Euler--Euler or volume-averaged Navier--Stokes equations are used in various applications to model systems with two or more interpenetrating phases. Each fluid obeys its own momentum and mass equations, and the phases are typically coupled via drag forces and a shared pressure. Monolithic solvers can therefore be very expensive and difficult to implement, so there is great computational appeal for decoupled methods. However, splitting the subproblems requires treating the coupling terms (pressure and drag) explicitly, which must be done carefully to avoid time-step restrictions. In this context, we derive a new first-order pressure-correction method based on the incompressibility of the mean velocity field, combined with an explicit treatment of the drag forces. Furthermore, both the convective and viscous terms are treated semi-implicitly. This gives us an implicit-explicit (IMEX) method that is very robust not only due to its unconditional energy stability, but also because it does not require any type of fixed-point iterations. Each time step has only linear, scalar transport equations and a single pressure Poisson problem as building blocks. We rigorously prove temporal stability without any CFL-like conditions, and the theory is confirmed through two-phase numerical examples.
Pacheco et al. (Mon,) studied this question.