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Consider some fluid (as nice as possible) in a plastic bag with holes illustrated by the image below (of course no holes have been drawn in this picture)

enter image description here

What is the corresponding PDE to model the mass density and the flow velocity (as well as the boundary conditions)?

Now let us prick holes with a needle dynamically (even randomly) such that the distribution of holes evolves with respect to time. What is the corresponding PDE?

I'm interested in the remaining mass in the bag. Are there references on this topic?

PS : Following the remarks of Jonathan J. it might be better to replace the word "plastic" by any general material. Then we may model the "bag" by some bounded open set $\Omega\subset \mathbb R^d$ with $d=2$ or $d=3$. Here I would like to emphasize that in general "bag" is deformable, but here $\Omega$ can be deformable or not and I'm interested in these two cases.

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  • $\begingroup$ How much do you want to model this specific bag? Would it be alright if we used an idealized shape, such as a sphere and poked holes in that? $\endgroup$
    – Jonathan J.
    Commented Jan 26 at 21:46
  • $\begingroup$ @JonathanJ. Thanks for specifying. More precisely, I most focus on dimension $d=2$ and $d=3$. If $d=2$, we may take the bag of form square, circle, or any regular and connected open set. If $d=3$, we take the bad of form sphere, cube $\endgroup$
    – GJC20
    Commented Jan 26 at 22:56

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