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9.4 Viscosity

In this section we consider the viscosity of the crust, which can be in a liquid or a solid phase. For strongly nonideal (Γ ≫ 1) and solid plasma the transport is mediated mainly by electrons. We will limit ourselves to this case only. For solid crust, we will assume a polycrystal structure, so that on a macroscopic scale the crust will behave as an isotropic medium.

Let us denote a stationary macroscopic hydrodynamic velocity field, imposed on the plasma, by UUU(rrr). In an isotropic plasma the viscous part of the stress tensor can be written as

( ) Πvis = η ∂Ui- + ∂Uj- − 2δ ∇∇∇ ⋅UUU + ζδ ∇∇∇ ⋅ UUU , (209 ) ij ∂xj ∂xi 3 ij ij
where η is the shear viscosity and ζ is the bulk viscosity. The viscous component of the stress tensor enters the equations of neutron star hydrodynamics and is relevant for neutron star pulsations.
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Figure 56: Electron shear viscosity of the crust and the upper layer of the core for log10T [K ] = 7,8,9. Calculated by Chugunov & Yakovlev [101Jump To The Next Citation Point] with a smooth composition model of the ground-state (Appendix B of Haensel, Potekhin, and Yakovlev [184Jump To The Next Citation Point]).
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Figure 57: Electron contribution to crust viscosity and effect of impurities. Solid lines – perfect one-component plasma. Dashed line – admixture of impurities with n = 0.05n imp i and |Z − Zimp | = 2. Curves are labeled by log10T [K]. Impurity contribution to ηe becomes dominant for T ≪ Tpi: this is visible for a log10 T[K ] = 7 curve. Ground-state crust model of Negele & Vautherin [303Jump To The Next Citation Point] is used. Figure made by A.Y. Potekhin.

First, let us consider volume preserving flows, characterized by ∇∇∇ ⋅UUU = 0. A schematic view of such a flow in the solid crust, characteristic of torsional oscillations of the crust, is shown in Figure 58View Image. The dissipation resulting in the entropy production is determined by the shear viscosity η. In the outer crust, η is a sum of the electron and nuclei contributions, η = ηe + ηN, but for ρ > 105 g cm −3 ηe ≫ ηN and η ≈ ηe.13. In the inner crust, an additional contribution from the normal component of the gas of dripped neutrons should be added.

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Figure 58: Schematic picture of torsional oscillations in neutron star crust. Left: equilibrium structure of the solid crust, represented as a two-dimensional square lattice. Right: shear flow in the crust (the shear velocity is indicated by arrows).

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 νηe = νηeN + νηimp + νeηe. However, as long as the temperature is not too low, the approximation νe ≈ νeN can be used.

To calculate ηe from the BE for electrons, we have to determine (0) δf = f − f due to the presence of a weak plasma velocity field, UUU. The solution of the BE, linearized in UUU and in δf, has the form

∂f(0) δf = A η(ε)[vvv ⋅∇∇∇ (ppp ⋅ UUU)]-----, (210 ) ∂ε
where Aη(ε) is a function to be determined from the BE. In the relaxation time approximation and for strongly degenerate electrons A η = τη = 1∕νeN eN is the effective relaxation time due to eN collisions, calculated at the electron Fermi surface.

The scattering frequency νeN, in turn, can be expressed in terms of the effective Coulomb logarithm ΛeN by

Z2e4n νηeN = 12π --2---NΛeN . (211 ) pFevFe
Having calculated the Coulomb logarithm, we get the electron viscosity using a standard formula
nepFevFe- ηe = 5νeN , (212 )
At low temperature, impurities can become the main scatterers of electrons, with
12πe4 ∑ νimp = -2----- (Z − Zimp)2nimpΛeimp . (213 ) pFevFe imp
In the inner crust, one must also consider the gas of dripped neutrons.

Calculations of the shear viscosity for the liquid phase were done by Flowers & Itoh [146147] 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 56View Image. 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 T, when electron-lattice scattering (via phonons) is suppressed by quantum effects. This is visualized in Figure 57View Image. Recently, the contribution to ηe resulting from the ee 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 ηee for ultra-relativistic electrons, and modifies the temperature dependence of ηee.

We are not aware of any calculations of the bulk viscosity of the crust, ζ. We just mention that it is generally assumed that the bulk viscosity of the crust is much smaller than the shear one, ζ ≪ η.

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