# Fluid Flow Characteristics of Healthy and Calcified Aortic Valves Using Three-Dimensional Lagrangian Coherent Structures Analysis

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Model Geometry

#### 2.2. Boundary Conditions

#### 2.3. Governing Equations in the Fluid and Solid Domains

^{−1}for the large arterial diameters; therefore, the blood could be modeled as a Newtonian fluid with a constant dynamic viscosity of 3.5 cP. For the shear strain rate beyond 50 s

^{−1}, the blood behaved as a homogeneous fluid with almost constant viscosity due to the high shear environment [5,45]. The mass density of the blood was used as 1056 kg/m

^{3}[8].

^{−5}for the solution convergence.

^{3}[9] and Poisson’s ratio of 0.45 [23] were employed for the healthy and calcified leaflets.

#### 2.4. FSI Coupling

#### 2.5. Finite-Time Lyapunov Exponent (FTLE) Analysis

^{−5}m in the horizontal and −0.01093 to 0.01133 m in the vertical directions to include both upstream and downstream in the domain. The element size for the computations was selected to be 0.7 × 10

^{−4}; therefore, a resolution of 388 × 318 elements was employed for the analysis.

## 3. Results

#### 3.1. Flow Streamlines and Pressure Contours

#### 3.2. FTLE Analysis Results

## 4. Discussion

## 5. Limitations

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) 3D aortic valve model. The grey surface is the cross-sectional plane in the longitudinal direction. The arrows show the direction of flow at the inlet and outlet. (

**b**) 2D view of the cross-sectional plane. Only the top half of the model is shown due to the symmetry. The dashed line represents the symmetry line. (

**c**) Boundary conditions shown on the cross-sectional plane. (

**d**) Transient inlet velocity profile of the blood flow. (

**e**) Mesh of the fluid domain. (

**f**) Mesh of the solid domain.

**Figure 2.**Velocity streamlines on the 3D aortic valve model. The configuration of the maximum valve opening for the healthy aortic valve is given at the bottom of the figure.

**Figure 3.**Pressure on the selected fluid particles for healthy, mildly calcified, and severely calcified aortic valves. The results are shown on the top half of the cross-sectional plane given in Figure 1. The flow was from right to left. The values presented are pressures relative to the valve outlet. The pressure was set to zero at the outlet boundary.

**Figure 4.**Backward finite-time Lyapunov exponent (FTLE) field plots of healthy (

**first column**), mildly calcified (

**second column**), and severely calcified (

**third column**) aortic valves.

**Figure 5.**Forward finite-time Lyapunov exponent (FTLE) field plots of healthy (

**first column**), mildly calcified (

**second column**), and severely calcified (

**third column**) aortic valves.

Healthy Aortic Valve | Mildly Calcified Aortic Valve | Severely Calcified Aortic Valve | |
---|---|---|---|

400th step (added for the first time) | 104,291 | 104,291 | 104,291 |

339th step (Total with readded particles) | 261,418 | 398,350 | 434,337 |

330th step (remaining particles) | 153,900 | 285,259 | 325,046 |

329th step (total with readded particles) | 254,945 | 382,472 | 421,736 |

320th step (remaining particles) | 155,146 | 268,018 | 305,521 |

319th step (total with re-added particles) | 255,743 | 367,599 | 403,394 |

310th step (remaining particles) | 165,389 | 276,347 | 309,397 |

309th step (total with readded particles) | 263,070 | 376,494 | 407977 |

300th step (remaining particles) | 185,322 | 295,500 | 323,787 |

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**MDPI and ACS Style**

Mutlu, O.; Salman, H.E.; Yalcin, H.C.; Olcay, A.B.
Fluid Flow Characteristics of Healthy and Calcified Aortic Valves Using Three-Dimensional Lagrangian Coherent Structures Analysis. *Fluids* **2021**, *6*, 203.
https://doi.org/10.3390/fluids6060203

**AMA Style**

Mutlu O, Salman HE, Yalcin HC, Olcay AB.
Fluid Flow Characteristics of Healthy and Calcified Aortic Valves Using Three-Dimensional Lagrangian Coherent Structures Analysis. *Fluids*. 2021; 6(6):203.
https://doi.org/10.3390/fluids6060203

**Chicago/Turabian Style**

Mutlu, Onur, Huseyin Enes Salman, Huseyin Cagatay Yalcin, and Ali Bahadir Olcay.
2021. "Fluid Flow Characteristics of Healthy and Calcified Aortic Valves Using Three-Dimensional Lagrangian Coherent Structures Analysis" *Fluids* 6, no. 6: 203.
https://doi.org/10.3390/fluids6060203