# A Model for the Generalised Dispersion of Synovial Fluids on Nutritional Transport with Joint Impacts of Electric and Magnetic Field

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*Math. Comput. Appl.*

**2023**,

*28*(1), 3; https://doi.org/10.3390/mca28010003 (registering DOI)

## Abstract

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## 1. Introduction

## 2. Formulation of the Problem

- A 2D, electrically conducting, viscous and incompressible synovial fluid is considered.
- Flow of fluid is laminar and steady.
- A constant magnetic field of strength ${B}_{0}$ is applied in the transverse direction.

## 3. Method of Solution

#### 3.1. Velocity Distribution

#### 3.2. Generalized Dispersion Model (GDM)

## 4. Results and Discussion

## 5. Conclusions

- Dispersion is accelerated by electromagnetic fields and other physical factors.
- In contrast to electromagnetic fields and other physical factors, the mean concentration drops as axial distance and time increase.
- Cells in the centre receive more nutrients than those in the periphery.
- The dispersion mechanism formula is used by orthopaedic surgeons to assess how well joints function.

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

$(\widehat{u},\widehat{v})$ | Horizontal and normal components of the fluid velocity |

$(\widehat{x},\widehat{y})$ | Cartesian coordinates |

$\widehat{p}$ | Pressure |

$\tilde{u}$ | average velocity |

${B}_{0}$ | Magnetic induction |

k | Permeability of porous medium |

$\widehat{{E}_{x}}$ | x component of electric field |

$\widehat{C}$ | Species concentration |

$\widehat{{C}_{0}}$ | Initial species concentration |

$\widehat{D}$ | Diffusion coefficient |

${K}_{k}$ | Dispersion coefficient |

M | Hartmann number |

$We$ | Electric number |

$Re$ | Reynolds number |

$Pe$ | Peclet number |

$Da$ | Darcy number |

Greek Symbols | |

$\mu $ | Dynamic viscosity |

$\widehat{\eta}$ | Kinematic viscosity |

$\alpha $ | Slip parameter |

${\sigma}_{0}$ | Electrical conductivity |

$\sigma $ | Porous parameter |

$\u03f5$ | Viscoelastic parameter |

$\varphi $ | Concentration |

${\varphi}_{m}$ | Mean concentration |

${\rho}_{e}$ | Dimensionless charge density |

$\tau $ | Dimensionless time |

$\xi $ | Dimensionless axial distance |

## References

- Alshehri, M.; Sharma, S.K. Computational model for the generalised dispersion of synovial fluid. Int. J. Adv. Comput. Sci. Appl.
**2017**, 8, 134–138. [Google Scholar] [CrossRef] [Green Version] - Zahn, M.; Shenton, K.E. Magnetic Fluids Bibliography. IEEE Trans.
**1980**, 16, 387–415. [Google Scholar] [CrossRef] - Akbar, N.S.; Nadeem, S.; Noor, N.F.M. Free convective MHD peristaltic flow of a Jeffrey nanofluid with convective surface boundary condition: A biomedicine–Nano model. Curr. Nanosci.
**2014**, 10, 432–440. [Google Scholar] [CrossRef] - Faghiri, S.; Akbari, S.; Shafii, M.B.; Hosseinzadeh, K. Hydrothermal analysis of non-Newtonian fluid flow (blood) through the circular tube under prescribed non-uniform wall heat flux. Theor. Appl. Mech. Lett.
**2022**, 12, 100360. [Google Scholar] [CrossRef] - Gulzar, M.M.; Aslam, A.; Waqas, M.; Javed, M.A.; Hosseinzadeh, K. A nonlinear mathematical analysis for magneto-hyperbolic-tangent liquid featuring simultaneous aspects of magnetic field, heat source and thermal stratification. Appl. Nanosci.
**2020**, 10, 4513–4518. [Google Scholar] [CrossRef] - Khan, U.; Zaib, A.; Shah, Z.; Baleanu, D.; Sherif, E.S.M. Impact of magnetic field on boundary-layer flow of Sisko liquid comprising nanomaterials migration through radially shrinking/stretching surface with zero mass flux. J. Mater. Res. Technol.
**2020**, 9, 3699–3709. [Google Scholar] [CrossRef] - Attar, M.A.; Roshani, M.; Hosseinzadeh, K.; Ganji, D.D. Analytical solution of fractional differential equations by Akbari–Ganji’s method. Partial. Differ. Equations Appl. Math.
**2022**, 6, 100450. [Google Scholar] [CrossRef] - Hossain, M.S.; Fayz-Al-Asad, M.; Mallik, M.S.I.; Yavuz, M.; Alim, M.A.; Khairul Basher, K.M. Numerical Study of the Effect of a Heated Cylinder on Natural Convection in a Square Cavity in the Presence of a Magnetic Field. Math. Comput. Appl.
**2022**, 27, 58. [Google Scholar] [CrossRef] - Bhuvaneswari, M.; Sivasankaran, S.; Kim, Y.J. Magnetoconvection in a square enclosure with sinusoidal temperature distributions on both side walls. Numer. Heat Transf. Part A Appl.
**2011**, 59, 167–184. [Google Scholar] [CrossRef] - Sivasankaran, S.; Malleswaran, A.; Bhuvaneswari, M.; Balan, P. Hydro-magnetic mixed convection in a lid-driven cavity with partially thermally active walls. Sci. Iran.
**2017**, 24, 153–163. [Google Scholar] [CrossRef] - Narrein, K.; Sivanandam, S.; Ganesan, P. Influence of transverse magnetic field on microchannel heat sink performance. J. Appl. Fluid Mech.
**2016**, 9, 3159–3166. [Google Scholar] [CrossRef] - Sivasankaran, S.; Narrein, K. Influence of geometry and magnetic field on convective flow of nanofluids in trapezoidal microchannel heat sink. Iran. J. Sci. Technol. Trans. Mech. Eng.
**2020**, 44, 373–382. [Google Scholar] [CrossRef] - Sivasankaran, S.; Bhuvaneswari, M.; Alzahrani, A.K. Numerical study on influence of magnetic field and discrete heating on free convection in a porous container. Sci. Iran.
**2022**. [Google Scholar] [CrossRef] - Sivasankaran, S.; Lee, J.; Bhuvaneswari, M. Effect of a partition on hydro-magnetic convection in an enclosure. Arab. J. Sci. Eng.
**2011**, 36, 1393–1406. [Google Scholar] [CrossRef] - Bindhu, R.; Sai SundaraKrishnan, G.; Sivasankaran, S.; Bhuvaneswari, M. Magneto-convection of water near its maximum density in a cavity with partially thermally active walls. Energy Environ.
**2019**, 30, 833–853. [Google Scholar] [CrossRef] - Bhuvaneswari, M.; Sivasankaran, S.; Karthikeyan, S.; Rajan, S. Stratification and Cross Diffusion Effects on Magneto-Convection Stagnation-Point Flow in a Porous Medium with Chemical Reaction, Radiation, and Slip Effects. In Applied Mathematics and Scientific Computing; Birkhäuser: Cham, Switzerland, 2019; pp. 245–253. [Google Scholar] [CrossRef]
- Rashad, A.M.; Sivasankaran, S.; Mansour, M.A.; Bhuvaneswari, M. Magneto-convection of nanofluids in a lid-driven trapezoidal cavity with internal heat generation and discrete heating. Numer. Heat Transf. Part A Appl.
**2017**, 71, 1223–1234. [Google Scholar] [CrossRef] - Niranjan, H.; Sivasankaran, S.; Bhuvaneswari, M. Analytical and numerical study on magnetoconvection stagnation-point flow in a porous medium with chemical reaction, radiation, and slip effects. Math. Probl. Eng.
**2016**, 2016, 4017076. [Google Scholar] [CrossRef] [Green Version] - Akbar, N.S.; Maraj, E.N.; Noor, N.F.M.; Habib, M.B. Exact solutions of an unsteady thermal conductive pressure driven peristaltic transport with temperature-dependent nanofluid viscosity. Case Stud. Therm. Eng.
**2022**, 35, 102124. [Google Scholar] [CrossRef] - Khan, U.; Zaib, A.; Khan, I.; Nisar, K.S. Insight into the dynamics of transient blood conveying gold nanoparticles when entropy generation and Lorentz force are significant. Int. Commun. Heat Mass Transf.
**2021**, 127, 105415. [Google Scholar] [CrossRef] - Kune, R.; Naik, H.S.; Reddy, B.S.; Chesneau, C. Role of Nanoparticles and Heat Source/Sink on MHD Flow of Cu-H2O Nanofluid Flow Past a Vertical Plate with Soret and Dufour Effects. Math. Comput. Appl.
**2022**, 27, 102. [Google Scholar] [CrossRef] - Sivasankaran, S.; Ananthan, S.S.; Abdul Hakeem, A.K. Mixed convection in a lid-driven cavity with sinusoidal boundary temperature at the bottom wall in the presence of magnetic field. Sci. Iran.
**2016**, 23, 1027–1036. [Google Scholar] [CrossRef] [Green Version] - Sivasankaran, S.; Malleswaran, A.; Lee, J.; Sundar, P. Hydro-magnetic combined convection in a lid-driven cavity with sinusoidal boundary conditions on both sidewalls. Int. J. Heat Mass Transf.
**2011**, 54, 512–525. [Google Scholar] [CrossRef] - Sivasankaran, S.; Bhuvaneswari, M.; Kim, Y.J.; Ho, C.J.; Pan, K.L. Numerical study on magneto-convection of cold water in an open cavity with variable fluid properties. Int. J. Heat Fluid Flow
**2011**, 32, 932–942. [Google Scholar] [CrossRef] - Sivasankaran, S.; Bhuvaneswari, M. Effect of thermally active zones and direction of magnetic field on hydromagnetic convection in an enclosure. Therm. Sci.
**2011**, 15, 367–382. [Google Scholar] [CrossRef] - Tandon, P.N.; Chaurasia, A.; Jain, V.K.; Gupta, T. Application of magnetic fields to synovial joints. Comput. Math. Appl.
**1991**, 22, 33–45. [Google Scholar] [CrossRef] [Green Version] - Rudraiah, N.; Kasiviswanathan, S.R.; Kaloni, P.N. Generalized dispersion in a synovial fluid of human joints. Biorheology
**1991**, 28, 207–219. [Google Scholar] [CrossRef] - Gill, W.N.; Sankarasubramanian, R. Exact analysis of unsteady convective diffusion. Proc. R. Soc. Lond. A Math. Phys. Sci.
**1970**, 316, 341–350. [Google Scholar] [CrossRef] - Ng, C.O.; Rudraiah, N.; Naiaraj, C.; Nagaraj, H.N. Electrohydrodynamic dispersion of macromolecular components in nanostructured biological bearing. J. Energy Heat Mass Transf.
**2005**, 27, 39–64. [Google Scholar] - Khan, A.A.; Farooq, A.; Vafai, K. Impact of induced magnetic field on synovial fluid with peristaltic flow in an asymmetric channel. J. Magn. Magn. Mater.
**2018**, 446, 54–67. [Google Scholar] [CrossRef] - Nagaraj, C.; Dinesh, P.A.; Kalavathi, G.K. Combined effects of electric field and magnetic field on electro hydrodynamic dispersion of macromolecular components in biological bearing. Defect Diffus. Forum
**2018**, 388, 361–377. [Google Scholar] [CrossRef] - Redeker, J.I.; Schmitt, B.; Grigull, N.P.; Braun, C.; Büttner, A.; Jansson, V.; Mayer-Wagner, S. Effect of electromagnetic fields on human osteoarthritic and non-osteoarthritic chondrocytes. BMC Complement. Altern. Med.
**2017**, 17, 1–8. [Google Scholar] [CrossRef] [Green Version] - Racine, J. Effects of electromagnetic fields on articular cells and osteoarthritis. Orthop. Proc.
**2018**, 100 (Supp. 15), 72. [Google Scholar] [CrossRef] - Ramakrishnan, K.; Swetha, N.N. Influence of porous parameter and thickness of the porous plate on the flow of synovial fluid in Human Joints. Int. J. Pure Appl. Math.
**2018**, 119, 221–229. [Google Scholar] - Beretta, G.; Mastorgio, A.F.; Pedrali, L.; Saponaro, S.; Sezenna, E. The effects of electric, magnetic and electromagnetic fields on microorganisms in the perspective of bioremediation. Rev. Environ. Sci. Bio/Technol.
**2019**, 18, 29–75. [Google Scholar] [CrossRef] [Green Version] - VijayaKumar, R.; Ratchagar, N.P. Mathematical Modeling of Synovial Joints with Chemical Reaction. J. Phys. Conf. Ser.
**2021**, 1724, 012051. [Google Scholar] [CrossRef] - Tandon, P.N.; Nirmala, P.; Pal, T.S.; Agarwal, R. Rheological study of lubricant gelling in synovial joints during articulation. Appl. Math. Model.
**1988**, 12, 72–77. [Google Scholar] [CrossRef] - Bali, R.; Shukla, A.K. Rheological effects of synovial fluid on nutritional transport. Tribol. Lett.
**2001**, 9, 233–239. [Google Scholar] [CrossRef] - Beavers, G.S.; Joseph, D.D. Boundary conditions at a naturally permeable wall. J. Fluid Mech.
**1967**, 30, 197–207. [Google Scholar] [CrossRef] - Rao K., S. Introduction to Partial Differential Equations, 3rd ed.; Prentice-Hall of India: New Delhi, India, 2010. [Google Scholar]

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**MDPI and ACS Style**

Kumar, B.R.; Vijayakumar, R.; Rani, A.J.
A Model for the Generalised Dispersion of Synovial Fluids on Nutritional Transport with Joint Impacts of Electric and Magnetic Field. *Math. Comput. Appl.* **2023**, *28*, 3.
https://doi.org/10.3390/mca28010003

**AMA Style**

Kumar BR, Vijayakumar R, Rani AJ.
A Model for the Generalised Dispersion of Synovial Fluids on Nutritional Transport with Joint Impacts of Electric and Magnetic Field. *Mathematical and Computational Applications*. 2023; 28(1):3.
https://doi.org/10.3390/mca28010003

**Chicago/Turabian Style**

Kumar, B. Rushi, R. Vijayakumar, and A. Jancy Rani.
2023. "A Model for the Generalised Dispersion of Synovial Fluids on Nutritional Transport with Joint Impacts of Electric and Magnetic Field" *Mathematical and Computational Applications* 28, no. 1: 3.
https://doi.org/10.3390/mca28010003