This study presents the implementation and validation of a second-order accurate solver for the 4-equation multi-fluid method in a cell-vertex context, to handle aeronautical air-assisted liquid sheet configurations. Validations include Laplace tests, droplets oscillations, and a two-dimensional configuration reminiscent of an aeronautical airblast injector. Promising results are obtained in the last case, especially when the pressure is increased, as relevant for aeronautical applications. Being fully compatible of the reactive flow formulation of the cell-vertex solver AVBP, this study paves the way to future monolithic simulations of airblast injectors for aeronautical combustion chambers, including both the multi-phase and the reactive regions.