# Effect of Significant Parameters on Squeeze Film Characteristics in Pathological Synovial Joints

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## Abstract

**:**

## 1. Introduction

## 2. Mathematical Model

#### 2.1. Governing Equations

**Region I:**- Fluid Film Region $0\le y\le h$$$\mu \frac{{\partial}^{2}{u}_{1}}{\partial {y}^{2}}-\frac{\partial p}{\partial x}=0$$$$\mu \frac{{\partial}^{2}{v}_{1}}{\partial {y}^{2}}-\frac{\partial p}{\partial z}=0$$$$\frac{\partial p}{\partial y}=0$$

**Region II:**- Cartilage Region $h\le y\le h+H$

#### 2.2. Boundary and Matching Conditions

#### 2.3. Solution of the Problem

## 3. Numerical Results and Discussions

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

${u}_{1,}{v}_{1}$ | Velocity of Synovial Fluid in Fluid Film |

${u}_{2,}{v}_{2}$ | Velocity of Synovial Fluid in Cartilage |

${V}_{x,}{V}_{z}$ | Velocity of Synovial Fluid at Interface |

${\phi}_{1,}{\phi}_{2}$ | Permeability of Articular Cartilage |

${Q}_{x,}{Q}_{z}$ | Synovial Fluid Flux |

$x,y,z$ | Cartesian Co-ordinate |

$\overline{p}\left(z\right)$ | Dimensionless Pressure |

$\overline{{W}_{z}}$ | Dimensionless Load Capacity |

$\overline{{T}_{z}}$ | Dimensionless Squeeze Time |

$h$ | Fluid Film Thickness |

$H$ | Articular Cartilage Thickness |

$w$ | Velocity of Synovial Fluid along Film Thickness |

${V}_{s}$ | Squeeze Velocity |

${L}_{z}$ | Width of Articular Cartilage |

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**Figure 5.**Pressure Variation with Axial Distance for Different Values of Articular Cartilage Permeability.

**Figure 8.**Variation of Load Capacity with Fluid Film Thickness for Different Values of Squeeze Velocity.

**Figure 9.**Variation of Load Capacity with Fluid Film Thickness for Different Values of Synovial Fluid Viscosity.

**Figure 10.**Variation of Load Capacity with Fluid Film Thickness for Different Values of Permeability.

**Figure 11.**Variation of Squeeze Time with Articular Cartilage Length for Different Values Synovial Fluid Viscosity.

**Figure 12.**Variation of Squeeze Time with Articular Cartilage Thickness for Different Values Synovial Fluid Viscosity.

Parameter | Values |
---|---|

Viscosity ($\mu $) | 1.0–3.5 poise |

Apparent Viscosity (${\mu}^{\prime}$) | 0.5–2.0 poise |

Cartilage Thickness $\left(H\right)$ | 0.1–0.7 cm |

Fluid Film Thickness $\left(h\right)$ | 0–5 × ${10}^{-3}$ cm |

Squeeze Velocity $\left({V}_{s}\right)$ | 0.1–0.6 cm/s |

Permeability in x-direction (${\phi}_{1})$ Permeability in z-direction $({\phi}_{2}$) | 1–8 × ${10}^{-16}$ cm/N-S 2–8 × ${10}^{-15}$ cm/N-S |

Cartilage Length $\left({L}_{x}\right)$ | 3–5 cm |

Cartilage Width (${L}_{z}$) | 2–3.5 cm |

Pressure Gradient $\left(\frac{\partial p}{\partial z}\right)$ | 0.02–0.774 Pas/cm |

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**MDPI and ACS Style**

Sadique, M.; Shah, S.R.; Sharma, S.K.; Islam, S.M.N.
Effect of Significant Parameters on Squeeze Film Characteristics in Pathological Synovial Joints. *Mathematics* **2023**, *11*, 1468.
https://doi.org/10.3390/math11061468

**AMA Style**

Sadique M, Shah SR, Sharma SK, Islam SMN.
Effect of Significant Parameters on Squeeze Film Characteristics in Pathological Synovial Joints. *Mathematics*. 2023; 11(6):1468.
https://doi.org/10.3390/math11061468

**Chicago/Turabian Style**

Sadique, Mo, Sapna Ratan Shah, Sunil Kumar Sharma, and Sardar M. N. Islam.
2023. "Effect of Significant Parameters on Squeeze Film Characteristics in Pathological Synovial Joints" *Mathematics* 11, no. 6: 1468.
https://doi.org/10.3390/math11061468