Coupled system, Navier-Stokes and porous medium

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You might want to add some of your code to get help as well. For the coupled Navier Stokes system (and not IPCS or other incremental schemes) you might want to look at this 2D Stationary Navier-Stokes equation at @dokken s response. Here you should just add the time dependency to have the NS instead of Stokes. Solving everything at the same time is more convenient than doing projections and corrections both within the fluid and between the fluid and the porous medium.