### Planar channel flow in Braginskii magnetohydrodynamics

P. J. Dellar (2011) Planar channel
flow in Braginskii magnetohydrodynamics J.
Fluid Mech. 667 520-543

Preprint available ( BraginskiiJFM.pdf
267K)

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Abstract

#### Braginskii magnetohydrodynamics 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 magnetohydrodynamics. 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 magnetohydrodynamics. 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.

Related paper

P. J. Dellar (2010) Lattice
Boltzmann formulation for Braginskii magnetohydrodynamics (in press)
Computers & Fluids