ | ||
---|---|---|---|---|---|---|

Cu Nanofluid | Hybrid Nanofluid | Cu Nanofluid | Hybrid Nanofluid | |||

0.0 | 1.0 | 3.0 | $1.64793574$ | $1.647935744$ | $0.23902529$ | $0.144332354$ |

1.0 | $1.64825439$ | $1.64814824$ | $0.23902712$ | $0.14433235$ | ||

2.0 | $1.64921003$ | $1.64878559$ | $0.23903264$ | $0.14433235$ | ||

0.0 | $1.65017632$ | $1.64943017$ | $0.23899158$ | $0.14431076$ | ||

1.0 | $1.64921003$ | $1.64878559$ | $0.23903264$ | $0.14433236$ | ||

2.0 | $1.64849017$ | $1.64830548$ | $0.23903597$ | $0.14433582$ | ||

2.5 | $1.65630099$ | $1.65612141$ | $0.24164339$ | $0.14724496$ | ||

5.0 | $1.63224142$ | $1.63204609$ | $0.23360486$ | $0.13824367$ | ||

$\infty $ | $1.60630293$ | $1.60609039$ | $0.22491558$ | $0.12840275$ |

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**MDPI and ACS Style**

Abbasi, A.; Farooq, W.; Tag-ElDin, E.S.M.; Khan, S.U.; Khan, M.I.; Guedri, K.; Elattar, S.; Waqas, M.; Galal, A.M.
Heat Transport Exploration for Hybrid Nanoparticle (Cu, Fe_{3}O_{4})—Based Blood Flow via Tapered Complex Wavy Curved Channel with Slip Features. *Micromachines* **2022**, *13*, 1415.
https://doi.org/10.3390/mi13091415

**AMA Style**

Abbasi A, Farooq W, Tag-ElDin ESM, Khan SU, Khan MI, Guedri K, Elattar S, Waqas M, Galal AM.
Heat Transport Exploration for Hybrid Nanoparticle (Cu, Fe_{3}O_{4})—Based Blood Flow via Tapered Complex Wavy Curved Channel with Slip Features. *Micromachines*. 2022; 13(9):1415.
https://doi.org/10.3390/mi13091415

**Chicago/Turabian Style**

Abbasi, A., W. Farooq, El Sayed Mohamed Tag-ElDin, Sami Ullah Khan, M. Ijaz Khan, Kamel Guedri, Samia Elattar, M. Waqas, and Ahmed M. Galal.
2022. "Heat Transport Exploration for Hybrid Nanoparticle (Cu, Fe_{3}O_{4})—Based Blood Flow via Tapered Complex Wavy Curved Channel with Slip Features" *Micromachines* 13, no. 9: 1415.
https://doi.org/10.3390/mi13091415