Relativistic Hydrodynamics 2026, 07: Perfect Multifluids & Hyperbolic PDE Systems x
A multifluid is a fluid consisting of different types particles. Opposed to simple fluids, the different kinds of particles will interact in different manners. For perfect fluids, this isn't too complicated yet. For sake of simplicity, the mixture of fluid will be described as several different fluids occupying the same space.
If perfect fluids are "coupled", the momentum exchange through collisions take place among the different constituents, leading to a uniform 4-velocity. The energy-momentum tensor are of a single species.
The rest-mass density, specific internal energy and pressure of the ensemble are linear combination. The global conservation equation is kept very simple by applying the known formulas to this primitive linear combination. The energy-momentum tensor decomposes into the ordinary part (denoted with index M) and a radiation field (denoted with index R), where N are the particle numbers, and I is the specific intensity of the radiation. From the radiation, emerge a radiation energy density e, radiation flux F and radiation stress tensor P, as well as a radiation 4-force density.
The coupling between matter and radiation fluids corresponds to a isotropic field condition for a comoving frame, with zero radiation flux. They can be summed up as an optically thick regime. Decoupled fluids then can be modelled by an optically thin regime. The radiation-hydrodynamic equations are moments of the radiation transfer equation, in analogue to the hydrodynamic equations being moments of the Maxwell-Boltzmann equations.
Interacting multifluids has intrinsic differences to that from coupled ones. The two number densities or particle currents need to be noted separately, as well as the 4-velocities. The 4-velocities both need to satisfy normalization. For coupled multifluids or single perfect fluids, the conjugate 4-momenta are given by the fluid 4-momentum per unit mass and unit volume, positioned parallel to the fluid 4-velocity. Fluid interaction breaks this condition, and instead expresses the conjugate 4-momenta in terms of a 2x2 symmetric matrix. The multifluid relative velocity Δ can be introduced for a more direct physical interpretation as the norm of the 4-velocity of one fluid in the rest frame of another. These concepts can be summed up into the first law of TD, for total energy density E, and entrainment function / coupling constant A
The equations of motion read mostly separately for each fluid
The Newtonian equations for conservation of mass, momentum and energy in the absence of external forces can be written in compact matrix form where U is the state vector.
A conservative formulation can lean on A(U) = ∂F/∂U and the flux vector F(U) so that ∂U/∂t + ∇F = 0. In this formulation, U is the vector of conservd variables. Define the characteristic vector W = R⁻¹U, to get the characteristic equations.
