Zucon flows

By NTZung, on April 11th, 2011

I’m constructing here an example of what I want to call “a zucon flow” of incompressible fluid. It is not a solution of anything (or more precisely, you’ll need an appropriate external force to get such a flow). But nevertheless it’s interesting to imagine such flows.

Zucon means “petit voyou”, and it has an “on” at the end, similarly to soliton, peakon, etc., that’s why I like this word.

Call the flow $\Phi_t$ . It will be the limit of smooth flows $\Phi_t^n$ when $n$ tends to infinity. The flows $\Phi_t^n$ will be constructed inductively, by iteration. $\Phi_t^n$ will be smooth, but $\Phi_t$ will have a single point in space-time where it’s not smooth. This is its “petit voyou”, or zucon, feature.

Construction of $\Phi^1$ :

Take a solid torus $T_1$ . The flow will $\Phi^1_t$ will be idle (equal to the identity map) when $t < - a_1$ and $t > a_1$ , and also idle outside the solid torus for all $t$ , but it will be a kind of rotational flow for $t \in [- a_1, a_1]$ , achieving highest rotational speed $\rho_1$ when $t=0$ . (The movement slows down near the border of the solid torus, but near the center of the solid torus it’s rotation for each $t$ ).

Construction of $\Phi^2$ :

$\Phi^2$ will be constructed relative to $\Phi^1$ : take a solid torus $T_2$ which lies inside $T_1$ , in such a way that the center circle of $T_2$ lies on a section of $T_1$ (a disk). Imagine this $T_2$ as a container which moves together with the flow $\Phi^1$ . $\Phi^2$ will move things inside this solid torus container, while the whole thing moves by $\Phi^1$ (a composition of flows). The movement created by $\Phi^2$ inside $T_2$ is similar to the movement created by $\Phi^1$ inside $T_1$ .

and so on and so on

We have sequences of:

* Times $a_1, a_2, \hdots$ : we will assume that this sequence is decreasing and converges to 0.

* Rotational speeds $\rho_1,\rho_2,\hdots$

* Solid torus sizes $R_1, R_2,\hdots$ : of course this sequence is decreasing because each solid torus is contained in the previous one. We make take, for example, $R_{n+1} = \alpha R_{n}$ where $\alpha$ is a positive constants.

Proposition: With the above assumptions, that the flow $\Phi$ can have at most 1 singular point in space-time.

In fact, the only possible singular time is 0. When $t=0$ , the only possible singular spatial point is

$x_0 = \bigcap_{n=0}^\infty T_n(0)$

where $T_n(0)$ denotes the position of $T_n$ at time 0.

The point $(x_0, 0)$ will actually be a singular point if the (relative) speeds are chosen well enough, so that they add up to infinity.

One may view $\Phi$ as a kind of vortex inside a vortex inside a vortex inside a vortex …

The above zucon flow example doesn’t prove anything, but it hints at the possibility of singular points in incompressible flows.

Vortex stretching and vortex germs ?

Vortex stretching (formation) is when a slow rotational movement becomes a fast moving vortex during a short period of time. There are papers which show that this is possible.

Once we understand better vortex formation, we may plant “vortex germs” into vector fields. Each vortex germ can become a vortex during a short period of time. All these vortex germs together can produce a vortex inside a vortex inside a vortex … We can choose vortex germs small enough so that their sum converges, but when they become vortexes, their speeds add up to infinity at the singular point ?!

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