TITLE: Vortex Voronoi diagram?
QUESTION [9 upvotes]: 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).



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!

REPLY [5 votes]: 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:

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.

(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.)