Hostname: page-component-8448b6f56d-wq2xx Total loading time: 0 Render date: 2024-04-25T06:11:35.968Z Has data issue: false hasContentIssue false
  • English
  • Français

Planar channel flow in Braginskii magnetohydrodynamics

Published online by Cambridge University Press:  14 January 2011

PAUL J. DELLAR
PAUL J. DELLAR*
Affiliation:
OCIAM, Mathematical Institute, 24–29 St Giles', Oxford OX1 3LB, UK
*
†Email address for correspondence: dellar@maths.ox.ac.uk
Get access Rights & Permissions [Opens in a new window]

Abstract

Braginskii magnetohydrodynamics (MHD) is a single-fluid description of large-scale motions in strongly magnetised plasmas. The ion Larmor radius in these plasmas is much shorter than the mean free path between collisions, so momentum transport across magnetic field lines is strongly suppressed. The relation between the strain rate and the viscous stress becomes highly anisotropic, with the viscous stress being predominantly aligned parallel to the magnetic field. We present an analytical study of the steady planar flow across an imposed uniform magnetic field driven by a uniform pressure gradient along a straight channel, the configuration known as Hartmann flow, in Braginskii MHD. The global momentum balance cannot be satisfied by just the parallel viscous stress, so we include the viscous stress perpendicular to magnetic field lines as well. The ratio of perpendicular to parallel viscosities is the key small parameter in our analysis. When another parameter, the Hartmann number, is large the flow is uniform across most of the channel, with boundary layers on either wall that are modifications of the Hartmann layers in standard isotropic MHD. However, the Hartmann layer solution predicts an infinite current and infinite shear at the wall, consistent with a local series solution of the underlying differential equation that is valid for all Hartmann numbers. These singularities are resolved by inner boundary layers whose width scales as the three-quarters power of the viscosity ratio, while the maximum velocity scales as the inverse one-quarter power of the viscosity ratio. The inner wall layers fit between the Hartmann layers, if present, and the walls. The solution thus does not approach a limit as the viscosity ratio tends to zero. Essential features of the solution, such as the maximum current and maximum velocity, are determined by the size of the viscosity ratio, which is the regularising small parameter.

JFM classification

Boundary Layers: Boundary Layers MHD and Electrohydrodynamics: High-Hartmann-number flows MHD and Electrohydrodynamics: Plasmas
Type
Papers
Information
Journal of Fluid Mechanics , Volume 667 , 25 January 2011 , pp. 520 - 543
Copyright
Copyright © Cambridge University Press 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Abramowitz, M. & Stegun, I. A. 1965 Handbook of Mathematical Functions. Dover.Google Scholar
Balbus, S. A. 2004 Viscous shear instability in weakly magnetized, dilute plasmas. Astrophys. J. 616, 857–864.CrossRefGoogle Scholar
Balescu, R. 1988 Transport Processes in Plasmas: Classical Transport Theory. North-Holland.Google Scholar
Bender, C. M. & Orszag, S. A. 1999 Advanced Mathematical Methods for Scientists and Engineers. Springer.CrossRefGoogle Scholar
Biskamp, D. 2000 Magnetic Reconnection in Plasmas. Cambridge University Press.CrossRefGoogle Scholar
Braginskii, S. I. 1965 Transport processes in a plasma. Rev. Plasma Phys. 1, 205–311.Google Scholar
Carilli, C. L. & Taylor, G. B. 2002 Cluster magnetic fields. Annu. Rev. Astron. Astrophys. 40, 319–348.CrossRefGoogle Scholar
Cash, J. R. & Mazzia, F. 2005 A new mesh selection algorithm, based on conditioning, for two-point boundary value codes. J. Comput. Appl. Math. 184, 362–381.CrossRefGoogle Scholar
Cercignani, C. 1988 The Boltzmann Equation and its Applications. Springer.CrossRefGoogle Scholar
Chapman, S. & Cowling, T. G. 1970 The Mathematical Theory of Non-Uniform Gases, 3rd edn. Cambridge University Press.Google Scholar
Chew, G. F., Goldberger, M. L. & Low, F. E. 1956 The Boltzmann equation and the one-fluid hydromagnetic equations in the absence of particle collisions. Proc. R. Soc. Lond. Ser. A 236, 112–118.Google Scholar
Craig, I. J. D. & Litvinenko, Y. E. 2009 Anisotropic viscous dissipation in three-dimensional magnetic merging solutions. Astron. Astrophys. 501, 755–760.CrossRefGoogle Scholar
Dong, R. & Stone, J. M. 2009 Buoyant bubbles in intracluster gas: effects of magnetic fields and anisotropic viscosity. Astrophys. J. 704, 1309–1320.CrossRefGoogle Scholar
Dorf, L., Sun, X., Intrator, T., Hendryx, J., Wurden, G., Furno, I. & Lapenta, G. 2007 Experimental verification of Braginskii's viscosity in MHD plasma jet of reconnection scaling experiment. Bull. APS 52, PM4.00007.Google Scholar
Epperlein, E. M. & Haines, M. G. 1986 Plasma transport coefficients in a magnetic field by direct numerical solution of the Fokker–Planck equation. Phys. Fluids 29, 1029–1041.CrossRefGoogle Scholar
Furno, I., Intrator, T., Torbert, E., Carey, C., Cash, M. D., Campbell, J. K., Fienup, W. J., Werley, C. A., Wurden, G. A. & Fiksel, G. 2003 Reconnection scaling experiment: a new device for three-dimensional magnetic reconnection studies. Rev. Sci. Instrum. 74, 2324–2331.CrossRefGoogle Scholar
Grad, H. 1949 On the kinetic theory of rarefied gases. Comm. Pure Appl. Math. 2, 331–407.CrossRefGoogle Scholar
Grad, H. 1958 Principles of the kinetic theory of gases. In Thermodynamik der Gase (ed. Flügge, S.), Handbuch der Physik, vol. 12, pp. 205–294. Springer.Google Scholar
Hogan, J. T. 1984 Collisional transport of momentum in axisymmetric configurations. Phys. Fluids 27, 2308–2312.CrossRefGoogle Scholar
Hollweg, J. V. 1986 Viscosity and the Chew–Goldberger–Low equations in the solar corona. Astrophys. J. 306, 730–739.CrossRefGoogle Scholar
Hunt, J. C. R. & Shercliff, J. A. 1971 Magnetohydrodynamics at high Hartmann number. Annu. Rev. Fluid Mech. 3, 37–62.CrossRefGoogle Scholar
Islam, T. & Balbus, S. 2005 Dynamics of the magnetoviscous instability. Astrophys. J. 633, 328–333.CrossRefGoogle Scholar
Kaufman, A. N. 1960 Plasma viscosity in a magnetic field. Phys. Fluids 3, 610–616.CrossRefGoogle Scholar
Landau, L. D. & Lifshitz, E. M. 1960 Electrodynamics of Continuous Media, Pergamon.Google Scholar
Lifshitz, E. M. & Pitaevskii, L. P. 1981 Physical Kinetics. Pergamon.Google Scholar
Lyutikov, M. 2008 Hartmann flow with Braginsky viscosity: a test problem for plasma in the intracluster medium. Astrophys. J. Lett. 673, L115–L117.CrossRefGoogle Scholar
Newcomb, W. A. 1966 Dynamics of a gyroviscous plasma. In Dynamics of Fluids and Plasmas (ed. Pai, S. I.), pp. 405–429. Academic.Google Scholar
Parrish, I. J., Stone, J. M. & Lemaster, N. 2008 The magnetothermal instability in the intracluster medium. Astrophys. J. 688, 905–917.CrossRefGoogle Scholar
Sanders, J. S., Fabian, A. C., Smith, R. K. & Peterson, J. R. 2010 A direct limit on the turbulent velocity of the intracluster medium in the core of Abell 1835 from XMM–Newton. Mon. Not. R. Astron. Soc. 402, L11–L15.CrossRefGoogle Scholar
Schekochihin, A., Cowley, S., Kulsrud, R., Hammett, G. & Sharma, P. 2005 Magnetised plasma turbulence in clusters of galaxies. In The Magnetized Plasma in Galaxy Evolution (ed. Chyzy, K. T., Otmianowska-Mazur, K., Soida, M. & Dettmar, R.-J.), pp. 86–92. Jagiellonian University, Krakow, Poland.Google Scholar
Sharma, P., Quataert, E., Hammett, G. W., & Stone, J. M. 2007 Electron heating in hot accretion flows. Astrophys. J. 667, 714–723.CrossRefGoogle Scholar
Spitzer, L. 1962 Physics of Fully Ionized Gases. Wiley.Google Scholar