Let us denote a stationary macroscopic hydrodynamic velocity field, imposed on the plasma, by .
In an isotropic plasma the viscous part of the stress tensor can be written as
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First, let us consider volume preserving flows, characterized by . A schematic view of such a flow
in the solid crust, characteristic of torsional oscillations of the crust, is shown in Figure 58
. The dissipation
resulting in the entropy production is determined by the shear viscosity
. In the outer crust,
is a
sum of the electron and nuclei contributions,
, but for
and
.13.
In the inner crust, an additional contribution from the normal component of the gas of dripped neutrons
should be added.
The electrons are scattered on nuclei, on impurity nuclei, and on themselves, so that the effective
frequency of their scattering is given by the sum . However, as long as the
temperature is not too low, the approximation
can be used.
To calculate from the BE for electrons, we have to determine
due to the presence of
a weak plasma velocity field,
. The solution of the BE, linearized in
and in
, has the form
The scattering frequency , in turn, can be expressed in terms of the effective Coulomb logarithm
by
Calculations of the shear viscosity for the liquid phase were done by Flowers & Itoh [146, 147] and
Nandkumar & Pethick [300]. Recently, calculations of the shear viscosity of the neutron star crust were
done for both the liquid and the crystal phases, by Chugunov & Yakovlev [101]; their results are
displayed in Figure 56. These authors also give analytic fitting formulae for the effective Coulomb
logarithms, which can be used for different models of the crust. The electron-impurity scattering
becomes dominant at low
, when electron-lattice scattering (via phonons) is suppressed by
quantum effects. This is visualized in Figure 57
. Recently, the contribution to
resulting from
the
scattering, was recalculated by Shternin [377], who took into account the Landau
damping of transverse plasmons. The Landau damping of transverse plasmons leads to a significant
suppression of
for ultra-relativistic electrons, and modifies the temperature dependence of
.
We are not aware of any calculations of the bulk viscosity of the crust, . We just mention that it
is generally assumed that the bulk viscosity of the crust is much smaller than the shear one,
.
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