Magnetohydrodynamic flow in an infinite channel.

*(English)*Zbl 0597.76116Summary: The magnetohydrodynamic (MHD) flow of an incompressible, viscous, electrically conducting fluid in an infinite channel, under an applied magnetic field has been investigated. The MHD flow between two parallel walls is of considerable practical importance because of the utility of induction flowmeters. The walls of the channel are taken perpendicular to the magnetic field and one of them is insulated, the other is partly insulated, partly conducting.

An analytical solution has been developed for the velocity field and magnetic field by reducing the problem to the solution of a Fredholm integral equation of the second kind, which has been solved numerically. Solutions have been obtained for Hartmann numbers M up to 200. All the infinite integrals obtained are transformed into finite integrals which contain modified Bessel functions of the second kind. So, the difficulties associated with the computation of infinite integrals with oscillating integrands which arise for large M have been avoided. It is found that, as M increases, boundary layers are formed near the non- conducting boundaries and in the interface region for both velocity and magnetic fields, and a stagnant region in front of the conducting boundary is developed for the velocity field. Selected graphs are given showing these behaviours.

An analytical solution has been developed for the velocity field and magnetic field by reducing the problem to the solution of a Fredholm integral equation of the second kind, which has been solved numerically. Solutions have been obtained for Hartmann numbers M up to 200. All the infinite integrals obtained are transformed into finite integrals which contain modified Bessel functions of the second kind. So, the difficulties associated with the computation of infinite integrals with oscillating integrands which arise for large M have been avoided. It is found that, as M increases, boundary layers are formed near the non- conducting boundaries and in the interface region for both velocity and magnetic fields, and a stagnant region in front of the conducting boundary is developed for the velocity field. Selected graphs are given showing these behaviours.

##### MSC:

76W05 | Magnetohydrodynamics and electrohydrodynamics |

65R20 | Numerical methods for integral equations |

76M99 | Basic methods in fluid mechanics |

45B05 | Fredholm integral equations |

##### Keywords:

magnetohydrodynamic flow; incompressible, viscous, electrically conducting fluid; infinite channel; magnetic field; two parallel walls; induction flowmeters; analytical solution; Fredholm integral equation of the second kind; modified Bessel functions of the second kind; infinite integrals with oscillating integrands; boundary layers; non-conducting boundaries; interface region; stagnant region; conducting boundary
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\textit{M. Sezgin}, Int. J. Numer. Methods Fluids 6, 593--609 (1986; Zbl 0597.76116)

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##### References:

[1] | Shercliff, Proc. Camb. Phil. Soc. 49 pp 136– (1953) |

[2] | Chang, Z. Angew. Math. Phys. 12 pp 100– (1961) |

[3] | Gold, J. Fluid Mech. 13 pp 505– (1962) |

[4] | Hunt, J. Fluid Mech. 21 pp 577– (1965) |

[5] | Grinberg, PMM 25 pp 1024– (1961) |

[6] | Grinberg, PMM 26 pp 80– (1962) |

[7] | Hunt, J. Fluid Mech. 23 pp 563– (1965) |

[8] | Chiang, Z. Angew. Math. Phys. 18 pp 92– (1967) |

[9] | Singh, Z. Angew. Math. Phys. 35 pp 760– (1984) |

[10] | Hunt, J. Fluid Mech. 31 pp 705– (1968) |

[11] | Wenger, J. Fluid Mech. 43 pp 211– (1970) |

[12] | Wu, Int. J. numer. method eng. 6 pp 3– (1973) |

[13] | Singh, Int. J. numer. methods eng. 18 pp 1091– (1982) |

[14] | Singh, Ind. J. Pure Appl. Math. 14 pp 1473– (1983) |

[15] | Singh, Int. J. numer. methods fluids 4 pp 291– (1984) |

[16] | Magnetofluid Dynamics, Abacus Press, 1975. |

[17] | and , Tables of Integrals. Series and Products, Academic Press Inc., New York and London, 1965. |

[18] | Mixed Boundary Value Problems in Potential Theory, North-Holland, Wiley, New York, 1966. · Zbl 0139.28801 |

[19] | Tables of Bessel Transforms, Springer-Verlag, 1972. · Zbl 0261.65003 · doi:10.1007/978-3-642-65462-6 |

[20] | Eason, Phil. Trans. Roy. Soc., Series A 247 pp 529– (1955) |

[21] | and , Magnetofluid Dynamics for Engineers and Applied Physicists, McGraw-Hill, 1973. |

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