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10.1 Variational formulation of multi-fluid hydrodynamics

In the convective variational approach of hydrodynamics developed by Carter [7080], and recently reviewed by Gourgoulhon [174] and Andersson & Comer [15], the dynamic equations are obtained from an action principle by considering variations of the fluid particle trajectories. First developed in the context of general relativity, this formalism has been adapted to the Newtonian framework in the usual 3+1 spacetime decomposition by Prix [342343]. As shown by Carter & Chamel in a series of papers [74Jump To The Next Citation Point75Jump To The Next Citation Point76Jump To The Next Citation Point], the Newtonian hydrodynamic equations can be written in a very concise and elegant form in a fully-4D covariant framework. Apart from facilitating the comparison between relativistic and nonrelativistic fluids, this approach sheds a new light on Newtonian hydrodynamics following the steps of Elie Cartan, who demonstrated in the 1920’s that the effects of gravitation in Newtonian theory can be expressed in geometric terms as in general relativity.

The variational formalism of Carter provides a very general framework for deriving the dynamic equations of any fluid mixture and for obtaining conservation laws, using exterior calculus. In particular, this formalism is very well suited to describing superfluid systems, like laboratory superfluids or neutron star interiors, by making a clear distinction between particle velocities and the corresponding momenta (see the discussion in Section 8.3.6).

The dynamics of the system (either in relativity or in the Newtonian limit) is thus governed by a Lagrangian density Λ, which depends on the particle 4-currents n μX = nXu μX, where nX and uμX are the particle number density and the 4-velocity of the constituent X, respectively. We will use Greek letters for spacetime indices with the Einstein summation convention for repeated indices. The index X runs over the different constituents in the system. Note that repeated chemical indices X will not mean summation unless explicitly specified.

The dynamic equations for a mixture of several interacting fluids (coupled by entrainment effects) can be obtained by requiring that the action integral

∫ 𝒮 = Λd 𝒱 (220 )
(where 𝒱 is the 4-volume element) be stationary under variations of the 4-currents nμX. These variations are not arbitrary because they have to conserve the number of particles. In classical mechanics of point-like particles, the equations of motion can be deduced from an action integral by considering displacements of the particle trajectories. Likewise, considering variations of the 4-currents induced by displacements of the fluid-particle worldlines yield
ν ν nX ϖXνμ + πXμ ∇νn X = fXμ , (221 )
where ∇ μ denotes the covariant derivative. X πμ, defined by
πX = -∂Λ- , (222 ) μ ∂n μX
is the 4-momentum per particle associated with the 4-current μ nX, X ϖ νμ is the vorticity 2-form defined by the exterior derivative of the corresponding 4-momentum
X X X X ϖ μν = 2∇[μπν] = ∇ μπν − ∇ νπμ , (223 )
and fX μ is the (nongravitational) 4-force density acting on the constituent X. Equation (222View Equation) is the generalization to fluids of the definition introduced in classical Lagrangian mechanics for the momentum of point-like particles. This equation shows that momentum and velocity are intrinsically different mathematical objects since the former is a co-vector while the latter is a vector. The vorticity 2-form is closely analogous to the electromagnetic 2-form F μν. Equation (221View Equation) is the covariant generalization of Euler’s equation to multi-fluid systems. The stress-energy-momentum tensor of this multi-fluid system is given by
μ μ ∑ μ X T ν = Ψδν + nX πν , (224 ) X
where the generalized pressure Ψ is defined by
∑ Ψ = Λ − n μπX . (225 ) X μ X
Note that so far we have made no assumptions regarding the spacetime geometry so that Equations (221View Equation), (224View Equation) and (225View Equation) are valid for both relativistic and nonrelativistic fluids. The presence of a frozen-in magnetic field and the elasticity of the solid crust can be taken into account within the same variational framework both in (special and general) relativity [73Jump To The Next Citation Point85Jump To The Next Citation Point] and in the Newtonian limit [73Jump To The Next Citation Point72Jump To The Next Citation Point].
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