Derivation of the fluid expansion model

We model the UNP as a three-species plasma composed of electrons and two ionic species. The species momentum equations are obtained from the velocity moments of the underlying kinetic description and are given by

where α = {e, Ca, Yb}. Apart from the time derivative, this model describes advection, pressure forces, external and internal forces, and collisions. Approximations relevant to our ultracold plasma experimental conditions can be made, and include steady-state electrons (∂u_e/∂t ≈ 0 on the ion time scales of interest here), small velocities among all species (u_α ⋅ ∇ u_α ≈ 0), vanishing pressure for the (ultracold) ions (p_Ca, p_{Y b} ≈ 0), internal electrostatic forces and a friction term between ionic species. With these approximations, Eq. (²⁵) become

where the subscripts 1 = Ca and 2 = Yb. These equations are coupled to the species continuity equations and a Poisson equation for the electric fields. Assuming the electrons are isothermal and Z = 1, we obtain