Divergence $0$ everywhere implies Flux $0$? I was told this by a college mate, and I was pretty unsure as to why/if this is true.
Consider the vectorfield $F=(1,1,1)$, it clearly has divergence $0$ at every point, but, picturing it in my head, I think the flux should not be $0$ for at least some surfaces (a plane normal to the vectors, for example).
I'm saying this because of my intuition about these concepts: consider said vector field representing some liquid moving, then If I place a grid (plane said above) I clearly have some liquid flowing through it, thus the flux should not be 0...
Where am I wrong?
 A: If the divergence of a vector field is zero ($\nabla \cdot \vec{v} = 0$), then the flux of that vector field through any closed surface is zero.  This is a consequence of the divergence theorem:  for any vector field $\vec{v}$,
$$
\iiint_\mathcal{V} (\nabla \cdot \vec{v}) \,dV = \iint_\mathcal{S} \vec{v} \cdot d\vec{a},
$$where $\mathcal{V}$ is any "nice" sub-volume of $\mathbb{R}^3$ and $\mathcal{S}$ is the boundary of $\mathcal{V}$.  Roughly speaking, the boundary surface of a volume does not have any "edges", and so is a "closed surface."  If you think of this in terms of "flow lines" of a fluid, then if $\nabla \cdot \vec{v} = 0$, there will be as many flow lines going into $\mathcal{V}$ as coming out of $\mathcal{V}$, and so the net flux of fluid through $\mathcal{S}$ is zero.
This logic does not hold for "open surfaces" (i.e., surfaces with an "edge") like you're describing;  and as you note, the flux of a divergence-free vector field through an open surface can easily be non-zero.
A: The flux of the vector field $F$ is not zero through every surface. However, there are two kind of surfaces which the flux through them can be zero by your vector field.
1) Consider a closed surface surrounding some region. Then, due to divergence theorem we have
$$\oint\limits_{\partial \Omega } {F.nda}  = \int\limits_\Omega  {\nabla .Fdv} $$
and since $\nabla .F = 0$, we have $\oint\limits_{\partial \Omega } {F.nda}=0 $ for any closed surface. Note that the surface is a closed one.
2) Suppose that $F.n=0$, i.e., a plane which is parallel to your vector field. Then the flux is zero through this plane.
