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Planar channel flow in Braginskii magnetohydrodynamics

  • PAUL J. DELLAR (a1)

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.

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Abramowitz, M. & Stegun, I. A. 1965 Handbook of Mathematical Functions. Dover.
Balbus, S. A. 2004 Viscous shear instability in weakly magnetized, dilute plasmas. Astrophys. J. 616, 857864.
Balescu, R. 1988 Transport Processes in Plasmas: Classical Transport Theory. North-Holland.
Bender, C. M. & Orszag, S. A. 1999 Advanced Mathematical Methods for Scientists and Engineers. Springer.
Biskamp, D. 2000 Magnetic Reconnection in Plasmas. Cambridge University Press.
Braginskii, S. I. 1965 Transport processes in a plasma. Rev. Plasma Phys. 1, 205311.
Carilli, C. L. & Taylor, G. B. 2002 Cluster magnetic fields. Annu. Rev. Astron. Astrophys. 40, 319348.
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, 362381.
Cercignani, C. 1988 The Boltzmann Equation and its Applications. Springer.
Chapman, S. & Cowling, T. G. 1970 The Mathematical Theory of Non-Uniform Gases, 3rd edn. Cambridge University Press.
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, 112118.
Craig, I. J. D. & Litvinenko, Y. E. 2009 Anisotropic viscous dissipation in three-dimensional magnetic merging solutions. Astron. Astrophys. 501, 755760.
Dong, R. & Stone, J. M. 2009 Buoyant bubbles in intracluster gas: effects of magnetic fields and anisotropic viscosity. Astrophys. J. 704, 13091320.
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.
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, 10291041.
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, 23242331.
Grad, H. 1949 On the kinetic theory of rarefied gases. Comm. Pure Appl. Math. 2, 331407.
Grad, H. 1958 Principles of the kinetic theory of gases. In Thermodynamik der Gase (ed. Flügge, S.), Handbuch der Physik, vol. 12, pp. 205294. Springer.
Hogan, J. T. 1984 Collisional transport of momentum in axisymmetric configurations. Phys. Fluids 27, 23082312.
Hollweg, J. V. 1986 Viscosity and the Chew–Goldberger–Low equations in the solar corona. Astrophys. J. 306, 730739.
Hunt, J. C. R. & Shercliff, J. A. 1971 Magnetohydrodynamics at high Hartmann number. Annu. Rev. Fluid Mech. 3, 3762.
Islam, T. & Balbus, S. 2005 Dynamics of the magnetoviscous instability. Astrophys. J. 633, 328333.
Kaufman, A. N. 1960 Plasma viscosity in a magnetic field. Phys. Fluids 3, 610616.
Landau, L. D. & Lifshitz, E. M. 1960 Electrodynamics of Continuous Media, Pergamon.
Lifshitz, E. M. & Pitaevskii, L. P. 1981 Physical Kinetics. Pergamon.
Lyutikov, M. 2008 Hartmann flow with Braginsky viscosity: a test problem for plasma in the intracluster medium. Astrophys. J. Lett. 673, L115L117.
Newcomb, W. A. 1966 Dynamics of a gyroviscous plasma. In Dynamics of Fluids and Plasmas (ed. Pai, S. I.), pp. 405429. Academic.
Parrish, I. J., Stone, J. M. & Lemaster, N. 2008 The magnetothermal instability in the intracluster medium. Astrophys. J. 688, 905917.
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, L11L15.
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. 8692. Jagiellonian University, Krakow, Poland.
Sharma, P., Quataert, E., Hammett, G. W., & Stone, J. M. 2007 Electron heating in hot accretion flows. Astrophys. J. 667, 714723.
Spitzer, L. 1962 Physics of Fully Ionized Gases. Wiley.
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