# Linking definition of incompressible fluid to physical intuition

So the picture I have in my head of an incompressible fluid is one which if I fill a canister to the brim with the fluid, and if I try to press down on the top of the canister, the container will bulge and break.

The math definition I am comfortable with is that the fluid flow $\vec{u}$ is a divergence free vector field, i.e. $$\nabla\cdot \vec{u}=0$$

I am trying to connect the two. My idea is given any volume canister (I guess technically there should be a time dimension to my canister but it shouldn't matter i think) $\Omega$, we have by the divergence theorem $$0=\int\int\int_\Omega\nabla\cdot \vec{u}\mathrm dv=\int\int_{\partial \Omega}\vec{u}\cdot\vec{n}ds$$ But I still don't see how having net flux of zero through any canister correspond to the physical picture?

Let us assume that your canister $\Omega$ is a cube for simplicity with its open top side parallel to the $x,y$-plane. Let us assume that by pressing onto its top surface, you induce a downwards flow in $z$-direction on the surface. Using the divergence theorem that you have written down, we can write: $$\int_{\partial \Omega} \vec{u} \cdot d\vec{s} = 0$$ We now know that from the top part of the cube, we have a negative contribution as there is an inward flow in the opposite direction of the surface normal. To balance this negative term, there needs to be a flow in the direction of the surface normal (to obtain a positive dot product) on one or all of the other surfaces. That is the flow that in your intuition makes your canister bulge or break. This picture does of course also apply to cylindrical canisters or basically any other shape.