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Consider the two images in this link: ejecting vector field image.

In the upper picture: a fluid is smoothly flowing in a two dimensional pipe (i.e. a velocity vector field passing without any interruption).

In the lower picture: the same fluid is flowing in a similar pipe, but now there is a hole (i.e. a part of the velocity vector field has now ejected out from that hole).

I'm currently not concerned in the physical reason behind it. Rather what I want to know is that,

  1. Which mathematical property of a velocity field (a vector field) will depict such a situation that a potion of that field will come out from the main stream in the lower case, but not in the upper case? Or in other words, what will be the exceptional (mathematical) property of the velocity field around the hole that lets the velocity field to leave the main stream and come out?

  2. If I'm given a vector field (for the sake of convenience, let's say the velocity field), can I predict (by pen-paper calculation) that if the velocity field will flow smoothly everywhere or at certain point(s), some part of it will be ejected out from the main stream (like the second case)?

My speculation: Is something to do here with the curl or the divergence of the vector field at the vicinity points of the hole? But what about the general (special) property of the vector field, that is required for being ejected?

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    $\begingroup$ Note that Stokes theorem (the divergence theorem) relates the flux along the boundary (i.e. the pipe i.e. a leak) to the divergence of the field inside the pipe. So you are right to think about the divergence. If you want the flux along the pipe to be zero, then with some reasonable conditions/assumptions about fluid flow, this forces the divergence to be zero. And clearly divergence being zero is sufficient for the flux to be zero. $\endgroup$
    – Osama Ghani
    Commented Nov 27, 2022 at 15:51
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    $\begingroup$ @OsamaGhani: Why would a divergence-free condition be sufficient for the flux through the hole in the pipe surface be zero? To begin with the velocity field is always divergence free for an incompressible fluid. By the divergence theorem the net flux through any surface enclosing a region of flow must be zero, but that does not necessarily hold for any portion of the surface such as where the hole is located. Imagine a rectangular region with pipe walls as the upper and lower segments, inflow at the left segment and outflow at the right segment. $\endgroup$
    – RRL
    Commented Nov 28, 2022 at 20:11
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    $\begingroup$ Suppose there is a hole in the upper segment. All that is required is that the net flux in at the left is equal to the net flux out at the right plus the net flux out through the hole. If there is a fully developed flow through the pipe and the wall is perforated at a location where the fluid pressure is much higher than the ambient pressure, do you believe there will be no fluid escaping through the hole, because the velocity field must be divergence- free $\endgroup$
    – RRL
    Commented Nov 28, 2022 at 20:15
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    $\begingroup$ @SCh: This has nothing to do with the curl or divergence of the undisturbed velocity field prior to the introduction of a hole in the pipe wall. Try the following experiment -- fully open a high-pressure water source connected to a rubber hose. Now punch a hole in the hose. Will there be no water escaping from the hole because the velocity field for an incompressible fluid has zero divergence? $\endgroup$
    – RRL
    Commented Nov 28, 2022 at 20:21
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    $\begingroup$ @OsamaGhani: No problem. What is interesting here is the question of what type of flow pattern would result from the presence of the hole. $\endgroup$
    – RRL
    Commented Nov 28, 2022 at 22:20

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