zero divergence means that the amount going into a region equals the amount coming out. in other words, nothing is lost. so for example the divergence of the density of a fluid is (usually) zero because you can't (unless there's a “source” or “sink”) create (or destroy) mass.

What if the divergence is zero?

The gas may be moving, but the volume rate of gas flowing into any closed surface must equal the volume rate flowing out, so the net flux is zero. Thus the gas velocity has zero divergence everywhere. A field which has zero divergence everywhere is called solenoidal. ... Therefore the divergence at any other point is zero.

What happens when the divergence of a vector field is zero?

If this divergence is ≡0 throughout the domain Ω of v this means that no new fluid is produced anywhere, and this entails that for any body B⊂Ω the total flux through its surface ∂B is zero.