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This proceeding is introducing a fractional step algorithm for diffusion-transport allowing computation of flows with sharp variations of viscosity. This splitting also allows to perform transport with Lagrangian methods and diffusion with Eulerian methods, using hybrid grid-particle formulation. This splitting algorithm is globally second order. It is applied to computation of mucus mobility in human lungs, where epithelium ciliated cells are beating. A sufficient mobility is required to havedoi:10.1016/j.piutam.2015.11.012 fatcat:k67bpf7v55ft5eimiy4gsa5m7e