Self answer
Since I started to think about this, I instantly realize it is easy to explain in Newtonian way, I wanted to understand it in fluid dynamical way though.
Let us regard small cells of fluid as solid materials like metal balls.
If one throws balls into a ball pool from a small hole, the stream of balls should be shaped because pushed ball should push the ball ahead repeatedly, and there is no force by which these balls spread out.
On the other hand, if one picks one ball at the hole out of the pool, the stream should not be shaped because a ball can not pull other balls, and if balls are moving little randomly, the empty hole will be filled by them slowly and this might be replaced by pressure of fluid.
It seems to make sense, but I could not sure how this push/pull difference is represented in the Navier-Stokes equations and the Continuity equation.
To make the problem easy, I construct a small model which has 4 openings like this.
Putting a fan at the bottom and starting to inhale air, there are two possibilities:
- Air spreads out all directions
- A straight flow is shaped
Both of them satisfies the continuity equation. Therefore we should check only whether these situations are stable and satify the Navier-Stokes equations or not.
The wrong one is former one of course, but why? Now we estimate the situation 1 is stable, and will reach a conflict. considering
incompressible 2D-Navier-Stokes equations:
$$
\frac{\partial u}{\partial t} + u_{x}\frac{\partial u}{\partial x} + u_{y}\frac{\partial u}{\partial y} = -\frac{1}{\rho}\nabla p
$$
We estimate it stable so don't need the first term:
$$
u_{x}\frac{\partial u}{\partial x} + u_{y}\frac{\partial u}{\partial y} = -\frac{1}{\rho}\nabla p
$$
we can see the convective term and pressure field form directly each other. Roughly speaking, air is accelerate/decelerate according to pressure gradient like this (red is higher):
But this has a problem. Because the outside pressure of these opening are same, there must be pressure gradient at top opening:
therefore, it can't be stable. The high pressure inside the pipe releases by accelerating air due to the pressure gradient at the top, then eventually it reaches the situation 2.
When it comes to extracting, also two options are here.
- Air gathers from all directions
- A straight flow is shaped
The wrong one is later this time. in the situation 2, the air at top opening should be accelerated without the fan, and only available resource is pressure gradient. Therefore, there must be pressure gradient outside around the top:
As shown in the figure, this condition has to make pressure gradient at side as well which excites the flow from outside to inside, and it goes to situation 1.
These are almost not mathematical analysis at all. Maybe wrong.
