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11.3 Two fluid case

The two-fluid problem is qualitatively different from the previous two cases, since there are now two independent density currents. This fact impacts the analysis in two crucial ways: (i) The master function Λ depends on n2 n, n2 s, and n2 = n2 ns sn (i.e. entrainment is present), and (ii) the equations of motion, after taking into account the transverse flow condition, are doubled to n s δfμ = 0 = δfμ. As we will see, the key point is that there are now two sound speeds that must be determined.

A variation of the chemical potential covectors that leaves the metric fixed takes the form

x ( x xx) ν ( xy xy) ν δμμ = ℬ μν + 𝒜μν δnx + 𝒳 μν + 𝒜μν δny, (199 )
where the complicated terms
( x[ ] xy ) 𝒜xx = − gμσgνρ -∂ℬ-- n ρnσ + nσn ρ + ∂-𝒜-- nσnρ , (200 ) μν ∂n2xy x y x y ∂n2xy y y ( x y xy ) 𝒜xμyν = 𝒜xyg μν − gμσgνρ ∂-ℬ--nσxn ρx +-∂ℬ--nσy nρy + ∂𝒜---nσyn ρx , (201 ) ∂n2xy ∂n2xy ∂n2xy
exist solely because of entrainment. The same procedure as in the previous two examples leads to the dispersion relation
( ) ( ) ( ) n n nn kiki s s ss kjkj ns ns ns kiki 2 ℬ − [ℬ00 + 𝒜 00] --2- ℬ − [ℬ00 + 𝒜 00] --2-- − 𝒜 − [𝒳00 + 𝒜 00] --2- = 0. (202 ) k0 k 0 k 0
This is quadratic in kiki∕k20 meaning there are two sound speeds. This is a natural result of the doubling of fluid degrees of freedom.

The sound speed analysis is local, but its results are seen globally in the analysis of modes of oscillation of a fluid body. For a neutron star, the full spectrum of modes is quite impressive (see McDermott et al. [77]): polar (or spheroidal) f-, p-, and g-modes, and the axial (or toroidal) r-modes. Epstein [41] was the first to suggest that there should be even more modes in superfluid neutron stars because the superfluidity allows the neutrons to move independently of the protons. Mendell [78] developed this idea further by using an analogy with coupled pendulums. He argued that the new modes should feature a counter-motion between the neutrons and protons, i.e. as the neutrons move out radially, say, the protons will move in. This is in contrast to ordinary fluid motion that would have the neutrons and protons move in more or less “lock-step”. Analytical and numerical studies [7074385Jump To The Next Citation Point] have confirmed this basic picture and the new modes of oscillation are known as superfluid modes.

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