# Effect of Micropolar Fluid Properties on the Blood Flow in a Human Carotid Model

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Numerical Model

#### 2.2. Simulation Details

#### 2.3. Model Validation

## 3. Results

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

$\mathbf{u}$ | Fluid velocity | m/s |

${\mu}_{\nu}$ | dynamic viscosity | kg m/s |

${\kappa}_{\nu}$ | Rotational viscosity | m^{2}/s |

$\omega $ | angular velocity | 1/s |

$\gamma $ | Material coefficient | g cm/s |

j | Micro inertia | m^{2} |

m | Vortex viscosity | - |

N | Spin-gradient viscosity | - |

Re | Reynolds number | - |

$\rho $ | Density of micropolar fluid | kg/m^{3} |

${U}_{ref}$ | Cross sectional average velocity | m/s |

D | Diameter | m |

r | Radius | m |

$\widehat{\mathbf{u}}$ | Non dimensional linear velocity | - |

$\widehat{\omega}$ | Non dimensional angular velocity | - |

J | Non dimensional micro inertia | - |

p | Non dimensional pressure | - |

$\Delta U$ | Percentage difference between micropolar and Newtonian velocities | - |

$\Delta {\tau}_{w}$ | Percentage difference between micropolar and Newtonian shear stresses | - |

$\Delta \mathsf{\Omega}$ | Percentage difference between the base micropolar case $(m=0.1)$ and the cases with higher | - |

h | width | m |

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**Figure 1.**Side view of the computational geometry of carotid and indication of measurement locations (red lines).

**Figure 2.**Velocity in the outlet 1 of the carotid model under different number of computational cells for the representative case of $m=0.5$.

**Figure 4.**Numerical and analytical results for (

**a**) velocity, and (

**b**) microrotation for the case G = 2.70, A = 6.53, a = 1.96, h = 0.01 [32].

**Figure 5.**(

**a**) Idealized stenosis geometry and (

**b**) Comparison of the velocity profiles between the present study and Ref. [33].

**Figure 7.**Blood (

**a**) velocity, (

**b**) microrotation, (

**c**) velocity in the center of the tube, and (

**d**) velocity near common carotid’s walls.

**Figure 8.**Difference among the Newtonian fluid $(m=0)$ and the micropolar cases, for (

**a**) velocity (the first data point from each graph correspond to $r/R=0.05$), and (

**b**) shear stress.

**Figure 10.**$\Delta U$ in the center of the tube (black squares) and near the boundaries (red dots). Green and blue lines indicates fitting of data.

**Figure 11.**$\Delta U$ along different locations in the internal carotid artery: (

**a**) ${11}^{\prime}$, (

**b**) ${22}^{\prime}$, (

**c**) ${33}^{\prime}$, and (

**d**) ${44}^{\prime}$.

**Figure 12.**$\Delta \mathsf{\Omega}$ along the monitoring locations in the internal carotid artery: (

**a**) ${11}^{\prime}$, (

**b**) ${22}^{\prime}$, (

**c**) 33, and (

**d**) ${44}^{\prime}$.

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Karvelas, E.; Sofiadis, G.; Papathanasiou, T.; Sarris, I.
Effect of Micropolar Fluid Properties on the Blood Flow in a Human Carotid Model. *Fluids* **2020**, *5*, 125.
https://doi.org/10.3390/fluids5030125

**AMA Style**

Karvelas E, Sofiadis G, Papathanasiou T, Sarris I.
Effect of Micropolar Fluid Properties on the Blood Flow in a Human Carotid Model. *Fluids*. 2020; 5(3):125.
https://doi.org/10.3390/fluids5030125

**Chicago/Turabian Style**

Karvelas, Evangelos, Giorgos Sofiadis, Thanasis Papathanasiou, and Ioannis Sarris.
2020. "Effect of Micropolar Fluid Properties on the Blood Flow in a Human Carotid Model" *Fluids* 5, no. 3: 125.
https://doi.org/10.3390/fluids5030125