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We present a multiphase model for incompressible flows of two immiscible fluids. Our model solves one shared set of incompressible Navier–Stokes equations for the two phase flow and an additional equation: the Cahn–Hilliard equation, for the evolution of the two fluids distribution. The introduced model is discretised in space using a high-order Discontinuous Galerkin Spectral Element Method (DGSEM). Time discretisation is performed by means of an efficient implicit-explicit approach that enables to maintain the time step restriction of a typical one phase Navier–Stokes solver. We show the validity and efficiency of our model and implementation in two classical two-dimensional test cases: the spinodal decomposition and the rising bubble.

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