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What are the equations that describe the flow of a fluid in a porous medium? Is there a variation of the Navier-Stokes equations?

I would like to model the flow of air through a sponge-like structure. Instead of modelling the structure of the sponge, I would like to treat it like a homogenous area, where I solve appropriate equations for conservation of momentum, mass and energy.

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Situationes where Stokes (without Navier) rules, are well known, eg hydrogeology (ground water) or oil geology (secondary or tertiaty recovery)

A word which might be usefull to search for: percolation.

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The keyword for porous flows is Darcy's flow, which is based on the Darcy law guiding the field of mass flux instead of velocity:

$$\vec{q}=-K\nabla p$$

$\vec{q}:=\vec{u} \cdot \text{porosity}$, so it is equivalent to normal flow only for homogeneous media.

This equation comes from averaging the NS over porous volume; however, as you can see, it is very different from NS -- and this is just inevitable that such averaging will change the mathematical form of the problem. And, as in your case, when some more complex theory is needed, one should get through the averaging of NS with a set of assumptions that suit your problem (clues for instance here).

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