# A Model of Synovial Fluid with a Hyaluronic Acid Source: A Numerical Challenge

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## Abstract

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## 1. Introduction

## 2. Model

#### 2.1. Equations

#### 2.2. Viscosity Model

#### 2.3. Analysis of the Model: Steady State

#### 2.4. Stability Consideration

## 3. Results

## 4. Discussion about Numerics

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

c | concentration | $\mathrm{kg}/{\mathrm{m}}^{3}$ |

$\tilde{c}$ | dimensionless concentration | - |

D | diffusion coefficient | ${\mathrm{m}}^{2}/\mathrm{s}$ |

h | half distance between the plates | m |

K | stress | $\mathrm{kg}/\mathrm{m}/{\mathrm{s}}^{2}$ |

$\tilde{K}$ | dimensionless stress | - |

n | flow behavior index | - |

r | strength of the source term | 1/s |

$\tilde{r}$ | dimensionless strength of the source term | - |

t | time | s |

$\tilde{t}$ | dimensionless time | - |

u | velocity | m/s |

$\tilde{u}$ | dimensionless velocity | - |

${\mathrm{U}}_{\mathrm{w}}$ | wall velocity | m/s |

$\eta $ | viscosity | kg/m/s |

${\eta}_{0}$ | zero-shear viscosity | kg/m/s |

$\tilde{\eta}$ | dimensionless viscosity | - |

$\lambda $ | Cross time constant | s |

$\rho $ | density | $\mathrm{kg}/{\mathrm{m}}^{3}$ |

## Abbreviations

SF | Synovial fluid |

HA | Hyaluronic acid |

CSC | Chebyshev spectral collocation |

Re | Reynolds number |

Pe | Péclet number |

## Appendix A. Numerical Treatment of the Problem

#### Appendix A.1. Spatial Discretization

#### Appendix A.2. Temporal Discretization

## References

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**Figure 1.**Physical system: a synovial fluid is confined between two infinitely long parallel plates separated by a distance $2h$. The top plate moves at constant speed ${U}_{w}$.

**Figure 3.**Steady dimensionless concentration profiles, $\tilde{c}$ scaled with the concentration at the centerline, $\tilde{c}(\tilde{y}=0)$ for $\tilde{r}=[0,0.5,1.5,2.2,2.45]$. The values of $\tilde{c}(\tilde{y}=0)$ for the given $\tilde{r}$ are $[0.155,0.204,0.457,2.86,27.9]$.

**Figure 4.**Steady dimensionless velocity profiles, $\tilde{u}$ for $\tilde{r}=[0,0.5,1.5,2.2,2.45]$. As $\tilde{r}$ is increased, the velocity becomes flatter and flatter at the core. For $\tilde{r}=2.45$, the velocity profile exhibits almost a step shape.

**Figure 5.**Steady dimensionless viscosity profiles, $\tilde{\eta}$ scaled with the viscosity at the centerline, $\tilde{\eta}(\tilde{y}=0)$ for $\tilde{r}=[0,0.5,1.5,2.2,2.45]$. The values of $\tilde{\eta}(\tilde{y}=0)$ for the given $\tilde{r}$ are $[55.5,1.96\times {10}^{2},1.39\times {10}^{5},1.54\times {10}^{32},2.35\times {10}^{314}]$.

**Figure 7.**(

**a**) Steady values of the dimensionless stress $\tilde{K}$ as a function of $\tilde{r}$. (

**b**) Steady strain rate (blue), viscosity (orange), and stress (green) as functions of $\tilde{r}$. Note that the ordinate is in ${\mathrm{log}}_{10}$ base.

**Figure 8.**The minimum cut-off $N\left(\tilde{r}\right)$ for obtaining satisfactory numerical solutions to Equation (3) using double precision.

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**MDPI and ACS Style**

Ozan, S.C.; Labrosse, G.; Uguz, A.K. A Model of Synovial Fluid with a Hyaluronic Acid Source: A Numerical Challenge. *Fluids* **2021**, *6*, 152.
https://doi.org/10.3390/fluids6040152

**AMA Style**

Ozan SC, Labrosse G, Uguz AK. A Model of Synovial Fluid with a Hyaluronic Acid Source: A Numerical Challenge. *Fluids*. 2021; 6(4):152.
https://doi.org/10.3390/fluids6040152

**Chicago/Turabian Style**

Ozan, S. Canberk, Gérard Labrosse, and A. Kerem Uguz. 2021. "A Model of Synovial Fluid with a Hyaluronic Acid Source: A Numerical Challenge" *Fluids* 6, no. 4: 152.
https://doi.org/10.3390/fluids6040152