# In the continuum mesh of infinitesimally parcels, has a parcel unit always the same neighbors?

Background: The closed path "P" can be defined at any initial time anywhere in the fluid. Following the fluid motion of the individual fluid parcels on "P", "P" becomes a material property of the parcel group on "P". Circulation can be defined around path "P" and for non divergent and homogeneous flows its conserved.

Now, it implicitly means that the parcels will have always the same parcels as neighbors, since if new parcels come in the place between two parcels on path "P" so that after a while there would be not closed path "P" possible anymore tracking the same initial group, "P" is not anymore a material property of the initial parcel group.

It is kind of weird, because the evolution of the distance between two water parcels is also a material property given as the dot product of the vector distance between them and the gradient of velocity tensor. (both informations are on the same book)

If this is true, how can the distance between parcels change if they are always neighbors and infinitesimally small. And if they slide on each other making possible that parcels occupy the position of other parcels without creating holes in the continuum mesh how can the circulation theorem as a material property be true?

I think your question is equivalent to: if a material loop in the flow can change its length (by stretching/contracting), how can it do so without breaking the loop?

Talking about continuous material loops you have already assumed continuum model of fluid. Now continuum model means simply that the material loop is a real line characterized by a parameter $s$ along its length. Now you can stretch a real line as much as you want without breaking it; more precisely you can always find a one-to-one correspondence between the initial line and the deformed line (this may interest you). All this shows is that continuum model does not rule out the possibility of the material loop remaining intact as it deforms in the flow, although it doesn't prove it.

Now if you take two fluid points on the loop separated by a small finite distance then you sure have a difference in velocity (in general) which causes the separation between them to change subsequently. However as you take the limit of separation between the two points on the loop going to zero, velocity difference between them also goes to zero. This shows why the loop will not break; it is because of the assumption that velocity field is continuous over the region of fluid on which it is defined.