# Exact Solutions to the Navier-Stokes Equations [closed]

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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## closed as too broad by Kyle Kanos, HDE 226868, Norbert Schuch, Gert, MAFIA36790Dec 25 '15 at 2:39

There are either too many possible answers, or good answers would be too long for this format. Please add details to narrow the answer set or to isolate an issue that can be answered in a few paragraphs.If this question can be reworded to fit the rules in the help center, please edit the question.

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
This appears to be a "big list" type question, which is considered off-topic here. – Kyle Kanos Dec 24 '15 at 18:10

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 '14 at 3:58