Effect of Wall Conductivity on an Electric Conducting Fluid Flow Between Rotating and Stationary Coaxial Disks in the Presence of a Uniform Axial Magnetic Field

Toshio Tagawa

doi:doi:10.11648/j.fm.20160202.11

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Effect of Wall Conductivity on an Electric Conducting Fluid Flow Between Rotating and Stationary Coaxial Disks in the Presence of a Uniform Axial Magnetic Field

Toshio Tagawa

Published in Fluid Mechanics (Volume 2, Issue 2)

Received: 20 September 2016 Accepted: 14 October 2016 Published: 8 November 2016

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Abstract

Numerical analyses have been carried out for magnetohydrodynamic flow between a rotating and a stationary disk, whose radii are sufficiently large in comparison with the gap between the two parallel coaxial disks. The gap is filled with an electric conducting fluid and a uniform axial magnetic field is imposed. The magnetic Prandtl number is assumed to be so small that the influence of the induced magnetic field is neglected. The flow depends on both the rotational Reynolds number and the Hartmann number as well as the wall conductance ratios of upper and lower disks. As the Reynolds number increases, the core region of rigid body rotation having slight axial component of velocity is observed between the two boundary layers, whose thickness becomes thinner in proportional to the square root of the Reynolds number. On the other hand, as the Hartmann number increases, the Lorentz force tends to suppress the secondary flow significantly and boundary layer thickness of the azimuthal component of velocity is proportional to the inverse of the Hartmann number. The derived boundary condition for the normal component of electric current density at the interface allows us to obtain similarity solutions for various combinations of each wall conductance ratio and its influence on the flow is quite significant.

Published in	Fluid Mechanics (Volume 2, Issue 2)
DOI	10.11648/j.fm.20160202.11
Page(s)	13-27
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2016. Published by Science Publishing Group

Keywords

Magnetohydrodynamics, Similarity Solution, Secondary Flow, Hartmann Number, Rotating Disk, Wall Conductance Ratio

References

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[2]	W. G. Cochran, "The flow due to a rotating disc, "Mathematical Proceedings of the Cambridge Philosophical Society. Vol. 30. No. 03. Cambridge University Press, p. 1934.
[3]	U. T. Bödewadt, "Die drehströmung über festem grunde," ZAMM‐Journal of Applied Mathematics and Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik 20.5 (1940): pp. 241-253.
[4]	G. K. Batchelor, "Note on a class of solutions of the Navier-Stokes equations representing steady rotationally-symmetric flow," The Quarterly Journal of Mechanics and Applied Mathematics 4.1 (1951): pp. 29-41.
[5]	K. Stewartson, "On the flow between two rotating coaxial disks," Mathematical Proceedings of the Cambridge Philosophical Society. Vol. 49. No. 02. Cambridge University Press, 1953.
[6]	G. N. Lance and M. H. Rogers, "The Axially Symmetric Flow of a Viscous Fluid Between two Infinite Rotating Disk," Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences. Vol. 266. No. 1324. The Royal Society, 1962.
[7]	Carl E. Pearson, "Numerical solutions for the time-dependent viscous flow between two rotating coaxial disks," Journal of Fluid Mechanics 21.04 (1965): pp. 623-633.
[8]	G. L. Mellor, P. J. Chapple and V. K. Stokes, "On the flow between a rotating and a stationary disk," Journal of Fluid Mechanics 31.01 (1968): pp. 95-112.
[9]	Donald. Greenspan, "Numerical studies of flow between rotating coaxial disks," IMA Journal of Applied Mathematics 9.3 (1972): pp. 370-377.
[10]	Lynn O. Wilson and N. L. Schryer, "Flow between a stationary and a rotating disk with suction," Journal of Fluid Mechanics 85.03 (1978): pp. 479-496.
[11]	D. Dijkstra and G. J. F. Van Heijst, "The flow between two finite rotating disks enclosed by a cylinder," Journal of Fluid Mechanics 128 (1983): pp. 123-154.
[12]	John F. Brady and Louis Durlofsky, "On rotating disk flow," Journal of fluid mechanics 175 (1987): pp. 363-394.
[13]	G., P. Gauthier Gondret and M. Rabaud, "Axisymmetric propagating vortices in the flow between a stationary and a rotating disk enclosed by a cylinder," Journal of Fluid Mechanics 386 (1999): pp. 105-126.
[14]	P. A. Davidson and A. Pothérat, "A note on Bödewadt–Hartmann layers," European Journal of Mechanics-B/Fluids 21.5 (2002): pp. 545-559.
[15]	P. Moresco and T. Alboussière, "Stability of Bödewadt–Hartmann layers," European Journal of Mechanics-B/Fluids 23.6 (2004): pp. 851-859.
[16]	C. J. Stephenson, "Magnetohydrodynamic flow between rotating coaxial disks," Journal of Fluid Mechanics 38.02 (1969): pp. 335-352.
[17]	S. Kamiyama and A. Sato, "Magnetohydrodynamic FIow between Parallel Rotating Disks: Report 1, Influence of Finite Wall-Conductance," Bulletin of JSME 15.86 (1972): pp. 941-948.
[18]	R. S. Agarwal and Rama Bhargava, "A numerical study of magnetohydrodynamic flow between a rotating and a stationary porous coaxial discs," Proceedings of the Indian Academy of Sciences-Section A. Part 3, Mathematical Sciences 88.5 (1979): pp. 399-407.
[19]	S. Kishore Kumar, William I. Thacker and Layne T. Watson, "Magnetohydrodynamic flow between a solid rotating disk and a porous stationary disk," Applied Mathematical Modelling 13.8 (1989): pp. 494-500.

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Toshio Tagawa. (2016). Effect of Wall Conductivity on an Electric Conducting Fluid Flow Between Rotating and Stationary Coaxial Disks in the Presence of a Uniform Axial Magnetic Field. Fluid Mechanics, 2(2), 13-27. https://doi.org/10.11648/j.fm.20160202.11

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ACS Style

Toshio Tagawa. Effect of Wall Conductivity on an Electric Conducting Fluid Flow Between Rotating and Stationary Coaxial Disks in the Presence of a Uniform Axial Magnetic Field. Fluid Mech. 2016, 2(2), 13-27. doi: 10.11648/j.fm.20160202.11

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AMA Style

Toshio Tagawa. Effect of Wall Conductivity on an Electric Conducting Fluid Flow Between Rotating and Stationary Coaxial Disks in the Presence of a Uniform Axial Magnetic Field. Fluid Mech. 2016;2(2):13-27. doi: 10.11648/j.fm.20160202.11

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@article{10.11648/j.fm.20160202.11,
  author = {Toshio Tagawa},
  title = {Effect of Wall Conductivity on an Electric Conducting Fluid Flow Between Rotating and Stationary Coaxial Disks in the Presence of a Uniform Axial Magnetic Field},
  journal = {Fluid Mechanics},
  volume = {2},
  number = {2},
  pages = {13-27},
  doi = {10.11648/j.fm.20160202.11},
  url = {https://doi.org/10.11648/j.fm.20160202.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.fm.20160202.11},
  abstract = {Numerical analyses have been carried out for magnetohydrodynamic flow between a rotating and a stationary disk, whose radii are sufficiently large in comparison with the gap between the two parallel coaxial disks. The gap is filled with an electric conducting fluid and a uniform axial magnetic field is imposed. The magnetic Prandtl number is assumed to be so small that the influence of the induced magnetic field is neglected. The flow depends on both the rotational Reynolds number and the Hartmann number as well as the wall conductance ratios of upper and lower disks. As the Reynolds number increases, the core region of rigid body rotation having slight axial component of velocity is observed between the two boundary layers, whose thickness becomes thinner in proportional to the square root of the Reynolds number. On the other hand, as the Hartmann number increases, the Lorentz force tends to suppress the secondary flow significantly and boundary layer thickness of the azimuthal component of velocity is proportional to the inverse of the Hartmann number. The derived boundary condition for the normal component of electric current density at the interface allows us to obtain similarity solutions for various combinations of each wall conductance ratio and its influence on the flow is quite significant.},
 year = {2016}
}

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TY  - JOUR
T1  - Effect of Wall Conductivity on an Electric Conducting Fluid Flow Between Rotating and Stationary Coaxial Disks in the Presence of a Uniform Axial Magnetic Field
AU  - Toshio Tagawa
Y1  - 2016/11/08
PY  - 2016
N1  - https://doi.org/10.11648/j.fm.20160202.11
DO  - 10.11648/j.fm.20160202.11
T2  - Fluid Mechanics
JF  - Fluid Mechanics
JO  - Fluid Mechanics
SP  - 13
EP  - 27
PB  - Science Publishing Group
SN  - 2575-1816
UR  - https://doi.org/10.11648/j.fm.20160202.11
AB  - Numerical analyses have been carried out for magnetohydrodynamic flow between a rotating and a stationary disk, whose radii are sufficiently large in comparison with the gap between the two parallel coaxial disks. The gap is filled with an electric conducting fluid and a uniform axial magnetic field is imposed. The magnetic Prandtl number is assumed to be so small that the influence of the induced magnetic field is neglected. The flow depends on both the rotational Reynolds number and the Hartmann number as well as the wall conductance ratios of upper and lower disks. As the Reynolds number increases, the core region of rigid body rotation having slight axial component of velocity is observed between the two boundary layers, whose thickness becomes thinner in proportional to the square root of the Reynolds number. On the other hand, as the Hartmann number increases, the Lorentz force tends to suppress the secondary flow significantly and boundary layer thickness of the azimuthal component of velocity is proportional to the inverse of the Hartmann number. The derived boundary condition for the normal component of electric current density at the interface allows us to obtain similarity solutions for various combinations of each wall conductance ratio and its influence on the flow is quite significant.
VL  - 2
IS  - 2
ER  -

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Author Information

Toshio Tagawa

Department of Aerospace Engineering, Tokyo Metropolitan University, Tokyo, Japan

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Plain Text BibTeX RIS

APA Style

Toshio Tagawa. (2016). Effect of Wall Conductivity on an Electric Conducting Fluid Flow Between Rotating and Stationary Coaxial Disks in the Presence of a Uniform Axial Magnetic Field. Fluid Mechanics, 2(2), 13-27. https://doi.org/10.11648/j.fm.20160202.11

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ACS Style

Toshio Tagawa. Effect of Wall Conductivity on an Electric Conducting Fluid Flow Between Rotating and Stationary Coaxial Disks in the Presence of a Uniform Axial Magnetic Field. Fluid Mech. 2016, 2(2), 13-27. doi: 10.11648/j.fm.20160202.11

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AMA Style

Toshio Tagawa. Effect of Wall Conductivity on an Electric Conducting Fluid Flow Between Rotating and Stationary Coaxial Disks in the Presence of a Uniform Axial Magnetic Field. Fluid Mech. 2016;2(2):13-27. doi: 10.11648/j.fm.20160202.11

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@article{10.11648/j.fm.20160202.11,
  author = {Toshio Tagawa},
  title = {Effect of Wall Conductivity on an Electric Conducting Fluid Flow Between Rotating and Stationary Coaxial Disks in the Presence of a Uniform Axial Magnetic Field},
  journal = {Fluid Mechanics},
  volume = {2},
  number = {2},
  pages = {13-27},
  doi = {10.11648/j.fm.20160202.11},
  url = {https://doi.org/10.11648/j.fm.20160202.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.fm.20160202.11},
  abstract = {Numerical analyses have been carried out for magnetohydrodynamic flow between a rotating and a stationary disk, whose radii are sufficiently large in comparison with the gap between the two parallel coaxial disks. The gap is filled with an electric conducting fluid and a uniform axial magnetic field is imposed. The magnetic Prandtl number is assumed to be so small that the influence of the induced magnetic field is neglected. The flow depends on both the rotational Reynolds number and the Hartmann number as well as the wall conductance ratios of upper and lower disks. As the Reynolds number increases, the core region of rigid body rotation having slight axial component of velocity is observed between the two boundary layers, whose thickness becomes thinner in proportional to the square root of the Reynolds number. On the other hand, as the Hartmann number increases, the Lorentz force tends to suppress the secondary flow significantly and boundary layer thickness of the azimuthal component of velocity is proportional to the inverse of the Hartmann number. The derived boundary condition for the normal component of electric current density at the interface allows us to obtain similarity solutions for various combinations of each wall conductance ratio and its influence on the flow is quite significant.},
 year = {2016}
}

Copy | Download

TY  - JOUR
T1  - Effect of Wall Conductivity on an Electric Conducting Fluid Flow Between Rotating and Stationary Coaxial Disks in the Presence of a Uniform Axial Magnetic Field
AU  - Toshio Tagawa
Y1  - 2016/11/08
PY  - 2016
N1  - https://doi.org/10.11648/j.fm.20160202.11
DO  - 10.11648/j.fm.20160202.11
T2  - Fluid Mechanics
JF  - Fluid Mechanics
JO  - Fluid Mechanics
SP  - 13
EP  - 27
PB  - Science Publishing Group
SN  - 2575-1816
UR  - https://doi.org/10.11648/j.fm.20160202.11
AB  - Numerical analyses have been carried out for magnetohydrodynamic flow between a rotating and a stationary disk, whose radii are sufficiently large in comparison with the gap between the two parallel coaxial disks. The gap is filled with an electric conducting fluid and a uniform axial magnetic field is imposed. The magnetic Prandtl number is assumed to be so small that the influence of the induced magnetic field is neglected. The flow depends on both the rotational Reynolds number and the Hartmann number as well as the wall conductance ratios of upper and lower disks. As the Reynolds number increases, the core region of rigid body rotation having slight axial component of velocity is observed between the two boundary layers, whose thickness becomes thinner in proportional to the square root of the Reynolds number. On the other hand, as the Hartmann number increases, the Lorentz force tends to suppress the secondary flow significantly and boundary layer thickness of the azimuthal component of velocity is proportional to the inverse of the Hartmann number. The derived boundary condition for the normal component of electric current density at the interface allows us to obtain similarity solutions for various combinations of each wall conductance ratio and its influence on the flow is quite significant.
VL  - 2
IS  - 2
ER  -

Copy | Download