Plasma-sheath transition in multi-fluid models with inertial terms under low pressure conditions: Comparison with the classical and kinetic theory

We present high-order accuracy simulations of the plasma-sheath transition with an ion-electron multi-fluid model that considers the electron inertial terms and the ion pressure gradient. By means of a third-order accuracy time-dependent finite volume scheme, we solve the isothermal multi-fluid equations with realistic ion-to-electron mass ratios. We propose a numerical procedure and boundary conditions that retrieve a steady-state solution of a planar 1D floating sheath. The classical solution to this problem neglects the ion temperature since the model equations present a singularity for a finite ion temperature. The multi-fluid simulations provide a rigorous solution to the plasma-sheath transition without singularities. We compare our solution to the classical theory, finding perfect agreement when the ion-to-electron temperature tends to zero. We discuss the effect of the ion temperature, the electron inertia, and the elastic collisions with neutrals in low-pressure plasmas. Finally, relying on a kinetic approach, we derive analytical expressions for the electron macroscopic quantities inside a steady sheath that assumes a truncated Maxwellian electron velocity distribution function (EVDF) with a wall that collects the electron random flux. The derived analytical expressions are beyond the classical isothermal solution with Boltzmann electrons. The isothermal multi-fluid solution captures properly the first two moments of the EVDF inside the sheath. Our analytical expressions show the need for solving for higher-order moments to fully explain the electron physics inside the sheath, as opposite to the classical isothermal assumption. This work demonstrates that the multi-fluid simulations are able to capture the plasma-sheath transition under weakly collisional conditions with a solution that is consistent with the classical and the kinetic theory.