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There are a number of exact solutions to the Navier-Stokes equations. How many exact solutions are currently known? Is it possible to enumerate all of the solutions to the Navier-Stokes equations?

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One can only enumerate the exact solutions known at a certain point in time, and even that is quite tedious since exact solutions depend on the precise formulation of the problem (changing the shape of the section of a pipe has an important impact, for example), and researchers have found solutions through various methods at various moments, often ignoring each others contributions. But you can get a decent list. –  Christoph B. Apr 10 '13 at 8:36

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Frank White's Viscous Fluid Flow book contains a good list of these "exact" solutions. I am not sure if it is complete though. I've provided links to a few of the solutions.

  1. Steady flow between a fixed and moving plate
  2. Axially moving concentric cylinders
  3. Flow between rotating concentric cylinders
  4. Hagan-Poiseuille flow
  5. Combined Couette-Poiseuille flow between plates
  6. Noncircular ducts -- fully developed flow
  7. Starting flow in a circular pipe
  8. Pipe flow due to an oscillating pressure gradient
  9. Suddenly accelerating plate
  10. Oscillating plate/oscillating freestream
  11. Steady Couette flow where the moving wall suddenly stops
  12. Unsteady Couette flow between a fixed and an oscillating plate
  13. Radial outflow from a porous cylinder
  14. Radial outflow between two circular plates
  15. Combined Poiseuille and Couette flow in a tube or annulus
  16. Gravity-driven thin fluid films
  17. Decay of a line Oseen-Lamb vortex
  18. The Taylor vortex profile
  19. Uniform suction on a plane
  20. Flow between plates with bottom injection and top suction
  21. Start up of wind driven surface water
  22. The Ekman Spiral
  23. Plane stagnation flow
  24. Axisymmetric stagnation flow
  25. Flow near an infinite rotating disk
  26. Jeffrey-Hamel flow in a wedge-shaped region
  27. Stokes' Solution for an Immersed Sphere -- Creeping Flow
  28. Creeping flow past a fluid sphere
  29. Blasius boundary layer
  30. Falkner-Skan-Cooke boundary layer
  31. Compressible self-similar boundary layer
  32. Free-shear flows
  33. Plane laminar wake -- far field
  34. Plane laminar jet
  35. Flat-plate with uniform wall-suction
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Ah wonderful! I was wondering if such a compilation existed as well. Most fluids texts have some subset of the analytic solutions but they are usually scattered through some giant tome of a book. Thank you! –  user44430 Apr 9 '13 at 16:39
    
C. Y. Wangs' 1991 « Exact Solutions of the Steady-State Navier-Stokes Equation » paper published in Annual Review of Fluid Mechanics, gives an overview of exact solutions in the steady case as its name suggests: annualreviews.org/doi/abs/10.1146/annurev.fl.23.010191.001111 It doesn't go into great depths (and is over 20 years old, but White's book was first published in 1974 and then updated in the nineties), but it gives extensive references to other works. I guess one could extract a few additional examples to complete the above list, such as Burgers' vortex and Beltrami flows. –  Christoph B. Apr 10 '13 at 8:31
    
This helped me some, with my question: spinning fluid inside a sphere. I would just add that you can approximate many things from these canned solutions. I have done: - fluid pushed through a cone - fluid between two spheres coming together - fluid between a slowly varying surface. –  The Polywell Guy Jun 4 at 3:58

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