##
**Simulation of thermomagnetic convection in a cavity using the lattice Boltzmann model.**
*(English)*
Zbl 1217.76060

Summary: Thermomagnetic convection in a differentially heated square cavity with an infinitely long third dimension is numerically simulated using the single relaxation time lattice Boltzmann method (LBM). This problem is of considerable interest when dealing with cooling of microelectronic devices, in situations where natural convection does not meet the cooling requirements, and forced convection is not viable due to the difficulties associated with pumping a ferrofluid. Therefore, circulation is achieved by imposing a magnetic field, which is created and controlled by placing a dipole at the bottom of the enclosure. The magnitude of the magnetic force is controlled by changing the electrical current through the dipole. In this study, the effects of combined natural convection and magnetic convection, which is commonly known as “thermomagnetic convection”, are analysed in terms of the flow modes and heat transfer characteristics of a magnetic fluid.

### MSC:

76M28 | Particle methods and lattice-gas methods |

76W05 | Magnetohydrodynamics and electrohydrodynamics |

PDF
BibTeX
XML
Cite

\textit{M. Hadavand} and \textit{A. C. M. Sousa}, J. Appl. Math. 2011, Article ID 538637, 14 p. (2011; Zbl 1217.76060)

Full Text:
DOI

### References:

[1] | A. E. Gill, “The boundary-layer regime for convection in a rectangular cavity,” Journal of Fluid Mechanics, vol. 26, pp. 515-536, 1966. |

[2] | S. Ostrach, “Natural convection in enclosures,” Journal of Heat Transfer, vol. 110, no. 4, pp. 1175-1190, 1988. |

[3] | V. Mariani and I. Moura Belo, “Numerical studies of natural convection in a square cavity,” Thermal Engineering, vol. 5, pp. 68-72, 2006. |

[4] | X. Shi and J. M. Khodadadi, “Laminar natural convection heat transfer in a differentially heated square cavity due to a thin fin on the hot wall,” Journal of Heat Transfer, vol. 125, no. 4, pp. 624-634, 2003. |

[5] | S. Odenbach, “Recent progress in magnetic fluid research,” Journal of Physics Condensed Matter, vol. 16, no. 32, pp. R1135-R1150, 2004. |

[6] | K. Raj and R. Moskowitz, “Commercial applications of ferrofluids,” Journal of Magnetism and Magnetic Materials, vol. 85, no. 1-3, pp. 233-245, 1990. |

[7] | B. A. Finlayson, “Convective instability of ferromagnetic fluids,” Journal of Fluid Mechanics, vol. 40, no. 4, pp. 753-767, 1970. · Zbl 0191.56703 |

[8] | R. Ganguly, S. Sen, and I. K. Puri, “Thermomagnetic convection in a square enclosure using a line dipole,” Physics of Fluids, vol. 16, no. 7, pp. 2228-2236, 2004. · Zbl 1186.76190 |

[9] | H. Yamaguchi, I. Kobori, Y. Uehata, and K. Shimada, “Natural convection of magnetic fluid in a rectangular box,” Journal of Magnetism and Magnetic Materials, vol. 201, no. 1-3, pp. 264-267, 1999. |

[10] | N. M. Al-Najem, K. M. Khanafer, and M. M. El-Refaee, “Numerical study of laminar natural convection in tilted cavity with transverse magnetic field,” International Journal of Numerical Methods for Heat & Fluid Flow, vol. 8, pp. 651-672, 1998. · Zbl 0962.76580 |

[11] | M. S. Krakov and I. V. Nikiforov, “To the influence of uniform magnetic field on thermomagnetic convection in square cavity,” Journal of Magnetism and Magnetic Materials, vol. 252, no. 1-3, pp. 209-211, 2002. |

[12] | T. Sawada, H. Kikura, A. Saito, and T. Tanahashi, “Natural convection of a magnetic fluid in concentric horizontal annuli under nonuniform magnetic fields,” Experimental Thermal and Fluid Science, vol. 7, no. 3, pp. 212-220, 1993. |

[13] | H. Kikura, T. Sawada, and T. Tanahashi, “Natural convection of a magnetic fluid in a cubic enclosure,” Journal of Magnetism and Magnetic Materials, vol. 122, no. 1-3, pp. 315-318, 1993. |

[14] | K. Nakatsuka, B. Jeyadevan, S. Neveu, and H. Koganezawa, “The magnetic fluid for heat transfer applications,” Journal of Magnetism and Magnetic Materials, vol. 252, no. 1-3, pp. 360-362, 2002. |

[15] | S. M. Snyder, T. Cader, and B. A. Finlayson, “Finite element model of magnetoconvection of a ferrofluid,” Journal of Magnetism and Magnetic Materials, vol. 262, no. 2, pp. 269-279, 2003. |

[16] | M. Hadavand, A. Nabovati, and A. C. M. Sousa, “Numerical simulation of thermomagnetic convection in an enclosure using the Lattice Boltzmann Method,” in Proceedings of the 8th International Conference on Nanochannels, Micorchannels and Minichannels (FEDSM-ICNMM ’10), Montreal, Canada, 2010. · Zbl 1217.80090 |

[17] | X. Q. Wang and A. S. Mujumdar, “Heat transfer characteristics of nanofluids: a review,” International Journal of Thermal Sciences, vol. 46, no. 1, pp. 1-19, 2007. |

[18] | D. J. Griffiths, Introduction to Electrodynamics, Prentice Hall, 3rd edition, 2002. |

[19] | A. Mukhopadhyay, R. Ganguly, S. Sen, and I. K. Puri, “A scaling analysis to characterize thermomagnetic convection,” International Journal of Heat and Mass Transfer, vol. 48, no. 17, pp. 3485-3492, 2005. · Zbl 1189.76499 |

[20] | A. Nabovati, Pore level simulation of single and two phase flow in porous media using Lattice Boltzmann method, Ph.D. dissertation, University of New Brunswick, New Brunswick, NJ, USA, 2009. |

[21] | P. L. Bhatnagar, E. P. Gross, and M. Krook, “A model for collision processes in gases. I. Small amplitude processes in charged and neutral one-component systems,” Physical Review, vol. 94, no. 3, pp. 511-525, 1954. · Zbl 0055.23609 |

[22] | X. He, S. Chen, and G. D. Doolen, “A novel thermal model for the Lattice Boltzmann method in incompressible limit,” Journal of Computational Physics, vol. 146, no. 1, pp. 282-300, 1998. · Zbl 0919.76068 |

[23] | Y. Shi, T. S. Zhao, and Z. L. Guo, “Thermal lattice Bhatnagar-Gross-Krook model for flows with viscous heat dissipation in the incompressible limit,” Physical Review E, vol. 70, no. 6, Article ID 066310, 2004. |

[24] | P. H. Kao and R. J. Yang, “Simulating oscillatory flows in Raleigh-Bénard convection using the lattice Boltzmann method,” International Journal of Heat and Mass Transfer, vol. 50, pp. 3315-3328, 2008. · Zbl 1119.76048 |

[25] | Y. H. Qian, D. d’Humieres, and P. Lallemand, “Lattice BGK model for Navier-Stokes equation,” Europhysics Letters, vol. 17, no. 6, pp. 479-484, 1992. · Zbl 1116.76419 |

[26] | G. de Vahl Davis, “Natural convection of air in a square cavity: a benchmark numerical solution,” International Journal for Numerical Methods in Fluids, vol. 3, no. 3, pp. 249-264, 1983. · Zbl 0538.76075 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.