An energy estimate and a stabilized Lagrange–Galerkin scheme for a multiphase flow model

Multiphase flow models are commonly employed for understanding complex fluid flows, while few mathematical discussions exist. For a general multiphase flow model in Gidaspow (1994), an energy decay property is proved. A stabilized Lagrange–Galerkin scheme for the model and its stability property are presented. Here, a hyperbolic tangent transformation is employed to preserve the boundedness of the volume fraction. A novel artificial term is added to obtain the stability property. Two-dimensional numerical examples exhibit the experimental order of convergence and applicability in modelling sedimentation phenomena. •The energy decay property of a multiphase flow model is proved.•A stabilized Lagrange–Galerkin scheme for the model is presented, and its stability property is studied.•It is observed from two-dimensional numerical examples that the scheme works well.