In this paper, a hybrid computational technique for the Direct Numerical Simulation of hydrodynamics and (convective) mixing in complex two-phase systems is presented. The hybrid Immersed Boundary and Volume of Fluid method is based upon a single-fluid approach in combination with Henry’s law for the concentration jump. The convective species fluxes are evaluated using PLIC-based geometrical advection, which is consistent with the advection of the local fluid volume fractions. Complex geometries can be accounted for and are treated with the Immersed Boundary method. Following the verification for single phase and two-phase systems, the model was used to investigate the influence of the container geometry on the mixing of a tracer due to tilting of the container. For both cylindrical and rectangular containers, an increasing tilting angle enhances the convective motion and consequently the mixing efficiency. The cylindrical geometry was found to possess a better mixing efficiency than the rectangular geometry.