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
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 . Recently, calculations of the shear viscosity of the neutron star crust were done for both the liquid and the crystal phases, by Chugunov & Yakovlev ; 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 , 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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