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Suppose there are a finite number of disjoint unit-radii disks in the plane, each spinning clockwise or counterclockwise at the same angular velocity. The plane is filled with a thin fluid layer, and the disks can be viewed as spinning fan blades determining vectors of fluid motion tangent to the disks. Is the resulting flow and vector field throughout the plane known? My initial intuition is that there should be something like a Voronoi diagram demarcating boundaries of regions of influence. But in exploring a bit I find it may even be nontrivial to determine the flow between just two counter-rotating vortices. For example, the following image was computed by Paul Nylander based on a paper by O.S. Kerr and J.W. Dold, "Periodic Steady Vortices in a Stagnation Point Flow," J. Fluid Mech., 276, 307-325 (1994).
alt text

As I am quite unschooled in this topic, pointers to relevant literature might suffice. Thanks!

Edit1. I've now asked a revised version of this question on Math Overflow, incorporating the clarifying suggestions of Rahul. I might hit a fluid dynamics expert there.

Edit2. Thanks to Rahul and David Bar Moshe here, and Willie Wong and Bob Terrell on MO, I have a much broader understanding of the problem, and could likely compute a numerical solution if needed. I appreciate the help!

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That image is... hypnotic... :) –  J. M. Oct 4 '10 at 23:14
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Strictly as a fluid dynamics problem, this is incompletely specified. But for the simplest additional conditions (incompressible, inviscid, irrotational flow), I believe this can be posed as a Laplace problem on the stream function (en.wikipedia.org/wiki/Stream_function) of the flow. I'll try to flesh this out as an answer when I have the chance. –  Rahul Oct 5 '10 at 1:06
    
@Rahul: Thanks, yes, specifying any conditions that would lead to a solvable formulation would be useful. Thanks for the ideas! –  Joseph O'Rourke Oct 5 '10 at 1:14

2 Answers 2

up vote 3 down vote accepted

Within the framework of Potential flow,the velocity field of a two dimensional potential flow is the gradient of the real part of a complex function $w(z)$ which is analytical outside regions of singularity. The imaginary part of the complex potential is the stream function, whose constant loci are the streamlines, i.e., the integral curves of the velocity field. For example for a single vortex loacted at $z_0$, the potential function is given by:

$w(z) = -\frac{i \Gamma}{2 \pi} log(z-z_0)$.

($\Gamma$ is (the constant) vorticity). It is easy to verify that this poential function gives rize to an angular flow around $z_0$ of velocity inversly proportional to the distance from $z_0$.

The real part of the complex potential saisfies the two dimensional Laplace equation and by linearity, new solutions can be obtained by superposition. In the case of the multiple vortices, the complex potential function takes the form

$w(z) = \sum_n -\frac{(-1)^n i\Gamma}{2 \pi} log(z-z_n)$,

where $z_n$ are the vortex centers.

Now, I can think of a possible candidate of a Voronoi diagram for the vortices by defining the cell walls as the loci of stagnation points in which the fluid velocity is zero. These points are characterized by the condition:

$\frac{dw(z)}{dz}=0$.

This suggestion will result a simple solution in the case of vortices rotating in the same direction, but not in the case of counter-rotating vortices.

Remark: From a superficial reading of the referred article, the vortex solution is a perturbation to a stagnation point flow. This is a flow towards a wall or a corner. To take into account the existence of this wall, one must add vortices from the other side of the wall according to the method of images. May be the stagnation point loci of the complete solution (including the images) will have the required Voronoi cell structure.

Edit:

Thanks to the correct comment by Rahul, the stagnation points are certainly discrete and do not have one dimensional loci. I think that that the suggestion can be salvaged if we define the Voronoi cell walls as passing through the stagnation points and having a normal in the direction of the line connecting the two vortices.

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"the loci of stagnation points in which the fluid velocity is zero." Yes, that is exactly what I had in mind. Thanks for the mention of the method of images! –  Joseph O'Rourke Oct 5 '10 at 14:29
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Can you clarify what you mean by the "loci of stagnation points"? I think the stagnation points themselves are going to be finite in number. –  Rahul Oct 5 '10 at 16:02

Edit: It appears an identical idea has, with far greater detail, already been given to you by jvkersch. I am humbled. I should also point out that my example below, which was only meant as an illustration, would not be a steady-state solution in a physical fluid, because the interaction between the vortices themselves would cause them to move.

David Bar Moshe's idea reminded me of some work in vector field visualization which does indeed use stagnation points to divide the fluid domain into something like "regions of influence". I believe the initial paper which introduced the idea was Helman and Hesselink's "Visualizing Vector Field Topology in Fluid Flows" (PDF copy).

In our case, because we assumed that the flow is incompressible and irrotational, in a generic configuration the velocity can only be zero at saddle points, where the flow points inward along two directions and outward along two directions. Streamlines along these directions are called the separatrices of the saddle point. If you place two particles close together on different sides of a separatrix and let them following the flow, they will diverge at the saddle point and follow disparate long-term trajectories. So these separatrices divide space into regions where the global topology of the streamlines is different.

Here's an example I cooked up in Matlab because I thought it would look pretty. There are four point vortices in a square, whose circulations are -1, -1, -2 and 1 going clockwise from top left. Here's what the direction of the velocity field looks like:

quiver plot

In the diagram below, the separatrices divide space into regions around vortices (and clusters thereof). In each distinct region, the streamlines wind around a particular set of vortices. You can see the saddle points as the points where four arcs come together. I believe this sort of diagram is what you were hoping for when you asked for something like a Voronoi diagram around the vortices.

separatrices

(For a complete picture, there ought to be arrows on the separatrices to indicate the direction of flow, but I couldn't figure out how to do that in Matlab.)

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